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Volume 9
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Mathematics, Volume 9, Issue 1 (January-1 2021) – 107 articles

Cover Story (view full-size image): Often the positivity of models used in financial mathematics is a desired property. Whether the trajectories of the stochastic model reach positive or negative values depends on the model parameters. For example, the positivity of a certain class of Chan–Karolyi–Longstaff–Sanders models is dependent on the initial value of a process. View this paper.
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13 pages, 302 KiB  
Article
Characterizations of Pareto-Nash Equilibria for Multiobjective Potential Population Games
by Guanghui Yang, Chanchan Li, Jinxiu Pi, Chun Wang, Wenjun Wu and Hui Yang
Mathematics 2021, 9(1), 99; https://doi.org/10.3390/math9010099 - 5 Jan 2021
Viewed by 2739
Abstract
This paper studies the characterizations of (weakly) Pareto-Nash equilibria for multiobjective population games with a vector-valued potential function called multiobjective potential population games, where agents synchronously maximize multiobjective functions with finite strategies via a partial order on the criteria-function set. In such games, [...] Read more.
This paper studies the characterizations of (weakly) Pareto-Nash equilibria for multiobjective population games with a vector-valued potential function called multiobjective potential population games, where agents synchronously maximize multiobjective functions with finite strategies via a partial order on the criteria-function set. In such games, multiobjective payoff functions are equal to the transpose of the Jacobi matrix of its potential function. For multiobjective potential population games, based on Kuhn-Tucker conditions of multiobjective optimization, a strongly (weakly) Kuhn-Tucker state is introduced for its vector-valued potential function and it is proven that each strongly (weakly) Kuhn-Tucker state is one (weakly) Pareto-Nash equilibrium. The converse is obtained for multiobjective potential population games with two strategies by utilizing Tucker’s Theorem of the alternative and Motzkin’s one of linear systems. Precisely, each (weakly) Pareto-Nash equilibrium is equivalent to a strongly (weakly) Kuhn-Tucker state for multiobjective potential population games with two strategies. These characterizations by a vector-valued approach are more comprehensive than an additive weighted method. Multiobjective potential population games are the extension of population potential games from a single objective to multiobjective cases. These novel results provide a theoretical basis for further computing (weakly) Pareto-Nash equilibria of multiobjective potential population games and their practical applications. Full article
(This article belongs to the Section Computational and Applied Mathematics)
17 pages, 1510 KiB  
Article
A Random Walk Model for Spatial Galaxy Distribution
by Vladimir V. Uchaikin, Vladimir A. Litvinov, Elena V. Kozhemyakina and Ilya I. Kozhemyakin
Mathematics 2021, 9(1), 98; https://doi.org/10.3390/math9010098 - 5 Jan 2021
Cited by 1 | Viewed by 2445
Abstract
A new statistical model of spatial distribution of observed galaxies is described. Statistical correlations are involved by means of Markov chain ensembles, whose parameters are extracted from the observable power spectrum by adopting of the Uchaikin–Zolotarev ansatz. Markov chain trajectories with the Lévy–Feldheim [...] Read more.
A new statistical model of spatial distribution of observed galaxies is described. Statistical correlations are involved by means of Markov chain ensembles, whose parameters are extracted from the observable power spectrum by adopting of the Uchaikin–Zolotarev ansatz. Markov chain trajectories with the Lévy–Feldheim distributed step lengths form the set of nodes imitating the positions of galaxy. The model plausibly reproduces the two-point correlation functions, cell-count data and some other important properties. It can effectively be used in the post-processing of astronomical data for cosmological studies. Full article
(This article belongs to the Special Issue Analytical Methods and Convergence in Probability with Applications)
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25 pages, 7445 KiB  
Article
Inelastic Deformable Image Registration (i-DIR): Capturing Sliding Motion through Automatic Detection of Discontinuities
by Carlos I. Andrade and Daniel E. Hurtado
Mathematics 2021, 9(1), 97; https://doi.org/10.3390/math9010097 - 5 Jan 2021
Cited by 1 | Viewed by 2553
Abstract
Deformable image registration (DIR) is an image-analysis method with a broad range of applications in biomedical sciences. Current applications of DIR on computed-tomography (CT) images of the lung and other organs under deformation suffer from large errors and artifacts due to the inability [...] Read more.
