Axioms Latest open access articles published in Axioms at https://www.mdpi.com/journal/axioms https://www.mdpi.com/journal/axioms MDPI en Creative Commons Attribution (CC-BY) MDPI support@mdpi.com Axioms, Vol. 10, Pages 163: The Existence of Nontrivial Solution for a Class of Kirchhoff-Type Equation of General Convolution Nonlinearity without Any Growth Conditions https://www.mdpi.com/2075-1680/10/3/163 In this paper, we consider the following Kirchhoff-type equation:{u∈H1(RN),−(a+b∫RN|∇u|2dx)Δu+V(x)u=(Iα*F(u))f(u)+λg(u),inRN, where a&amp;gt;0, b≥0, λ&amp;gt;0, α∈(N−2,N), N≥3, V:RN→R is a potential function and Iα is a Riesz potential of order α∈(N−2,N). Under certain assumptions on V(x), f(u) and g(u), we prove that the equation has at least one nontrivial solution by variational methods. Axioms, Vol. 10, Pages 163: The Existence of Nontrivial Solution for a Class of Kirchhoff-Type Equation of General Convolution Nonlinearity without Any Growth Conditions

Authors: Li Zhou Chuanxi Zhu

In this paper, we consider the following Kirchhoff-type equation:{u∈H1(RN),−(a+b∫RN|∇u|2dx)Δu+V(x)u=(Iα*F(u))f(u)+λg(u),inRN, where a&amp;gt;0, b≥0, λ&amp;gt;0, α∈(N−2,N), N≥3, V:RN→R is a potential function and Iα is a Riesz potential of order α∈(N−2,N). Under certain assumptions on V(x), f(u) and g(u), we prove that the equation has at least one nontrivial solution by variational methods.

]]>
The Existence of Nontrivial Solution for a Class of Kirchhoff-Type Equation of General Convolution Nonlinearity without Any Growth Conditions Li Zhou Chuanxi Zhu doi: 10.3390/axioms10030163 Axioms 2021-07-27 Axioms 2021-07-27 10 3 Article 163 10.3390/axioms10030163 https://www.mdpi.com/2075-1680/10/3/163
Axioms, Vol. 10, Pages 164: Rough Approximation Operators on a Complete Orthomodular Lattice https://www.mdpi.com/2075-1680/10/3/164 This paper studies rough approximation via join and meet on a complete orthomodular lattice. Different from Boolean algebra, the distributive law of join over meet does not hold in orthomodular lattices. Some properties of rough approximation rely on the distributive law. Furthermore, we study the relationship among the distributive law, rough approximation and orthomodular lattice-valued relation. Axioms, Vol. 10, Pages 164: Rough Approximation Operators on a Complete Orthomodular Lattice

Authors: Songsong Dai

This paper studies rough approximation via join and meet on a complete orthomodular lattice. Different from Boolean algebra, the distributive law of join over meet does not hold in orthomodular lattices. Some properties of rough approximation rely on the distributive law. Furthermore, we study the relationship among the distributive law, rough approximation and orthomodular lattice-valued relation.

]]>
Rough Approximation Operators on a Complete Orthomodular Lattice Songsong Dai doi: 10.3390/axioms10030164 Axioms 2021-07-27 Axioms 2021-07-27 10 3 Article 164 10.3390/axioms10030164 https://www.mdpi.com/2075-1680/10/3/164
Axioms, Vol. 10, Pages 162: Critical Indices and Self-Similar Power Transform https://www.mdpi.com/2075-1680/10/3/162 “Odd” factor approximants of the special form suggested by Gluzman and Yukalov (J. Math. Chem. 2006, 39, 47) are amenable to optimization by power transformation and can be successfully applied to critical phenomena. The approach is based on the idea that the critical index by itself should be optimized through the parameters of power transform to be calculated from the minimal sensitivity (derivative) optimization condition. The critical index is a product of the algebraic self-similar renormalization which contributes to the expressions the set of control parameters typical to the algebraic self-similar renormalization, and of the power transform which corrects them even further. The parameter of power transformation is, in a nutshell, the multiplier connecting the critical exponent and the correction-to-scaling exponent. We mostly study the minimal model of critical phenomena based on expansions with only two coefficients and critical points. The optimization appears to bring quite accurate, uniquely defined results given by simple formulas. Many important cases of critical phenomena are covered by the simple formula. For the longer series, the optimization condition possesses multiple solutions, and additional constraints should be applied. In particular, we constrain the sought solution by requiring it to be the best in prediction of the coefficients not employed in its construction. In principle, the error/measure of such prediction can be optimized by itself, with respect to the parameter of power transform. Methods of calculation based on optimized power-transformed factors are applied and results presented for critical indices of several key models of conductivity and viscosity of random media, swelling of polymers, permeability in two-dimensional channels. Several quantum mechanical problems are discussed as well. Axioms, Vol. 10, Pages 162: Critical Indices and Self-Similar Power Transform

Authors: Simon Gluzman

“Odd” factor approximants of the special form suggested by Gluzman and Yukalov (J. Math. Chem. 2006, 39, 47) are amenable to optimization by power transformation and can be successfully applied to critical phenomena. The approach is based on the idea that the critical index by itself should be optimized through the parameters of power transform to be calculated from the minimal sensitivity (derivative) optimization condition. The critical index is a product of the algebraic self-similar renormalization which contributes to the expressions the set of control parameters typical to the algebraic self-similar renormalization, and of the power transform which corrects them even further. The parameter of power transformation is, in a nutshell, the multiplier connecting the critical exponent and the correction-to-scaling exponent. We mostly study the minimal model of critical phenomena based on expansions with only two coefficients and critical points. The optimization appears to bring quite accurate, uniquely defined results given by simple formulas. Many important cases of critical phenomena are covered by the simple formula. For the longer series, the optimization condition possesses multiple solutions, and additional constraints should be applied. In particular, we constrain the sought solution by requiring it to be the best in prediction of the coefficients not employed in its construction. In principle, the error/measure of such prediction can be optimized by itself, with respect to the parameter of power transform. Methods of calculation based on optimized power-transformed factors are applied and results presented for critical indices of several key models of conductivity and viscosity of random media, swelling of polymers, permeability in two-dimensional channels. Several quantum mechanical problems are discussed as well.

]]>
Critical Indices and Self-Similar Power Transform Simon Gluzman doi: 10.3390/axioms10030162 Axioms 2021-07-26 Axioms 2021-07-26 10 3 Article 162 10.3390/axioms10030162 https://www.mdpi.com/2075-1680/10/3/162
Axioms, Vol. 10, Pages 161: Semilocal Convergence of the Extension of Chun’s Method https://www.mdpi.com/2075-1680/10/3/161 In this work, we use the technique of recurrence relations to prove the semilocal convergence in Banach spaces of the multidimensional extension of Chun’s iterative method. This is an iterative method of fourth order, that can be transferred to the multivariable case by using the divided difference operator. We obtain the domain of existence and uniqueness by taking a suitable starting point and imposing a Lipschitz condition to the first Fréchet derivative in the whole domain. Moreover, we apply the theoretical results obtained to a nonlinear integral equation of Hammerstein type, showing the applicability of our results. Axioms, Vol. 10, Pages 161: Semilocal Convergence of the Extension of Chun’s Method

Authors: Alicia Cordero Javier G. Maimó Eulalia Martínez Juan R. Torregrosa María P. Vassileva

In this work, we use the technique of recurrence relations to prove the semilocal convergence in Banach spaces of the multidimensional extension of Chun’s iterative method. This is an iterative method of fourth order, that can be transferred to the multivariable case by using the divided difference operator. We obtain the domain of existence and uniqueness by taking a suitable starting point and imposing a Lipschitz condition to the first Fréchet derivative in the whole domain. Moreover, we apply the theoretical results obtained to a nonlinear integral equation of Hammerstein type, showing the applicability of our results.

]]>
Semilocal Convergence of the Extension of Chun’s Method Alicia Cordero Javier G. Maimó Eulalia Martínez Juan R. Torregrosa María P. Vassileva doi: 10.3390/axioms10030161 Axioms 2021-07-26 Axioms 2021-07-26 10 3 Article 161 10.3390/axioms10030161 https://www.mdpi.com/2075-1680/10/3/161
Axioms, Vol. 10, Pages 160: Properties of Certain Multivalent Analytic Functions Associated with the Lemniscate of Bernoulli https://www.mdpi.com/2075-1680/10/3/160 Using differential subordination, we consider conditions of β so that some multivalent analytic functions are subordinate to (1+z)γ (0&amp;lt;γ≤1). Notably, these results are applied to derive sufficient conditions for f∈A to satisfy the condition zf′(z)f(z)2−1&amp;lt;1. Several previous results are extended. Axioms, Vol. 10, Pages 160: Properties of Certain Multivalent Analytic Functions Associated with the Lemniscate of Bernoulli

Authors: Likai Liu Jin-Lin Liu

Using differential subordination, we consider conditions of β so that some multivalent analytic functions are subordinate to (1+z)γ (0&amp;lt;γ≤1). Notably, these results are applied to derive sufficient conditions for f∈A to satisfy the condition zf′(z)f(z)2−1&amp;lt;1. Several previous results are extended.

]]>
Properties of Certain Multivalent Analytic Functions Associated with the Lemniscate of Bernoulli Likai Liu Jin-Lin Liu doi: 10.3390/axioms10030160 Axioms 2021-07-26 Axioms 2021-07-26 10 3 Article 160 10.3390/axioms10030160 https://www.mdpi.com/2075-1680/10/3/160
Axioms, Vol. 10, Pages 159: Popularity Prediction of Online Contents via Cascade Graph and Temporal Information https://www.mdpi.com/2075-1680/10/3/159 Predicting the popularity of online content is an important task for content recommendation, social influence prediction and so on. Recent deep learning models generally utilize graph neural networks to model the complex relationship between information cascade graph and future popularity, and have shown better prediction results compared with traditional methods. However, existing models adopt simple graph pooling strategies, e.g., summation or average, which prone to generate inefficient cascade graph representation and lead to unsatisfactory prediction results. Meanwhile, they often overlook the temporal information in the diffusion process which has been proved to be a salient predictor for popularity prediction. To focus attention on the important users and exclude noises caused by other less relevant users when generating cascade graph representation, we learn the importance coefficient of users and adopt sample mechanism in graph pooling process. In order to capture the temporal features in the diffusion process, we incorporate the inter-infection duration time information into our model by using LSTM neural network. The results show that temporal information rather than cascade graph information is a better predictor for popularity. The experimental results on real datasets show that our model significantly improves the prediction accuracy compared with other state-of-the-art methods. Axioms, Vol. 10, Pages 159: Popularity Prediction of Online Contents via Cascade Graph and Temporal Information

Authors: Yingdan Shang Bin Zhou Ye Wang Aiping Li Kai Chen Yichen Song Changjian Lin

Predicting the popularity of online content is an important task for content recommendation, social influence prediction and so on. Recent deep learning models generally utilize graph neural networks to model the complex relationship between information cascade graph and future popularity, and have shown better prediction results compared with traditional methods. However, existing models adopt simple graph pooling strategies, e.g., summation or average, which prone to generate inefficient cascade graph representation and lead to unsatisfactory prediction results. Meanwhile, they often overlook the temporal information in the diffusion process which has been proved to be a salient predictor for popularity prediction. To focus attention on the important users and exclude noises caused by other less relevant users when generating cascade graph representation, we learn the importance coefficient of users and adopt sample mechanism in graph pooling process. In order to capture the temporal features in the diffusion process, we incorporate the inter-infection duration time information into our model by using LSTM neural network. The results show that temporal information rather than cascade graph information is a better predictor for popularity. The experimental results on real datasets show that our model significantly improves the prediction accuracy compared with other state-of-the-art methods.

]]>
Popularity Prediction of Online Contents via Cascade Graph and Temporal Information Yingdan Shang Bin Zhou Ye Wang Aiping Li Kai Chen Yichen Song Changjian Lin doi: 10.3390/axioms10030159 Axioms 2021-07-23 Axioms 2021-07-23 10 3 Article 159 10.3390/axioms10030159 https://www.mdpi.com/2075-1680/10/3/159
Axioms, Vol. 10, Pages 158: Gauss–Newton–Secant Method for Solving Nonlinear Least Squares Problems under Generalized Lipschitz Conditions https://www.mdpi.com/2075-1680/10/3/158 We develop a local convergence of an iterative method for solving nonlinear least squares problems with operator decomposition under the classical and generalized Lipschitz conditions. We consider the case of both zero and nonzero residuals and determine their convergence orders. We use two types of Lipschitz conditions (center and restricted region conditions) to study the convergence of the method. Moreover, we obtain a larger radius of convergence and tighter error estimates than in previous works. Hence, we extend the applicability of this method under the same computational effort. Axioms, Vol. 10, Pages 158: Gauss–Newton–Secant Method for Solving Nonlinear Least Squares Problems under Generalized Lipschitz Conditions

Authors: Ioannis K. Argyros Stepan Shakhno Roman Iakymchuk Halyna Yarmola Michael I. Argyros

We develop a local convergence of an iterative method for solving nonlinear least squares problems with operator decomposition under the classical and generalized Lipschitz conditions. We consider the case of both zero and nonzero residuals and determine their convergence orders. We use two types of Lipschitz conditions (center and restricted region conditions) to study the convergence of the method. Moreover, we obtain a larger radius of convergence and tighter error estimates than in previous works. Hence, we extend the applicability of this method under the same computational effort.

