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# Editor’s Choice Articles

Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.

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21 pages, 1852 KiB
Article
An Analytic Solution for 2D Heat Conduction Problems with General Dirichlet Boundary Conditions
by Heng-Pin Hsu, Te-Wen Tu and Jer-Rong Chang
Axioms 2023, 12(5), 416; https://doi.org/10.3390/axioms12050416 - 24 Apr 2023
Cited by 3 | Viewed by 3448
Abstract
This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case [...] Read more.
This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case of space–time dependence. First, the physical system is decomposed into two one-dimensional subsystems, each of which can be solved by combining the proposed shifting function method with the eigenfunction expansion theorem. Therefore, through the superposition of the solutions of the two subsystems, the complete solution in the form of series can be obtained. Two numerical examples are used to investigate the analytic solution of the 2D heat conduction problems with space–time-dependent boundary conditions. The considered space–time-dependent functions are separable in the space–time domain for convenience. The space-dependent function is specified as a sine function and/or a parabolic function, and the time-dependent function is specified as an exponential function and/or a cosine function. In order to verify the correctness of the proposed method, the case of the space-dependent sinusoidal function and time-dependent exponential function is studied, and the consistency between the derived solution and the literature solution is verified. The parameter influence of the time-dependent function of the boundary conditions on the temperature variation is also investigated, and the time-dependent function includes harmonic type and exponential type. Full article
(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)
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20 pages, 534 KiB
Article
by Olawale Kazeem Oyewole and Simeon Reich
Axioms 2023, 12(3), 256; https://doi.org/10.3390/axioms12030256 - 1 Mar 2023
Cited by 5 | Viewed by 1452
Abstract
In this work, we are concerned with the iterative approximation of solutions to equilibrium problems in the framework of Hadamard manifolds. We introduce a subgradient extragradient type method with a self-adaptive step size. The use of a step size which is allowed to [...] Read more.
In this work, we are concerned with the iterative approximation of solutions to equilibrium problems in the framework of Hadamard manifolds. We introduce a subgradient extragradient type method with a self-adaptive step size. The use of a step size which is allowed to increase per iteration is to avoid the dependence of our method on the Lipschitz constant of the underlying operator as has been the case in recent articles in this direction. In general, operators satisfying weak monotonicity conditions seem to be more applicable in practice. By using inertial and viscosity techniques, we establish a convergence result for solving a pseudomonotone equilibrium problem under some appropriate conditions. As applications, we use our method to solve some theoretical optimization problems. Finally, we present some numerical illustrations in order to demonstrate the quantitative efficacy and superiority of our proposed method over a previous method present in the literature. Full article
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10 pages, 307 KiB
Article
Initial Coefficients Estimates and Fekete–Szegö Inequality Problem for a General Subclass of Bi-Univalent Functions Defined by Subordination
by Mohamed Illafe, Feras Yousef, Maisarah Haji Mohd and Shamani Supramaniam
Axioms 2023, 12(3), 235; https://doi.org/10.3390/axioms12030235 - 23 Feb 2023
Cited by 15 | Viewed by 1486
Abstract
In the present work, we aim to introduce and investigate a novel comprehensive subclass of normalized analytic bi-univalent functions involving Gegenbauer polynomials and the zero-truncated Poisson distribution. For functions in the aforementioned class, we find upper estimates of the second and third Taylor–Maclaurin [...] Read more.
In the present work, we aim to introduce and investigate a novel comprehensive subclass of normalized analytic bi-univalent functions involving Gegenbauer polynomials and the zero-truncated Poisson distribution. For functions in the aforementioned class, we find upper estimates of the second and third Taylor–Maclaurin coefficients, and then we solve the Fekete–Szegö functional problem. Moreover, by setting the values of the parameters included in our main results, we obtain several links to some of the earlier known findings. Full article
16 pages, 356 KiB
Article
Boundary Value Problems for Fractional Differential Equations of Caputo Type and Ulam Type Stability: Basic Concepts and Study
by Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan
Axioms 2023, 12(3), 226; https://doi.org/10.3390/axioms12030226 - 21 Feb 2023
Cited by 7 | Viewed by 2090
Abstract
Boundary value problems are very applicable problems for different types of differential equations and stability of solutions, which are an important qualitative question in the theory of differential equations. There are various types of stability, one of which is the so called Ulam-type [...] Read more.
Boundary value problems are very applicable problems for different types of differential equations and stability of solutions, which are an important qualitative question in the theory of differential equations. There are various types of stability, one of which is the so called Ulam-type stability, and it is a special type of data dependence of solutions of differential equations. For boundary value problems, this type of stability requires some additional understanding, and, in connection with this, we discuss the Ulam-Hyers stability for different types of differential equations, such as ordinary differential equations and generalized proportional Caputo fractional differential equations. To propose an appropriate idea of Ulam-type stability, we consider a boundary condition with a parameter, and the value of the parameter depends on the chosen arbitrary solution of the corresponding differential inequality. Several examples are given to illustrate the theoretical considerations. Full article
14 pages, 286 KiB
Article
A New Advanced Class of Convex Functions with Related Results
Axioms 2023, 12(2), 195; https://doi.org/10.3390/axioms12020195 - 13 Feb 2023
Cited by 6 | Viewed by 1794
Abstract
It is the purpose of this paper to propose a novel class of convex functions associated with strong $\eta$-convexity. A relationship between the newly defined function and an earlier generalized class of convex functions is hereby established. To point out the importance [...] Read more.
It is the purpose of this paper to propose a novel class of convex functions associated with strong $\eta$-convexity. A relationship between the newly defined function and an earlier generalized class of convex functions is hereby established. To point out the importance of the new class of functions, some examples are presented. Additionally, the famous Hermite–Hadamard inequality is derived for this generalized family of convex functions. Furthermore, some inequalities and results for strong $\eta$-convex function are also derived. We anticipate that this new class of convex functions will open the research door to more investigations in this direction. Full article
15 pages, 269 KiB
Article
On Intuitionistic Fuzzy Temporal Topological Structures
by Krassimir Atanassov
Axioms 2023, 12(2), 182; https://doi.org/10.3390/axioms12020182 - 9 Feb 2023
Cited by 6 | Viewed by 1240
Abstract
In the present paper, four different intuitionistic fuzzy temporal topological structures are introduced, and some of their properties are discussed. These topological structures are based on the intuitionistic fuzzy topological operators and on the temporal intuitionistic fuzzy topological operators, which exist in intuitionistic [...] Read more.
In the present paper, four different intuitionistic fuzzy temporal topological structures are introduced, and some of their properties are discussed. These topological structures are based on the intuitionistic fuzzy topological operators and on the temporal intuitionistic fuzzy topological operators, which exist in intuitionistic fuzzy sets theory. The new structures are direct extensions of the IFTSs and will be the basis for introducing of a new type of topological structures. Full article
(This article belongs to the Special Issue Fuzzy Logic and Application in Multi-Criteria Decision-Making (MCDM))
20 pages, 2236 KiB
Article
Two Novel Models for Traffic Sign Detection Based on YOLOv5s
by Wei Bai, Jingyi Zhao, Chenxu Dai, Haiyang Zhang, Li Zhao, Zhanlin Ji and Ivan Ganchev
Axioms 2023, 12(2), 160; https://doi.org/10.3390/axioms12020160 - 3 Feb 2023
Cited by 26 | Viewed by 3052
Abstract
Object detection and image recognition are some of the most significant and challenging branches in the field of computer vision. The prosperous development of unmanned driving technology has made the detection and recognition of traffic signs crucial. Affected by diverse factors such as [...] Read more.
