Axioms

11 pages, 285 KiB

Open AccessArticle

On Solutions and Stability of Stochastic Functional Equations Emerging in Psychological Theory of Learning

by Ali Turab, Janusz Brzdęk and Wajahat Ali

Axioms 2022, 11(3), 143; https://doi.org/10.3390/axioms11030143 - 21 Mar 2022

Cited by 4 | Viewed by 2126

We show how to apply the well-known fixed-point approach in the study of the existence, uniqueness, and stability of solutions to some particular types of functional equations. Moreover, we also obtain the Ulam stability result for them. The functional equations that we consider [...] Read more.

We show how to apply the well-known fixed-point approach in the study of the existence, uniqueness, and stability of solutions to some particular types of functional equations. Moreover, we also obtain the Ulam stability result for them. The functional equations that we consider can be used to explain various experiments in mathematical psychology and arise in a natural way in the stochastic approach to the processes of perception, learning, reasoning, and cognition. Full article

(This article belongs to the Special Issue Advances in Mathematics and Its Applications)

9 pages, 1022 KiB

Open AccessArticle

Exploring the Limits of Euler–Bernoulli Theory in Micromechanics

by Chrysoula K. Manoli, Styliani Papatzani and Dionysios E. Mouzakis

Axioms 2022, 11(3), 142; https://doi.org/10.3390/axioms11030142 - 19 Mar 2022

Cited by 3 | Viewed by 2532

Abstract

In this study, the limits of the Euler–Bernoulli theory in micromechanics are explored. Raman spectroscopy, which is extremely accurate and reliable, is employed to study the bending of a microbeam of a length of 191 μm. It is found that at the micro-scale, [...] Read more.

In this study, the limits of the Euler–Bernoulli theory in micromechanics are explored. Raman spectroscopy, which is extremely accurate and reliable, is employed to study the bending of a microbeam of a length of 191 μm. It is found that at the micro-scale, the Euler–Bernoulli theory remains an exact and consistent tool, and, possibly, other elasticity theories (such as micropolar theory, gradient elasticity theory, and couple stress theory) are not always required to study this phenomenon. More specifically, good correlation was achieved between the theoretical and experimental results, the former acquired via the theoretical equations and the latter obtained with the use of atomic force microscopy and Raman spectroscopy. The exact predicted strain of an atomic force microscope microbeam under bending, by Euler–Bernoulli equations is confirmed by Raman spectroscopy. Full article

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

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14 pages, 298 KiB

Open AccessArticle

Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity

by Dana Bibulova, Burkhan Kalimbetov and Valeriy Safonov

Axioms 2022, 11(3), 141; https://doi.org/10.3390/axioms11030141 - 18 Mar 2022

Cited by 4 | Viewed by 1737

Abstract

This article investigates an equation with a rapidly oscillating inhomogeneity and with a rapidly decreasing kernel of an integral operator of Fredholm type. Earlier, differential problems of this type were studied in which the integral term was either absent or had the form [...] Read more.

This article investigates an equation with a rapidly oscillating inhomogeneity and with a rapidly decreasing kernel of an integral operator of Fredholm type. Earlier, differential problems of this type were studied in which the integral term was either absent or had the form of a Volterra-type integral. The presence of an integral operator and its type significantly affect the development of an algorithm for asymptotic solutions, in the implementation of which it is necessary to take into account essential singularities generated by the rapidly decreasing kernel of the integral operator. It is shown in tise work that when passing the structure of essentially singular singularities changes from an integral operator of Volterra type to an operator of Fredholm type. If in the case of the Volterra operator they change with a change in the independent variable, then the singularities generated by the kernel of the integral Fredholm-type operators are constant and depend only on a small parameter. All these effects, as well as the effects introduced by the rapidly oscillating inhomogeneity, are necessary to take into account when developing an algorithm for constructing asymptotic solutions to the original problem, which is implemented in this work. Full article

(This article belongs to the Special Issue Singularly Perturbed Problems: Asymptotic Analysis and Approximate Solution)

33 pages, 5053 KiB

Open AccessArticle

Development of a Model for Evaluating the Efficiency of Transport Companies: PCA–DEA–MCDM Model

by Željko Stević, Smiljka Miškić, Dragan Vojinović, Eldina Huskanović, Miomir Stanković and Dragan Pamučar

Axioms 2022, 11(3), 140; https://doi.org/10.3390/axioms11030140 - 18 Mar 2022

Cited by 32 | Viewed by 4352

Abstract

The efficiency of transport companies is a very important factor for the companies themselves, as well as for the entire economic system. The main goal of this paper is to develop an integrated model for determining the efficiency of representative transport companies over [...] Read more.