Deformable image registration (DIR) is an image-analysis method with a broad range of applications in biomedical sciences. Current applications of DIR on computed-tomography (CT) images of the lung and other organs under deformation suffer from large errors and artifacts due to the inability of standard DIR methods to capture sliding between interfaces, as standard transformation models cannot adequately handle discontinuities. In this work, we aim at creating a novel inelastic deformable image registration (i-DIR) method that automatically detects sliding surfaces and that is capable of handling sliding discontinuous motion. Our method relies on the introduction of an inelastic regularization term in the DIR formulation, where sliding is characterized as an inelastic shear strain. We validate the i-DIR by studying synthetic image datasets with strong sliding motion, and compare its results against two other elastic DIR formulations using landmark analysis. Further, we demonstrate the applicability of the i-DIR method to medical CT images by registering lung CT images. Our results show that the i-DIR method delivers accurate estimates of a local lung strain that are similar to fields reported in the literature, and that do not exhibit spurious oscillatory patterns typically observed in elastic DIR methods. We conclude that the i-DIR method automatically locates regions of sliding that arise in the dorsal pleural cavity, delivering significantly smaller errors than traditional elastic DIR methods. Full article
(This article belongs to the Special Issue Mathematical Modeling in Biomechanics and Mechanobiology)
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15 pages, 7840 KiB  
Article
Mathematical Modelling, Analysis and Control of a Three to Five-Phase Matrix Converter for Minimal Switching Losses
by Kotb B. Tawfiq, Mohamed N. Ibrahim, Hegazy Rezk, Elwy E. El-kholy and Peter Sergeant
Mathematics 2021, 9(1), 96; https://doi.org/10.3390/math9010096 - 5 Jan 2021
Cited by 13 | Viewed by 3027
Abstract
The interest in motor drive systems with a number of phases greater than three has increased, mainly in high-power industrial fields due to their advantages compared with three-phase drive systems. In this paper, comprehensive mathematical modeling of a five-phase matrix converter (MC) is [...] Read more.
The interest in motor drive systems with a number of phases greater than three has increased, mainly in high-power industrial fields due to their advantages compared with three-phase drive systems. In this paper, comprehensive mathematical modeling of a five-phase matrix converter (MC) is introduced. Besides that, the direct and indirect space vector modulation (SVM) control methods are compared and analyzed. Furthermore, a mathematical model for the MC with the transformation between the indirect and direct topology is constructed. The indirect technique is used to control the five-phase MC with minimum switching losses. In this technique, SVM deals with a five-phase MC as a virtual two-stage converter with a virtual DC link (i.e., rectifier and inverter stages). The voltage gain is limited to a value of 0.79. Moreover, to analyze the effectiveness of the control technique and the advantages of the MC, a static R-L load is employed. However, the load can also be an industrial load, such as hospital pumping or vehicular applications. The presented analysis proves that the MC gives a wide range of output frequencies, and it has the ability to control the input displacement factor and the output voltage magnitude. In addition, the absence of the massive DC link capacitors is an essential feature for the MC, resulting in increased reliability and a reduced size converter. Eventually, an experimental validation is conducted on a static load to validate the presented model and the control method. It is observed that good matching between the simulation and the experimental results is achieved. Full article
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21 pages, 6601 KiB  
Article
Investment Selection Based on Bonferroni Mean under Generalized Probabilistic Hesitant Fuzzy Environments
by Wenying Wu, Zhiwei Ni, Feifei Jin, Jian Wu, Ying Li and Ping Li
Mathematics 2021, 9(1), 107; https://doi.org/10.3390/math9010107 - 5 Jan 2021
Cited by 4 | Viewed by 2141
Abstract
In investment selection problems, the existence of contingency and uncertainty may result in the loss of attribute information. Then, how to make proper investment decision-making will be a tricky proposition. In this work, a multiattribute group decision making (MAGDM) method based on the [...] Read more.
In investment selection problems, the existence of contingency and uncertainty may result in the loss of attribute information. Then, how to make proper investment decision-making will be a tricky proposition. In this work, a multiattribute group decision making (MAGDM) method based on the generalized probabilistic hesitant fuzzy Bonferroni mean (GPHFBM) operator is constructed, which enables decision-makers to select the proper parameters in decision-making process. Firstly, the GPHFBM operator is proposed by combining the Bonferroni mean operator and Archimedean norm. Secondly, five excellent properties of the GPHFBM operator are discussed in detail. In view of applications, we further develop some special aggregation operators for GPHFBM with the various values of parameters b, d and additive operators g(t). Finally, we propose a probabilistic hesitant fuzzy MAGDM method based on the GPHFBM operator to analyze the aggregated information. A case study of the investment of social insurance funds is given to depict the validity and reasonability of the proposed method. Ultimately, the company X4 is selected as the investment company with the best comprehensive indicator. Full article
(This article belongs to the Section Fuzzy Sets, Systems and Decision Making)
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40 pages, 643 KiB  
Review
A Guide to Special Functions in Fractional Calculus
by Virginia Kiryakova
Mathematics 2021, 9(1), 106; https://doi.org/10.3390/math9010106 - 5 Jan 2021
Cited by 43 | Viewed by 4937
Abstract
Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project. Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as [...] Read more.
Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project. Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as a “Guide to the Functions”. This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G-function and, specially, of the generalized hypergeometric functions pFq. These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to “new” classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC). Generally, under this notion, we have in mind the Fox H-functions, their most widely used cases of the Wright generalized hypergeometric functions pΨq and, in particular, the Mittag–Leffler type functions, among them the “Queen function of fractional calculus”, the Mittag–Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author’s works for more than 30 years, and support the wide spreading and important role of these functions by several examples. Full article
(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)
10 pages, 260 KiB  
Review
Highly Efficient Robust and Stable M-Estimates of Location
by Georgy Shevlyakov
Mathematics 2021, 9(1), 105; https://doi.org/10.3390/math9010105 - 5 Jan 2021
Cited by 3 | Viewed by 1975
Abstract
This article is partially a review and partially a contribution. The classical two approaches to robustness, Huber’s minimax and Hampel’s based on influence functions, are reviewed with the accent on distribution classes of a non-neighborhood nature. Mainly, attention is paid to the minimax [...] Read more.