Object detection and image recognition are some of the most significant and challenging branches in the field of computer vision. The prosperous development of unmanned driving technology has made the detection and recognition of traffic signs crucial. Affected by diverse factors such as light, the presence of small objects, and complicated backgrounds, the results of traditional traffic sign detection technology are not satisfactory. To solve this problem, this paper proposes two novel traffic sign detection models, called YOLOv5-DH and YOLOv5-TDHSA, based on the YOLOv5s model with the following improvements (YOLOv5-DH uses only the second improvement): (1) replacing the last layer of the ‘Conv + Batch Normalization + SiLU’ (CBS) structure in the YOLOv5s backbone with a transformer self-attention module (T in the YOLOv5-TDHSA’s name), and also adding a similar module to the last layer of its neck, so that the image information can be used more comprehensively, (2) replacing the YOLOv5s coupled head with a decoupled head (DH in both models’ names) so as to increase the detection accuracy and speed up the convergence, and (3) adding a small-object detection layer (S in the YOLOv5-TDHSA’s name) and an adaptive anchor (A in the YOLOv5-TDHSA’s name) to the YOLOv5s neck to improve the detection of small objects. Based on experiments conducted on two public datasets, it is demonstrated that both proposed models perform better than the original YOLOv5s model and three other state-of-the-art models (Faster R-CNN, YOLOv4-Tiny, and YOLOv5n) in terms of the mean accuracy (mAP) and F1 score, achieving mAP values of 77.9% and 83.4% and F1 score values of 0.767 and 0.811 on the TT100K dataset, and mAP values of 68.1% and 69.8% and F1 score values of 0.71 and 0.72 on the CCTSDB2021 dataset, respectively, for YOLOv5-DH and YOLOv5-TDHSA. This was achieved, however, at the expense of both proposed models having a bigger size, greater number of parameters, and slower processing speed than YOLOv5s, YOLOv4-Tiny and YOLOv5n, surpassing only Faster R-CNN in this regard. The results also confirmed that the incorporation of the T and SA improvements into YOLOv5s leads to further enhancement, represented by the YOLOv5-TDHSA model, which is superior to the other proposed model, YOLOv5-DH, which avails of only one YOLOv5s improvement (i.e., DH). Full article
(This article belongs to the Special Issue Various Deep Learning Algorithms in Computational Intelligence)
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15 pages, 326 KiB
Article
Third-Order Differential Subordinations Using Fractional Integral of Gaussian Hypergeometric Function
by Georgia Irina Oros, Gheorghe Oros and Lavinia Florina Preluca
Axioms 2023, 12(2), 133; https://doi.org/10.3390/axioms12020133 - 28 Jan 2023
Cited by 5 | Viewed by 1247
Abstract
Sanford S. Miller and Petru T. Mocanu’s theory of second-order differential subordinations was extended for the case of third-order differential subordinations by José A. Antonino and Sanford S. Miller in 2011. In this paper, new results are proved regarding third-order differential subordinations that [...] Read more.
Sanford S. Miller and Petru T. Mocanu’s theory of second-order differential subordinations was extended for the case of third-order differential subordinations by José A. Antonino and Sanford S. Miller in 2011. In this paper, new results are proved regarding third-order differential subordinations that extend the ones involving the classical second-order differential subordination theory. A method for finding a dominant of a third-order differential subordination is provided when the behavior of the function is not known on the boundary of the unit disc. Additionally, a new method for obtaining the best dominant of a third-order differential subordination is presented. This newly proposed method essentially consists of finding the univalent solution for the differential equation that corresponds to the differential subordination considered in the investigation; previous results involving third-order differential subordinations have been obtained mainly by investigating specific classes of admissible functions. The fractional integral of the Gaussian hypergeometric function, previously associated with the theory of fuzzy differential subordination, is used in this paper to obtain an interesting third-order differential subordination by involving a specific convex function. The best dominant is also provided, and the example presented proves the importance of the theoretical results involving the fractional integral of the Gaussian hypergeometric function. Full article
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)
23 pages, 363 KiB
Article
Boundary-Value Problem for Nonlinear Fractional Differential Equations of Variable Order with Finite Delay via Kuratowski Measure of Noncompactness
by Benoumran Telli, Mohammed Said Souid and Ivanka Stamova
Axioms 2023, 12(1), 80; https://doi.org/10.3390/axioms12010080 - 12 Jan 2023
Cited by 9 | Viewed by 1668
Abstract
This paper is devoted to boundary-value problems for Riemann–Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is first studied using a Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness. Secondly, the Ulam–Hyers stability criteria are examined. [...] Read more.
This paper is devoted to boundary-value problems for Riemann–Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is first studied using a Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness. Secondly, the Ulam–Hyers stability criteria are examined. All of the results in this study are established with the help of generalized intervals and piecewise constant functions. We convert the Riemann–Liouville fractional variable-order problem to equivalent standard Riemann–Liouville problems of fractional-constant orders. Finally, two examples are constructed to illustrate the validity of the observed results. Full article
10 pages, 247 KiB
Article
Fixed Point Theorems for Generalized Classes of Operators
by Cristiana Ionescu
Axioms 2023, 12(1), 69; https://doi.org/10.3390/axioms12010069 - 9 Jan 2023
Cited by 2 | Viewed by 1766
Abstract
In this work, we consider weakly generalized operators, which extend the Geraghty mappings that are studied with regard to the existence and uniqueness of their fixed points, in the setting offered by strong b-metric spaces. Classic results are obtained as corollaries. An [...] Read more.
In this work, we consider weakly generalized operators, which extend the Geraghty mappings that are studied with regard to the existence and uniqueness of their fixed points, in the setting offered by strong b-metric spaces. Classic results are obtained as corollaries. An example is provided to support these outcomes. Full article
10 pages, 369 KiB
Article
New Classes of Degenerate Unified Polynomials
by Daniel Bedoya, Clemente Cesarano, Stiven Díaz and William Ramírez
Axioms 2023, 12(1), 21; https://doi.org/10.3390/axioms12010021 - 25 Dec 2022
Cited by 10 | Viewed by 1492
Abstract
In this paper, we introduce a class of new classes of degenerate unified polynomials and we show some algebraic and differential properties. This class includes the Appell-type classical polynomials and their most relevant generalizations. Most of the results are proved by using generating [...] Read more.
In this paper, we introduce a class of new classes of degenerate unified polynomials and we show some algebraic and differential properties. This class includes the Appell-type classical polynomials and their most relevant generalizations. Most of the results are proved by using generating function methods and we illustrate our results with some examples. Full article
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24 pages, 371 KiB
Article
On Fuzzy Implications Derived from General Overlap Functions and Their Relation to Other Classes
by Jocivania Pinheiro, Helida Santos, Graçaliz P. Dimuro, Benjamin Bedregal, Regivan H. N. Santiago, Javier Fernandez and Humberto Bustince
Axioms 2023, 12(1), 17; https://doi.org/10.3390/axioms12010017 - 24 Dec 2022
Cited by 3 | Viewed by 1620
Abstract
There are distinct techniques to generate fuzzy implication functions. Despite most of them using the combination of associative aggregators and fuzzy negations, other connectives such as (general) overlap/grouping functions may be a better strategy. Since these possibly non-associative operators have been successfully used [...] Read more.
There are distinct techniques to generate fuzzy implication functions. Despite most of them using the combination of associative aggregators and fuzzy negations, other connectives such as (general) overlap/grouping functions may be a better strategy. Since these possibly non-associative operators have been successfully used in many applications, such as decision making, classification and image processing, the idea of this work is to continue previous studies related to fuzzy implication functions derived from general overlap functions. In order to obtain a more general and flexible context, we extend the class of implications derived by fuzzy negations and t-norms, replacing the latter by general overlap functions, obtaining the so-called $\left(\mathcal{GO},N\right)$-implication functions. We also investigate their properties, the aggregation of $\left(\mathcal{GO},N\right)$-implication functions, their characterization and the intersections with other classes of fuzzy implication functions. Full article
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16 pages, 663 KiB
Article
The Use of a Log-Normal Prior for the Student t-Distribution
by Se Yoon Lee
Axioms 2022, 11(9), 462; https://doi.org/10.3390/axioms11090462 - 8 Sep 2022
Cited by 4 | Viewed by 3317
Abstract
It is typically difficult to estimate the number of degrees of freedom due to the leptokurtic nature of the Student t-distribution. Particularly in studies with small sample sizes, special care is needed concerning prior choice in order to ensure that the analysis [...] Read more.