]]>
Gauss–Newton–Secant Method for Solving Nonlinear Least Squares Problems under Generalized Lipschitz Conditions Ioannis K. Argyros Stepan Shakhno Roman Iakymchuk Halyna Yarmola Michael I. Argyros doi: 10.3390/axioms10030158 Axioms 2021-07-21 Axioms 2021-07-21 10 3 Article 158 10.3390/axioms10030158 https://www.mdpi.com/2075-1680/10/3/158
Axioms, Vol. 10, Pages 157: New Results on the SSIE with an Operator of the form FΔ⊂E+Fx′ Involving the Spaces of Strongly Summable and Convergent Sequences Using the Cesàro Method https://www.mdpi.com/2075-1680/10/3/157 Given any sequence a=(an)n≥1 of positive real numbers and any set E of complex sequences, we can use Ea to represent the set of all sequences y=(yn)n≥1 such that y/a=(yn/an)n≥1∈E. In this paper, we use the spaces w∞, w0 and w of strongly bounded, summable to zero and summable sequences, which are the sets of all sequences y such that n−1∑k=1nykn is bounded and tends to zero, and such that y−le∈w0, for some scalarl . These sets were used in the statistical convergence. Then we deal with the solvability of each of the SSIE FΔ⊂E+Fx′, where E is a linear space of sequences, F=c0, c, ℓ∞, w0, w or w∞, and F′=c0, c or ℓ∞. For instance, the solvability of the SSIE wΔ⊂w0+sxc relies on determining the set of all sequences x=xnn≥1∈U+ that satisfy the following statement. For every sequence y that satisfies the condition limn→∞n−1∑k=1nyk−yk−1−l=0, there are two sequences u and v, with y=u+v such that limn→∞n−1∑k=1nuk=0 and limn→∞vn/xn=L for some scalars l and L. Axioms, Vol. 10, Pages 157: New Results on the SSIE with an Operator of the form FΔ⊂E+Fx′ Involving the Spaces of Strongly Summable and Convergent Sequences Using the Cesàro Method

Authors: Bruno de Malafosse

Given any sequence a=(an)n≥1 of positive real numbers and any set E of complex sequences, we can use Ea to represent the set of all sequences y=(yn)n≥1 such that y/a=(yn/an)n≥1∈E. In this paper, we use the spaces w∞, w0 and w of strongly bounded, summable to zero and summable sequences, which are the sets of all sequences y such that n−1∑k=1nykn is bounded and tends to zero, and such that y−le∈w0, for some scalarl . These sets were used in the statistical convergence. Then we deal with the solvability of each of the SSIE FΔ⊂E+Fx′, where E is a linear space of sequences, F=c0, c, ℓ∞, w0, w or w∞, and F′=c0, c or ℓ∞. For instance, the solvability of the SSIE wΔ⊂w0+sxc relies on determining the set of all sequences x=xnn≥1∈U+ that satisfy the following statement. For every sequence y that satisfies the condition limn→∞n−1∑k=1nyk−yk−1−l=0, there are two sequences u and v, with y=u+v such that limn→∞n−1∑k=1nuk=0 and limn→∞vn/xn=L for some scalars l and L.

]]>
New Results on the SSIE with an Operator of the form FΔ⊂E+Fx′ Involving the Spaces of Strongly Summable and Convergent Sequences Using the Cesàro Method Bruno de Malafosse doi: 10.3390/axioms10030157 Axioms 2021-07-19 Axioms 2021-07-19 10 3 Article 157 10.3390/axioms10030157 https://www.mdpi.com/2075-1680/10/3/157
Axioms, Vol. 10, Pages 156: Stable Convergence Theorems for Products of Uniformly Continuous Mappings in Metric Spaces https://www.mdpi.com/2075-1680/10/3/156 We study the behavior of inexact products of uniformly continuous self-mappings of a complete metric space that is uniformly continuous and bounded on bounded sets. It is shown that previously established convergence theorems for products of non-expansive mappings continue to hold even under the presence of computational errors. Axioms, Vol. 10, Pages 156: Stable Convergence Theorems for Products of Uniformly Continuous Mappings in Metric Spaces

Authors: Alexander J. Zaslavski

We study the behavior of inexact products of uniformly continuous self-mappings of a complete metric space that is uniformly continuous and bounded on bounded sets. It is shown that previously established convergence theorems for products of non-expansive mappings continue to hold even under the presence of computational errors.

]]>
Stable Convergence Theorems for Products of Uniformly Continuous Mappings in Metric Spaces Alexander J. Zaslavski doi: 10.3390/axioms10030156 Axioms 2021-07-19 Axioms 2021-07-19 10 3 Article 156 10.3390/axioms10030156 https://www.mdpi.com/2075-1680/10/3/156
Axioms, Vol. 10, Pages 155: Long Colimits of Topological Groups III: Homeomorphisms of Products and Coproducts https://www.mdpi.com/2075-1680/10/3/155 The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in a number of ways, including as a colimit of homeomorphism groups with a given compact support or as a subgroup of the homeomorphism group of its Stone-Čech compactification. A space is said to have the Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. The authors provide necessary and sufficient conditions for finite products of ordinals equipped with the order topology to have CSHP. In addition, necessary conditions are presented for finite products and coproducts of spaces to have CSHP. Axioms, Vol. 10, Pages 155: Long Colimits of Topological Groups III: Homeomorphisms of Products and Coproducts

Authors: Rafael Dahmen Gábor Lukács

The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in a number of ways, including as a colimit of homeomorphism groups with a given compact support or as a subgroup of the homeomorphism group of its Stone-Čech compactification. A space is said to have the Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. The authors provide necessary and sufficient conditions for finite products of ordinals equipped with the order topology to have CSHP. In addition, necessary conditions are presented for finite products and coproducts of spaces to have CSHP.

]]>
Long Colimits of Topological Groups III: Homeomorphisms of Products and Coproducts Rafael Dahmen Gábor Lukács doi: 10.3390/axioms10030155 Axioms 2021-07-19 Axioms 2021-07-19 10 3 Article 155 10.3390/axioms10030155 https://www.mdpi.com/2075-1680/10/3/155
Axioms, Vol. 10, Pages 154: Water Particles Monitoring in the Atacama Desert: SPC Approach Based on Proportional Data https://www.mdpi.com/2075-1680/10/3/154 Statistical monitoring tools are well established in the literature, creating organizational cultures such as Six Sigma or Total Quality Management. Nevertheless, most of this literature is based on the normality assumption, e.g., based on the law of large numbers, and brings limitations towards truncated processes as open questions in this field. This work was motivated by the register of elements related to the water particles monitoring (relative humidity), an important source of moisture for the Copiapó watershed, and the Atacama region of Chile (the Atacama Desert), and presenting high asymmetry for rates and proportions data. This paper proposes a new control chart for interval data about rates and proportions (symbolic interval data) when they are not results of a Bernoulli process. The unit-Lindley distribution has many interesting properties, such as having only one parameter, from which we develop the unit-Lindley chart for both classical and symbolic data. The performance of the proposed control chart is analyzed using the average run length (ARL), median run length (MRL), and standard deviation of the run length (SDRL) metrics calculated through an extensive Monte Carlo simulation study. Results from the real data applications reveal the tool’s potential to be adopted to estimate the control limits in a Statistical Process Control (SPC) framework. Axioms, Vol. 10, Pages 154: Water Particles Monitoring in the Atacama Desert: SPC Approach Based on Proportional Data

Authors: Anderson Fonseca Paulo Henrique Ferreira Diego Carvalho do Nascimento Rosemeire Fiaccone Christopher Ulloa-Correa Ayón García-Piña Francisco Louzada

Statistical monitoring tools are well established in the literature, creating organizational cultures such as Six Sigma or Total Quality Management. Nevertheless, most of this literature is based on the normality assumption, e.g., based on the law of large numbers, and brings limitations towards truncated processes as open questions in this field. This work was motivated by the register of elements related to the water particles monitoring (relative humidity), an important source of moisture for the Copiapó watershed, and the Atacama region of Chile (the Atacama Desert), and presenting high asymmetry for rates and proportions data. This paper proposes a new control chart for interval data about rates and proportions (symbolic interval data) when they are not results of a Bernoulli process. The unit-Lindley distribution has many interesting properties, such as having only one parameter, from which we develop the unit-Lindley chart for both classical and symbolic data. The performance of the proposed control chart is analyzed using the average run length (ARL), median run length (MRL), and standard deviation of the run length (SDRL) metrics calculated through an extensive Monte Carlo simulation study. Results from the real data applications reveal the tool’s potential to be adopted to estimate the control limits in a Statistical Process Control (SPC) framework.

]]>
Water Particles Monitoring in the Atacama Desert: SPC Approach Based on Proportional Data Anderson Fonseca Paulo Henrique Ferreira Diego Carvalho do Nascimento Rosemeire Fiaccone Christopher Ulloa-Correa Ayón García-Piña Francisco Louzada doi: 10.3390/axioms10030154 Axioms 2021-07-13 Axioms 2021-07-13 10 3 Article 154 10.3390/axioms10030154 https://www.mdpi.com/2075-1680/10/3/154
Axioms, Vol. 10, Pages 152: A Tseng-Type Algorithm with Self-Adaptive Techniques for Solving the Split Problem of Fixed Points and Pseudomonotone Variational Inequalities in Hilbert Spaces https://www.mdpi.com/2075-1680/10/3/152 In this paper, we survey the split problem of fixed points of two pseudocontractive operators and variational inequalities of two pseudomonotone operators in Hilbert spaces. We present a Tseng-type iterative algorithm for solving the split problem by using self-adaptive techniques. Under certain assumptions, we show that the proposed algorithm converges weakly to a solution of the split problem. An application is included. Axioms, Vol. 10, Pages 152: A Tseng-Type Algorithm with Self-Adaptive Techniques for Solving the Split Problem of Fixed Points and Pseudomonotone Variational Inequalities in Hilbert Spaces

Authors: Li-Jun Zhu Yeong-Cheng Liou

In this paper, we survey the split problem of fixed points of two pseudocontractive operators and variational inequalities of two pseudomonotone operators in Hilbert spaces. We present a Tseng-type iterative algorithm for solving the split problem by using self-adaptive techniques. Under certain assumptions, we show that the proposed algorithm converges weakly to a solution of the split problem. An application is included.

]]>
A Tseng-Type Algorithm with Self-Adaptive Techniques for Solving the Split Problem of Fixed Points and Pseudomonotone Variational Inequalities in Hilbert Spaces Li-Jun Zhu Yeong-Cheng Liou doi: 10.3390/axioms10030152 Axioms 2021-07-10 Axioms 2021-07-10 10 3 Article 152 10.3390/axioms10030152 https://www.mdpi.com/2075-1680/10/3/152
Axioms, Vol. 10, Pages 153: Analysis of Novel Oscillations of Quantized Mechanical Energy in Mass-Accreting Nano-Oscillator Systems https://www.mdpi.com/2075-1680/10/3/153 Quantum characteristics of a mass-accreting oscillator are investigated using the invariant operator theory, which is a rigorous mathematical tool for unfolding quantum theory for time-dependent Hamiltonian systems. In particular, the quantum energy of the system is analyzed in detail and compared to the classical one. We focus on two particular cases; one is a linearly mass-accreting oscillator and the other is an exponentially mass-accreting one. It is confirmed that the quantum energy is in agreement with the classical one in the limit ℏ→0. We showed that not only the classical but also the quantum energy oscillates with time. It is carefully analyzed why the energy oscillates with time, and a reasonable explanation for that outcome is given. Axioms, Vol. 10, Pages 153: Analysis of Novel Oscillations of Quantized Mechanical Energy in Mass-Accreting Nano-Oscillator Systems

Authors: Jeong Ryeol Choi

Quantum characteristics of a mass-accreting oscillator are investigated using the invariant operator theory, which is a rigorous mathematical tool for unfolding quantum theory for time-dependent Hamiltonian systems. In particular, the quantum energy of the system is analyzed in detail and compared to the classical one. We focus on two particular cases; one is a linearly mass-accreting oscillator and the other is an exponentially mass-accreting one. It is confirmed that the quantum energy is in agreement with the classical one in the limit ℏ→0. We showed that not only the classical but also the quantum energy oscillates with time. It is carefully analyzed why the energy oscillates with time, and a reasonable explanation for that outcome is given.

]]>
Analysis of Novel Oscillations of Quantized Mechanical Energy in Mass-Accreting Nano-Oscillator Systems Jeong Ryeol Choi doi: 10.3390/axioms10030153 Axioms 2021-07-10 Axioms 2021-07-10 10 3 Article 153 10.3390/axioms10030153 https://www.mdpi.com/2075-1680/10/3/153
Axioms, Vol. 10, Pages 151: Distinguished Property in Tensor Products and Weak* Dual Spaces https://www.mdpi.com/2075-1680/10/3/151 A local convex space E is said to be distinguished if its strong dual Eβ′ has the topology β(E′,(Eβ′)′), i.e., if Eβ′ is barrelled. The distinguished property of the local convex space CpX of real-valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and Cp-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space CpX is distinguished if and only if any function f∈RX belongs to the pointwise closure of a pointwise bounded set in CX. The extensively studied distinguished properties in the injective tensor products CpX⊗εE and in Cp(X,E) contrasts with the few distinguished properties of injective tensor products related to the dual space LpX of CpX endowed with the weak* topology, as well as to the weak* dual of Cp(X,E). To partially fill this gap, some distinguished properties in the injective tensor product space LpX⊗εE are presented and a characterization of the distinguished property of the weak* dual of Cp(X,E) for wide classes of spaces X and E is provided. Axioms, Vol. 10, Pages 151: Distinguished Property in Tensor Products and Weak* Dual Spaces

Authors: Salvador López-Alfonso Manuel López-Pellicer Santiago Moll-López

A local convex space E is said to be distinguished if its strong dual Eβ′ has the topology β(E′,(Eβ′)′), i.e., if Eβ′ is barrelled. The distinguished property of the local convex space CpX of real-valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and Cp-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space CpX is distinguished if and only if any function f∈RX belongs to the pointwise closure of a pointwise bounded set in CX. The extensively studied distinguished properties in the injective tensor products CpX⊗εE and in Cp(X,E) contrasts with the few distinguished properties of injective tensor products related to the dual space LpX of CpX endowed with the weak* topology, as well as to the weak* dual of Cp(X,E). To partially fill this gap, some distinguished properties in the injective tensor product space LpX⊗εE are presented and a characterization of the distinguished property of the weak* dual of Cp(X,E) for wide classes of spaces X and E is provided.

]]>
Distinguished Property in Tensor Products and Weak* Dual Spaces Salvador López-Alfonso Manuel López-Pellicer Santiago Moll-López doi: 10.3390/axioms10030151 Axioms 2021-07-08 Axioms 2021-07-08 10 3 Article 151 10.3390/axioms10030151 https://www.mdpi.com/2075-1680/10/3/151
Axioms, Vol. 10, Pages 150: Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability https://www.mdpi.com/2075-1680/10/3/150 In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained. Axioms, Vol. 10, Pages 150: Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Authors: Andriy Zagorodnyuk Anna Hihliuk

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

]]>
Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability Andriy Zagorodnyuk Anna Hihliuk doi: 10.3390/axioms10030150 Axioms 2021-07-07 Axioms 2021-07-07 10 3 Article 150 10.3390/axioms10030150 https://www.mdpi.com/2075-1680/10/3/150
Axioms, Vol. 10, Pages 149: Unveiling the Dynamics of the European Entrepreneurial Framework Conditions over the Last Two Decades: A Cluster Analysis https://www.mdpi.com/2075-1680/10/3/149 Entrepreneurship is a theme of global interest, and it is the subject of investigations conducted by many researchers and projects. In particular, the Global Entrepreneurship Monitor project is a global project that involves several countries and years of surveys on entrepreneurship indicators. This study focuses on the 12 indicators of the entrepreneurial ecosystem defined by the Entrepreneurial Framework Conditions (EFCs). The EFCs are specifically related to the quality of the entrepreneurial ecosystem. Using clustering techniques, the present study analyzes how European experts’ perceptions on the EFCs of their home country have changed between 2000 and 2019. The main finding is the existence of significant differences between the clusters obtained over the years and between countries. Therefore, in theoretical terms, this dynamical behavior in relation to the entrepreneurial conditions of economies should be considered in future works, namely, those concerning the definition of the number of clusters, which, according to the internal validation measures computed in this work, should be two. Axioms, Vol. 10, Pages 149: Unveiling the Dynamics of the European Entrepreneurial Framework Conditions over the Last Two Decades: A Cluster Analysis

Authors: Eliana Costa e Silva Aldina Correia Ana Borges

Entrepreneurship is a theme of global interest, and it is the subject of investigations conducted by many researchers and projects. In particular, the Global Entrepreneurship Monitor project is a global project that involves several countries and years of surveys on entrepreneurship indicators. This study focuses on the 12 indicators of the entrepreneurial ecosystem defined by the Entrepreneurial Framework Conditions (EFCs). The EFCs are specifically related to the quality of the entrepreneurial ecosystem. Using clustering techniques, the present study analyzes how European experts’ perceptions on the EFCs of their home country have changed between 2000 and 2019. The main finding is the existence of significant differences between the clusters obtained over the years and between countries. Therefore, in theoretical terms, this dynamical behavior in relation to the entrepreneurial conditions of economies should be considered in future works, namely, those concerning the definition of the number of clusters, which, according to the internal validation measures computed in this work, should be two.