This article is partially a review and partially a contribution. The classical two approaches to robustness, Huber’s minimax and Hampel’s based on influence functions, are reviewed with the accent on distribution classes of a non-neighborhood nature. Mainly, attention is paid to the minimax Huber’s M-estimates of location designed for the classes with bounded quantiles and Meshalkin-Shurygin’s stable M-estimates. The contribution is focused on the comparative performance evaluation study of these estimates, together with the classical robust M-estimates under the normal, double-exponential (Laplace), Cauchy, and contaminated normal (Tukey gross error) distributions. The obtained results are as follows: (i) under the normal, double-exponential, Cauchy, and heavily-contaminated normal distributions, the proposed robust minimax M-estimates outperform the classical Huber’s and Hampel’s M-estimates in asymptotic efficiency; (ii) in the case of heavy-tailed double-exponential and Cauchy distributions, the Meshalkin-Shurygin’s radical stable M-estimate also outperforms the classical robust M-estimates; (iii) for moderately contaminated normal, the classical robust estimates slightly outperform the proposed minimax M-estimates. Several directions of future works are enlisted. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)
14 pages, 297 KiB  
Article
An Inverse Mixed Impedance Scattering Problem in a Chiral Medium
by Evagelia S. Athanasiadou
Mathematics 2021, 9(1), 104; https://doi.org/10.3390/math9010104 - 5 Jan 2021
Cited by 5 | Viewed by 1910
Abstract
An inverse scattering problem of time-harmonic chiral electromagnetic waves for a buried partially coated object was studied. The buried object was embedded in a piecewise isotropic homogeneous background chiral material. On the boundary of the scattering object, the total electromagnetic field satisfied perfect [...] Read more.
An inverse scattering problem of time-harmonic chiral electromagnetic waves for a buried partially coated object was studied. The buried object was embedded in a piecewise isotropic homogeneous background chiral material. On the boundary of the scattering object, the total electromagnetic field satisfied perfect conductor and impedance boundary conditions. A modified linear sampling method, which originated from the chiral reciprocity gap functional, was employed for reconstruction of the shape of the buried object without requiring any a priori knowledge of the material properties of the scattering object. Furthermore, a characterization of the impedance of the object’s surface was determined. Full article
(This article belongs to the Special Issue Numerical Linear Algebra and the Applications)
17 pages, 4208 KiB  
Article
Intelligent Agents in Co-Evolving Knowledge Networks
by Evangelos Ioannidis, Nikos Varsakelis and Ioannis Antoniou
Mathematics 2021, 9(1), 103; https://doi.org/10.3390/math9010103 - 5 Jan 2021
Cited by 6 | Viewed by 3674
Abstract
We extend the agent-based models for knowledge diffusion in networks, restricted to random mindless interactions and to “frozen” (static) networks, in order to take into account intelligent agents and network co-evolution. Intelligent agents make decisions under bounded rationality. This is the [...] Read more.
We extend the agent-based models for knowledge diffusion in networks, restricted to random mindless interactions and to “frozen” (static) networks, in order to take into account intelligent agents and network co-evolution. Intelligent agents make decisions under bounded rationality. This is the key distinction of intelligent interacting agents compared to mindless colliding molecules, involved in the usual diffusion mechanism resulting from accidental collisions. The co-evolution of link weights and knowledge levels is modeled at the local microscopic level of “agent-to-agent” interaction. Our network co-evolution model is actually a “learning mechanism”, where weight updates depend on the previous values of both weights and knowledge levels. The goal of our work is to explore the impact of (a) the intelligence of the agents, modeled by the selection-decision rule for knowledge acquisition, (b) the innovation rate of the agents, (c) the number of “top innovators” and (d) the network size. We find that rational intelligent agents transform the network into a “centralized world”, reducing the entropy of their selections-decisions for knowledge acquisition. In addition, we find that the average knowledge, as well as the “knowledge inequality”, grow exponentially. Full article
(This article belongs to the Section Network Science)
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15 pages, 369 KiB  
Article
Kähler–Einstein Metrics on Smooth Fano Symmetric Varieties with Picard Number One
by Jae-Hyouk Lee, Kyeong-Dong Park and Sungmin Yoo
Mathematics 2021, 9(1), 102; https://doi.org/10.3390/math9010102 - 5 Jan 2021
Cited by 3 | Viewed by 6571
Abstract
Symmetric varieties are normal equivarient open embeddings of symmetric homogeneous spaces, and they are interesting examples of spherical varieties. We prove that all smooth Fano symmetric varieties with Picard number one admit Kähler–Einstein metrics by using a combinatorial criterion for K-stability of Fano [...] Read more.