The efficiency of transport companies is a very important factor for the companies themselves, as well as for the entire economic system. The main goal of this paper is to develop an integrated model for determining the efficiency of representative transport companies over a period of eight years. An original model was developed that includes the integration of DEA (Data Envelopment Analysis), PCA (Principal Component Analysis), CRITIC (Criteria Importance Through Inter criteria Correlatio), Entropy and MARCOS (Measurement Alternatives and Ranking according to the COmpromise Solution) methods in order to determine the final efficiency of transport companies based on 10 input–output parameters. The results showed that the most efficient business performance was achieved in the period 2014–2017, followed by slightly less efficient results. Then, extensive sensitivity analysis and comparative analysis were performed, which confirmed, to some extent, the previously obtained results. In the sensitivity analysis, 30 scenarios with changes in the weights of criteria were created, while the comparative analysis was carried out with three other MCDM (Multi-Criteria Decision-Making) methods. Finally, the rank correlation index was determined using the Spearman and WS (Wojciech Salabun) correlation coefficients. According to the final results, very efficient years can be separated that can be the benchmark for furthering the business. Full article

(This article belongs to the Special Issue Multiple-Criteria Decision Making II)

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22 pages, 623 KiB

Open AccessFeature PaperArticle

Constant-Sign Green’s Function of a Second-Order Perturbed Periodic Problem

by Alberto Cabada, Lucía López-Somoza and Mouhcine Yousfi

Axioms 2022, 11(3), 139; https://doi.org/10.3390/axioms11030139 - 17 Mar 2022

Cited by 1 | Viewed by 2068

Abstract

In this paper, we were interested in obtaining the exact expression and studying the regions of constant sign of Green’s function related to a second-order perturbed periodic problem coupled with integral boundary conditions at the extremes of the interval of the definition. To [...] Read more.

In this paper, we were interested in obtaining the exact expression and studying the regions of constant sign of Green’s function related to a second-order perturbed periodic problem coupled with integral boundary conditions at the extremes of the interval of the definition. To obtain the expression of Green’s function related to this problem, we used the theory presented in a previous paper of the authors for general non-local perturbed boundary-value problems. Moreover, we characterized the parameter set where such a Green’s function has a constant sign. To this end, we needed to consider first a related second-order problem without integral boundary conditions, obtaining the properties of its Green’s function and then using them to compute the sign of the one related to the main problem. Full article

(This article belongs to the Special Issue Advances in Nonlinear Boundary Value Problems: Theory and Applications)

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25 pages, 550 KiB

Open AccessArticle

Deformed Mathematical Objects Stemming from the q-Logarithm Function

by Ernesto P. Borges and Bruno G. da Costa

Axioms 2022, 11(3), 138; https://doi.org/10.3390/axioms11030138 - 16 Mar 2022

Cited by 6 | Viewed by 2696

Abstract

Generalized numbers, arithmetic operators, and derivative operators, grouped in four classes based on symmetry features, are introduced. Their building element is the pair of q-logarithm/q-exponential inverse functions. Some of the objects were previously described in the literature, while others are [...] Read more.

Generalized numbers, arithmetic operators, and derivative operators, grouped in four classes based on symmetry features, are introduced. Their building element is the pair of q-logarithm/q-exponential inverse functions. Some of the objects were previously described in the literature, while others are newly defined. Commutativity, associativity, and distributivity, and also a pair of linear/nonlinear derivatives, are observed within each class. Two entropic functionals emerge from the formalism, and one of them is the nonadditive Tsallis entropy. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)

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10 pages, 810 KiB

Open AccessArticle

Toroidal Spectral Drawing

by Ming-Hsuan Kang and Jing-Wen Gu

Axioms 2022, 11(3), 137; https://doi.org/10.3390/axioms11030137 - 16 Mar 2022

Viewed by 2241

Abstract

We give a deterministic drawing algorithm to draw a graph onto a torus, which is based on the usual spectral drawing algorithm. For most of the well-known toroidal vertex-transitive graphs, the result drawings give an embedding of the graphs onto the torus. Full article

(This article belongs to the Special Issue Linear Algebra: Matrix Theory, Graph Theory and Applications)

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18 pages, 340 KiB

Open AccessFeature PaperArticle

Monotonicity Arguments for Variational–Hemivariational Inequalities in Hilbert Spaces

by Mircea Sofonea

Axioms 2022, 11(3), 136; https://doi.org/10.3390/axioms11030136 - 16 Mar 2022

Cited by 1 | Viewed by 1842

Abstract

We consider a variational–hemivariational inequality in a real Hilbert space, which depends on two parameters. We prove that the inequality is governed by a maximal monotone operator, then we deduce various existence, uniqueness and equivalence results. The proofs are based on the theory [...] Read more.

We consider a variational–hemivariational inequality in a real Hilbert space, which depends on two parameters. We prove that the inequality is governed by a maximal monotone operator, then we deduce various existence, uniqueness and equivalence results. The proofs are based on the theory of maximal monotone operators, fixed point arguments and the properties of the subdifferential, both in the sense of Clarke and in the sense of convex analysis. These results lay the background in the study of various classes of inequalities. We use them to prove existence, uniqueness and continuous dependence results for the solution of elliptic and history-dependent variational–hemivariational inequalities. We also present some iterative methods in solving these inequalities, together with various convergence results. Full article

(This article belongs to the Special Issue Advances in General Topology and Its Application)

21 pages, 6473 KiB

Open AccessArticle

Analysis of the Term Structure of Major Currencies Using Principal Component Analysis and Autoencoders

by Soo Chang Chae and Sun-Yong Choi

Axioms 2022, 11(3), 135; https://doi.org/10.3390/axioms11030135 - 15 Mar 2022

Cited by 4 | Viewed by 3111

Abstract

Recently, machine-learning algorithms and existing financial data-analysis methods have been actively studied. Although the term structure of government bonds has been well-researched, the majority of studies only analyze the characteristics of one country in detail using one method. In this paper, we analyze [...] Read more.