]]>
Unveiling the Dynamics of the European Entrepreneurial Framework Conditions over the Last Two Decades: A Cluster Analysis Eliana Costa e Silva Aldina Correia Ana Borges doi: 10.3390/axioms10030149 Axioms 2021-07-06 Axioms 2021-07-06 10 3 Article 149 10.3390/axioms10030149 https://www.mdpi.com/2075-1680/10/3/149
Axioms, Vol. 10, Pages 148: Robust Stability of Time-Varying Markov Jump Linear Systems with Respect to a Class of Structured, Stochastic, Nonlinear Parametric Uncertainties https://www.mdpi.com/2075-1680/10/3/148 This note is devoted to a robust stability analysis, as well as to the problem of the robust stabilization of a class of continuous-time Markovian jump linear systems subject to block-diagonal stochastic parameter perturbations. The considered parametric uncertainties are of multiplicative white noise type with unknown intensity. In order to effectively address the multi-perturbations case, we use scaling techniques. These techniques allow us to obtain an estimation of the lower bound of the stability radius. A first characterization of a lower bound of the stability radius is obtained in terms of the unique bounded and positive semidefinite solutions of adequately defined parameterized backward Lyapunov differential equations. A second characterization is given in terms of the existence of positive solutions of adequately defined parameterized backward Lyapunov differential inequalities. This second result is then exploited in order to solve a robust control synthesis problem. Axioms, Vol. 10, Pages 148: Robust Stability of Time-Varying Markov Jump Linear Systems with Respect to a Class of Structured, Stochastic, Nonlinear Parametric Uncertainties

Authors: Vasile Dragan Samir Aberkane

This note is devoted to a robust stability analysis, as well as to the problem of the robust stabilization of a class of continuous-time Markovian jump linear systems subject to block-diagonal stochastic parameter perturbations. The considered parametric uncertainties are of multiplicative white noise type with unknown intensity. In order to effectively address the multi-perturbations case, we use scaling techniques. These techniques allow us to obtain an estimation of the lower bound of the stability radius. A first characterization of a lower bound of the stability radius is obtained in terms of the unique bounded and positive semidefinite solutions of adequately defined parameterized backward Lyapunov differential equations. A second characterization is given in terms of the existence of positive solutions of adequately defined parameterized backward Lyapunov differential inequalities. This second result is then exploited in order to solve a robust control synthesis problem.

]]>
Robust Stability of Time-Varying Markov Jump Linear Systems with Respect to a Class of Structured, Stochastic, Nonlinear Parametric Uncertainties Vasile Dragan Samir Aberkane doi: 10.3390/axioms10030148 Axioms 2021-07-05 Axioms 2021-07-05 10 3 Article 148 10.3390/axioms10030148 https://www.mdpi.com/2075-1680/10/3/148
Axioms, Vol. 10, Pages 147: Inertial Accelerated Algorithm for Fixed Point of Asymptotically Nonexpansive Mapping in Real Uniformly Convex Banach Spaces https://www.mdpi.com/2075-1680/10/3/147 In this work, we introduce a new inertial accelerated Mann algorithm for finding a point in the set of fixed points of asymptotically nonexpansive mapping in a real uniformly convex Banach space. We also establish weak and strong convergence theorems of the scheme. Finally, we give a numerical experiment to validate the performance of our algorithm and compare with some existing methods. Our results generalize and improve some recent results in the literature. Axioms, Vol. 10, Pages 147: Inertial Accelerated Algorithm for Fixed Point of Asymptotically Nonexpansive Mapping in Real Uniformly Convex Banach Spaces

Authors: Murtala Haruna Harbau Godwin Chidi Ugwunnadi Lateef Olakunle Jolaoso Ahmad Abdulwahab

In this work, we introduce a new inertial accelerated Mann algorithm for finding a point in the set of fixed points of asymptotically nonexpansive mapping in a real uniformly convex Banach space. We also establish weak and strong convergence theorems of the scheme. Finally, we give a numerical experiment to validate the performance of our algorithm and compare with some existing methods. Our results generalize and improve some recent results in the literature.

]]>
Inertial Accelerated Algorithm for Fixed Point of Asymptotically Nonexpansive Mapping in Real Uniformly Convex Banach Spaces Murtala Haruna Harbau Godwin Chidi Ugwunnadi Lateef Olakunle Jolaoso Ahmad Abdulwahab doi: 10.3390/axioms10030147 Axioms 2021-07-03 Axioms 2021-07-03 10 3 Article 147 10.3390/axioms10030147 https://www.mdpi.com/2075-1680/10/3/147
Axioms, Vol. 10, Pages 146: On the Colored and the Set-Theoretical Yang–Baxter Equations https://www.mdpi.com/2075-1680/10/3/146 This paper is related to several articles published in AXIOMS, SCI, etc. The main concepts of the current paper are the colored Yang–Baxter equation and the set-theoretical Yang–Baxter equation. The Euler formula, colagebra structures, and means play an important role in our study. We show that some new solutions for a certain system of equations lead to colored Yang–Baxter operators, which are related to an Euler formula for matrices, and the set-theoretical solutions to the Yang–Baxter equation are related to means. A new coalgebra is obtained and studied. Axioms, Vol. 10, Pages 146: On the Colored and the Set-Theoretical Yang–Baxter Equations

Authors: Laszlo Barna Iantovics Florin Felix Nichita

This paper is related to several articles published in AXIOMS, SCI, etc. The main concepts of the current paper are the colored Yang–Baxter equation and the set-theoretical Yang–Baxter equation. The Euler formula, colagebra structures, and means play an important role in our study. We show that some new solutions for a certain system of equations lead to colored Yang–Baxter operators, which are related to an Euler formula for matrices, and the set-theoretical solutions to the Yang–Baxter equation are related to means. A new coalgebra is obtained and studied.

]]>
On the Colored and the Set-Theoretical Yang–Baxter Equations Laszlo Barna Iantovics Florin Felix Nichita doi: 10.3390/axioms10030146 Axioms 2021-07-02 Axioms 2021-07-02 10 3 Article 146 10.3390/axioms10030146 https://www.mdpi.com/2075-1680/10/3/146
Axioms, Vol. 10, Pages 145: Approach to Multi-Attribute Decision-Making Methods for Performance Evaluation Process Using Interval-Valued T-Spherical Fuzzy Hamacher Aggregation Information https://www.mdpi.com/2075-1680/10/3/145 Interval-valued T-spherical fuzzy set (IVTSFS) handles uncertain and vague information by discussing their membership degree (MD), abstinence degree (AD), non-membership degree (NMD), and refusal degree (RD). MD, AD, NMD, and RD are defined in terms of closed subintervals of [0,&amp;nbsp;1] that reduce information loss compared to the T-spherical fuzzy set (TSFS), which takes crisp values from [0,&amp;nbsp;1] intervals; hence, some information may be lost. The purpose of this manuscript is to develop some Hamacher aggregation operators (HAOs) in the environment of IVTSFSs. To do so, some Hamacher operational laws based on Hamacher t-norms (HTNs) and Hamacher t-conorms (HTCNs) are introduced. Using Hamacher operational laws, we develop some aggregation operators (AOs), including an interval-valued T-spherical fuzzy Hamacher (IVTSFH) weighted averaging (IVTSFHWA) operator, an IVTSFH-ordered weighted averaging (IVTSFHOWA) operator, an IVTSFH hybrid averaging (IVTSFHHA) operator, an IVTSFH-weighted geometric (IVTSFHWG) operator, an IVTSFH-ordered weighted geometric (IVTSFHOWG) operator, and an IVTSFH hybrid geometric (IVTSFHHG) operator. The validation of the newly developed HAOs is investigated, and their basic properties are examined. In view of some restrictions, the generalization and proposed HAOs are shown, and a multi-attribute decision-making (MADM) procedure is explored based on the HAOs, which are further exemplified. Finally, a comparative analysis of the proposed work is also discussed with previous literature to show the superiority of our work. Axioms, Vol. 10, Pages 145: Approach to Multi-Attribute Decision-Making Methods for Performance Evaluation Process Using Interval-Valued T-Spherical Fuzzy Hamacher Aggregation Information

Authors: Yun Jin Zareena Kousar Kifayat Ullah Tahir Mahmood Nimet Yapici Pehlivan Zeeshan Ali

Interval-valued T-spherical fuzzy set (IVTSFS) handles uncertain and vague information by discussing their membership degree (MD), abstinence degree (AD), non-membership degree (NMD), and refusal degree (RD). MD, AD, NMD, and RD are defined in terms of closed subintervals of [0,&amp;nbsp;1] that reduce information loss compared to the T-spherical fuzzy set (TSFS), which takes crisp values from [0,&amp;nbsp;1] intervals; hence, some information may be lost. The purpose of this manuscript is to develop some Hamacher aggregation operators (HAOs) in the environment of IVTSFSs. To do so, some Hamacher operational laws based on Hamacher t-norms (HTNs) and Hamacher t-conorms (HTCNs) are introduced. Using Hamacher operational laws, we develop some aggregation operators (AOs), including an interval-valued T-spherical fuzzy Hamacher (IVTSFH) weighted averaging (IVTSFHWA) operator, an IVTSFH-ordered weighted averaging (IVTSFHOWA) operator, an IVTSFH hybrid averaging (IVTSFHHA) operator, an IVTSFH-weighted geometric (IVTSFHWG) operator, an IVTSFH-ordered weighted geometric (IVTSFHOWG) operator, and an IVTSFH hybrid geometric (IVTSFHHG) operator. The validation of the newly developed HAOs is investigated, and their basic properties are examined. In view of some restrictions, the generalization and proposed HAOs are shown, and a multi-attribute decision-making (MADM) procedure is explored based on the HAOs, which are further exemplified. Finally, a comparative analysis of the proposed work is also discussed with previous literature to show the superiority of our work.

]]>
Approach to Multi-Attribute Decision-Making Methods for Performance Evaluation Process Using Interval-Valued T-Spherical Fuzzy Hamacher Aggregation Information Yun Jin Zareena Kousar Kifayat Ullah Tahir Mahmood Nimet Yapici Pehlivan Zeeshan Ali doi: 10.3390/axioms10030145 Axioms 2021-07-01 Axioms 2021-07-01 10 3 Article 145 10.3390/axioms10030145 https://www.mdpi.com/2075-1680/10/3/145
Axioms, Vol. 10, Pages 144: On the Natural Density of Sets Related to Generalized Fibonacci Numbers of Order r https://www.mdpi.com/2075-1680/10/3/144 For r≥2 and a≥1 integers, let (tn(r,a))n≥1 be the sequence of the (r,a)-generalized Fibonacci numbers which is defined by the recurrence tn(r,a)=tn−1(r,a)+⋯+tn−r(r,a) for n&amp;gt;r, with initial values ti(r,a)=1, for all i∈[1,r−1] and tr(r,a)=a. In this paper, we shall prove (in particular) that, for any given r≥2, there exists a positive proportion of positive integers which can not be written as tn(r,a) for any (n,a)∈Z≥r+2×Z≥1. Axioms, Vol. 10, Pages 144: On the Natural Density of Sets Related to Generalized Fibonacci Numbers of Order r

Authors: Pavel Trojovský

For r≥2 and a≥1 integers, let (tn(r,a))n≥1 be the sequence of the (r,a)-generalized Fibonacci numbers which is defined by the recurrence tn(r,a)=tn−1(r,a)+⋯+tn−r(r,a) for n&amp;gt;r, with initial values ti(r,a)=1, for all i∈[1,r−1] and tr(r,a)=a. In this paper, we shall prove (in particular) that, for any given r≥2, there exists a positive proportion of positive integers which can not be written as tn(r,a) for any (n,a)∈Z≥r+2×Z≥1.

]]>
On the Natural Density of Sets Related to Generalized Fibonacci Numbers of Order r Pavel Trojovský doi: 10.3390/axioms10030144 Axioms 2021-07-01 Axioms 2021-07-01 10 3 Article 144 10.3390/axioms10030144 https://www.mdpi.com/2075-1680/10/3/144
Axioms, Vol. 10, Pages 141: Symmetry Breaking of a Time-2D Space Fractional Wave Equation in a Complex Domain https://www.mdpi.com/2075-1680/10/3/141 (1) Background: symmetry breaking (self-organized transformation of symmetric stats) is a global phenomenon that arises in an extensive diversity of essentially symmetric physical structures. We investigate the symmetry breaking of time-2D space fractional wave equation in a complex domain; (2) Methods: a fractional differential operator is used together with a symmetric operator to define a new fractional symmetric operator. Then by applying the new operator, we formulate a generalized time-2D space fractional wave equation. We shall utilize the two concepts: subordination and majorization to present our results; (3) Results: we obtain different formulas of analytic solutions using the geometric analysis. The solution suggests univalent (1-1) in the open unit disk. Moreover, under certain conditions, it was starlike and dominated by a chaotic function type sine. In addition, the authors formulated a fractional time wave equation by using the Atangana–Baleanu fractional operators in terms of the Riemann–Liouville and Caputo derivatives. Axioms, Vol. 10, Pages 141: Symmetry Breaking of a Time-2D Space Fractional Wave Equation in a Complex Domain

Authors: Rabha W. Ibrahim Dumitru Baleanu

(1) Background: symmetry breaking (self-organized transformation of symmetric stats) is a global phenomenon that arises in an extensive diversity of essentially symmetric physical structures. We investigate the symmetry breaking of time-2D space fractional wave equation in a complex domain; (2) Methods: a fractional differential operator is used together with a symmetric operator to define a new fractional symmetric operator. Then by applying the new operator, we formulate a generalized time-2D space fractional wave equation. We shall utilize the two concepts: subordination and majorization to present our results; (3) Results: we obtain different formulas of analytic solutions using the geometric analysis. The solution suggests univalent (1-1) in the open unit disk. Moreover, under certain conditions, it was starlike and dominated by a chaotic function type sine. In addition, the authors formulated a fractional time wave equation by using the Atangana–Baleanu fractional operators in terms of the Riemann–Liouville and Caputo derivatives.