Symmetric varieties are normal equivarient open embeddings of symmetric homogeneous spaces, and they are interesting examples of spherical varieties. We prove that all smooth Fano symmetric varieties with Picard number one admit Kähler–Einstein metrics by using a combinatorial criterion for K-stability of Fano spherical varieties obtained by Delcroix. For this purpose, we present their algebraic moment polytopes and compute the barycenter of each moment polytope with respect to the Duistermaat–Heckman measure. Full article
(This article belongs to the Section Algebra, Geometry and Topology)
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1 pages, 162 KiB  
Correction
Correction: Ricceri, B. A Class of Equations with Three Solutions. Mathematics 2020, 8, 478
by Biagio Ricceri
Mathematics 2021, 9(1), 101; https://doi.org/10.3390/math9010101 - 5 Jan 2021
Viewed by 1550
Abstract
The author wishes to make the following correction to this paper [...] Full article
(This article belongs to the Special Issue Nonlinear Functional Analysis and Its Applications)
9 pages, 460 KiB  
Article
A Stochastic Lomax Diffusion Process: Statistical Inference and Application
by Ahmed Nafidi, Ilyasse Makroz and Ramón Gutiérrez Sánchez
Mathematics 2021, 9(1), 100; https://doi.org/10.3390/math9010100 - 5 Jan 2021
Cited by 4 | Viewed by 2781
Abstract
In this paper, we discuss a new stochastic diffusion process in which the trend function is proportional to the Lomax density function. This distribution arises naturally in the studies of the frequency of extremely rare events. We first consider the probabilistic characteristics of [...] Read more.
In this paper, we discuss a new stochastic diffusion process in which the trend function is proportional to the Lomax density function. This distribution arises naturally in the studies of the frequency of extremely rare events. We first consider the probabilistic characteristics of the proposed model, including its analytic expression as the unique solution to a stochastic differential equation, the transition probability density function together with the conditional and unconditional trend functions. Then, we present a method to address the problem of parameter estimation using maximum likelihood with discrete sampling. This estimation requires the solution of a non-linear equation, which is achieved via the simulated annealing method. Finally, we apply the proposed model to a real-world example concerning adolescent fertility rate in Morocco. Full article
(This article belongs to the Section Probability and Statistics)
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16 pages, 337 KiB  
Article
On Admissible Orders on the Set of Discrete Fuzzy Numbers for Application in Decision Making Problems
by Juan Vicente Riera, Sebastia Massanet, Humberto Bustince and Javier Fernandez
Mathematics 2021, 9(1), 95; https://doi.org/10.3390/math9010095 - 4 Jan 2021
Cited by 6 | Viewed by 2106
Abstract
The study of orders is a constantly evolving topic, not only for its interest from a theoretical point of view, but also for its possible applications. Recently, one of the hot lines of research has been the construction of admissible orders in different [...] Read more.
The study of orders is a constantly evolving topic, not only for its interest from a theoretical point of view, but also for its possible applications. Recently, one of the hot lines of research has been the construction of admissible orders in different frameworks. Following this direction, this paper presents a new representation theorem in the field of discrete fuzzy numbers that enables the construction of two families of admissible orders in the set of discrete fuzzy numbers whose support is a closed interval of a finite chain, leading to the first admissible orders introduced in this framework. Full article
(This article belongs to the Special Issue Analytical and Algebraic Aspects of Decision Making)
10 pages, 357 KiB  
Article
Approximating Correlation Matrices Using Stochastic Lie Group Methods
by Michelle Muniz, Matthias Ehrhardt and Michael Günther
Mathematics 2021, 9(1), 94; https://doi.org/10.3390/math9010094 - 4 Jan 2021
Cited by 8 | Viewed by 3498
Abstract
Specifying time-dependent correlation matrices is a problem that occurs in several important areas of finance and risk management. The goal of this work is to tackle this problem by applying techniques of geometric integration in financial mathematics, i.e., to combine two fields of [...] Read more.
Specifying time-dependent correlation matrices is a problem that occurs in several important areas of finance and risk management. The goal of this work is to tackle this problem by applying techniques of geometric integration in financial mathematics, i.e., to combine two fields of numerical mathematics that have not been studied yet jointly. Based on isospectral flows we create valid time-dependent correlation matrices, so called correlation flows, by solving a stochastic differential equation (SDE) that evolves in the special orthogonal group. Since the geometric structure of the special orthogonal group needs to be preserved we use stochastic Lie group integrators to solve this SDE. An application example is presented to illustrate this novel methodology. Full article
(This article belongs to the Section Financial Mathematics)
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17 pages, 535 KiB  
Article
Towards Better Concordance among Contextualized Evaluations in FAST-GDM Problems
by Marcelo Loor, Ana Tapia-Rosero and Guy De Tré
Mathematics 2021, 9(1), 93; https://doi.org/10.3390/math9010093 - 4 Jan 2021
Cited by 3 | Viewed by 2544
Abstract
A flexible attribute-set group decision-making (FAST-GDM) problem consists in finding the most suitable option(s) out of the options under consideration, with a general agreement among a heterogeneous group of experts who can focus on different attributes to evaluate those options. An open challenge [...] Read more.