Recently, machine-learning algorithms and existing financial data-analysis methods have been actively studied. Although the term structure of government bonds has been well-researched, the majority of studies only analyze the characteristics of one country in detail using one method. In this paper, we analyze the term structure and determine the common factors using principal component analysis (PCA) and an autoencoder (AE). We collected data on the government bonds of three countries with major currencies (the US, the UK, and Japan), extracted features, and compared them. In the PCA-based analysis, we reduced the number of dimensions by converting the normalized data into a covariance matrix and checked the first five principal components visually using graphs. In the AE-based analysis, the model consisted of two encoder layers, one middle layer, and two decoder layers, and the number of nodes in the middle layer was adjusted from one to five. As a result, no significant similarity was found for each country in the dataset, and it was appropriate to extract three features in both methods. Each feature extracted by PCA and the AE had a completely different form, and this appears to be due to the differences in the feature extraction methods. In the case of PCA, the volatility of the datasets affected the features, but in the case of AE, the results seemed to be more affected by the size of the dataset. Based on the findings of this study, this topic can be expanded to compare the results of other machine-learning algorithms or countries. Full article

(This article belongs to the Special Issue Mathematical Analysis for Financial Modelling)

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19 pages, 1708 KiB

Open AccessArticle

The Role of Risk Forecast and Risk Tolerance in Portfolio Management: A Case Study of the Chinese Financial Sector

by Jianxu Liu, Yangnan Cheng, Xiaoqing Li and Songsak Sriboonchitta

Axioms 2022, 11(3), 134; https://doi.org/10.3390/axioms11030134 - 15 Mar 2022

Cited by 5 | Viewed by 4291

Abstract

Portfolio decisions are affected by the volatility of financial markets and investors’ risk tolerance levels. To better allocate portfolios; we introduce risk tolerance into the portfolio management problem by considering the risk contribution of portfolio components. In this paper, portfolio weights are allocated [...] Read more.

Portfolio decisions are affected by the volatility of financial markets and investors’ risk tolerance levels. To better allocate portfolios; we introduce risk tolerance into the portfolio management problem by considering the risk contribution of portfolio components. In this paper, portfolio weights are allocated to two stages. In the first stage, the portfolio risks and the risk contribution of each share are forecasted. In the second stage, we put forward three weighting techniques—“aggressive”, “moderate” and “conservative”, according to three standard levels of risk tolerance. In addition, a new risk measure called “joint extreme risk probability” (JERP), with risk tolerance taken into account, is proposed. A case study of the Chinese financial industry is conducted to verify the performance of our methods. The empirical results demonstrate that weighting techniques constrained by risk tolerance lead to higher gains in a normal market and less loss when a market is risky. Compared with risk-tolerance-adjusted strategies, the relationship between the performance of the traditional conditional value at risk (CVaR) minimization method and the market risk level is less obviously demonstrated. Viewed from the results, JERP functions as an effective signal that helps investors to deal with potential market risks. Full article

(This article belongs to the Special Issue Applied Mathematics in Finance and Economics)

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14 pages, 1011 KiB

Open AccessArticle

An Effective Approximation Algorithm for Second-Order Singular Functional Differential Equations

by Mohammad Izadi, Hari M. Srivastava and Waleed Adel

Axioms 2022, 11(3), 133; https://doi.org/10.3390/axioms11030133 - 14 Mar 2022

Cited by 11 | Viewed by 2358

Abstract

In this research study, a novel computational algorithm for solving a second-order singular functional differential equation as a generalization of the well-known Lane–Emden and differential-difference equations is presented by using the Bessel bases. This technique depends on transforming the problem into a system [...] Read more.

In this research study, a novel computational algorithm for solving a second-order singular functional differential equation as a generalization of the well-known Lane–Emden and differential-difference equations is presented by using the Bessel bases. This technique depends on transforming the problem into a system of algebraic equations and by solving this system the unknown Bessel coefficients are determined and the solution will be known. The method is tested on several test examples and proves to provide accurate results as compared to other existing methods from the literature. The simplicity and robustness of the proposed technique drive us to investigate more of their applications to several similar problems in the future. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications III)

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9 pages, 251 KiB

Open AccessArticle

On Some New Ostrowski–Mercer-Type Inequalities for Differentiable Functions

by Ifra Bashir Sial, Nichaphat Patanarapeelert, Muhammad Aamir Ali, Hüseyin Budak and Thanin Sitthiwirattham

Axioms 2022, 11(3), 132; https://doi.org/10.3390/axioms11030132 - 14 Mar 2022

Cited by 10 | Viewed by 2249

Abstract

In this paper, we establish a new integral identity involving differentiable functions, and then we use the newly established identity to prove some Ostrowski–Mercer-type inequalities for differentiable convex functions. It is also demonstrated that the newly established inequalities are generalizations of some of [...] Read more.