]]>
Symmetry Breaking of a Time-2D Space Fractional Wave Equation in a Complex Domain Rabha W. Ibrahim Dumitru Baleanu doi: 10.3390/axioms10030141 Axioms 2021-06-30 Axioms 2021-06-30 10 3 Article 141 10.3390/axioms10030141 https://www.mdpi.com/2075-1680/10/3/141
Axioms, Vol. 10, Pages 142: Null Darboux Curve Pairs in Minkowski 3-Space https://www.mdpi.com/2075-1680/10/3/142 Based on the fundamental theories of null curves in Minkowski 3-space, the null Darboux mate curves of a null curve are defined which can be regarded as a kind of extension for Bertrand curves and Mannheim curves in Minkowski 3-space. The relationships of null Darboux curve pairs are explored and their expression forms are presented explicitly. Axioms, Vol. 10, Pages 142: Null Darboux Curve Pairs in Minkowski 3-Space

Authors: Jinhua Qian Mingyu Sun Pei Yin Young-Ho Kim

Based on the fundamental theories of null curves in Minkowski 3-space, the null Darboux mate curves of a null curve are defined which can be regarded as a kind of extension for Bertrand curves and Mannheim curves in Minkowski 3-space. The relationships of null Darboux curve pairs are explored and their expression forms are presented explicitly.

]]>
Null Darboux Curve Pairs in Minkowski 3-Space Jinhua Qian Mingyu Sun Pei Yin Young-Ho Kim doi: 10.3390/axioms10030142 Axioms 2021-06-30 Axioms 2021-06-30 10 3 Article 142 10.3390/axioms10030142 https://www.mdpi.com/2075-1680/10/3/142
Axioms, Vol. 10, Pages 143: Investigation of Proximal Spaces Using Relators https://www.mdpi.com/2075-1680/10/3/143 In this paper, we define uniformities and proximities as relators and show the equivalences of these definitions with classical ones. For this, we summarize the essential properties of relators, using their theory from earlier works of Á. Száz. Moreover, we prove implications between important topological properties of relators and disprove others. Finally, we add an analogous definition for uniformly and proximally filtered properties. Axioms, Vol. 10, Pages 143: Investigation of Proximal Spaces Using Relators

Authors: Gergely Pataki

In this paper, we define uniformities and proximities as relators and show the equivalences of these definitions with classical ones. For this, we summarize the essential properties of relators, using their theory from earlier works of Á. Száz. Moreover, we prove implications between important topological properties of relators and disprove others. Finally, we add an analogous definition for uniformly and proximally filtered properties.

]]>
Investigation of Proximal Spaces Using Relators Gergely Pataki doi: 10.3390/axioms10030143 Axioms 2021-06-30 Axioms 2021-06-30 10 3 Article 143 10.3390/axioms10030143 https://www.mdpi.com/2075-1680/10/3/143
Axioms, Vol. 10, Pages 140: The Key Success Factors for Attracting Foreign Investment in the Post-Epidemic Era https://www.mdpi.com/2075-1680/10/3/140 The global economy has been hit by the unexpected COVID-19 outbreak, and foreign investment has been seen as one of the most important tools to boost the economy. However, in the highly uncertain post-epidemic era, determining how to attract foreign investment is the key to revitalizing the economy. What are the important factors for governments to attract investment, and how to improve them? This will be an important decision in the post-epidemic era. Therefore, this study develops a novel decision-making model to explore the key factors in attracting foreign investment. The model first uses fuzzy Delphi to explore the key factors of attracting foreign investment in the post-epidemic era, and then uses DEMATEL to construct the causal relationships among these factors. To overcome the uncertainty of various information sources and inconsistent messages from decision-makers, this study combined neutrosophic set theory to conduct quantitative analysis. The results of the study show that the model is suitable for analyzing the key factors of investment attraction in the post-epidemic period. Based on the results of the study, we also propose strategies that will help the relevant policy-making departments to understand the root causes of the problem and to formulate appropriate investment strategies in advance. In addition, the model is also used for comparative analysis, which reveals that this novel approach can integrate more incomplete information and present expert opinions in a more objective way. Axioms, Vol. 10, Pages 140: The Key Success Factors for Attracting Foreign Investment in the Post-Epidemic Era

Authors: Sun-Weng Huang James J. H. Liou Shih-Hsiung Cheng William Tang Jessica C. Y. Ma Gwo-Hshiung Tzeng

The global economy has been hit by the unexpected COVID-19 outbreak, and foreign investment has been seen as one of the most important tools to boost the economy. However, in the highly uncertain post-epidemic era, determining how to attract foreign investment is the key to revitalizing the economy. What are the important factors for governments to attract investment, and how to improve them? This will be an important decision in the post-epidemic era. Therefore, this study develops a novel decision-making model to explore the key factors in attracting foreign investment. The model first uses fuzzy Delphi to explore the key factors of attracting foreign investment in the post-epidemic era, and then uses DEMATEL to construct the causal relationships among these factors. To overcome the uncertainty of various information sources and inconsistent messages from decision-makers, this study combined neutrosophic set theory to conduct quantitative analysis. The results of the study show that the model is suitable for analyzing the key factors of investment attraction in the post-epidemic period. Based on the results of the study, we also propose strategies that will help the relevant policy-making departments to understand the root causes of the problem and to formulate appropriate investment strategies in advance. In addition, the model is also used for comparative analysis, which reveals that this novel approach can integrate more incomplete information and present expert opinions in a more objective way.

]]>
The Key Success Factors for Attracting Foreign Investment in the Post-Epidemic Era Sun-Weng Huang James J. H. Liou Shih-Hsiung Cheng William Tang Jessica C. Y. Ma Gwo-Hshiung Tzeng doi: 10.3390/axioms10030140 Axioms 2021-06-30 Axioms 2021-06-30 10 3 Article 140 10.3390/axioms10030140 https://www.mdpi.com/2075-1680/10/3/140
Axioms, Vol. 10, Pages 139: Optimization of Convolutional Neural Networks Architectures Using PSO for Sign Language Recognition https://www.mdpi.com/2075-1680/10/3/139 This paper presents an approach to design convolutional neural network architectures, using the particle swarm optimization algorithm. The adjustment of the hyper-parameters and finding the optimal network architecture of convolutional neural networks represents an important challenge. Network performance and achieving efficient learning models for a particular problem depends on setting hyper-parameter values and this implies exploring a huge and complex search space. The use of heuristic-based searches supports these types of problems; therefore, the main contribution of this research work is to apply the PSO algorithm to find the optimal parameters of the convolutional neural networks which include the number of convolutional layers, the filter size used in the convolutional process, the number of convolutional filters, and the batch size. This work describes two optimization approaches; the first, the parameters obtained by PSO are kept under the same conditions in each convolutional layer, and the objective function evaluated by PSO is given by the classification rate; in the second, the PSO generates different parameters per layer, and the objective function is composed of the recognition rate in conjunction with the Akaike information criterion, the latter helps to find the best network performance but with the minimum parameters. The optimized architectures are implemented in three study cases of sign language databases, in which are included the Mexican Sign Language alphabet, the American Sign Language MNIST, and the American Sign Language alphabet. According to the results, the proposed methodologies achieved favorable results with a recognition rate higher than 99%, showing competitive results compared to other state-of-the-art approaches. Axioms, Vol. 10, Pages 139: Optimization of Convolutional Neural Networks Architectures Using PSO for Sign Language Recognition

Authors: Jonathan Fregoso Claudia I. Gonzalez Gabriela E. Martinez

This paper presents an approach to design convolutional neural network architectures, using the particle swarm optimization algorithm. The adjustment of the hyper-parameters and finding the optimal network architecture of convolutional neural networks represents an important challenge. Network performance and achieving efficient learning models for a particular problem depends on setting hyper-parameter values and this implies exploring a huge and complex search space. The use of heuristic-based searches supports these types of problems; therefore, the main contribution of this research work is to apply the PSO algorithm to find the optimal parameters of the convolutional neural networks which include the number of convolutional layers, the filter size used in the convolutional process, the number of convolutional filters, and the batch size. This work describes two optimization approaches; the first, the parameters obtained by PSO are kept under the same conditions in each convolutional layer, and the objective function evaluated by PSO is given by the classification rate; in the second, the PSO generates different parameters per layer, and the objective function is composed of the recognition rate in conjunction with the Akaike information criterion, the latter helps to find the best network performance but with the minimum parameters. The optimized architectures are implemented in three study cases of sign language databases, in which are included the Mexican Sign Language alphabet, the American Sign Language MNIST, and the American Sign Language alphabet. According to the results, the proposed methodologies achieved favorable results with a recognition rate higher than 99%, showing competitive results compared to other state-of-the-art approaches.

]]>
Optimization of Convolutional Neural Networks Architectures Using PSO for Sign Language Recognition Jonathan Fregoso Claudia I. Gonzalez Gabriela E. Martinez doi: 10.3390/axioms10030139 Axioms 2021-06-29 Axioms 2021-06-29 10 3 Article 139 10.3390/axioms10030139 https://www.mdpi.com/2075-1680/10/3/139
Axioms, Vol. 10, Pages 138: Delay-Dependent Stability, Integrability and Boundedeness Criteria for Delay Differential Systems https://www.mdpi.com/2075-1680/10/3/138 This paper deals with non-perturbed and perturbed systems of nonlinear differential systems of first order with multiple time-varying delays. Here, for the considered systems, easily verifiable and applicable uniformly asymptotic stability, integrability, and boundedness criteria are obtained via defining an appropriate Lyapunov–Krasovskiĭ functional (LKF) and using the Lyapunov–Krasovskiĭ method (LKM). Comparisons with a former result that can be found in the literature illustrate the novelty of the stability theorem and show new contributions to the qualitative theory of solutions. A discussion of two illustrative examples and the obtained results are presented. Axioms, Vol. 10, Pages 138: Delay-Dependent Stability, Integrability and Boundedeness Criteria for Delay Differential Systems

Authors: Osman Tunç Cemil Tunç Yuanheng Wang

This paper deals with non-perturbed and perturbed systems of nonlinear differential systems of first order with multiple time-varying delays. Here, for the considered systems, easily verifiable and applicable uniformly asymptotic stability, integrability, and boundedness criteria are obtained via defining an appropriate Lyapunov–Krasovskiĭ functional (LKF) and using the Lyapunov–Krasovskiĭ method (LKM). Comparisons with a former result that can be found in the literature illustrate the novelty of the stability theorem and show new contributions to the qualitative theory of solutions. A discussion of two illustrative examples and the obtained results are presented.

]]>
Delay-Dependent Stability, Integrability and Boundedeness Criteria for Delay Differential Systems Osman Tunç Cemil Tunç Yuanheng Wang doi: 10.3390/axioms10030138 Axioms 2021-06-29 Axioms 2021-06-29 10 3 Article 138 10.3390/axioms10030138 https://www.mdpi.com/2075-1680/10/3/138
Axioms, Vol. 10, Pages 136: Generalizations of Hermite–Hadamard Type Integral Inequalities for Convex Functions https://www.mdpi.com/2075-1680/10/3/136 In the paper, with the help of two known integral identities and by virtue of the classical Hölder integral inequality, the authors establish several new integral inequalities of the Hermite–Hadamard type for convex functions. These newly established inequalities generalize some known results. Axioms, Vol. 10, Pages 136: Generalizations of Hermite–Hadamard Type Integral Inequalities for Convex Functions

Authors: Ying Wu Hong-Ping Yin Bai-Ni Guo

In the paper, with the help of two known integral identities and by virtue of the classical Hölder integral inequality, the authors establish several new integral inequalities of the Hermite–Hadamard type for convex functions. These newly established inequalities generalize some known results.

]]>
Generalizations of Hermite–Hadamard Type Integral Inequalities for Convex Functions Ying Wu Hong-Ping Yin Bai-Ni Guo doi: 10.3390/axioms10030136 Axioms 2021-06-28 Axioms 2021-06-28 10 3 Article 136 10.3390/axioms10030136 https://www.mdpi.com/2075-1680/10/3/136
Axioms, Vol. 10, Pages 137: Two Inverse Problems Solution by Feedback Tracking Control https://www.mdpi.com/2075-1680/10/3/137 Two inverse ill-posed problems are considered. The first problem is an input restoration of a linear system. The second one is a restoration of time-dependent coefficients of a linear ordinary differential equation. Both problems are reformulated as auxiliary optimal control problems with regularizing cost functional. For the coefficients restoration problem, two control models are proposed. In the first model, the control coefficients are approximated by the output and the estimates of its derivatives. This model yields an approximating linear-quadratic optimal control problem having a known explicit solution. The derivatives are also obtained as auxiliary linear-quadratic tracking controls. The second control model is accurate and leads to a bilinear-quadratic optimal control problem. The latter is tackled in two ways: by an iterative procedure and by a feedback linearization. Simulation results show that a bilinear model provides more accurate coefficients estimates. Axioms, Vol. 10, Pages 137: Two Inverse Problems Solution by Feedback Tracking Control

Two inverse ill-posed problems are considered. The first problem is an input restoration of a linear system. The second one is a restoration of time-dependent coefficients of a linear ordinary differential equation. Both problems are reformulated as auxiliary optimal control problems with regularizing cost functional. For the coefficients restoration problem, two control models are proposed. In the first model, the control coefficients are approximated by the output and the estimates of its derivatives. This model yields an approximating linear-quadratic optimal control problem having a known explicit solution. The derivatives are also obtained as auxiliary linear-quadratic tracking controls. The second control model is accurate and leads to a bilinear-quadratic optimal control problem. The latter is tackled in two ways: by an iterative procedure and by a feedback linearization. Simulation results show that a bilinear model provides more accurate coefficients estimates.

]]>
Two Inverse Problems Solution by Feedback Tracking Control Vladimir Turetsky doi: 10.3390/axioms10030137 Axioms 2021-06-28 Axioms 2021-06-28 10 3 Article 137 10.3390/axioms10030137 https://www.mdpi.com/2075-1680/10/3/137
Axioms, Vol. 10, Pages 135: Mathematical Analysis of a Fractional COVID-19 Model Applied to Wuhan, Spain and Portugal https://www.mdpi.com/2075-1680/10/3/135 We propose a qualitative analysis of a recent fractional-order COVID-19 model. We start by showing that the model is mathematically and biologically well posed. Then, we give a proof on the global stability of the disease free equilibrium point. Finally, some numerical simulations are performed to ensure stability and convergence of the disease free equilibrium point. Axioms, Vol. 10, Pages 135: Mathematical Analysis of a Fractional COVID-19 Model Applied to Wuhan, Spain and Portugal

Authors: Faïçal Ndaïrou Delfim F. M. Torres

We propose a qualitative analysis of a recent fractional-order COVID-19 model. We start by showing that the model is mathematically and biologically well posed. Then, we give a proof on the global stability of the disease free equilibrium point. Finally, some numerical simulations are performed to ensure stability and convergence of the disease free equilibrium point.