It is typically difficult to estimate the number of degrees of freedom due to the leptokurtic nature of the Student t-distribution. Particularly in studies with small sample sizes, special care is needed concerning prior choice in order to ensure that the analysis is not overly dominated by any prior distribution. In this article, popular priors used in the existing literature are examined by characterizing their distributional properties on an effective support where it is desirable to concentrate on most of the prior probability mass. Additionally, we suggest a log-normal prior as a viable prior option. We show that the Bayesian estimator based on a log-normal prior compares favorably to other Bayesian estimators based on the priors previously proposed via simulation studies and financial applications. Full article
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29 pages, 8785 KiB
Article
Design of Type-3 Fuzzy Systems and Ensemble Neural Networks for COVID-19 Time Series Prediction Using a Firefly Algorithm
by Patricia Melin, Daniela Sánchez, Juan R. Castro and Oscar Castillo
Axioms 2022, 11(8), 410; https://doi.org/10.3390/axioms11080410 - 17 Aug 2022
Cited by 29 | Viewed by 2348
Abstract
In this work, information on COVID-19 confirmed cases is utilized as a dataset to perform time series predictions. We propose the design of ensemble neural networks (ENNs) and type-3 fuzzy inference systems (FISs) for predicting COVID-19 data. The answers for each ENN module [...] Read more.
In this work, information on COVID-19 confirmed cases is utilized as a dataset to perform time series predictions. We propose the design of ensemble neural networks (ENNs) and type-3 fuzzy inference systems (FISs) for predicting COVID-19 data. The answers for each ENN module are combined using weights provided by the type-3 FIS, in which the ENN is also designed using the firefly algorithm (FA) optimization technique. The proposed method, called ENNT3FL-FA, is applied to the COVID-19 data for confirmed cases from 12 countries. The COVID-19 data have shown to be a complex time series due to unstable behavior in certain periods of time. For example, it is unknown when a new wave will exist and how it will affect each country due to the increase in cases due to many factors. The proposed method seeks mainly to find the number of modules of the ENN and the best possible parameters, such as lower scale and lower lag of the type-3 FIS. Each module of the ENN produces an individual prediction. Each prediction error is an input for the type-3 FIS; moreover, outputs provide a weight for each prediction, and then the final prediction can be calculated. The type-3 fuzzy weighted average (FWA) integration method is compared with the type-2 FWA to verify its ability to predict future confirmed cases by using two data periods. The achieved results show how the proposed method allows better results for the real prediction of 20 future days for most of the countries used in this study, especially when the number of data points increases. In countries such as Germany, India, Italy, Mexico, Poland, Spain, the United Kingdom, and the United States of America, on average, the proposed ENNT3FL-FA achieves a better performance for the prediction of future days for both data points. The proposed method proves to be more stable with complex time series to predict future information such as the one utilized in this study. Intelligence techniques and their combination in the proposed method are recommended for time series with many data points. Full article
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24 pages, 485 KiB
Article
A Theoretical Dynamical Noninteracting Model for General Manipulation Systems Using Axiomatic Geometric Structures
by Paolo Mercorelli
Axioms 2022, 11(7), 309; https://doi.org/10.3390/axioms11070309 - 25 Jun 2022
Cited by 4 | Viewed by 1641
Abstract
This paper presents a new theoretical approach to the study of robotics manipulators dynamics. It is based on the well-known geometric approach to system dynamics, according to which some axiomatic definitions of geometric structures concerning invariant subspaces are used. In such a framework, [...] Read more.
This paper presents a new theoretical approach to the study of robotics manipulators dynamics. It is based on the well-known geometric approach to system dynamics, according to which some axiomatic definitions of geometric structures concerning invariant subspaces are used. In such a framework, certain typical problems in robotics are mathematically formalised and analysed in axiomatic form. The outcomes are sufficiently general that it is possible to discuss the structural properties of robotic manipulation. A generalized theoretical linear model is used, and a thorough analysis is made. The noninteracting nature of this model is also proven through a specific theorem. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
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18 pages, 320 KiB
Article
Numerical Processes for Approximating Solutions of Nonlinear Equations
by Samundra Regmi, Ioannis K. Argyros, Santhosh George and Christopher I. Argyros
Axioms 2022, 11(7), 307; https://doi.org/10.3390/axioms11070307 - 24 Jun 2022
Cited by 5 | Viewed by 1668
Abstract
In this article, we present generalized conditions of three-step iterative schemes for solving nonlinear equations. The convergence order is shown using Taylor series, but the existence of high-order derivatives is assumed. However, only the first derivative appears on these schemes. Therefore, the hypotheses [...] Read more.
In this article, we present generalized conditions of three-step iterative schemes for solving nonlinear equations. The convergence order is shown using Taylor series, but the existence of high-order derivatives is assumed. However, only the first derivative appears on these schemes. Therefore, the hypotheses limit the utilization of the schemes to operators that are at least nine times differentiable, although the schemes may converge. To the best of our knowledge, no semi-local convergence has been given in the setting of a Banach space. Our goal is to extend the applicability of these schemes in both the local and semi-local convergence cases. Moreover, we use our idea of recurrent functions and conditions only on the derivative or divided differences of order one that appear in these schemes. This idea can be applied to extend other high convergence multipoint and multistep schemes. Numerical applications where the convergence criteria are tested complement this article. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)
9 pages, 291 KiB
Article
Some Cardinal and Geometric Properties of the Space of Permutation Degree
by Ljubiša D. R. Kočinac, Farkhod G. Mukhamadiev and Anvar K. Sadullaev
Axioms 2022, 11(6), 290; https://doi.org/10.3390/axioms11060290 - 14 Jun 2022
Cited by 11 | Viewed by 1883
Abstract
This paper is devoted to the investigation of cardinal invariants such as the hereditary density, hereditary weak density, and hereditary Lindelöf number. The relation between the spread and the extent of the space ${\mathsf{SP}}^{\mathsf{2}}\left(\mathbb{R},\tau \left(A\right)\right)$ [...] Read more.
This paper is devoted to the investigation of cardinal invariants such as the hereditary density, hereditary weak density, and hereditary Lindelöf number. The relation between the spread and the extent of the space ${\mathsf{SP}}^{\mathsf{2}}\left(\mathbb{R},\tau \left(A\right)\right)$ of permutation degree of the Hattori space is discussed. In particular, it is shown that the space ${\mathsf{SP}}^{\mathsf{2}}\left(\mathbb{R},{\tau }_{S}\right)$ contains a closed discrete subset of cardinality $\mathfrak{c}$. Moreover, it is shown that the functor ${\mathsf{SP}}_{\mathsf{G}}^{\mathsf{n}}$ preserves the homotopy and the retraction of topological spaces. In addition, we prove that if the spaces X and Y are homotopically equivalent, then the spaces ${\mathsf{SP}}_{\mathsf{G}}^{\mathsf{n}}X$ and ${\mathsf{SP}}_{\mathsf{G}}^{\mathsf{n}}Y$ are also homotopically equivalent. As a result, it has been proved that the functor ${\mathsf{SP}}_{\mathsf{G}}^{\mathsf{n}}$ is a covariant homotopy functor. Full article
17 pages, 1386 KiB
Review
Continuous-Stage Runge–Kutta Approximation to Differential Problems
by Pierluigi Amodio, Luigi Brugnano and Felice Iavernaro
Axioms 2022, 11(5), 192; https://doi.org/10.3390/axioms11050192 - 21 Apr 2022
Cited by 9 | Viewed by 2214
Abstract
In recent years, the efficient numerical solution of Hamiltonian problems has led to the definition of a class of energy-conserving Runge–Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Such methods admit an interesting interpretation in terms of continuous-stage Runge–Kutta methods. In [...] Read more.