In this paper, we establish a new integral identity involving differentiable functions, and then we use the newly established identity to prove some Ostrowski–Mercer-type inequalities for differentiable convex functions. It is also demonstrated that the newly established inequalities are generalizations of some of the Ostrowski inequalities established inside the literature. There are also some applications to the special means of real numbers given. Full article

(This article belongs to the Special Issue Current Research on Mathematical Inequalities)

20 pages, 373 KiB

Open AccessFeature PaperArticle

Splitting Extensions of Nonassociative Algebras and Modules with Metagroup Relations

by Sergey Victor Ludkowski

Axioms 2022, 11(3), 131; https://doi.org/10.3390/axioms11030131 - 14 Mar 2022

Cited by 1 | Viewed by 1776

Abstract

A class of nonassociative algebras is investigated with mild relations induced from metagroup structures. Modules over nonassociative algebras are studied. For a class of modules over nonassociative algebras, their extensions and splitting extensions are scrutinized. For this purpose tensor products of modules and [...] Read more.

A class of nonassociative algebras is investigated with mild relations induced from metagroup structures. Modules over nonassociative algebras are studied. For a class of modules over nonassociative algebras, their extensions and splitting extensions are scrutinized. For this purpose tensor products of modules and induced modules over nonassociative algebras are investigated. Moreover, a developed cohomology theory on them is used. Full article

(This article belongs to the Special Issue Algebra, Logic and Applications)

18 pages, 737 KiB

Open AccessArticle

Generalization of Fuzzy Connectives

by Stefanos Makariadis and Basil Papadopoulos

Axioms 2022, 11(3), 130; https://doi.org/10.3390/axioms11030130 - 12 Mar 2022

Cited by 2 | Viewed by 2951

Abstract

This paper is centered around the creation of new fuzzy connectives using automorphism functions. The fuzzy connectives theory has been implemented in many problems and fields. In particular, the N-negations, t-norms, S-conorms and I-implications concepts played crucial roles in forming the theory and [...] Read more.

This paper is centered around the creation of new fuzzy connectives using automorphism functions. The fuzzy connectives theory has been implemented in many problems and fields. In particular, the N-negations, t-norms, S-conorms and I-implications concepts played crucial roles in forming the theory and applications of the fuzzy sets. Thus far, there are multiple strategies for producing fuzzy connectives. The purpose of this paper is to provide a new strategy that is more flexible and fast in comparison with the rest. In order to create this method, automorphism and additive generator functions were utilized. The general formulas created with this method can provide new fuzzy connectives. The main conclusion is that new fuzzy connectives can be created faster and with more flexibility with our strategy. Full article

(This article belongs to the Special Issue Fuzzy Logic as the Foundation for Theories of Fuzzy Mathematical Structures)

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25 pages, 1505 KiB

Open AccessArticle

A Non-Standard Finite Difference Scheme for a Diffusive HIV-1 Infection Model with Immune Response and Intracellular Delay

by Xiao-Lan Liu and Cheng-Cheng Zhu

Axioms 2022, 11(3), 129; https://doi.org/10.3390/axioms11030129 - 12 Mar 2022

Cited by 7 | Viewed by 2227

Abstract

In this paper, we propose and study a diffusive HIV infection model with infected cells delay, virus mature delay, abstract function incidence rate and a virus diffusion term. By introducing the reproductive numbers for viral infection

R_{0}

and for CTL immune response [...] Read more.

In this paper, we propose and study a diffusive HIV infection model with infected cells delay, virus mature delay, abstract function incidence rate and a virus diffusion term. By introducing the reproductive numbers for viral infection

R_{0}

and for CTL immune response number

R_{1}

, we show that

R_{0}

and

R_{1}

act as threshold parameter for the existence and stability of equilibria. If

R_{0} \leq 1

, the infection-free equilibrium

E_{0}

is globally asymptotically stable, and the viruses are cleared; if

R_{1} \leq 1 < R_{0}

, the CTL-inactivated equilibrium

E_{1}

is globally asymptotically stable, and the infection becomes chronic but without persistent CTL response; if

R_{1} > 1

, the CTL-activated equilibrium

E_{2}

is globally asymptotically stable, and the infection is chronic with persistent CTL response. Next, we study the dynamic of the discreted system of our model by using non-standard finite difference scheme. We find that the global stability of the equilibria of the continuous model and the discrete model is not always consistent. That is, if

R_{0} \leq 1

, or

R_{1} \leq 1 < R_{0}

, the global stability of the two kinds model is consistent. However, if

R_{1} > 1

, the global stability of the two kinds model is not consistent. Finally, numerical simulations are carried out to illustrate the theoretical results and show the effects of diffusion factors on the time-delay virus model. 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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11 pages, 319 KiB

Open AccessArticle

Some Korovkin-Type Approximation Theorems Associated with a Certain Deferred Weighted Statistical Riemann-Integrable Sequence of Functions

by Hari Mohan Srivastava, Bidu Bhusan Jena and Susanta Kumar Paikray

Axioms 2022, 11(3), 128; https://doi.org/10.3390/axioms11030128 - 12 Mar 2022

Cited by 4 | Viewed by 2296

Abstract

Here, in this article, we introduce and systematically investigate the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability for sequences of functions. We begin by proving an inclusion theorem that establishes a relation between these two potentially useful [...] Read more.