]]>
Mathematical Analysis of a Fractional COVID-19 Model Applied to Wuhan, Spain and Portugal Faïçal Ndaïrou Delfim F. M. Torres doi: 10.3390/axioms10030135 Axioms 2021-06-27 Axioms 2021-06-27 10 3 Article 135 10.3390/axioms10030135 https://www.mdpi.com/2075-1680/10/3/135
Axioms, Vol. 10, Pages 134: L-Fuzzy Rough Approximation Operators Based on Co-Implication and Their (Single) Axiomatic Characterizations https://www.mdpi.com/2075-1680/10/3/134 For L a complete co-residuated lattice and R an L-fuzzy relation, an L-fuzzy upper approximation operator based on co-implication adjoint with L is constructed and discussed. It is proved that, when L is regular, the new approximation operator is the dual operator of the Qiao–Hu L-fuzzy lower approximation operator defined in 2018. Then, the new approximation operator is characterized by using an axiom set (in particular, by single axiom). Furthermore, the L-fuzzy upper approximation operators generated by serial, symmetric, reflexive, mediate, transitive, and Euclidean L-fuzzy relations and their compositions are characterize through axiom set (single axiom), respectively. Axioms, Vol. 10, Pages 134: L-Fuzzy Rough Approximation Operators Based on Co-Implication and Their (Single) Axiomatic Characterizations

Authors: Qiu Jin Lingqiang Li

For L a complete co-residuated lattice and R an L-fuzzy relation, an L-fuzzy upper approximation operator based on co-implication adjoint with L is constructed and discussed. It is proved that, when L is regular, the new approximation operator is the dual operator of the Qiao–Hu L-fuzzy lower approximation operator defined in 2018. Then, the new approximation operator is characterized by using an axiom set (in particular, by single axiom). Furthermore, the L-fuzzy upper approximation operators generated by serial, symmetric, reflexive, mediate, transitive, and Euclidean L-fuzzy relations and their compositions are characterize through axiom set (single axiom), respectively.

]]>
L-Fuzzy Rough Approximation Operators Based on Co-Implication and Their (Single) Axiomatic Characterizations Qiu Jin Lingqiang Li doi: 10.3390/axioms10030134 Axioms 2021-06-27 Axioms 2021-06-27 10 3 Article 134 10.3390/axioms10030134 https://www.mdpi.com/2075-1680/10/3/134
Axioms, Vol. 10, Pages 133: Variational-Like Inequality Problem Involving Generalized Cayley Operator https://www.mdpi.com/2075-1680/10/3/133 This article deals with the study of a variational-like inequality problem which involves the generalized Cayley operator. We compare our problem with a fixed point equation, and based on it we construct an iterative algorithm to obtain the solution of our problem. Convergence analysis as well as stability analysis are studied. Axioms, Vol. 10, Pages 133: Variational-Like Inequality Problem Involving Generalized Cayley Operator

This article deals with the study of a variational-like inequality problem which involves the generalized Cayley operator. We compare our problem with a fixed point equation, and based on it we construct an iterative algorithm to obtain the solution of our problem. Convergence analysis as well as stability analysis are studied.

]]>
Variational-Like Inequality Problem Involving Generalized Cayley Operator Zahoor Ahmad Rather Rais Ahmad Ching-Feng Wen doi: 10.3390/axioms10030133 Axioms 2021-06-26 Axioms 2021-06-26 10 3 Article 133 10.3390/axioms10030133 https://www.mdpi.com/2075-1680/10/3/133
Axioms, Vol. 10, Pages 132: Nash Equilibrium Sequence in a Singular Two-Person Linear-Quadratic Differential Game https://www.mdpi.com/2075-1680/10/3/132 A finite-horizon two-person non-zero-sum differential game is considered. The dynamics of the game is linear. Each of the players has a quadratic functional on its own disposal, which should be minimized. The case where weight matrices in control costs of one player are singular in both functionals is studied. Hence, the game under the consideration is singular. A novel definition of the Nash equilibrium in this game (a Nash equilibrium sequence) is proposed. The game is solved by application of the regularization method. This method yields a new differential game, which is a regular Nash equilibrium game. Moreover, the new game is a partial cheap control game. An asymptotic analysis of this game is carried out. Based on this analysis, the Nash equilibrium sequence of the pairs of the players’ state-feedback controls in the singular game is constructed. The expressions for the optimal values of the functionals in the singular game are obtained. Illustrative examples are presented. Axioms, Vol. 10, Pages 132: Nash Equilibrium Sequence in a Singular Two-Person Linear-Quadratic Differential Game

Authors: Valery Y. Glizer

A finite-horizon two-person non-zero-sum differential game is considered. The dynamics of the game is linear. Each of the players has a quadratic functional on its own disposal, which should be minimized. The case where weight matrices in control costs of one player are singular in both functionals is studied. Hence, the game under the consideration is singular. A novel definition of the Nash equilibrium in this game (a Nash equilibrium sequence) is proposed. The game is solved by application of the regularization method. This method yields a new differential game, which is a regular Nash equilibrium game. Moreover, the new game is a partial cheap control game. An asymptotic analysis of this game is carried out. Based on this analysis, the Nash equilibrium sequence of the pairs of the players’ state-feedback controls in the singular game is constructed. The expressions for the optimal values of the functionals in the singular game are obtained. Illustrative examples are presented.

]]>
Nash Equilibrium Sequence in a Singular Two-Person Linear-Quadratic Differential Game Valery Y. Glizer doi: 10.3390/axioms10030132 Axioms 2021-06-25 Axioms 2021-06-25 10 3 Article 132 10.3390/axioms10030132 https://www.mdpi.com/2075-1680/10/3/132
Axioms, Vol. 10, Pages 131: A Decision Support Model for Measuring Technological Progress and Productivity Growth: The Case of Commercial Banks in Vietnam https://www.mdpi.com/2075-1680/10/3/131 The interactive relationship between the banking system and enterprise makes up the role that affects a national economy. Significantly, the relationship between banking and technology has become tighter over the past few decades. An assessment of bank performance is critical for understanding their position and provides valuable information to practitioners. In this paper, we assess the performance of the top 18 commercial banks in Vietnam during 2015–2019. The assessment utilizes two data envelopment analysis (DEA) models while involving the banks’ performance in six dimensions, including assets, deposits, operating expenses, liabilities as inputs, while treating loans and net income as outputs. Using the Malmquist measurement, the total productivity growth indexes of the banks are obtained, which are decomposed into technical and technological evolutions. Window analysis is used to compute the efficiencies of the banks in every single year in 2015–2019. From the results of Malmquist, most banks are found to decrease their Malmquist productivity indexes from 2015 to 2019, wherein both of their technical and technological indexes declined. Window analysis indicates B6-SHBank, B1-Vietinbank, and B18-PetrolimexGroup as the most efficient banks during 2015–2019, and in the interim, B16-BaoVietBank, B11-NationalCitizen, and B13-VietnamMaritime ranked on the bottom line. The managerial implications of this research help to reflect the comprehensive insights of the top Vietnamese commercial bank performance and offer a strategic guideline for decision-makers toward sustainable development in the banking industry. Axioms, Vol. 10, Pages 131: A Decision Support Model for Measuring Technological Progress and Productivity Growth: The Case of Commercial Banks in Vietnam

Authors: Chia-Nan Wang Ngoc-Ai-Thy Nguyen Thanh-Tuan Dang Thi-Thuy-Quynh Trinh

The interactive relationship between the banking system and enterprise makes up the role that affects a national economy. Significantly, the relationship between banking and technology has become tighter over the past few decades. An assessment of bank performance is critical for understanding their position and provides valuable information to practitioners. In this paper, we assess the performance of the top 18 commercial banks in Vietnam during 2015–2019. The assessment utilizes two data envelopment analysis (DEA) models while involving the banks’ performance in six dimensions, including assets, deposits, operating expenses, liabilities as inputs, while treating loans and net income as outputs. Using the Malmquist measurement, the total productivity growth indexes of the banks are obtained, which are decomposed into technical and technological evolutions. Window analysis is used to compute the efficiencies of the banks in every single year in 2015–2019. From the results of Malmquist, most banks are found to decrease their Malmquist productivity indexes from 2015 to 2019, wherein both of their technical and technological indexes declined. Window analysis indicates B6-SHBank, B1-Vietinbank, and B18-PetrolimexGroup as the most efficient banks during 2015–2019, and in the interim, B16-BaoVietBank, B11-NationalCitizen, and B13-VietnamMaritime ranked on the bottom line. The managerial implications of this research help to reflect the comprehensive insights of the top Vietnamese commercial bank performance and offer a strategic guideline for decision-makers toward sustainable development in the banking industry.

]]>
A Decision Support Model for Measuring Technological Progress and Productivity Growth: The Case of Commercial Banks in Vietnam Chia-Nan Wang Ngoc-Ai-Thy Nguyen Thanh-Tuan Dang Thi-Thuy-Quynh Trinh doi: 10.3390/axioms10030131 Axioms 2021-06-24 Axioms 2021-06-24 10 3 Article 131 10.3390/axioms10030131 https://www.mdpi.com/2075-1680/10/3/131
Axioms, Vol. 10, Pages 130: Non-Instantaneous Impulsive Boundary Value Problems Containing Caputo Fractional Derivative of a Function with Respect to Another Function and Riemann–Stieltjes Fractional Integral Boundary Conditions https://www.mdpi.com/2075-1680/10/3/130 In the present article we study existence and uniqueness results for a new class of boundary value problems consisting by non-instantaneous impulses and Caputo fractional derivative of a function with respect to another function, supplemented with Riemann–Stieltjes fractional integral boundary conditions. The existence of a unique solution is obtained via Banach’s contraction mapping principle, while an existence result is established by using Leray–Schauder nonlinear alternative. Examples illustrating the main results are also constructed. Axioms, Vol. 10, Pages 130: Non-Instantaneous Impulsive Boundary Value Problems Containing Caputo Fractional Derivative of a Function with Respect to Another Function and Riemann–Stieltjes Fractional Integral Boundary Conditions

In the present article we study existence and uniqueness results for a new class of boundary value problems consisting by non-instantaneous impulses and Caputo fractional derivative of a function with respect to another function, supplemented with Riemann–Stieltjes fractional integral boundary conditions. The existence of a unique solution is obtained via Banach’s contraction mapping principle, while an existence result is established by using Leray–Schauder nonlinear alternative. Examples illustrating the main results are also constructed.

]]>
Non-Instantaneous Impulsive Boundary Value Problems Containing Caputo Fractional Derivative of a Function with Respect to Another Function and Riemann–Stieltjes Fractional Integral Boundary Conditions Suphawat Asawasamrit Yasintorn Thadang Sotiris Ntouyas Jessada Tariboon doi: 10.3390/axioms10030130 Axioms 2021-06-23 Axioms 2021-06-23 10 3 Article 130 10.3390/axioms10030130 https://www.mdpi.com/2075-1680/10/3/130
Axioms, Vol. 10, Pages 129: Tenth Volume of Axioms and Why Axioms Was Launched https://www.mdpi.com/2075-1680/10/3/129 This year we publish the tenth volume of Axioms (ISSN 2075-1680) [...] Axioms, Vol. 10, Pages 129: Tenth Volume of Axioms and Why Axioms Was Launched

Authors: Shu-Kun Lin

This year we publish the tenth volume of Axioms (ISSN 2075-1680) [...]

]]>
Tenth Volume of Axioms and Why Axioms Was Launched Shu-Kun Lin doi: 10.3390/axioms10030129 Axioms 2021-06-23 Axioms 2021-06-23 10 3 Editorial 129 10.3390/axioms10030129 https://www.mdpi.com/2075-1680/10/3/129
Axioms, Vol. 10, Pages 128: Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics https://www.mdpi.com/2075-1680/10/3/128 Aristotelian diagrams, such as the square of opposition, are well-known in the context of normal modal logics (i.e., systems of modal logic which can be given a relational semantics in terms of Kripke models). This paper studies Aristotelian diagrams for non-normal systems of modal logic (based on neighborhood semantics, a topologically inspired generalization of relational semantics). In particular, we investigate the phenomenon of logic-sensitivity of Aristotelian diagrams. We distinguish between four different types of logic-sensitivity, viz. with respect to (i) Aristotelian families, (ii) logical equivalence of formulas, (iii) contingency of formulas, and (iv) Boolean subfamilies of a given Aristotelian family. We provide concrete examples of Aristotelian diagrams that illustrate these four types of logic-sensitivity in the realm of normal modal logic. Next, we discuss more subtle examples of Aristotelian diagrams, which are not sensitive with respect to normal modal logics, but which nevertheless turn out to be highly logic-sensitive once we turn to non-normal systems of modal logic. Axioms, Vol. 10, Pages 128: Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics

Authors: Lorenz Demey

Aristotelian diagrams, such as the square of opposition, are well-known in the context of normal modal logics (i.e., systems of modal logic which can be given a relational semantics in terms of Kripke models). This paper studies Aristotelian diagrams for non-normal systems of modal logic (based on neighborhood semantics, a topologically inspired generalization of relational semantics). In particular, we investigate the phenomenon of logic-sensitivity of Aristotelian diagrams. We distinguish between four different types of logic-sensitivity, viz. with respect to (i) Aristotelian families, (ii) logical equivalence of formulas, (iii) contingency of formulas, and (iv) Boolean subfamilies of a given Aristotelian family. We provide concrete examples of Aristotelian diagrams that illustrate these four types of logic-sensitivity in the realm of normal modal logic. Next, we discuss more subtle examples of Aristotelian diagrams, which are not sensitive with respect to normal modal logics, but which nevertheless turn out to be highly logic-sensitive once we turn to non-normal systems of modal logic.

]]>
Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics Lorenz Demey doi: 10.3390/axioms10030128 Axioms 2021-06-22 Axioms 2021-06-22 10 3 Article 128 10.3390/axioms10030128 https://www.mdpi.com/2075-1680/10/3/128
Axioms, Vol. 10, Pages 127: Some New Observations for F-Contractions in Vector-Valued Metric Spaces of Perov’s Type https://www.mdpi.com/2075-1680/10/2/127 The main purpose of this article is to improve, generalize and complement some recently established results for Perov’s type F-contractions. In our approach, we use only the property (F1) of Wardowski while other authors employed all three conditions. Working only with the fact that the function F is strictly increasing on 0,+∞m, we obtain as a consequence new families of contractive conditions in the realm of vector-valued metric spaces of Perov’s type. At the end of the article, we present an example that supports obtained theoretical results and genuinely generalizes several known results in existing literature. Axioms, Vol. 10, Pages 127: Some New Observations for F-Contractions in Vector-Valued Metric Spaces of Perov’s Type

The main purpose of this article is to improve, generalize and complement some recently established results for Perov’s type F-contractions. In our approach, we use only the property (F1) of Wardowski while other authors employed all three conditions. Working only with the fact that the function F is strictly increasing on 0,+∞m, we obtain as a consequence new families of contractive conditions in the realm of vector-valued metric spaces of Perov’s type. At the end of the article, we present an example that supports obtained theoretical results and genuinely generalizes several known results in existing literature.