A flexible attribute-set group decision-making (FAST-GDM) problem consists in finding the most suitable option(s) out of the options under consideration, with a general agreement among a heterogeneous group of experts who can focus on different attributes to evaluate those options. An open challenge in FAST-GDM problems is to design consensus reaching processes (CRPs) by which the participants can perform evaluations with a high level of consensus. To address this challenge, a novel algorithm for reaching consensus is proposed in this paper. By means of the algorithm, called FAST-CR-XMIS, a participant can reconsider his/her evaluations after studying the most influential samples that have been shared by others through contextualized evaluations. Since exchanging those samples may make participants’ understandings more like each other, an increase of the level of consensus is expected. A simulation of a CRP where contextualized evaluations of newswire stories are characterized as augmented intuitionistic fuzzy sets (AIFS) shows how FAST-CR-XMIS can increase the level of consensus among the participants during the CRP. Full article
(This article belongs to the Special Issue Intuitionistic Fuzzy Sets and Applications)
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8 pages, 259 KiB  
Article
Fixed Point Theory Using ψ Contractive Mapping in C ∗ -Algebra Valued B-Metric Space
by Rahmah Mustafa, Saleh Omran and Quang Ngoc Nguyen
Mathematics 2021, 9(1), 92; https://doi.org/10.3390/math9010092 - 4 Jan 2021
Cited by 9 | Viewed by 2882
Abstract
In this paper, fixed point theorems using ψ contractive mapping in C∗-algebra valued b-metric space are introduced. By stating multiple scenarios that illustrate the application domains, we demonstrate several applications from the obtained results. In particular, we begin with the definition [...] Read more.
In this paper, fixed point theorems using ψ contractive mapping in C∗-algebra valued b-metric space are introduced. By stating multiple scenarios that illustrate the application domains, we demonstrate several applications from the obtained results. In particular, we begin with the definition of the positive function and then recall some properties of the function that lay the fundamental basis for the research. We then study some fixed point theorems in the C∗-algebra valued b-metric space using a positive function. Full article
(This article belongs to the Special Issue Abstract Metric Spaces: Usefulness in Statistics and Fixed Point Theory)
23 pages, 595 KiB  
Article
Rigorous Mathematical Investigation of a Nonlocal and Nonlinear Second-Order Anisotropic Reaction-Diffusion Model: Applications on Image Segmentation
by Costică Moroşanu and Silviu Pavăl
Mathematics 2021, 9(1), 91; https://doi.org/10.3390/math9010091 - 4 Jan 2021
Cited by 11 | Viewed by 1970
Abstract
In this paper we are addressing two main topics, as follows. First, a rigorous qualitative study is elaborated for a second-order parabolic problem, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as non-homogeneous Cauchy-Neumann boundary conditions. Under certain assumptions on [...] Read more.
In this paper we are addressing two main topics, as follows. First, a rigorous qualitative study is elaborated for a second-order parabolic problem, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as non-homogeneous Cauchy-Neumann boundary conditions. Under certain assumptions on the input data: f(t,x), w(t,x) and v0(x), we prove the well-posedness (the existence, a priori estimates, regularity, uniqueness) of a solution in the Sobolev space Wp1,2(Q), facilitating for the present model to be a more complete description of certain classes of physical phenomena. The second topic refers to the construction of two numerical schemes in order to approximate the solution of a particular mathematical model (local and nonlocal case). To illustrate the effectiveness of the new mathematical model, we present some numerical experiments by applying the model to image segmentation tasks. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Image Analysis)
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13 pages, 1202 KiB  
Article
Mathematical Analysis of Maxwell Fluid Flow through a Porous Plate Channel Induced by a Constantly Accelerating or Oscillating Wall
by Constantin Fetecau, Rahmat Ellahi and Sadiq M. Sait
Mathematics 2021, 9(1), 90; https://doi.org/10.3390/math9010090 - 4 Jan 2021
Cited by 25 | Viewed by 3492
Abstract
Exact expressions for dimensionless velocity and shear stress fields corresponding to two unsteady motions of incompressible upper-convected Maxwell (UCM) fluids through a plate channel are analytically established. The porous effects are taken into consideration. The fluid motion is generated by one of the [...] Read more.