Here, in this article, we introduce and systematically investigate the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability for sequences of functions. We begin by proving an inclusion theorem that establishes a relation between these two potentially useful concepts. We also state and prove two Korovkin-type approximation theorems involving algebraic test functions by using our proposed concepts and methodologies. Furthermore, in order to demonstrate the usefulness of our findings, we consider an illustrative example involving a sequence of positive linear operators in conjunction with the familiar Bernstein polynomials. Finally, in the concluding section, we propose some directions for future research on this topic, which are based upon the core concept of statistical Lebesgue-measurable sequences of functions. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

19 pages, 452 KiB

Open AccessArticle

Optimal Consumption, Investment, and Housing Choice: A Dynamic Programming Approach

by Qi Li and Seryoong Ahn

Axioms 2022, 11(3), 127; https://doi.org/10.3390/axioms11030127 - 11 Mar 2022

Cited by 3 | Viewed by 2531

Abstract

We investigate a portfolio selection problem involving an agent’s realistic housing service choice, where the agent not only has to choose the size of house to live in, but also has to select between renting and purchasing a house. Adopting a dynamic programming [...] Read more.

We investigate a portfolio selection problem involving an agent’s realistic housing service choice, where the agent not only has to choose the size of house to live in, but also has to select between renting and purchasing a house. Adopting a dynamic programming approach, we derive a closed-form solution to obtain the optimal policies for the consumption, investment, housing service, and purchasing time for a house. We also present various numerical demonstrations showing the impacts of parameters in the financial and housing markets and the agent’s preference, which visually show the economic implications of our model. Our model makes a significant contribution because it is a pioneering model for the optimal time to purchase a house, which has not been investigated in depth in existing mathematical portfolio optimization models. Full article

(This article belongs to the Special Issue Mathematical Analysis for Financial Modelling)

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15 pages, 553 KiB

Open AccessArticle

The Spectral Distribution of Random Mixed Graphs

by Yue Guan, Bo Cheng, Minfeng Chen, Meili Liang, Jianxi Liu, Jinxun Wang, Chao Yang and Li Zeng

Axioms 2022, 11(3), 126; https://doi.org/10.3390/axioms11030126 - 11 Mar 2022

Viewed by 2117

Abstract

In this work, we propose a random mixed graph model

G_{n} (p (n), q (n))

that incorporates both the classical Erdős-Rényi’s random graph model and the random oriented graph model. We show that the empirical [...] Read more.

In this work, we propose a random mixed graph model

G_{n} (p (n), q (n))

that incorporates both the classical Erdős-Rényi’s random graph model and the random oriented graph model. We show that the empirical spectral distribution of

G_{n} (p (n), q (n))

converges to the standard semicircle law under some mild condition, and the Monte Carlo simulation highly agrees with our result. Full article

(This article belongs to the Special Issue Graph Theory with Applications)

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11 pages, 328 KiB

Open AccessArticle

On the Semi-Group Property of the Perpendicular Bisector in a Normed Space

by Gheorghiță Zbăganu

Axioms 2022, 11(3), 125; https://doi.org/10.3390/axioms11030125 - 10 Mar 2022

Viewed by 1950

Abstract

Let (X,d) be a metric linear space and a∈X. The point a divides the space into three sets: H_a = {x ∈ X: d(0,x) < d(x,a)}, M_a [...] Read more.

Let (X,d) be a metric linear space and a∈X. The point a divides the space into three sets: H_a = {x ∈ X: d(0,x) < d(x,a)}, M_a = {x ∈ X: d(0,x) = d(x,a)} and L_a = {x ∈ X: d(0,x) > d(x,a)}. If the distance is generated by a norm, H_a is called the Leibnizian halfspace of a, M_a is the perpendicular bisector of the segment 0,a and L_a is the remaining set L_a = X\(H_a∪ M_a). It is known that the perpendicular bisector of the segment [0,a] is an affine subspace of X for all a ∈ X if, and only if, X is an inner product space, that is, if and only if the norm is generated by an inner product. In this case, it is also true that if x,y ∈ L_a ∪ M_a, then x + y ∈ L_a∪ M_a. Otherwise written, the set L_a∪ M_a is a semi-group with respect to addition. We investigate the problem: for what kind of norms in X the pair (L_a ∪ M_a,+) is a semi-group for all a ∈ X? In that case, we say that “

(X, ‖ . ‖)

has the semi-group property” or that “the norm

‖ . ‖

has the semi-group property”. This is a threedimensional property, meaning that if all the three-dimensional subspaces of X have it, then X also has it. We prove that for two-dimensional spaces, (L_a,+) is a semi-group for any norm, that