]]>
Some New Observations for F-Contractions in Vector-Valued Metric Spaces of Perov’s Type Nikola Mirkov Stojan Radenović Slobodan Radojević doi: 10.3390/axioms10020127 Axioms 2021-06-21 Axioms 2021-06-21 10 2 Article 127 10.3390/axioms10020127 https://www.mdpi.com/2075-1680/10/2/127
Axioms, Vol. 10, Pages 126: Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in C2 https://www.mdpi.com/2075-1680/10/2/126 This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z), and the partial differential difference equations f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z12=eg(z),f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z2+∂f(z)∂z1+∂f(z)∂z22=eg(z). We establish some theorems about the forms of the finite order transcendental entire solutions of these functional equations. We also list a series of examples to explain the existence of the finite order transcendental entire solutions of such equations. Meantime, some examples show that there exists a very significant difference with the previous literature on the growth order of the finite order transcendental entire solutions. Our results show that some functional equations can admit the transcendental entire solutions with any positive integer order. These results make a few improvements of the previous theorems given by Xu and Cao, Liu and Yang. Axioms, Vol. 10, Pages 126: Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in C2

Authors: Hong Li Hongyan Xu

This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z), and the partial differential difference equations f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z12=eg(z),f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z2+∂f(z)∂z1+∂f(z)∂z22=eg(z). We establish some theorems about the forms of the finite order transcendental entire solutions of these functional equations. We also list a series of examples to explain the existence of the finite order transcendental entire solutions of such equations. Meantime, some examples show that there exists a very significant difference with the previous literature on the growth order of the finite order transcendental entire solutions. Our results show that some functional equations can admit the transcendental entire solutions with any positive integer order. These results make a few improvements of the previous theorems given by Xu and Cao, Liu and Yang.

]]>
Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in C2 Hong Li Hongyan Xu doi: 10.3390/axioms10020126 Axioms 2021-06-21 Axioms 2021-06-21 10 2 Article 126 10.3390/axioms10020126 https://www.mdpi.com/2075-1680/10/2/126
Axioms, Vol. 10, Pages 125: Explicit Formulas for Some Infinite 3F2(1)-Series https://www.mdpi.com/2075-1680/10/2/125 We establish two recurrence relations for some Clausen’s hypergeometric functions with unit argument. We solve them to give the explicit formulas. Additionally, we use the moments of Ramanujan’s generalized elliptic integrals to obtain these recurrence relations. Axioms, Vol. 10, Pages 125: Explicit Formulas for Some Infinite 3F2(1)-Series

Authors: Kwang-Wu Chen

We establish two recurrence relations for some Clausen’s hypergeometric functions with unit argument. We solve them to give the explicit formulas. Additionally, we use the moments of Ramanujan’s generalized elliptic integrals to obtain these recurrence relations.

]]>
Explicit Formulas for Some Infinite 3F2(1)-Series Kwang-Wu Chen doi: 10.3390/axioms10020125 Axioms 2021-06-19 Axioms 2021-06-19 10 2 Article 125 10.3390/axioms10020125 https://www.mdpi.com/2075-1680/10/2/125
Axioms, Vol. 10, Pages 124: An Innovative Grey Approach for Group Multi-Criteria Decision Analysis Based on the Median of Ratings by Using Python https://www.mdpi.com/2075-1680/10/2/124 Some decision-making problems, i.e., multi-criteria decision analysis (MCDA) problems, require taking into account the attitudes of a large number of decision-makers and/or respondents. Therefore, an approach to the transformation of crisp ratings, collected from respondents, in grey interval numbers form based on the median of collected scores, i.e., ratings, is considered in this article. In this way, the simplicity of collecting respondents’ attitudes using crisp values, i.e., by applying some form of Likert scale, is combined with the advantages that can be achieved by using grey interval numbers. In this way, a grey extension of MCDA methods is obtained. The application of the proposed approach was considered in the example of evaluating the websites of tourism organizations by using several MCDA methods. Additionally, an analysis of the application of the proposed approach in the case of a large number of respondents, done in Python, is presented. The advantages of the proposed method, as well as its possible limitations, are summarized. Axioms, Vol. 10, Pages 124: An Innovative Grey Approach for Group Multi-Criteria Decision Analysis Based on the Median of Ratings by Using Python

Authors: Dragiša Stanujkić Darjan Karabašević Gabrijela Popović Predrag S. Stanimirović Florentin Smarandache Muzafer Saračević Alptekin Ulutaş Vasilios N. Katsikis

Some decision-making problems, i.e., multi-criteria decision analysis (MCDA) problems, require taking into account the attitudes of a large number of decision-makers and/or respondents. Therefore, an approach to the transformation of crisp ratings, collected from respondents, in grey interval numbers form based on the median of collected scores, i.e., ratings, is considered in this article. In this way, the simplicity of collecting respondents’ attitudes using crisp values, i.e., by applying some form of Likert scale, is combined with the advantages that can be achieved by using grey interval numbers. In this way, a grey extension of MCDA methods is obtained. The application of the proposed approach was considered in the example of evaluating the websites of tourism organizations by using several MCDA methods. Additionally, an analysis of the application of the proposed approach in the case of a large number of respondents, done in Python, is presented. The advantages of the proposed method, as well as its possible limitations, are summarized.

]]>
An Innovative Grey Approach for Group Multi-Criteria Decision Analysis Based on the Median of Ratings by Using Python Dragiša Stanujkić Darjan Karabašević Gabrijela Popović Predrag S. Stanimirović Florentin Smarandache Muzafer Saračević Alptekin Ulutaş Vasilios N. Katsikis doi: 10.3390/axioms10020124 Axioms 2021-06-19 Axioms 2021-06-19 10 2 Article 124 10.3390/axioms10020124 https://www.mdpi.com/2075-1680/10/2/124
Axioms, Vol. 10, Pages 123: Strong Interacting Internal Waves in Rotating Ocean: Novel Fractional Approach https://www.mdpi.com/2075-1680/10/2/123 The main objective of the present study is to analyze the nature and capture the corresponding consequences of the solution obtained for the Gardner–Ostrovsky equation with the help of the q-homotopy analysis transform technique (q-HATT). In the rotating ocean, the considered equations exemplify strong interacting internal waves. The fractional operator employed in the present study is used in order to illustrate its importance in generalizing the models associated with kernel singular. The fixed-point theorem and the Banach space are considered to present the existence and uniqueness within the frame of the Caputo–Fabrizio (CF) fractional operator. Furthermore, for different fractional orders, the nature has been captured in plots. The realized consequences confirm that the considered procedure is reliable and highly methodical for investigating the consequences related to the nonlinear models of both integer and fractional order. Axioms, Vol. 10, Pages 123: Strong Interacting Internal Waves in Rotating Ocean: Novel Fractional Approach

Authors: Pundikala Veeresha Haci Mehmet Baskonus Wei Gao

The main objective of the present study is to analyze the nature and capture the corresponding consequences of the solution obtained for the Gardner–Ostrovsky equation with the help of the q-homotopy analysis transform technique (q-HATT). In the rotating ocean, the considered equations exemplify strong interacting internal waves. The fractional operator employed in the present study is used in order to illustrate its importance in generalizing the models associated with kernel singular. The fixed-point theorem and the Banach space are considered to present the existence and uniqueness within the frame of the Caputo–Fabrizio (CF) fractional operator. Furthermore, for different fractional orders, the nature has been captured in plots. The realized consequences confirm that the considered procedure is reliable and highly methodical for investigating the consequences related to the nonlinear models of both integer and fractional order.

]]>
Strong Interacting Internal Waves in Rotating Ocean: Novel Fractional Approach Pundikala Veeresha Haci Mehmet Baskonus Wei Gao doi: 10.3390/axioms10020123 Axioms 2021-06-16 Axioms 2021-06-16 10 2 Article 123 10.3390/axioms10020123 https://www.mdpi.com/2075-1680/10/2/123
Axioms, Vol. 10, Pages 122: New Procedures of a Fractional Order Model of Novel Coronavirus (COVID-19) Outbreak via Wavelets Method https://www.mdpi.com/2075-1680/10/2/122 Coronaviruses are a group of RNA (ribonucleic acid) viruses with the capacity for rapid mutation and recombination. Coronaviruses are known to cause respiratory or intestinal infections in humans and animals. In this paper, a biologically compatible set of nonlinear fractional differential equations governing the outbreak of the novel coronavirus is suggested based on a model previously proposed in the literature. Then, this set is numerically solved utilizing two new methods employing sine–cosine and Bernoulli wavelets and their operational matrices. Moreover, the convergence of the solution is experimentally studied. Furthermore, the accuracy of the solution is proved via comparing the results with those obtained in previous research for the primary model. Furthermore, the computational costs are compared by measuring the CPU running time. Finally, the effects of the fractional orders on the outbreak of the COVID-19 are investigated. Axioms, Vol. 10, Pages 122: New Procedures of a Fractional Order Model of Novel Coronavirus (COVID-19) Outbreak via Wavelets Method

Coronaviruses are a group of RNA (ribonucleic acid) viruses with the capacity for rapid mutation and recombination. Coronaviruses are known to cause respiratory or intestinal infections in humans and animals. In this paper, a biologically compatible set of nonlinear fractional differential equations governing the outbreak of the novel coronavirus is suggested based on a model previously proposed in the literature. Then, this set is numerically solved utilizing two new methods employing sine–cosine and Bernoulli wavelets and their operational matrices. Moreover, the convergence of the solution is experimentally studied. Furthermore, the accuracy of the solution is proved via comparing the results with those obtained in previous research for the primary model. Furthermore, the computational costs are compared by measuring the CPU running time. Finally, the effects of the fractional orders on the outbreak of the COVID-19 are investigated.

]]>
New Procedures of a Fractional Order Model of Novel Coronavirus (COVID-19) Outbreak via Wavelets Method Maryamsadat Hedayati Reza Ezzati Samad Noeiaghdam doi: 10.3390/axioms10020122 Axioms 2021-06-16 Axioms 2021-06-16 10 2 Article 122 10.3390/axioms10020122 https://www.mdpi.com/2075-1680/10/2/122
Axioms, Vol. 10, Pages 121: Classical Partition Function for Non-Relativistic Gravity https://www.mdpi.com/2075-1680/10/2/121 We considered the canonical gravitational partition function Z associated to the classical Boltzmann–Gibbs (BG) distribution e−βHZ. It is popularly thought that it cannot be built up because the integral involved in constructing Z diverges at the origin. Contrariwise, it was shown in (Physica A 497 (2018) 310), by appeal to sophisticated mathematics developed in the second half of the last century, that this is not so. Z can indeed be computed by recourse to (A) the analytical extension treatments of Gradshteyn and Rizhik and Guelfand and Shilov, that permit tackling some divergent integrals and (B) the dimensional regularization approach. Only one special instance was discussed in the above reference. In this work, we obtain the classical partition function for Newton’s gravity in the four cases that immediately come to mind. Axioms, Vol. 10, Pages 121: Classical Partition Function for Non-Relativistic Gravity

Authors: Mir Hameeda Angelo Plastino Mario Carlos Rocca Javier Zamora

We considered the canonical gravitational partition function Z associated to the classical Boltzmann–Gibbs (BG) distribution e−βHZ. It is popularly thought that it cannot be built up because the integral involved in constructing Z diverges at the origin. Contrariwise, it was shown in (Physica A 497 (2018) 310), by appeal to sophisticated mathematics developed in the second half of the last century, that this is not so. Z can indeed be computed by recourse to (A) the analytical extension treatments of Gradshteyn and Rizhik and Guelfand and Shilov, that permit tackling some divergent integrals and (B) the dimensional regularization approach. Only one special instance was discussed in the above reference. In this work, we obtain the classical partition function for Newton’s gravity in the four cases that immediately come to mind.

]]>
Classical Partition Function for Non-Relativistic Gravity Mir Hameeda Angelo Plastino Mario Carlos Rocca Javier Zamora doi: 10.3390/axioms10020121 Axioms 2021-06-16 Axioms 2021-06-16 10 2 Article 121 10.3390/axioms10020121 https://www.mdpi.com/2075-1680/10/2/121
Axioms, Vol. 10, Pages 120: A Two-Stage Closed-Loop Supply Chain Pricing Decision: Cross-Channel Recycling and Channel Preference https://www.mdpi.com/2075-1680/10/2/120 This paper focuses on the pricing problem of a two-stage closed-loop supply chain (CLSC) considering the cross-channel recycling and channel preference based on a single manufacturer and a single traditional retailer. The pricing decision problem raises from the manufacturer’s direct sales and the retailer’s retailing including recycling. Managers need to focus on intelligible management considering consumer channel preferences, cross-channel recovery and pricing strategies. According to game theory, centralized and decentralized CLSC decision models are used to provide an efficient solution to managers for the pricing problem. The centralized model consists of differential and uniform pricing strategy and the decentralized model consists of manufacturer-led Stackelberg, retailer-led Stackelberg and Nash equilibrium game, respectively. The impact of cross-channel recycling rate and channel preference on pricing and profitability in a two-stage CLSC system is explained elaborately in this study. The results show that cross-channel recovery rates and consumer channel preferences have a direct significant impact on pricing strategies including profit allocation decisions in CLSC. It demonstrated that different channel preferences leading to different pricing strategies and decision for manufacturers and retailers choices. Manufacturer’s pricing decreases when channel preferences are constant and cross-channel recovery rates increase. Retailer’s pricing remains stable as the cross-channel recovery rate has less affected on it. Furthermore, if the cross-channel recovery rates increase, then the manufacturers pricing decreases and retailers pricing increases. This information will be a helpful guideline for the manager to select suitable pricing strategies based on the company scenario. Axioms, Vol. 10, Pages 120: A Two-Stage Closed-Loop Supply Chain Pricing Decision: Cross-Channel Recycling and Channel Preference

Authors: Wenjun Pan Miao Lin

This paper focuses on the pricing problem of a two-stage closed-loop supply chain (CLSC) considering the cross-channel recycling and channel preference based on a single manufacturer and a single traditional retailer. The pricing decision problem raises from the manufacturer’s direct sales and the retailer’s retailing including recycling. Managers need to focus on intelligible management considering consumer channel preferences, cross-channel recovery and pricing strategies. According to game theory, centralized and decentralized CLSC decision models are used to provide an efficient solution to managers for the pricing problem. The centralized model consists of differential and uniform pricing strategy and the decentralized model consists of manufacturer-led Stackelberg, retailer-led Stackelberg and Nash equilibrium game, respectively. The impact of cross-channel recycling rate and channel preference on pricing and profitability in a two-stage CLSC system is explained elaborately in this study. The results show that cross-channel recovery rates and consumer channel preferences have a direct significant impact on pricing strategies including profit allocation decisions in CLSC. It demonstrated that different channel preferences leading to different pricing strategies and decision for manufacturers and retailers choices. Manufacturer’s pricing decreases when channel preferences are constant and cross-channel recovery rates increase. Retailer’s pricing remains stable as the cross-channel recovery rate has less affected on it. Furthermore, if the cross-channel recovery rates increase, then the manufacturers pricing decreases and retailers pricing increases. This information will be a helpful guideline for the manager to select suitable pricing strategies based on the company scenario.