Exact expressions for dimensionless velocity and shear stress fields corresponding to two unsteady motions of incompressible upper-convected Maxwell (UCM) fluids through a plate channel are analytically established. The porous effects are taken into consideration. The fluid motion is generated by one of the plates which is moving in its plane and the obtained solutions satisfy all imposed initial and boundary conditions. The starting solutions corresponding to the oscillatory motion are presented as sum of their steady-state and transient components. They can be useful for those who want to eliminate the transients from their experiments. For a check of the obtained results, their steady-state components are presented in different forms whose equivalence is graphically illustrated. Analytical solutions for the incompressible Newtonian fluids performing the same motions are recovered as limiting cases of the presented results. The influence of physical parameters on the fluid motion is graphically shown and discussed. It is found that the Maxwell fluids flow slower as compared to Newtonian fluids. The required time to reach the steady-state is also presented. It is found that the presence of porous medium delays the appearance of the steady-state. Full article
(This article belongs to the Special Issue Mathematics in Fluid Mechanics: Theory, Modelling and Analytical Methods)
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25 pages, 600 KiB  
Article
Changepoint in Error-Prone Relations
by Michal Pešta
Mathematics 2021, 9(1), 89; https://doi.org/10.3390/math9010089 - 4 Jan 2021
Cited by 2 | Viewed by 2457
Abstract
Linear relations, containing measurement errors in input and output data, are considered. Parameters of these so-called errors-in-variables models can change at some unknown moment. The aim is to test whether such an unknown change has occurred or not. For instance, detecting a change [...] Read more.
Linear relations, containing measurement errors in input and output data, are considered. Parameters of these so-called errors-in-variables models can change at some unknown moment. The aim is to test whether such an unknown change has occurred or not. For instance, detecting a change in trend for a randomly spaced time series is a special case of the investigated framework. The designed changepoint tests are shown to be consistent and involve neither nuisance parameters nor tuning constants, which makes the testing procedures effortlessly applicable. A changepoint estimator is also introduced and its consistency is proved. A boundary issue is avoided, meaning that the changepoint can be detected when being close to the extremities of the observation regime. As a theoretical basis for the developed methods, a weak invariance principle for the smallest singular value of the data matrix is provided, assuming weakly dependent and non-stationary errors. The results are presented in a simulation study, which demonstrates computational efficiency of the techniques. The completely data-driven tests are illustrated through problems coming from calibration and insurance; however, the methodology can be applied to other areas such as clinical measurements, dietary assessment, computational psychometrics, or environmental toxicology as manifested in the paper. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications)
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8 pages, 249 KiB  
Article
On the Direct Limit from Pseudo Jacobi Polynomials to Hermite Polynomials
by Elchin I. Jafarov, Aygun M. Mammadova and Joris Van der Jeugt
Mathematics 2021, 9(1), 88; https://doi.org/10.3390/math9010088 - 4 Jan 2021
Cited by 4 | Viewed by 2235
Abstract
In this short communication, we present a new limit relation that reduces pseudo-Jacobi polynomials directly to Hermite polynomials. The proof of this limit relation is based upon 2F1-type hypergeometric transformation formulas, which are applicable to even and odd polynomials separately. [...] Read more.
In this short communication, we present a new limit relation that reduces pseudo-Jacobi polynomials directly to Hermite polynomials. The proof of this limit relation is based upon 2F1-type hypergeometric transformation formulas, which are applicable to even and odd polynomials separately. This limit opens the way to studying new exactly solvable harmonic oscillator models in quantum mechanics in terms of pseudo-Jacobi polynomials. Full article
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18 pages, 1115 KiB  
Article
Properties and Applications of a New Family of Skew Distributions
by Emilio Gómez-Déniz, Barry C. Arnold, José M. Sarabia and Héctor W. Gómez
Mathematics 2021, 9(1), 87; https://doi.org/10.3390/math9010087 - 3 Jan 2021
Cited by 4 | Viewed by 3274
Abstract
We introduce two families of continuous distribution functions with not-necessarily symmetric densities, which contain a parent distribution as a special case. The two families proposed depend on two parameters and are presented as an alternative to the skew normal distribution and other proposals [...] Read more.
We introduce two families of continuous distribution functions with not-necessarily symmetric densities, which contain a parent distribution as a special case. The two families proposed depend on two parameters and are presented as an alternative to the skew normal distribution and other proposals in the statistical literature. The density functions of these new families are given by a closed expression which allows us to easily compute probabilities, moments and related quantities. The second family can exhibit bimodality and its standardized fourth central moment (kurtosis) can be lower than that of the Azzalini skew normal distribution. Since the second proposed family can be bimodal we fit two well-known data set with this feature as applications. We concentrate attention on the case in which the normal distribution is the parent distribution but some consideration is given to other parent distributions, such as the logistic distribution. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications)
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18 pages, 1069 KiB  
Article
Convergence and Stability of a Parametric Class of Iterative Schemes for Solving Nonlinear Systems
by Alicia Cordero, Eva G. Villalba, Juan R. Torregrosa and Paula Triguero-Navarro
Mathematics 2021, 9(1), 86; https://doi.org/10.3390/math9010086 - 3 Jan 2021
Cited by 19 | Viewed by 2687
Abstract
A new parametric class of iterative schemes for solving nonlinear systems is designed. The third- or fourth-order convergence, depending on the values of the parameter being proven. The analysis of the dynamical behavior of this class in the context of scalar nonlinear equations [...] Read more.