(X, ‖ . ‖)

has the semi-group property if, and only if, the norm is strictly convex, and, in higher dimensions, the property fails to be true even if the norm is strictly convex. Moreover, studying the L^p norms in higher dimensions, we prove that the semi-group property holds if, and only if, p = 2. This fact leads us to the conjecture that in dimensions greater than three, the semi-group property holds if, and only if, X is an inner-product space. Full article

(This article belongs to the Special Issue Symmetry of Nonlinear Operators)

19 pages, 647 KiB

Open AccessArticle

Modeling the Impact of the Imperfect Vaccination of the COVID-19 with Optimal Containment Strategy

by Lahbib Benahmadi, Mustapha Lhous, Abdessamad Tridane, Omar Zakary and Mostafa Rachik

Axioms 2022, 11(3), 124; https://doi.org/10.3390/axioms11030124 - 10 Mar 2022

Cited by 3 | Viewed by 2327

Abstract

Since the beginning of the COVID-19 pandemic, vaccination has been the main strategy to contain the spread of the coronavirus. However, with the administration of many types of vaccines and the constant mutation of viruses, the issue of how effective these vaccines are [...] Read more.

Since the beginning of the COVID-19 pandemic, vaccination has been the main strategy to contain the spread of the coronavirus. However, with the administration of many types of vaccines and the constant mutation of viruses, the issue of how effective these vaccines are in protecting the population is raised. This work aimed to present a mathematical model that investigates the imperfect vaccine and finds the additional measures needed to help reduce the burden of disease. We determine the

R_{0}

threshold of disease spread and use stability analysis to determine the condition that will result in disease eradication. We also fitted our model to COVID-19 data from Morocco to estimate the parameters of the model. The sensitivity analysis of the basic reproduction number, with respect to the parameters of the model, is simulated for the four possible scenarios of the disease progress. Finally, we investigate the optimal containment measures that could be implemented with vaccination. To illustrate our results, we perform the numerical simulations of optimal control. Full article

(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)

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10 pages, 230 KiB

Open AccessArticle

Forward Order Law for the Reflexive Inner Inverse of Multiple Matrix Products

by Wanna Zhou, Zhiping Xiong and Yingying Qin

Axioms 2022, 11(3), 123; https://doi.org/10.3390/axioms11030123 - 10 Mar 2022

Cited by 2 | Viewed by 1953

Abstract

The generalized inverse has numerous important applications in aspects of the theoretic research of matrices and statistics. One of the core problems of generalized inverse is finding the necessary and sufficient conditions for the reverse (or the forward) order laws for the generalized [...] Read more.

The generalized inverse has numerous important applications in aspects of the theoretic research of matrices and statistics. One of the core problems of generalized inverse is finding the necessary and sufficient conditions for the reverse (or the forward) order laws for the generalized inverse of matrix products. In this paper, by using the extremal ranks of the generalized Schur complement, some necessary and sufficient conditions are given for the forward order law for

A_{1} {1, 2} A_{2} {1, 2} \dots A_{n} {1, 2} \subseteq (A_{1} A_{2} \dots A_{n}) {1, 2}

. Full article

(This article belongs to the Special Issue Algebra, Logic and Applications)

13 pages, 421 KiB

Open AccessArticle

A Model in Which the Separation Principle Holds for a Given Effective Projective Sigma-Class

by Vladimir Kanovei and Vassily Lyubetsky

Axioms 2022, 11(3), 122; https://doi.org/10.3390/axioms11030122 - 10 Mar 2022

Cited by 4 | Viewed by 1935

Abstract

In this paper, we prove the following: If

n \geq 3

, there is a generic extension of

L

—the constructible universe—in which it is true that the Separation principle holds for both effective (lightface) classes

Σ_{n}^{1}

and

Π_{n}^{1}

[...] Read more.

In this paper, we prove the following: If

n \geq 3

, there is a generic extension of

L

—the constructible universe—in which it is true that the Separation principle holds for both effective (lightface) classes

Σ_{n}^{1}

and

Π_{n}^{1}

of sets of integers. The result was announced long ago by Leo Harrington with a sketch of the proof for

n = 3

; its full proof has never been presented. Our methods are based on a countable product of almost-disjoint forcing notions independent in the sense of Jensen–Solovay. Full article

(This article belongs to the Section Logic)

27 pages, 706 KiB

Open AccessArticle

Determining COVID-19 Dynamics Using Physics Informed Neural Networks

by Joseph Malinzi, Simanga Gwebu and Sandile Motsa

Axioms 2022, 11(3), 121; https://doi.org/10.3390/axioms11030121 - 10 Mar 2022

Cited by 7 | Viewed by 3985

Abstract

The Physics Informed Neural Networks framework is applied to the understanding of the dynamics of COVID-19. To provide the governing system of equations used by the framework, the Susceptible–Infected–Recovered–Death mathematical model is used. This study focused on finding the patterns of the dynamics [...] Read more.