In recent years, the efficient numerical solution of Hamiltonian problems has led to the definition of a class of energy-conserving Runge–Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Such methods admit an interesting interpretation in terms of continuous-stage Runge–Kutta methods. In this review paper, we recall this aspect and extend it to higher-order differential problems. Full article
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27 pages, 7638 KiB
Article
Improvement of the WENO-NIP Scheme for Hyperbolic Conservation Laws
by Ruo Li and Wei Zhong
Axioms 2022, 11(5), 190; https://doi.org/10.3390/axioms11050190 - 20 Apr 2022
Cited by 4 | Viewed by 2710
Abstract
The WENO-NIP scheme was obtained by developing a class of ${L}_{1}$-norm smoothness indicators based on Newton interpolation polynomial. It recovers the optimal convergence order in smooth regions regardless of critical points and achieves better resolution than the classical WENO-JS scheme. However, [...] Read more.
The WENO-NIP scheme was obtained by developing a class of ${L}_{1}$-norm smoothness indicators based on Newton interpolation polynomial. It recovers the optimal convergence order in smooth regions regardless of critical points and achieves better resolution than the classical WENO-JS scheme. However, the WENO-NIP scheme produces severe spurious oscillations when solving 1D linear advection problems with discontinuities at long output times, and it is also very oscillatory near discontinuities for 1D Riemann problems. In this paper, we find that the spectral property of WENO-NIP exhibits the negative dissipation characteristic, and this is the reason why WENO-NIP is unstable near discontinuities. Using this knowledge, we develop a way of improving the WENO-NIP scheme by introducing an additional term to eliminate the negative dissipation interval. The proposed scheme, denoted as WENO-NIP+, maintains the same convergence property, as well as the same low-dissipation property, as the corresponding WENO-NIP scheme. Numerical examples confirm that the proposed scheme is much more stable near discontinuities for 1D linear advection problems with large output times and 1D Riemann problems than the WENO-NIP scheme. Furthermore, the new scheme is far less dissipative in the region with high-frequency waves. In addition, the improved WENO-NIP+ scheme can remove or at least greatly decrease the post-shock oscillations that are commonly produced by the WENO-NIP scheme when simulating 2D Euler equations with strong shocks. Full article
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15 pages, 614 KiB
Article
Fractional Modelling and Optimal Control of COVID-19 Transmission in Portugal
by Silvério Rosa and Delfim F. M. Torres
Axioms 2022, 11(4), 170; https://doi.org/10.3390/axioms11040170 - 11 Apr 2022
Cited by 10 | Viewed by 2396
Abstract
A fractional-order compartmental model was recently used to describe real data of the first wave of the COVID-19 pandemic in Portugal [Chaos Solitons Fractals 144 (2021), Art. 110652]. Here, we modify that model in order to correct time dimensions and use it to [...] Read more.
A fractional-order compartmental model was recently used to describe real data of the first wave of the COVID-19 pandemic in Portugal [Chaos Solitons Fractals 144 (2021), Art. 110652]. Here, we modify that model in order to correct time dimensions and use it to investigate the third wave of COVID-19 that occurred in Portugal from December 2020 to February 2021, and that has surpassed all previous waves, both in number and consequences. A new fractional optimal control problem is then formulated and solved, with vaccination and preventive measures as controls. A cost-effectiveness analysis is carried out, and the obtained results are discussed. Full article
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11 pages, 305 KiB
Article
Positive Solutions for a System of Fractional Boundary Value Problems with r-Laplacian Operators, Uncoupled Nonlocal Conditions and Positive Parameters
by Alexandru Tudorache and Rodica Luca
Axioms 2022, 11(4), 164; https://doi.org/10.3390/axioms11040164 - 6 Apr 2022
Cited by 4 | Viewed by 1952
Abstract
In this paper, we investigate the existence and nonexistence of positive solutions for a system of Riemann–Liouville fractional differential equations with r-Laplacian operators, subject to nonlocal uncoupled boundary conditions that contain Riemann–Stieltjes integrals, various fractional derivatives and positive parameters. We first change [...] Read more.
In this paper, we investigate the existence and nonexistence of positive solutions for a system of Riemann–Liouville fractional differential equations with r-Laplacian operators, subject to nonlocal uncoupled boundary conditions that contain Riemann–Stieltjes integrals, various fractional derivatives and positive parameters. We first change the unknown functions such that the new boundary conditions have no positive parameters, and then, by using the corresponding Green functions, we equivalently write this new problem as a system of nonlinear integral equations. By constructing an appropriate operator $\mathcal{A}$, the solutions of the integral system are the fixed points of $\mathcal{A}$. Following some assumptions regarding the nonlinearities of the system, we show (by applying the Schauder fixed-point theorem) that operator $\mathcal{A}$ has at least one fixed point, which is a positive solution of our problem, when the positive parameters belong to some intervals. Then, we present intervals for the parameters for which our problem has no positive solution. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
16 pages, 322 KiB
Article
On the Bias in Confirmatory Factor Analysis When Treating Discrete Variables as Ordinal Instead of Continuous
by Alexander Robitzsch
Axioms 2022, 11(4), 162; https://doi.org/10.3390/axioms11040162 - 1 Apr 2022
Cited by 7 | Viewed by 2920
Abstract
Confirmatory factor analysis is some of the most widely used statistical techniques in the social sciences. Frequently, variables (i.e., items) stemming from questionnaires are analyzed. Two competing approaches for estimating confirmatory factor analysis can be distinguished. First, ordinal variables could be treated as [...] Read more.
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22 pages, 729 KiB
Article
Mathematical Modeling to Study Optimal Allocation of Vaccines against COVID-19 Using an Age-Structured Population
by Gilberto González-Parra, Myladis R. Cogollo and Abraham J. Arenas
Axioms 2022, 11(3), 109; https://doi.org/10.3390/axioms11030109 - 2 Mar 2022
Cited by 12 | Viewed by 3925
Abstract
Vaccination against the coronavirus disease 2019 (COVID-19) started in early December of 2020 in the USA. The efficacy of the vaccines vary depending on the SARS-CoV-2 variant. Some countries have been able to deploy strong vaccination programs, and large proportions of their populations [...] Read more.
Vaccination against the coronavirus disease 2019 (COVID-19) started in early December of 2020 in the USA. The efficacy of the vaccines vary depending on the SARS-CoV-2 variant. Some countries have been able to deploy strong vaccination programs, and large proportions of their populations have been fully vaccinated. In other countries, low proportions of their populations have been vaccinated, due to different factors. For instance, countries such as Afghanistan, Cameroon, Ghana, Haiti and Syria have less than $10%$ of their populations fully vaccinated at this time. Implementing an optimal vaccination program is a very complex process due to a variety of variables that affect the programs. Besides, science, policy and ethics are all involved in the determination of the main objectives of the vaccination program. We present two nonlinear mathematical models that allow us to gain insight into the optimal vaccination strategy under different situations, taking into account the case fatality rate and age-structure of the population. We study scenarios with different availabilities and efficacies of the vaccines. The results of this study show that for most scenarios, the optimal allocation of vaccines is to first give the doses to people in the 55+ age group. However, in some situations the optimal strategy is to first allocate vaccines to the 15–54 age group. This situation occurs whenever the SARS-CoV-2 transmission rate is relatively high and the people in the 55+ age group have a transmission rate 50% or less that of those in the 15–54 age group. This study and similar ones can provide scientific recommendations for countries where the proportion of vaccinated individuals is relatively small or for future pandemics. Full article
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18 pages, 306 KiB
Article
Mathematical Neural Networks
by Julia García Cabello
Axioms 2022, 11(2), 80; https://doi.org/10.3390/axioms11020080 - 17 Feb 2022
Cited by 4 | Viewed by 4906
Abstract
ANNs succeed in several tasks for real scenarios due to their high learning abilities. This paper focuses on theoretical aspects of ANNs to enhance the capacity of implementing those modifications that make ANNs absorb the defining features of each scenario. This work may [...] Read more.