A new parametric class of iterative schemes for solving nonlinear systems is designed. The third- or fourth-order convergence, depending on the values of the parameter being proven. The analysis of the dynamical behavior of this class in the context of scalar nonlinear equations is presented. This study gives us important information about the stability and reliability of the members of the family. The numerical results obtained by applying different elements of the family for solving the Hammerstein integral equation and the Fisher’s equation confirm the theoretical results. Full article
(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)
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19 pages, 385 KiB  
Article
An Overview of the Hamilton–Jacobi Theory: the Classical and Geometrical Approaches and Some Extensions and Applications
by Narciso Román-Roy
Mathematics 2021, 9(1), 85; https://doi.org/10.3390/math9010085 - 3 Jan 2021
Cited by 5 | Viewed by 3771
Abstract
This work is devoted to review the modern geometric description of the Lagrangian and Hamiltonian formalisms of the Hamilton–Jacobi theory. The relation with the “classical” Hamiltonian approach using canonical transformations is also analyzed. Furthermore, a more general framework for the theory is also [...] Read more.
This work is devoted to review the modern geometric description of the Lagrangian and Hamiltonian formalisms of the Hamilton–Jacobi theory. The relation with the “classical” Hamiltonian approach using canonical transformations is also analyzed. Furthermore, a more general framework for the theory is also briefly explained. It is also shown how, from this generic framework, the Lagrangian and Hamiltonian cases of the theory for dynamical systems are recovered, as well as how the model can be extended to other types of physical systems, such as higher-order dynamical systems and (first-order) classical field theories in their multisymplectic formulation. Full article
(This article belongs to the Special Issue New Trends in Hamilton-Jacobi Theory: Conservative and Dissipative Dynamics)
17 pages, 619 KiB  
Article
Optimality Conditions for Group Sparse Constrained Optimization Problems
by Wenying Wu and Dingtao Peng
Mathematics 2021, 9(1), 84; https://doi.org/10.3390/math9010084 - 1 Jan 2021
Viewed by 3008
Abstract
In this paper, optimality conditions for the group sparse constrained optimization (GSCO) problems are studied. Firstly, the equivalent characterizations of Bouligand tangent cone, Clarke tangent cone and their corresponding normal cones of the group sparse set are derived. Secondly, by using tangent cones [...] Read more.
In this paper, optimality conditions for the group sparse constrained optimization (GSCO) problems are studied. Firstly, the equivalent characterizations of Bouligand tangent cone, Clarke tangent cone and their corresponding normal cones of the group sparse set are derived. Secondly, by using tangent cones and normal cones, four types of stationary points for GSCO problems are given: TB-stationary point, NB-stationary point, TC-stationary point and NC-stationary point, which are used to characterize first-order optimality conditions for GSCO problems. Furthermore, both the relationship among the four types of stationary points and the relationship between stationary points and local minimizers are discussed. Finally, second-order necessary and sufficient optimality conditions for GSCO problems are provided. Full article
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15 pages, 415 KiB  
Article
A Picard-Type Iterative Scheme for Fredholm Integral Equations of the Second Kind
by José M. Gutiérrez and Miguel Á. Hernández-Verón
Mathematics 2021, 9(1), 83; https://doi.org/10.3390/math9010083 - 1 Jan 2021
Cited by 3 | Viewed by 2454
Abstract
In this work, we present an application of Newton’s method for solving nonlinear equations in Banach spaces to a particular problem: the approximation of the inverse operators that appear in the solution of Fredholm integral equations. Therefore, we construct an iterative method with [...] Read more.
In this work, we present an application of Newton’s method for solving nonlinear equations in Banach spaces to a particular problem: the approximation of the inverse operators that appear in the solution of Fredholm integral equations. Therefore, we construct an iterative method with quadratic convergence that does not use either derivatives or inverse operators. Consequently, this new procedure is especially useful for solving non-homogeneous Fredholm integral equations of the first kind. We combine this method with a technique to find the solution of Fredholm integral equations with separable kernels to obtain a procedure that allows us to approach the solution when the kernel is non-separable. Full article
(This article belongs to the Special Issue Application of Iterative Methods for Solving Nonlinear Equations)
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25 pages, 6637 KiB  
Article
Elicitation of the Factors Affecting Electricity Distribution Efficiency Using the Fuzzy AHP Method
by Vecihi Yiğit, Nazlı Nisa Demir, Hisham Alidrisi and Mehmet Emin Aydin
Mathematics 2021, 9(1), 82; https://doi.org/10.3390/math9010082 - 31 Dec 2020
Cited by 4 | Viewed by 2890
Abstract
Efficient and uninterrupted energy supply plays a crucial role in the quality of modern daily life, while it is obvious that the efficiency and performance of energy supply companies has a significant impact on energy supply itself and on determining and finetuning the [...] Read more.