The Physics Informed Neural Networks framework is applied to the understanding of the dynamics of COVID-19. To provide the governing system of equations used by the framework, the Susceptible–Infected–Recovered–Death mathematical model is used. This study focused on finding the patterns of the dynamics of the disease which involves predicting the infection rate, recovery rate and death rate; thus, predicting the active infections, total recovered, susceptible and deceased at any required time. The study used data that were collected on the dynamics of COVID-19 from the Kingdom of Eswatini between March 2020 and September 2021. The obtained results could be used for making future forecasts on COVID-19 in Eswatini. Full article

(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)

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13 pages, 1511 KiB

Open AccessArticle

On Large and Small Data Blow-Up Solutions in the Trojan Y Chromosome Model

by Eric M. Takyi, Matthew A. Beauregard, Thomas Griffin, Landrey Bobo and Rana D. Parshad

Axioms 2022, 11(3), 120; https://doi.org/10.3390/axioms11030120 - 9 Mar 2022

Cited by 2 | Viewed by 2132

Abstract

The Trojan Y Chromosome Strategy (TYC) is the only genetic biological control method in practice in North America for controlling invasive populations with an XX–XY sex determinism. Herein a modified organism, that is a supermale or feminised supermale, is introduced into an invasive [...] Read more.

The Trojan Y Chromosome Strategy (TYC) is the only genetic biological control method in practice in North America for controlling invasive populations with an XX–XY sex determinism. Herein a modified organism, that is a supermale or feminised supermale, is introduced into an invasive population to skew the sex ratio over time, causing local extinction. We consider the three species TYC reaction diffusion model, and show that introduction of supermales above certain thresholds, and for certain initial data, solutions can blow-up in finite time. Thus, in order to have biologically meaningful solutions, one needs to restrict parameter and initial data regimes, in TYC type models. Full article

(This article belongs to the Special Issue Advances in Reaction Diffusion Systems and Applications)

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19 pages, 464 KiB

Open AccessArticle

Estimation Methods of the Multiple-Group One-Dimensional Factor Model: Implied Identification Constraints in the Violation of Measurement Invariance

by Alexander Robitzsch

Axioms 2022, 11(3), 119; https://doi.org/10.3390/axioms11030119 - 9 Mar 2022

Cited by 8 | Viewed by 2568

Abstract

Factor analysis is one of the most important statistical tools for analyzing multivariate data (i.e., items) in the social sciences. An essential case is the comparison of multiple groups on a one-dimensional factor variable that can be interpreted as a summary of the [...] Read more.

Factor analysis is one of the most important statistical tools for analyzing multivariate data (i.e., items) in the social sciences. An essential case is the comparison of multiple groups on a one-dimensional factor variable that can be interpreted as a summary of the items. The assumption of measurement invariance is a frequently employed assumption that enables the comparison of the factor variable across groups. This article discusses different estimation methods of the multiple-group one-dimensional factor model under violations of measurement invariance (i.e., measurement noninvariance). In detail, joint estimation, linking methods, and regularized estimation approaches are treated. It is argued that linking approaches and regularization approaches can be equivalent to joint estimation approaches if appropriate (robust) loss functions are employed. Each of the estimation approaches defines identification constraints of parameters that quantify violations of measurement invariance. We argue in the discussion section that the fitted multiple-group one-dimensional factor analysis will likely be misspecified due to the violation of measurement invariance. Hence, because there is always indeterminacy in determining group comparisons of the factor variable under noninvariance, the preference of particular fitting strategies such as partial invariance over alternatives is unjustified. In contrast, researchers purposely define fitting functions that minimize the extent of model misspecification due to the choice of a particular (robust) loss function. Full article

(This article belongs to the Section Mathematical Analysis)

5 pages, 234 KiB

Open AccessArticle

Topological Transcendental Fields

by Taboka Prince Chalebgwa and Sidney A. Morris

Axioms 2022, 11(3), 118; https://doi.org/10.3390/axioms11030118 - 7 Mar 2022

Cited by 5 | Viewed by 2276

Abstract

This article initiates the study of topological transcendental fields

F

which are subfields of the topological field

C

of all complex numbers such that

F

only consists of rational numbers and a nonempty set of transcendental numbers.

F

, with the topology it [...] Read more.

This article initiates the study of topological transcendental fields

F

which are subfields of the topological field

C

of all complex numbers such that

F

only consists of rational numbers and a nonempty set of transcendental numbers.

F

, with the topology it inherits as a subspace of

C

, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is

Q (T)

, the extension of the field of rational numbers by a set T of transcendental numbers. It is proven that there exist precisely

2^{ℵ_{0}}

countably infinite topological transcendental fields and each is homeomorphic to the space

Q

of rational numbers with its usual topology. It is also shown that there is a class of

2^{2^{ℵ_{0}}}

of topological transcendental fields of the form

Q (T)

with T a set of Liouville numbers, no two of which are homeomorphic. Full article

(This article belongs to the Collection Topological Groups)

18 pages, 326 KiB

Open AccessArticle

New Relation-Theoretic Fixed Point Theorems in Fuzzy Metric Spaces with an Application to Fractional Differential Equations

by Samera M. Saleh, Waleed M. Alfaqih, Salvatore Sessa and Ferdinando Di Martino

Axioms 2022, 11(3), 117; https://doi.org/10.3390/axioms11030117 - 7 Mar 2022

Cited by 8 | Viewed by 2286

Abstract

In this paper, we introduce the notion of fuzzy

R - ψ -

contractive mappings and prove some relevant results on the existence and uniqueness of fixed points for this type of mappings in the setting of non-Archimedean fuzzy metric spaces. Several illustrative [...] Read more.