ANNs succeed in several tasks for real scenarios due to their high learning abilities. This paper focuses on theoretical aspects of ANNs to enhance the capacity of implementing those modifications that make ANNs absorb the defining features of each scenario. This work may be also encompassed within the trend devoted to providing mathematical explanations of ANN performance, with special attention to activation functions. The base algorithm has been mathematically decoded to analyse the required features of activation functions regarding their impact on the training process and on the applicability of the Universal Approximation Theorem. Particularly, significant new results to identify those activation functions which undergo some usual failings (gradient preserving) are presented here. This is the first paper—to the best of the author’s knowledge—that stresses the role of injectivity for activation functions, which has received scant attention in literature but has great incidence on the ANN performance. In this line, a characterization of injective activation functions has been provided related to monotonic functions which satisfy the classical contractive condition as a particular case of Lipschitz functions. A summary table on these is also provided, targeted at documenting how to select the best activation function for each situation. Full article
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12 pages, 406 KiB
Article
Positive Numerical Approximation of Integro-Differential Epidemic Model
by Eleonora Messina, Mario Pezzella and Antonia Vecchio
Axioms 2022, 11(2), 69; https://doi.org/10.3390/axioms11020069 - 9 Feb 2022
Cited by 7 | Viewed by 2638
Abstract
In this paper, we study a dynamically consistent numerical method for the approximation of a nonlinear integro-differential equation modeling an epidemic with age of infection. The discrete scheme is based on direct quadrature methods with Gregory convolution weights and preserves, with no restrictive [...] Read more.
In this paper, we study a dynamically consistent numerical method for the approximation of a nonlinear integro-differential equation modeling an epidemic with age of infection. The discrete scheme is based on direct quadrature methods with Gregory convolution weights and preserves, with no restrictive conditions on the step-length of integration h, some of the essential properties of the continuous system. In particular, the numerical solution is positive and bounded and, in cases of interest in applications, it is monotone. We prove an order of convergence theorem and show by numerical experiments that the discrete final size tends to its continuous equivalent as h tends to zero. Full article
(This article belongs to the Special Issue Differential Equations: Theories, Methods and Modern Applications)
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10 pages, 778 KiB
Article
Statistical Convergence via q-Calculus and a Korovkin’s Type Approximation Theorem
by Mohammad Ayman Mursaleen and Stefano Serra-Capizzano
Axioms 2022, 11(2), 70; https://doi.org/10.3390/axioms11020070 - 9 Feb 2022
Cited by 31 | Viewed by 3052
Abstract
In this paper, we define and study q-statistical limit point, q-statistical cluster point, q-statistically Cauchy, q-strongly Cesàro and statistically ${C}_{1}^{q}$-summable sequences. We establish relationships of q-statistical convergence with q-statistically Cauchy, q-strongly Cesàro and [...] Read more.
In this paper, we define and study q-statistical limit point, q-statistical cluster point, q-statistically Cauchy, q-strongly Cesàro and statistically ${C}_{1}^{q}$-summable sequences. We establish relationships of q-statistical convergence with q-statistically Cauchy, q-strongly Cesàro and statistically ${C}_{1}^{q}$-summable sequences. Further, we apply q-statistical convergence to prove a Korovkin type approximation theorem. Full article
31 pages, 486 KiB
Article
Riemann–Liouville Fractional Sobolev and Bounded Variation Spaces
by Antonio Leaci and Franco Tomarelli
Axioms 2022, 11(1), 30; https://doi.org/10.3390/axioms11010030 - 14 Jan 2022
Cited by 3 | Viewed by 2503
Abstract
We establish some properties of the bilateral Riemann–Liouville fractional derivative ${D}^{s}$. We set the notation, and study the associated Sobolev spaces of fractional order s, denoted by ${W}^{s,1}\left(a,b\right)$, and the fractional [...] Read more.
We establish some properties of the bilateral Riemann–Liouville fractional derivative ${D}^{s}$. We set the notation, and study the associated Sobolev spaces of fractional order s, denoted by ${W}^{s,1}\left(a,b\right)$, and the fractional bounded variation spaces of fractional order s, denoted by $B{V}^{s}\left(a,b\right)$. Examples, embeddings and compactness properties related to these spaces are addressed, aiming to set a functional framework suitable for fractional variational models for image analysis. Full article
17 pages, 5490 KiB
Article
Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals
by Wolf-Dieter Richter
Axioms 2021, 10(4), 340; https://doi.org/10.3390/axioms10040340 - 10 Dec 2021
Cited by 5 | Viewed by 2817
Abstract
We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal [...] Read more.
We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix multiplication. We also extend Euler’s formula and discuss solutions of quadratic equations for the p-norm-antinorm realization of the abstract complex algebraic structure. In addition, we prove an advanced invariance property of certain probability densities. Full article
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11 pages, 288 KiB
Article
An Extension of Beta Function by Using Wiman’s Function
by Rahul Goyal, Shaher Momani, Praveen Agarwal and Michael Th. Rassias
Axioms 2021, 10(3), 187; https://doi.org/10.3390/axioms10030187 - 16 Aug 2021
Cited by 13 | Viewed by 2645
Abstract
The main purpose of this paper is to study extension of the extended beta function by Shadab et al. by using 2-parameter Mittag-Leffler function given by Wiman. In particular, we study some functional relations, integral representation, Mellin transform and derivative formulas for this [...] Read more.
The main purpose of this paper is to study extension of the extended beta function by Shadab et al. by using 2-parameter Mittag-Leffler function given by Wiman. In particular, we study some functional relations, integral representation, Mellin transform and derivative formulas for this extended beta function. Full article
(This article belongs to the Special Issue Special Functions Associated with Fractional Calculus)
10 pages, 285 KiB
Article
Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability
by Andriy Zagorodnyuk and Anna Hihliuk
Axioms 2021, 10(3), 150; https://doi.org/10.3390/axioms10030150 - 7 Jul 2021
Cited by 7 | Viewed by 2390
Abstract
In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the [...] Read more.
In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained. Full article
(This article belongs to the Special Issue Analytic Functions and Nonlinear Functional Analysis)
6 pages, 243 KiB
Article
On the Natural Density of Sets Related to Generalized Fibonacci Numbers of Order r
by Pavel Trojovský
Axioms 2021, 10(3), 144; https://doi.org/10.3390/axioms10030144 - 1 Jul 2021
Cited by 4 | Viewed by 2770
Abstract
For $r\ge 2$ and $a\ge 1$ integers, let ${\left({t}_{n}^{\left(r,a\right)}\right)}_{n\ge 1}$ be the sequence of the $\left(r,a\right)$-generalized Fibonacci numbers which is defined by the recurrence [...] Read more.
For $r\ge 2$ and $a\ge 1$ integers, let ${\left({t}_{n}^{\left(r,a\right)}\right)}_{n\ge 1}$ be the sequence of the $\left(r,a\right)$-generalized Fibonacci numbers which is defined by the recurrence ${t}_{n}^{\left(r,a\right)}={t}_{n-1}^{\left(r,a\right)}+\cdots +{t}_{n-r}^{\left(r,a\right)}$ for $n>r$, with initial values ${t}_{i}^{\left(r,a\right)}=1$, for all $i\in \left[1,r-1\right]$ and ${t}_{r}^{\left(r,a\right)}=a$. In this paper, we shall prove (in particular) that, for any given $r\ge 2$, there exists a positive proportion of positive integers which can not be written as ${t}_{n}^{\left(r,a\right)}$ for any $\left(n,a\right)\in {\mathbb{Z}}_{\ge r+2}×{\mathbb{Z}}_{\ge 1}$. Full article
(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)
13 pages, 711 KiB
Article
Mathematical Analysis of a Fractional COVID-19 Model Applied to Wuhan, Spain and Portugal
by Faïçal Ndaïrou and Delfim F. M. Torres
Axioms 2021, 10(3), 135; https://doi.org/10.3390/axioms10030135 - 27 Jun 2021
Cited by 18 | Viewed by 3437
Abstract
We propose a qualitative analysis of a recent fractional-order COVID-19 model. We start by showing that the model is mathematically and biologically well posed. Then, we give a proof on the global stability of the disease free equilibrium point. Finally, some numerical simulations [...] Read more.