]]>
A Two-Stage Closed-Loop Supply Chain Pricing Decision: Cross-Channel Recycling and Channel Preference Wenjun Pan Miao Lin doi: 10.3390/axioms10020120 Axioms 2021-06-15 Axioms 2021-06-15 10 2 Article 120 10.3390/axioms10020120 https://www.mdpi.com/2075-1680/10/2/120
Axioms, Vol. 10, Pages 119: A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory https://www.mdpi.com/2075-1680/10/2/119 It is well known that Zermelo-Fraenkel Set Theory (ZF), despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the Löwenheim–Skolem theorem. This paper presents the axioms one has to accept to get rid of these two features. For that matter, some twenty axioms are formulated in a non-classical first-order language with countably many constants: to this collection of axioms is associated a universe of discourse consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function. The axioms of ZF are derived from this finite axiom schema, and it is shown that it does not have a countable model—if it has a model at all, that is. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics. Axioms, Vol. 10, Pages 119: A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

Authors: Marcoen J. T. F. Cabbolet

It is well known that Zermelo-Fraenkel Set Theory (ZF), despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the Löwenheim–Skolem theorem. This paper presents the axioms one has to accept to get rid of these two features. For that matter, some twenty axioms are formulated in a non-classical first-order language with countably many constants: to this collection of axioms is associated a universe of discourse consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function. The axioms of ZF are derived from this finite axiom schema, and it is shown that it does not have a countable model—if it has a model at all, that is. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics.

]]>
A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory Marcoen J. T. F. Cabbolet doi: 10.3390/axioms10020119 Axioms 2021-06-14 Axioms 2021-06-14 10 2 Article 119 10.3390/axioms10020119 https://www.mdpi.com/2075-1680/10/2/119
Axioms, Vol. 10, Pages 118: Vector Fields and Differential Forms on the Orbit Space of a Proper Action https://www.mdpi.com/2075-1680/10/2/118 In this paper, we study differential forms and vector fields on the orbit space of a proper action of a Lie group on a smooth manifold, defining them as multilinear maps on the generators of infinitesimal diffeomorphisms, respectively. This yields an intrinsic view of vector fields and differential forms on the orbit space. Axioms, Vol. 10, Pages 118: Vector Fields and Differential Forms on the Orbit Space of a Proper Action

Authors: Larry Bates Richard Cushman Jędrzej Śniatycki

In this paper, we study differential forms and vector fields on the orbit space of a proper action of a Lie group on a smooth manifold, defining them as multilinear maps on the generators of infinitesimal diffeomorphisms, respectively. This yields an intrinsic view of vector fields and differential forms on the orbit space.

]]>
Vector Fields and Differential Forms on the Orbit Space of a Proper Action Larry Bates Richard Cushman Jędrzej Śniatycki doi: 10.3390/axioms10020118 Axioms 2021-06-10 Axioms 2021-06-10 10 2 Article 118 10.3390/axioms10020118 https://www.mdpi.com/2075-1680/10/2/118
Axioms, Vol. 10, Pages 117: About Cogredient and Contragredient Linear Differential Equations https://www.mdpi.com/2075-1680/10/2/117 The notions of cogredience and contragredience, which have great importance to the question of algebraic independence of linear differential equation solutions, are discussed in the paper. Conditions of equivalence of two definitions of cogredience and contragredience are found. Axioms, Vol. 10, Pages 117: About Cogredient and Contragredient Linear Differential Equations

Authors: Vasily Gorelov

The notions of cogredience and contragredience, which have great importance to the question of algebraic independence of linear differential equation solutions, are discussed in the paper. Conditions of equivalence of two definitions of cogredience and contragredience are found.

]]>
About Cogredient and Contragredient Linear Differential Equations Vasily Gorelov doi: 10.3390/axioms10020117 Axioms 2021-06-10 Axioms 2021-06-10 10 2 Article 117 10.3390/axioms10020117 https://www.mdpi.com/2075-1680/10/2/117
Axioms, Vol. 10, Pages 116: A Note on the Computation of the Modular Inverse for Cryptography https://www.mdpi.com/2075-1680/10/2/116 In literature, there are a number of cryptographic algorithms (RSA, ElGamal, NTRU, etc.) that require multiple computations of modulo multiplicative inverses. In this paper, we describe the modulo operation and we recollect the main approaches to computing the modulus. Then, given a and n positive integers, we present the sequence (zj)j≥0, where zj=zj−1+aβj−n, a&amp;lt;n and GCD(a,n)=1. Regarding the above sequence, we show that it is bounded and admits a simple explicit, periodic solution. The main result is that the inverse of a modulo n is given by a−1=⌊im⌋+1 with m=n/a. The computational cost of such an index i is O(a), which is less than O(nlnn) of the Euler’s phi function. Furthermore, we suggest an algorithm for the computation of a−1 using plain multiplications instead of modular multiplications. The latter, still, has complexity O(a) versus complexity O(n) (naive algorithm) or complexity O(lnn) (extended Euclidean algorithm). Therefore, the above procedure is more convenient when a&amp;lt;&amp;lt;n (e.g., a&amp;lt;lnn). Axioms, Vol. 10, Pages 116: A Note on the Computation of the Modular Inverse for Cryptography

Authors: Michele Bufalo Daniele Bufalo Giuseppe Orlando

In literature, there are a number of cryptographic algorithms (RSA, ElGamal, NTRU, etc.) that require multiple computations of modulo multiplicative inverses. In this paper, we describe the modulo operation and we recollect the main approaches to computing the modulus. Then, given a and n positive integers, we present the sequence (zj)j≥0, where zj=zj−1+aβj−n, a&amp;lt;n and GCD(a,n)=1. Regarding the above sequence, we show that it is bounded and admits a simple explicit, periodic solution. The main result is that the inverse of a modulo n is given by a−1=⌊im⌋+1 with m=n/a. The computational cost of such an index i is O(a), which is less than O(nlnn) of the Euler’s phi function. Furthermore, we suggest an algorithm for the computation of a−1 using plain multiplications instead of modular multiplications. The latter, still, has complexity O(a) versus complexity O(n) (naive algorithm) or complexity O(lnn) (extended Euclidean algorithm). Therefore, the above procedure is more convenient when a&amp;lt;&amp;lt;n (e.g., a&amp;lt;lnn).

]]>
A Note on the Computation of the Modular Inverse for Cryptography Michele Bufalo Daniele Bufalo Giuseppe Orlando doi: 10.3390/axioms10020116 Axioms 2021-06-09 Axioms 2021-06-09 10 2 Article 116 10.3390/axioms10020116 https://www.mdpi.com/2075-1680/10/2/116
Axioms, Vol. 10, Pages 115: Analytic Representation of Maxwell—Boltzmann and Tsallis Thermonuclear Functions with Depleted Tail https://www.mdpi.com/2075-1680/10/2/115 The closed forms of the non-resonant thermonuclear function in the Maxwell–Boltzmann and Tsallis case with depleted tail are obtained in generalized special functions. The results are written in terms of H-function of two variables. The importance of the results in this paper lies in the fact that the reaction rate probability integrals in Maxwell-Boltzmann and Tsallis cases are not obtained by the conventional method of approximation or by means of a single variable transform technique but by means of a two variable transform method. The behaviour of the depleted non-resonant thermonuclear functions are examined using graphs. The results in the paper are of much interest to astrophysicists and statisticians in their future work in this area. Axioms, Vol. 10, Pages 115: Analytic Representation of Maxwell—Boltzmann and Tsallis Thermonuclear Functions with Depleted Tail

Authors: Dilip Kumar Hans Haubold

The closed forms of the non-resonant thermonuclear function in the Maxwell–Boltzmann and Tsallis case with depleted tail are obtained in generalized special functions. The results are written in terms of H-function of two variables. The importance of the results in this paper lies in the fact that the reaction rate probability integrals in Maxwell-Boltzmann and Tsallis cases are not obtained by the conventional method of approximation or by means of a single variable transform technique but by means of a two variable transform method. The behaviour of the depleted non-resonant thermonuclear functions are examined using graphs. The results in the paper are of much interest to astrophysicists and statisticians in their future work in this area.

]]>
Analytic Representation of Maxwell—Boltzmann and Tsallis Thermonuclear Functions with Depleted Tail Dilip Kumar Hans Haubold doi: 10.3390/axioms10020115 Axioms 2021-06-07 Axioms 2021-06-07 10 2 Article 115 10.3390/axioms10020115 https://www.mdpi.com/2075-1680/10/2/115
Axioms, Vol. 10, Pages 114: Numerical Solution of an Interval-Based Uncertain SIR (Susceptible–Infected–Recovered) Epidemic Model by Homotopy Analysis Method https://www.mdpi.com/2075-1680/10/2/114 This work proposes an interval-based uncertain Susceptible–Infected–Recovered (SIR) epidemic model. The interval model has been numerically solved by the homotopy analysis method (HAM). The SIR epidemic model is proposed and solved under different uncertain intervals by the HAM to obtain the numerical solution of the model. Furthermore, the SIR ODE model was transformed into a stochastic differential equation (SDE) model and the results of the stochastic and deterministic models were compared using numerical simulations. The results obtained were compared with the numerical solution and found to be in good agreement. Finally, various simulations were done to discuss the solution. Axioms, Vol. 10, Pages 114: Numerical Solution of an Interval-Based Uncertain SIR (Susceptible–Infected–Recovered) Epidemic Model by Homotopy Analysis Method

Authors: Emmanuel A. Bakare Snehashish Chakraverty Radovan Potucek

This work proposes an interval-based uncertain Susceptible–Infected–Recovered (SIR) epidemic model. The interval model has been numerically solved by the homotopy analysis method (HAM). The SIR epidemic model is proposed and solved under different uncertain intervals by the HAM to obtain the numerical solution of the model. Furthermore, the SIR ODE model was transformed into a stochastic differential equation (SDE) model and the results of the stochastic and deterministic models were compared using numerical simulations. The results obtained were compared with the numerical solution and found to be in good agreement. Finally, various simulations were done to discuss the solution.

]]>
Numerical Solution of an Interval-Based Uncertain SIR (Susceptible–Infected–Recovered) Epidemic Model by Homotopy Analysis Method Emmanuel A. Bakare Snehashish Chakraverty Radovan Potucek doi: 10.3390/axioms10020114 Axioms 2021-06-06 Axioms 2021-06-06 10 2 Article 114 10.3390/axioms10020114 https://www.mdpi.com/2075-1680/10/2/114
Axioms, Vol. 10, Pages 113: A Decision-Making Approach Based on New Aggregation Operators under Fermatean Fuzzy Linguistic Information Environment https://www.mdpi.com/2075-1680/10/2/113 Fermatean fuzzy linguistic (FFL) set theory provides an efficient tool for modeling a higher level of uncertain and imprecise information, which cannot be represented using intuitionistic fuzzy linguistic (IFL)/Pythagorean fuzzy linguistic (PFL) sets. On the other hand, the linguistic scale function (LSF) is the better way to consider the semantics of the linguistic terms during the evaluation process. It is worth noting that the existing operational laws and aggregation operators (AOs) for Fermatean fuzzy linguistic numbers (FFLNs) are not valid in many situations, which can generate errors in real-life applications. The present study aims to define new robust operational laws and AOs under Fermatean fuzzy linguistic environment. To do so, first, we define some new modified operational laws for FFLNs based on LSF and prove some important mathematical properties of them. Next, the work defines several new AOs, namely, the FFL-weighted averaging (FFLWA) operator, the FFL-weighted geometric (FFLWG) operator, the FFL-ordered weighted averaging (FFLOWA) operator, the FFL-ordered weighted geometric (FFLOWG) operator, the FFL-hybrid averaging (FFLHA) operator and the FFL-hybrid geometric (FFLHG) operator under Fermatean fuzzy linguistic environment. Several properties of these AOs are investigated in detail. Further, based on the proposed AOs, a new decision-making approach with Fermatean fuzzy linguistic information is developed to solve group decision-making problems with multiple attributes. Finally, to illustrate the effectiveness of the present approach, a real-life supplier selection problem is presented where the evaluation information of the alternatives is given in terms of FFLNs. Compared to the existing methods, the salient features of the developed approach are (1) it can solve decision-making problems with qualitative information data using FFLNs; (2) It can consider the attitudinal character of the decision-makers during the solution process; (3) It has a solid ability to distinguish the optimal alternative. Axioms, Vol. 10, Pages 113: A Decision-Making Approach Based on New Aggregation Operators under Fermatean Fuzzy Linguistic Information Environment

Authors: Rajkumar Verma

Fermatean fuzzy linguistic (FFL) set theory provides an efficient tool for modeling a higher level of uncertain and imprecise information, which cannot be represented using intuitionistic fuzzy linguistic (IFL)/Pythagorean fuzzy linguistic (PFL) sets. On the other hand, the linguistic scale function (LSF) is the better way to consider the semantics of the linguistic terms during the evaluation process. It is worth noting that the existing operational laws and aggregation operators (AOs) for Fermatean fuzzy linguistic numbers (FFLNs) are not valid in many situations, which can generate errors in real-life applications. The present study aims to define new robust operational laws and AOs under Fermatean fuzzy linguistic environment. To do so, first, we define some new modified operational laws for FFLNs based on LSF and prove some important mathematical properties of them. Next, the work defines several new AOs, namely, the FFL-weighted averaging (FFLWA) operator, the FFL-weighted geometric (FFLWG) operator, the FFL-ordered weighted averaging (FFLOWA) operator, the FFL-ordered weighted geometric (FFLOWG) operator, the FFL-hybrid averaging (FFLHA) operator and the FFL-hybrid geometric (FFLHG) operator under Fermatean fuzzy linguistic environment. Several properties of these AOs are investigated in detail. Further, based on the proposed AOs, a new decision-making approach with Fermatean fuzzy linguistic information is developed to solve group decision-making problems with multiple attributes. Finally, to illustrate the effectiveness of the present approach, a real-life supplier selection problem is presented where the evaluation information of the alternatives is given in terms of FFLNs. Compared to the existing methods, the salient features of the developed approach are (1) it can solve decision-making problems with qualitative information data using FFLNs; (2) It can consider the attitudinal character of the decision-makers during the solution process; (3) It has a solid ability to distinguish the optimal alternative.