Efficient and uninterrupted energy supply plays a crucial role in the quality of modern daily life, while it is obvious that the efficiency and performance of energy supply companies has a significant impact on energy supply itself and on determining and finetuning the future roadmap of the sector. In this study, the performance and efficiency of energy supply companies with respect to productivity is investigated with reference to a case study of an electricity distribution company in Turkey. The factors affecting the company’s performance and their corresponding weight have been determined and obtained using the analytical hierarchy process (AHP) and the Fuzzy AHP methods, two well-known multi-criteria decision-making methods, which are widely used in the literature. The results help demonstrate that the criteria obtained to evaluate the company’s energy supply performance play a crucial role in developing strategies, policies and action plans to achieve continuous improvement and consistent development. Full article
(This article belongs to the Special Issue Applications of Fuzzy Optimization and Fuzzy Decision Making)
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27 pages, 391 KiB  
Article
Weak Dependence Notions and Their Mutual Relationships
by Jorge Navarro, Franco Pellerey and Miguel A. Sordo
Mathematics 2021, 9(1), 81; https://doi.org/10.3390/math9010081 - 31 Dec 2020
Cited by 9 | Viewed by 2434
Abstract
New weak notions of positive dependence between the components X and Y of a random pair (X,Y) have been considered in recent papers that deal with the effects of dependence on conditional residual lifetimes and conditional inactivity times. The [...] Read more.
New weak notions of positive dependence between the components X and Y of a random pair (X,Y) have been considered in recent papers that deal with the effects of dependence on conditional residual lifetimes and conditional inactivity times. The purpose of this paper is to provide a structured framework for the definition and description of these notions, and other new ones, and to describe their mutual relationships. An exhaustive review of some well-know notions of dependence, with a complete description of the equivalent definitions and reciprocal relationships, some of them expressed in terms of the properties of the copula or survival copula of (X,Y), is also provided. Full article
(This article belongs to the Section Probability and Statistics)
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24 pages, 757 KiB  
Article
Application of Multi-Objective Evolutionary Algorithms for Planning Healthy and Balanced School Lunches
by Juan-Manuel Ramos-Pérez, Gara Miranda, Eduardo Segredo, Coromoto León and Casiano Rodríguez-León
Mathematics 2021, 9(1), 80; https://doi.org/10.3390/math9010080 - 31 Dec 2020
Cited by 10 | Viewed by 3665
Abstract
A multi-objective formulation of the Menu Planning Problem, which is termed the Multi-objective Menu Planning Problem, is presented herein. Menu planning is of great interest in the health field due to the importance of proper nutrition in today’s society, and particularly, in school [...] Read more.
A multi-objective formulation of the Menu Planning Problem, which is termed the Multi-objective Menu Planning Problem, is presented herein. Menu planning is of great interest in the health field due to the importance of proper nutrition in today’s society, and particularly, in school canteens. In addition to considering the cost of the meal plan as the classic objective to be minimized, we also introduce a second objective aimed at minimizing the degree of repetition of courses and food groups that a particular meal plan consists of. The motivation behind this particular multi-objective formulation is to offer a meal plan that is not only affordable but also varied and balanced from a nutritional standpoint. The plan is designed for a given number of days and ensures that the specific nutritional requirements of school-age children are satisfied. The main goal of the current work is to demonstrate the multi-objective nature of the said formulation, through a comprehensive experimental assessment carried out over a set of multi-objective evolutionary algorithms applied to different instances. At the same time, we are also interested in validating the multi-objective formulation by performing quantitative and qualitative analyses of the solutions attained when solving it. Computational results show the multi-objective nature of the said formulation, as well as that it allows suitable meal plans to be obtained. Full article
(This article belongs to the Special Issue Multiple Criteria Decision Making)
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9 pages, 296 KiB  
Article
Wiener Complexity versus the Eccentric Complexity
by Martin Knor and Riste Škrekovski
Mathematics 2021, 9(1), 79; https://doi.org/10.3390/math9010079 - 31 Dec 2020
Viewed by 1930
Abstract
Let wG(u) be the sum of distances from u to all the other vertices of G. The Wiener complexity, CW(G), is the number of different values of wG(u) in [...] Read more.
Let wG(u) be the sum of distances from u to all the other vertices of G. The Wiener complexity, CW(G), is the number of different values of wG(u) in G, and the eccentric complexity, Cec(G), is the number of different eccentricities in G. In this paper, we prove that for every integer c there are infinitely many graphs G such that CW(G)−Cec(G)=c. Moreover, we prove this statement using graphs with the smallest possible cyclomatic number. That is, if c≥0 we prove this statement using trees, and if c<0 we prove it using unicyclic graphs. Further, we prove that Cec(G)≤2CW(G)−1 if G is a unicyclic graph. In our proofs we use that the function wG(u) is convex on paths consisting of bridges. This property also promptly implies the already known bound for trees Cec(G)≤CW(G). Finally, we answer in positive an open question by finding infinitely many graphs G with diameter 3 such that Cec(G)<CW(G). Full article
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13 pages, 933 KiB  
Article
On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method
by Haifa Bin Jebreen and Fairouz Tchier
Mathematics 2021, 9(1), 78; https://doi.org/10.3390/math9010078 - 31 Dec 2020
Viewed by 2377
Abstract
Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a [...] Read more.
Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis. Full article
(This article belongs to the Special Issue Recent Advances in Differential Equations and Applications)
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