We propose a qualitative analysis of a recent fractional-order COVID-19 model. We start by showing that the model is mathematically and biologically well posed. Then, we give a proof on the global stability of the disease free equilibrium point. Finally, some numerical simulations are performed to ensure stability and convergence of the disease free equilibrium point. Full article
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9 pages, 298 KiB
Article
Singular Integral Neumann Boundary Conditions for Semilinear Elliptic PDEs
by Praveen Agarwal, Jochen Merker and Gregor Schuldt
Axioms 2021, 10(2), 74; https://doi.org/10.3390/axioms10020074 - 24 Apr 2021
Cited by 11 | Viewed by 2549
Abstract
In this article, we discuss semilinear elliptic partial differential equations with singular integral Neumann boundary conditions. Such boundary value problems occur in applications as mathematical models of nonlocal interaction between interior points and boundary points. Particularly, we are interested in the uniqueness of [...] Read more.
In this article, we discuss semilinear elliptic partial differential equations with singular integral Neumann boundary conditions. Such boundary value problems occur in applications as mathematical models of nonlocal interaction between interior points and boundary points. Particularly, we are interested in the uniqueness of solutions to such problems. For the sublinear and subcritical case, we calculate, on the one hand, illustrative, rather explicit solutions in the one-dimensional case. On the other hand, we prove in the general case the existence and—via the strong solution of an integro-PDE with a kind of fractional divergence as a lower order term—uniqueness up to a constant. Full article
(This article belongs to the Special Issue Special Functions Associated with Fractional Calculus)
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24 pages, 1814 KiB
Article
Optimal Control Analysis of Cholera Dynamics in the Presence of Asymptotic Transmission
by Emmanuel A. Bakare and Sarka Hoskova-Mayerova
Axioms 2021, 10(2), 60; https://doi.org/10.3390/axioms10020060 - 12 Apr 2021
Cited by 15 | Viewed by 3252
Abstract
Many mathematical models have explored the dynamics of cholera but none have been used to predict the optimal strategies of the three control interventions (the use of hygiene promotion and social mobilization; the use of treatment by drug/oral re-hydration solution; and the use [...] Read more.
Many mathematical models have explored the dynamics of cholera but none have been used to predict the optimal strategies of the three control interventions (the use of hygiene promotion and social mobilization; the use of treatment by drug/oral re-hydration solution; and the use of safe water, hygiene, and sanitation). The goal here is to develop (deterministic and stochastic) mathematical models of cholera transmission and control dynamics, with the aim of investigating the effect of the three control interventions against cholera transmission in order to find optimal control strategies. The reproduction number ${R}_{p}$ was obtained through the next generation matrix method and sensitivity and elasticity analysis were performed. The global stability of the equilibrium was obtained using the Lyapunov functional. Optimal control theory was applied to investigate the optimal control strategies for controlling the spread of cholera using the combination of control interventions. The Pontryagin’s maximum principle was used to characterize the optimal levels of combined control interventions. The models were validated using numerical experiments and sensitivity analysis was done. Optimal control theory showed that the combinations of the control intervention influenced disease progression. The characterisation of the optimal levels of the multiple control interventions showed the means for minimizing cholera transmission, mortality, and morbidity in finite time. The numerical experiments showed that there are fluctuations and noise due to its dependence on the corresponding population size and that the optimal control strategies to effectively control cholera transmission, mortality, and morbidity was through the combinations of all three control interventions. The developed models achieved the reduction, control, and/or elimination of cholera through incorporating multiple control interventions. Full article
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19 pages, 558 KiB
Article
Qualitative Analyses of Integro-Fractional Differential Equations with Caputo Derivatives and Retardations via the Lyapunov–Razumikhin Method
by Osman Tunç, Özkan Atan, Cemil Tunç and Jen-Chih Yao
Axioms 2021, 10(2), 58; https://doi.org/10.3390/axioms10020058 - 9 Apr 2021
Cited by 39 | Viewed by 2749
Abstract
The purpose of this paper is to investigate some qualitative properties of solutions of nonlinear fractional retarded Volterra integro-differential equations (FrRIDEs) with Caputo fractional derivatives. These properties include uniform stability, asymptotic stability, Mittag–Leffer stability and boundedness. The presented results are proved by defining [...] Read more.
The purpose of this paper is to investigate some qualitative properties of solutions of nonlinear fractional retarded Volterra integro-differential equations (FrRIDEs) with Caputo fractional derivatives. These properties include uniform stability, asymptotic stability, Mittag–Leffer stability and boundedness. The presented results are proved by defining an appropriate Lyapunov function and applying the Lyapunov–Razumikhin method (LRM). Hence, some results that are available in the literature are improved for the FrRIDEs and obtained under weaker conditions via the advantage of the LRM. In order to illustrate the results, two examples are provided. Full article
(This article belongs to the Special Issue Special Issue in Honor of the 60th Birthday of Professor Hong-Kun Xu)
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9 pages, 266 KiB
Article
Pseudo-Lucas Functions of Fractional Degree and Applications
by Clemente Cesarano, Pierpaolo Natalini and Paolo Emilio Ricci
Axioms 2021, 10(2), 51; https://doi.org/10.3390/axioms10020051 - 2 Apr 2021
Cited by 5 | Viewed by 2021
Abstract
In a recent article, the first and second kinds of multivariate Chebyshev polynomials of fractional degree, and the relevant integral repesentations, have been studied. In this article, we introduce the first and second kinds of pseudo-Lucas functions of fractional degree, and we show [...] Read more.
In a recent article, the first and second kinds of multivariate Chebyshev polynomials of fractional degree, and the relevant integral repesentations, have been studied. In this article, we introduce the first and second kinds of pseudo-Lucas functions of fractional degree, and we show possible applications of these new functions. For the first kind, we compute the fractional Newton sum rules of any orthogonal polynomial set starting from the entries of the Jacobi matrix. For the second kind, the representation formulas for the fractional powers of a $r×r$ matrix, already introduced by using the pseudo-Chebyshev functions, are extended to the Lucas case. Full article
(This article belongs to the Special Issue Special Functions Associated with Fractional Calculus)
25 pages, 388 KiB
Article
Fractional Newton–Raphson Method Accelerated with Aitken’s Method
by A. Torres-Hernandez, F. Brambila-Paz, U. Iturrarán-Viveros and R. Caballero-Cruz
Axioms 2021, 10(2), 47; https://doi.org/10.3390/axioms10020047 - 31 Mar 2021
Cited by 14 | Viewed by 4361
Abstract
In the following paper, we present a way to accelerate the speed of convergence of the fractional Newton–Raphson (F N–R) method, which seems to have an order of convergence at least linearly for the case in which the order $\alpha$ of the derivative [...] Read more.
In the following paper, we present a way to accelerate the speed of convergence of the fractional Newton–Raphson (F N–R) method, which seems to have an order of convergence at least linearly for the case in which the order $\alpha$ of the derivative is different from one. A simplified way of constructing the Riemann–Liouville (R–L) fractional operators, fractional integral and fractional derivative is presented along with examples of its application on different functions. Furthermore, an introduction to Aitken’s method is made and it is explained why it has the ability to accelerate the convergence of the iterative methods, in order to finally present the results that were obtained when implementing Aitken’s method in the F N–R method, where it is shown that F N–R with Aitken’s method converges faster than the simple F N–R. Full article
14 pages, 432 KiB
Article
A New Inverted Topp-Leone Distribution: Applications to the COVID-19 Mortality Rate in Two Different Countries
by Ehab M. Almetwally, Randa Alharbi, Dalia Alnagar and Eslam Hossam Hafez
Axioms 2021, 10(1), 25; https://doi.org/10.3390/axioms10010025 - 26 Feb 2021
Cited by 58 | Viewed by 3895
Abstract
This paper aims to find a statistical model for the COVID-19 spread in the United Kingdom and Canada. We used an efficient and superior model for fitting the COVID 19 mortality rates in these countries by specifying an optimal statistical model. A new [...] Read more.