In this paper, we introduce the notion of fuzzy

R - ψ -

contractive mappings and prove some relevant results on the existence and uniqueness of fixed points for this type of mappings in the setting of non-Archimedean fuzzy metric spaces. Several illustrative examples are also given to support our newly proven results. Furthermore, we apply our main results to prove the existence and uniqueness of a solution for Caputo fractional differential equations. Full article

(This article belongs to the Section Mathematical Analysis)

18 pages, 1602 KiB

Open AccessArticle

Dynamical Analysis of a Predator-Prey Model Incorporating Predator Cannibalism and Refuge

by Maya Rayungsari, Agus Suryanto, Wuryansari Muharini Kusumawinahyu and Isnani Darti

Axioms 2022, 11(3), 116; https://doi.org/10.3390/axioms11030116 - 7 Mar 2022

Cited by 14 | Viewed by 3501

Abstract

We consider a mathematical model to describe the interaction between predator and prey that includes predator cannibalism and refuge. We aim to study the dynamics and its long-term behavior of the proposed model, as well as to discuss the effects of crucial parameters [...] Read more.

We consider a mathematical model to describe the interaction between predator and prey that includes predator cannibalism and refuge. We aim to study the dynamics and its long-term behavior of the proposed model, as well as to discuss the effects of crucial parameters associated with the model. We first show the boundedness and positivity of the solution of the model. Then, we study the existence and stability of all possible equilibrium points. The local stability of the model around each equilibrium point is studied via the linearized system, while the global stability is performed by defining a Lyapunov function. The model has four equilibrium points. It is found that the equilibrium point representing the extinction of both prey and predator populations is always unstable, while the other equilibrium points are conditionally stable. In addition, there is forward bifurcation phenomena that occur under certain condition. To support our analytical findings, we perform some numerical simulations. Full article

(This article belongs to the Special Issue Recent Advances in Nonlinear Differential Equations: Theory, Methods and Applications)

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15 pages, 3625 KiB

Open AccessArticle

Balanced Medical Image Classification with Transfer Learning and Convolutional Neural Networks

by David Benavente, Gustavo Gatica and Jesús González-Feliu

Axioms 2022, 11(3), 115; https://doi.org/10.3390/axioms11030115 - 7 Mar 2022

Cited by 3 | Viewed by 2716

Abstract

This paper aims to propose a tool for image classification in medical diagnosis decision support, in a context where computational power is limited and then specific, high-speed computing infrastructures cannot be used (mainly for economic and energy consuming reasons). The proposed method combines [...] Read more.

This paper aims to propose a tool for image classification in medical diagnosis decision support, in a context where computational power is limited and then specific, high-speed computing infrastructures cannot be used (mainly for economic and energy consuming reasons). The proposed method combines a deep neural networks algorithm with medical imaging procedures and is implemented to allow an efficient use on affordable hardware. The convolutional neural network (CNN) procedure used VGG16 as its base architecture, using the transfer learning technique with the parameters obtained in the ImageNet competition. Two convolutional blocks and one dense block were added to this architecture. The tool was developed and calibrated on the basis of five common lung diseases using 5430 images from two public datasets and the transfer learning technique. The holdout ratios of 90% and 10% for training and testing, respectively, were obtained, and the regularization tools were dropout, early stopping, and Lasso regularization (L2). An accuracy (ACC) of 56% and an area under the receiver-operating characteristic curve (ROC—AUC) of 50% were reached in testing, which are suitable for decision support in a resource-constrained environment. Full article

(This article belongs to the Section Mathematical Analysis)

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16 pages, 475 KiB

Open AccessArticle

The Average Eccentricity of Block Graphs: A Block Order Sequence Perspective

by Xingfu Li, Guihai Yu and Kinkar Chandra Das

Axioms 2022, 11(3), 114; https://doi.org/10.3390/axioms11030114 - 3 Mar 2022

Cited by 4 | Viewed by 2363

Abstract

A graph is a block graph if its blocks are all cliques. In this paper, we study the average eccentricity of block graphs from the perspective of block order sequences. An equivalence relation is established under the block order sequence and used [...] Read more.

A graph is a block graph if its blocks are all cliques. In this paper, we study the average eccentricity of block graphs from the perspective of block order sequences. An equivalence relation is established under the block order sequence and used to prove the lower and upper bounds of the eccentricity on block graphs. The result is that the lower and upper bounds of the average eccentricity on block graphs are 1 and

\frac{1}{n} ⌊ \frac{3}{4} n^{2} - \frac{1}{2} n ⌋

, respectively, where n is the order of the block graph. Finally, we devise a linear time algorithm to calculate the block order sequence. Full article

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Axioms, Volume 11, Issue 3 (March 2022) – 61 articles

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