This paper aims to find a statistical model for the COVID-19 spread in the United Kingdom and Canada. We used an efficient and superior model for fitting the COVID 19 mortality rates in these countries by specifying an optimal statistical model. A new lifetime distribution with two-parameter is introduced by a combination of inverted Topp-Leone distribution and modified Kies family to produce the modified Kies inverted Topp-Leone (MKITL) distribution, which covers a lot of application that both the traditional inverted Topp-Leone and the modified Kies provide poor fitting for them. This new distribution has many valuable properties as simple linear representation, hazard rate function, and moment function. We made several methods of estimations as maximum likelihood estimation, least squares estimators, weighted least-squares estimators, maximum product spacing, Cram$e´$r-von Mises estimators, and Anderson-Darling estimators methods are applied to estimate the unknown parameters of MKITL distribution. A numerical result of the Monte Carlo simulation is obtained to assess the use of estimation methods. also, we applied different data sets to the new distribution to assess its performance in modeling data. Full article
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17 pages, 2825 KiB
Article
On Λ-Fractional Viscoelastic Models
by Anastassios K. Lazopoulos and Dimitrios Karaoulanis
Axioms 2021, 10(1), 22; https://doi.org/10.3390/axioms10010022 - 20 Feb 2021
Cited by 5 | Viewed by 2448
Abstract
Λ-Fractional Derivative (Λ-FD) is a new groundbreaking Fractional Derivative (FD) introduced recently in mechanics. This derivative, along with Λ-Transform (Λ-T), provides a reliable alternative to fractional differential equations’ current solving. To put it straightforwardly, Λ-Fractional Derivative might be the only authentic non-local derivative [...] Read more.
Λ-Fractional Derivative (Λ-FD) is a new groundbreaking Fractional Derivative (FD) introduced recently in mechanics. This derivative, along with Λ-Transform (Λ-T), provides a reliable alternative to fractional differential equations’ current solving. To put it straightforwardly, Λ-Fractional Derivative might be the only authentic non-local derivative that exists. In the present article, Λ-Fractional Derivative is used to describe the phenomenon of viscoelasticity, while the whole methodology is demonstrated meticulously. The fractional viscoelastic Zener model is studied, for relaxation as well as for creep. Interesting results are extracted and compared to other methodologies showing the value of the pre-mentioned method. Full article
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8 pages, 228 KiB
Article
Global Optimization and Common Best Proximity Points for Some Multivalued Contractive Pairs of Mappings
by Pradip Debnath and Hari Mohan Srivastava
Axioms 2020, 9(3), 102; https://doi.org/10.3390/axioms9030102 - 7 Sep 2020
Cited by 20 | Viewed by 2630
Abstract
In this paper, we study a problem of global optimization using common best proximity point of a pair of multivalued mappings. First, we introduce a multivalued Banach-type contractive pair of mappings and establish criteria for the existence of their common best proximity point. [...] Read more.
In this paper, we study a problem of global optimization using common best proximity point of a pair of multivalued mappings. First, we introduce a multivalued Banach-type contractive pair of mappings and establish criteria for the existence of their common best proximity point. Next, we put forward the concept of multivalued Kannan-type contractive pair and also the concept of weak $Δ$-property to determine the existence of common best proximity point for such a pair of maps. Full article
18 pages, 345 KiB
Article
An Accelerated Extragradient Method for Solving Pseudomonotone Equilibrium Problems with Applications
by Nopparat Wairojjana, Habib ur Rehman, Ioannis K. Argyros and Nuttapol Pakkaranang
Axioms 2020, 9(3), 99; https://doi.org/10.3390/axioms9030099 - 17 Aug 2020
Cited by 8 | Viewed by 3313
Abstract
Several methods have been put forward to solve equilibrium problems, in which the two-step extragradient method is very useful and significant. In this article, we propose a new extragradient-like method to evaluate the numerical solution of the pseudomonotone equilibrium in real Hilbert space. [...] Read more.
Several methods have been put forward to solve equilibrium problems, in which the two-step extragradient method is very useful and significant. In this article, we propose a new extragradient-like method to evaluate the numerical solution of the pseudomonotone equilibrium in real Hilbert space. This method uses a non-monotonically stepsize technique based on local bifunction values and Lipschitz-type constants. Furthermore, we establish the weak convergence theorem for the suggested method and provide the applications of our results. Finally, several experimental results are reported to see the performance of the proposed method. Full article
(This article belongs to the Special Issue Iterative Processes for Nonlinear Problems with Applications)
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18 pages, 2600 KiB
Article
The Modified Helmholtz Equation on a Regular Hexagon—The Symmetric Dirichlet Problem
by Konstantinos Kalimeris and Athanassios S. Fokas
Axioms 2020, 9(3), 89; https://doi.org/10.3390/axioms9030089 - 28 Jul 2020
Cited by 1 | Viewed by 3051
Abstract
Using the unified transform, also known as the Fokas method, we analyse the modified Helmholtz equation in the regular hexagon with symmetric Dirichlet boundary conditions; namely, the boundary value problem where the trace of the solution is given by the same function on [...] Read more.
Using the unified transform, also known as the Fokas method, we analyse the modified Helmholtz equation in the regular hexagon with symmetric Dirichlet boundary conditions; namely, the boundary value problem where the trace of the solution is given by the same function on each side of the hexagon. We show that if this function is odd, then this problem can be solved in closed form; numerical verification is also provided. Full article
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18 pages, 450 KiB
Article
Facility Location Selection for B-Schools in Indian Context: A Multi-Criteria Group Decision Based Analysis
by Sanjib Biswas and Dragan Pamucar
Axioms 2020, 9(3), 77; https://doi.org/10.3390/axioms9030077 - 8 Jul 2020
Cited by 28 | Viewed by 4417
Abstract
Facility location is one of the critical strategic decisions for any organization. It not only carries the organization’s identity but also connects the point of origin and point of consumption. In the case of higher educational institutions, specifically B-Schools, location is one of [...] Read more.
Facility location is one of the critical strategic decisions for any organization. It not only carries the organization’s identity but also connects the point of origin and point of consumption. In the case of higher educational institutions, specifically B-Schools, location is one of the primary concerns for potential students and their parents while selecting an institution for pursuing higher education. There has been a plethora of research conducted to investigate the factors influencing the B-School selection decision-making. However, location as a standalone factor has not been widely studied. This paper aims to explore various location selection criteria from the viewpoint of the candidates who aspire to enroll in B-Schools. We apply an integrated group decision-making framework of pivot pairwise relative criteria importance assessment (PIPRECIA), and level-based weight assessment LBWA is used wherein a group of student counselors, admission executives, and educators from India has participated. The factors which influence the location decision are identified through qualitative opinion analysis. The results show that connectivity and commutation are the dominant issues. Full article
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17 pages, 256 KiB
Article
Functors among Relational Variants of Categories Related to L-Fuzzy Partitions, L-Fuzzy Pretopological Spaces and L-Fuzzy Closure Spaces
by Jiří Močkoř
Axioms 2020, 9(2), 63; https://doi.org/10.3390/axioms9020063 - 2 Jun 2020
Cited by 6 | Viewed by 2204
Abstract
Various types of topological and closure operators are significantly used in fuzzy theory and applications. Although they are different operators, in some cases it is possible to transform an operator of one type into another. This in turn makes it possible to transform [...] Read more.
Various types of topological and closure operators are significantly used in fuzzy theory and applications. Although they are different operators, in some cases it is possible to transform an operator of one type into another. This in turn makes it possible to transform results relating to an operator of one type into results relating to another operator. In the paper relationships among 15 categories of modifications of topological L-valued operators, including Čech closure or interior L-valued operators, L-fuzzy pretopological and L-fuzzy co-pretopological operators, L-valued fuzzy relations, upper and lower F-transforms and spaces with fuzzy partitions are investigated. The common feature of these categories is that their morphisms are various L-fuzzy relations and not only maps. We prove the existence of 23 functors among these categories, which represent transformation processes of one operator into another operator, and we show how these transformation processes can be mutually combined. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Applications)
13 pages, 346 KiB
Article
Initial Value Problem For Nonlinear Fractional Differential Equations With ψ-Caputo Derivative Via Monotone Iterative Technique
by Choukri Derbazi, Zidane Baitiche, Mouffak Benchohra and Alberto Cabada
Axioms 2020, 9(2), 57; https://doi.org/10.3390/axioms9020057 - 21 May 2020
Cited by 36 | Viewed by 3795
Abstract
In this article, we discuss the existence and uniqueness of extremal solutions for nonlinear initial value problems of fractional differential equations involving the $ψ$ -Caputo derivative. Moreover, some uniqueness results are obtained. Our results rely on the standard tools of functional analysis. More [...] Read more.
In this article, we discuss the existence and uniqueness of extremal solutions for nonlinear initial value problems of fractional differential equations involving the $ψ$ -Caputo derivative. Moreover, some uniqueness results are obtained. Our results rely on the standard tools of functional analysis. More precisely we apply the monotone iterative technique combined with the method of upper and lower solutions to establish sufficient conditions for existence as well as the uniqueness of extremal solutions to the initial value problem. An illustrative example is presented to point out the applicability of our main results. Full article
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16 pages, 291 KiB
Article
Aristotle’s Syllogistic as a Deductive System
by Piotr Kulicki
Axioms 2020, 9(2), 56; https://doi.org/10.3390/axioms9020056 - 19 May 2020
Cited by 4 | Viewed by 6869
Abstract
Aristotle’s syllogistic is the first ever deductive system. After centuries, Aristotle’s ideas are still interesting for logicians who develop Aristotle’s work and draw inspiration from his results and even more from his methods. In the paper we discuss the essential elements of the [...] Read more.
Aristotle’s syllogistic is the first ever deductive system. After centuries, Aristotle’s ideas are still interesting for logicians who develop Aristotle’s work and draw inspiration from his results and even more from his methods. In the paper we discuss the essential elements of the Aristotelian system of syllogistic and Łukasiewicz’s reconstruction of it based on the tools of modern formal logic. We pay special attention to the notion of completeness of a deductive system as discussed by both authors. We describe in detail how completeness can be defined and proved with the use of an axiomatic refutation system. Finally, we apply this methodology to different axiomatizations of syllogistic presented by Łukasiewicz, Lemmon and Shepherdson. Full article
33 pages, 429 KiB
Review
The Generalized Hypergeometric Structure of the Ward Identities of CFT’s in Momentum Space in d > 2
by Claudio Corianò and Matteo Maria Maglio
Axioms 2020, 9(2), 54; https://doi.org/10.3390/axioms9020054 - 14 May 2020
Cited by 5 | Viewed by 2704
Abstract
We review the emergence of hypergeometric structures (of F4 Appell functions) from the conformal Ward identities (CWIs) in conformal field theories (CFTs) in dimensions d > 2. We illustrate the case of scalar 3- and 4-point functions. 3-point functions are associated to [...] Read more.
We review the emergence of hypergeometric structures (of F4 Appell functions) from the conformal Ward identities (CWIs) in conformal field theories (CFTs) in dimensions d > 2. We illustrate the case of scalar 3- and 4-point functions. 3-point functions are associated to hypergeometric systems with four independent solutions. For symmetric correlators, they can be expressed in terms of a single 3K integral—functions of quadratic ratios of momenta—which is a parametric integral of three modified Bessel K functions. In the case of scalar 4-point functions, by requiring the correlator to be conformal invariant in coordinate space as well as in some dual variables (i.e., dual conformal invariant), its explicit expression is also given by a 3K integral, or as a linear combination of Appell functions which are now quartic ratios of momenta. Similar expressions have been obtained in the past in the computation of an infinite class of planar ladder (Feynman) diagrams in perturbation theory, which, however, do not share the same (dual conformal/conformal) symmetry of our solutions. We then discuss some hypergeometric functions of 3 variables, which define 8 particular solutions of the CWIs and correspond to Lauricella functions. They can also be combined in terms of 4K integral and appear in an asymptotic description of the scalar 4-point function, in special kinematical limits. Full article
(This article belongs to the Special Issue Geometric Analysis and Mathematical Physics)
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13 pages, 272 KiB
Article
Integral Representation of Coherent Lower Previsions by Super-Additive Integrals
Axioms 2020, 9(2), 43; https://doi.org/10.3390/axioms9020043 - 23 Apr 2020
Cited by 8 | Viewed by 2580
Abstract
Coherent lower previsions generalize the expected values and they are defined on the class of all real random variables on a finite non-empty set. Well known construction of coherent lower previsions by means of lower probabilities, or by means of super-modular capacities-based Choquet [...] Read more.
Coherent lower previsions generalize the expected values and they are defined on the class of all real random variables on a finite non-empty set. Well known construction of coherent lower previsions by means of lower probabilities, or by means of super-modular capacities-based Choquet integrals, do not cover this important class of functionals on real random variables. In this paper, a new approach to the construction of coherent lower previsions acting on a finite space is proposed, exemplified and studied. It is based on special decomposition integrals recently introduced by Even and Lehrer, in our case the considered decomposition systems being single collections and thus called collection integrals. In special case when these integrals, defined for non-negative random variables only, are shift-invariant, we extend them to the class of all real random variables, thus obtaining so called super-additive integrals. Our proposed construction can be seen then as a normalized super-additive integral. We discuss and exemplify several particular cases, for example, when collections determine a coherent lower prevision for any monotone set function. For some particular collections, only particular set functions can be considered for our construction. Conjugated coherent upper previsions are also considered. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Applications)
14 pages, 463 KiB
Article
Quasinormal Modes of Charged Black Holes in Higher-Dimensional Einstein-Power-Maxwell Theory
by Grigoris Panotopoulos
Axioms 2020, 9(1), 33; https://doi.org/10.3390/axioms9010033 - 24 Mar 2020
Cited by 16 | Viewed by 3524
Abstract
We compute the quasinormal frequencies for scalar perturbations of charged black holes in five-dimensional Einstein-power-Maxwell theory. The impact on the spectrum of the electric charge of the black holes, of the angular degree, of the overtone number, and of the mass of the [...] Read more.
We compute the quasinormal frequencies for scalar perturbations of charged black holes in five-dimensional Einstein-power-Maxwell theory. The impact on the spectrum of the electric charge of the black holes, of the angular degree, of the overtone number, and of the mass of the test scalar field is investigated in detail. The quasinormal spectra in the eikonal limit are computed as well for several different space-time dimensionalities. Full article
(This article belongs to the Special Issue Theory and Mathematical Aspects of Black Holes)
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7 pages, 265 KiB
Article
On Grothendieck Sets
by Juan Carlos Ferrando, Salvador López-Alfonso and Manuel López-Pellicer
Axioms 2020, 9(1), 34; https://doi.org/10.3390/axioms9010034 - 24 Mar 2020
Cited by 5 | Viewed by 2581
Abstract
We call a subset $M$ of an algebra of sets $A$ a Grothendieck set for the Banach space $b a ( A )$ of bounded finitely additive scalar-valued measures on $A$ equipped with the variation norm if each sequence [...] Read more.
We call a subset $M$ of an algebra of sets $A$ a Grothendieck set for the Banach space $b a ( A )$ of bounded finitely additive scalar-valued measures on $A$ equipped with the variation norm if each sequence $μ n n = 1 ∞$ in $b a ( A )$ which is pointwise convergent on $M$ is weakly convergent in $b a ( A )$ , i.e., if there is $μ ∈ b a A$ such that $μ n A → μ A$ for every $A ∈ M$ then $μ n → μ$ weakly in $b a ( A )$ . A subset $M$ of an algebra of sets $A$ is called a Nikodým set for $b a ( A )$ if each sequence $μ n n = 1 ∞$ in $b a ( A )$ which is pointwise bounded on $M$ is bounded in $b a ( A )$ . We prove that if $Σ$ is a $σ$ -algebra of subsets of a set $Ω$ which is covered by an increasing sequence $Σ n : n ∈ N$ of subsets of $Σ$ there exists $p ∈ N$ such that $Σ p$ is a Grothendieck set for $b a ( A )$ . This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a $σ$ -algebra $Σ$ is covered by an increasing sequence $Σ n : n ∈ N$ of subsets, there is $p ∈ N$ such that $Σ p$ is a Nikodým set for $b a Σ$ . This also refines the Grothendieck result stating that for each $σ$ -algebra $Σ$ the Banach space $ℓ ∞ Σ$ is a Grothendieck space. Some applications to classic Banach space theory are given. Full article