]]>
A Decision-Making Approach Based on New Aggregation Operators under Fermatean Fuzzy Linguistic Information Environment Rajkumar Verma doi: 10.3390/axioms10020113 Axioms 2021-06-04 Axioms 2021-06-04 10 2 Article 113 10.3390/axioms10020113 https://www.mdpi.com/2075-1680/10/2/113
Axioms, Vol. 10, Pages 112: On a Class of Isoperimetric Constrained Controlled Optimization Problems https://www.mdpi.com/2075-1680/10/2/112 In this paper, we investigate the Lagrange dynamics generated by a class of isoperimetric constrained controlled optimization problems involving second-order partial derivatives and boundary conditions. More precisely, we derive necessary optimality conditions for the considered class of variational control problems governed by path-independent curvilinear integral functionals. Moreover, the theoretical results presented in the paper are accompanied by an illustrative example. Furthermore, an algorithm is proposed to emphasize the steps to be followed to solve a control problem such as the one studied in this paper. Axioms, Vol. 10, Pages 112: On a Class of Isoperimetric Constrained Controlled Optimization Problems

Authors: Savin Treanţă

In this paper, we investigate the Lagrange dynamics generated by a class of isoperimetric constrained controlled optimization problems involving second-order partial derivatives and boundary conditions. More precisely, we derive necessary optimality conditions for the considered class of variational control problems governed by path-independent curvilinear integral functionals. Moreover, the theoretical results presented in the paper are accompanied by an illustrative example. Furthermore, an algorithm is proposed to emphasize the steps to be followed to solve a control problem such as the one studied in this paper.

]]>
On a Class of Isoperimetric Constrained Controlled Optimization Problems Savin Treanţă doi: 10.3390/axioms10020112 Axioms 2021-06-03 Axioms 2021-06-03 10 2 Article 112 10.3390/axioms10020112 https://www.mdpi.com/2075-1680/10/2/112
Axioms, Vol. 10, Pages 111: B-Fredholm Spectra of Drazin Invertible Operators and Applications https://www.mdpi.com/2075-1680/10/2/111 In this article, we consider Drazin invertible operators for study of the relationship between their B-Fredholm spectra and the transfer between some of the spectral properties defined through B-Fredholm spectra of this class of operators. Among other results, we investigate the transfer of generalized a-Weyl’s theorem from T to their Drazin inverse S, if it exists. Axioms, Vol. 10, Pages 111: B-Fredholm Spectra of Drazin Invertible Operators and Applications

Authors: Elvis Aponte Jhixon Macías José Sanabria José Soto

In this article, we consider Drazin invertible operators for study of the relationship between their B-Fredholm spectra and the transfer between some of the spectral properties defined through B-Fredholm spectra of this class of operators. Among other results, we investigate the transfer of generalized a-Weyl’s theorem from T to their Drazin inverse S, if it exists.

]]>
B-Fredholm Spectra of Drazin Invertible Operators and Applications Elvis Aponte Jhixon Macías José Sanabria José Soto doi: 10.3390/axioms10020111 Axioms 2021-06-02 Axioms 2021-06-02 10 2 Article 111 10.3390/axioms10020111 https://www.mdpi.com/2075-1680/10/2/111
Axioms, Vol. 10, Pages 110: Nonlocal Problem for a Third-Order Equation with Multiple Characteristics with General Boundary Conditions https://www.mdpi.com/2075-1680/10/2/110 The article considers third-order equations with multiple characteristics with general boundary value conditions and non-local initial data. A regular solution to the problem with known methods is constructed here. The uniqueness of the solution to the problem is proved by the method of energy integrals. This uses the theory of non-negative quadratic forms. The existence of a solution to the problem is proved by reducing the problem to Fredholm integral equations of the second kind. In this case, the method of Green’s function and potential is used. Axioms, Vol. 10, Pages 110: Nonlocal Problem for a Third-Order Equation with Multiple Characteristics with General Boundary Conditions

Authors: Abdukomil Risbekovich Khashimov Dana Smetanová

The article considers third-order equations with multiple characteristics with general boundary value conditions and non-local initial data. A regular solution to the problem with known methods is constructed here. The uniqueness of the solution to the problem is proved by the method of energy integrals. This uses the theory of non-negative quadratic forms. The existence of a solution to the problem is proved by reducing the problem to Fredholm integral equations of the second kind. In this case, the method of Green’s function and potential is used.

]]>
Nonlocal Problem for a Third-Order Equation with Multiple Characteristics with General Boundary Conditions Abdukomil Risbekovich Khashimov Dana Smetanová doi: 10.3390/axioms10020110 Axioms 2021-06-02 Axioms 2021-06-02 10 2 Article 110 10.3390/axioms10020110 https://www.mdpi.com/2075-1680/10/2/110
Axioms, Vol. 10, Pages 109: A Self-Adaptive Algorithm for the Common Solution of the Split Minimization Problem and the Fixed Point Problem https://www.mdpi.com/2075-1680/10/2/109 In this paper, a new self-adaptive step size algorithm to approximate the solution of the split minimization problem and the fixed point problem of nonexpansive mappings was constructed, which combined the proximal algorithm and a modified Mann’s iterative method with the inertial extrapolation. The strong convergence theorem was provided in the framework of Hilbert spaces and then proven under some suitable conditions. Our result improved related results in the literature. Moreover, some numerical experiments were also provided to show our algorithm’s consistency, accuracy, and performance compared to the existing algorithms in the literature. Axioms, Vol. 10, Pages 109: A Self-Adaptive Algorithm for the Common Solution of the Split Minimization Problem and the Fixed Point Problem

Authors: Nattakarn Kaewyong Kanokwan Sitthithakerngkiet

In this paper, a new self-adaptive step size algorithm to approximate the solution of the split minimization problem and the fixed point problem of nonexpansive mappings was constructed, which combined the proximal algorithm and a modified Mann’s iterative method with the inertial extrapolation. The strong convergence theorem was provided in the framework of Hilbert spaces and then proven under some suitable conditions. Our result improved related results in the literature. Moreover, some numerical experiments were also provided to show our algorithm’s consistency, accuracy, and performance compared to the existing algorithms in the literature.

]]>
A Self-Adaptive Algorithm for the Common Solution of the Split Minimization Problem and the Fixed Point Problem Nattakarn Kaewyong Kanokwan Sitthithakerngkiet doi: 10.3390/axioms10020109 Axioms 2021-05-30 Axioms 2021-05-30 10 2 Article 109 10.3390/axioms10020109 https://www.mdpi.com/2075-1680/10/2/109
Axioms, Vol. 10, Pages 108: Relaxation Limit of the Aggregation Equation with Pointy Potential https://www.mdpi.com/2075-1680/10/2/108 This work was devoted to the study of a relaxation limit of the so-called aggregation equation with a pointy potential in one-dimensional space. The aggregation equation is today widely used to model the dynamics of a density of individuals attracting each other through a potential. When this potential is pointy, solutions are known to blow up in final time. For this reason, measure-valued solutions have been defined. In this paper, we investigated an approximation of such measure-valued solutions thanks to a relaxation limit in the spirit of Jin and Xin. We study the convergence of this approximation and give a rigorous estimate of the speed of convergence in one dimension with the Newtonian potential. We also investigated the numerical discretization of this relaxation limit by uniformly accurate schemes. Axioms, Vol. 10, Pages 108: Relaxation Limit of the Aggregation Equation with Pointy Potential

Authors: Benoît Fabrèges Frédéric Lagoutière Sébastien Tran Tien Nicolas Vauchelet

This work was devoted to the study of a relaxation limit of the so-called aggregation equation with a pointy potential in one-dimensional space. The aggregation equation is today widely used to model the dynamics of a density of individuals attracting each other through a potential. When this potential is pointy, solutions are known to blow up in final time. For this reason, measure-valued solutions have been defined. In this paper, we investigated an approximation of such measure-valued solutions thanks to a relaxation limit in the spirit of Jin and Xin. We study the convergence of this approximation and give a rigorous estimate of the speed of convergence in one dimension with the Newtonian potential. We also investigated the numerical discretization of this relaxation limit by uniformly accurate schemes.

]]>
Relaxation Limit of the Aggregation Equation with Pointy Potential Benoît Fabrèges Frédéric Lagoutière Sébastien Tran Tien Nicolas Vauchelet doi: 10.3390/axioms10020108 Axioms 2021-05-28 Axioms 2021-05-28 10 2 Article 108 10.3390/axioms10020108 https://www.mdpi.com/2075-1680/10/2/108
Axioms, Vol. 10, Pages 107: Spectral Transformations and Associated Linear Functionals of the First Kind https://www.mdpi.com/2075-1680/10/2/107 Given a quasi-definite linear functional u in the linear space of polynomials with complex coefficients, let us consider the corresponding sequence of monic orthogonal polynomials (SMOP in short) (Pn)n≥0. For a canonical Christoffel transformation u˜=(x−c)u with SMOP (P˜n)n≥0, we are interested to study the relation between u˜ and u(1)˜, where u(1) is the linear functional for the associated orthogonal polynomials of the first kind (Pn(1))n≥0, and u(1)˜=(x−c)u(1) is its Christoffel transformation. This problem is also studied for canonical Geronimus transformations. Axioms, Vol. 10, Pages 107: Spectral Transformations and Associated Linear Functionals of the First Kind

Authors: Juan Carlos García-Ardila Francisco Marcellán

Given a quasi-definite linear functional u in the linear space of polynomials with complex coefficients, let us consider the corresponding sequence of monic orthogonal polynomials (SMOP in short) (Pn)n≥0. For a canonical Christoffel transformation u˜=(x−c)u with SMOP (P˜n)n≥0, we are interested to study the relation between u˜ and u(1)˜, where u(1) is the linear functional for the associated orthogonal polynomials of the first kind (Pn(1))n≥0, and u(1)˜=(x−c)u(1) is its Christoffel transformation. This problem is also studied for canonical Geronimus transformations.

]]>
Spectral Transformations and Associated Linear Functionals of the First Kind Juan Carlos García-Ardila Francisco Marcellán doi: 10.3390/axioms10020107 Axioms 2021-05-28 Axioms 2021-05-28 10 2 Article 107 10.3390/axioms10020107 https://www.mdpi.com/2075-1680/10/2/107
Axioms, Vol. 10, Pages 106: Monitoring and Recognizing Enterprise Public Opinion from High-Risk Users Based on User Portrait and Random Forest Algorithm https://www.mdpi.com/2075-1680/10/2/106 With the rapid development of “We media” technology, netizens can freely express their opinions regarding enterprise products on a network platform. Consequently, online public opinion about enterprises has become a prominent issue. Negative comments posted by some netizens may trigger negative public opinion, which can have a significant impact on an enterprise’s image. From the perspective of helping enterprises deal with negative public opinion, this paper combines user portrait technology and a random forest algorithm to help enterprises identify high-risk users who have posted negative comments and thus may trigger negative public opinion. In this way, enterprises can monitor the public opinion of high-risk users to prevent negative public opinion events. Firstly, we crawled the information of users participating in discussions of product experience, and we constructed a portrait of enterprise public opinion users. Then, the characteristics of the portraits were quantified into indicators such as the user’s activity, the user’s influence, and the user’s emotional tendency, and the indicators were sorted. According to the order of the indicators, the users were divided into high-risk, moderate-risk, and low-risk categories. Next, a supervised high-risk user identification model for this classification was established, based on a random forest algorithm. In turn, the trained random forest identifier can be used to predict whether the authors of newly published public opinion information are high-risk users. Finally, a back propagation neural network algorithm was used to identify users and compared with the results of model recognition in this paper. The results showed that the average recognition accuracy of the back propagation neural network is only 72.33%, while the average recognition accuracy of the model constructed in this paper is as high as 98.49%, which verifies the feasibility and accuracy of the proposed random forest recognition method. Axioms, Vol. 10, Pages 106: Monitoring and Recognizing Enterprise Public Opinion from High-Risk Users Based on User Portrait and Random Forest Algorithm

Authors: Tinggui Chen Xiaohua Yin Lijuan Peng Jingtao Rong Jianjun Yang Guodong Cong

With the rapid development of “We media” technology, netizens can freely express their opinions regarding enterprise products on a network platform. Consequently, online public opinion about enterprises has become a prominent issue. Negative comments posted by some netizens may trigger negative public opinion, which can have a significant impact on an enterprise’s image. From the perspective of helping enterprises deal with negative public opinion, this paper combines user portrait technology and a random forest algorithm to help enterprises identify high-risk users who have posted negative comments and thus may trigger negative public opinion. In this way, enterprises can monitor the public opinion of high-risk users to prevent negative public opinion events. Firstly, we crawled the information of users participating in discussions of product experience, and we constructed a portrait of enterprise public opinion users. Then, the characteristics of the portraits were quantified into indicators such as the user’s activity, the user’s influence, and the user’s emotional tendency, and the indicators were sorted. According to the order of the indicators, the users were divided into high-risk, moderate-risk, and low-risk categories. Next, a supervised high-risk user identification model for this classification was established, based on a random forest algorithm. In turn, the trained random forest identifier can be used to predict whether the authors of newly published public opinion information are high-risk users. Finally, a back propagation neural network algorithm was used to identify users and compared with the results of model recognition in this paper. The results showed that the average recognition accuracy of the back propagation neural network is only 72.33%, while the average recognition accuracy of the model constructed in this paper is as high as 98.49%, which verifies the feasibility and accuracy of the proposed random forest recognition method.

]]>
Monitoring and Recognizing Enterprise Public Opinion from High-Risk Users Based on User Portrait and Random Forest Algorithm Tinggui Chen Xiaohua Yin Lijuan Peng Jingtao Rong Jianjun Yang Guodong Cong doi: 10.3390/axioms10020106 Axioms 2021-05-27 Axioms 2021-05-27 10 2 Article 106 10.3390/axioms10020106 https://www.mdpi.com/2075-1680/10/2/106
Axioms, Vol. 10, Pages 105: Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate https://www.mdpi.com/2075-1680/10/2/105 In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt). Axioms, Vol. 10, Pages 105: Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

Authors: Lokesh Singh Dhirendra Bahuguna

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

]]>
Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate Lokesh Singh Dhirendra Bahuguna doi: 10.3390/axioms10020105 Axioms 2021-05-25 Axioms 2021-05-25 10 2 Article 105 10.3390/axioms10020105 https://www.mdpi.com/2075-1680/10/2/105
Axioms, Vol. 10, Pages 104: An Intelligent System for Allocating Times to the Main Activities of Managers https://www.mdpi.com/2075-1680/10/2/104 Correctly allocating times to the main activities of a manager is a crucial task that directly affects the possibility of success for any company. Decision support based on state-of-the-art methods can lead to better performance in this activity. However, allocating times to managerial activities is not straightforward; the decision support should provide a flexible recommendation so the manager can make a final decision while ensuring robustness. This paper describes and assesses a novel approach where a search for the best distribution of the manager’s time is performed by an intelligent decision support system. The approach consists of eliciting manager preferences to define the value of the manager’s main activities and, by using a portfolio-like optimization based on differential evolution, obtaining the best time allocation. Aiming at applicability in practical scenarios, the approach can deal with many activities, group decisions, cope with imprecision, vagueness, ill-determination, and other types of uncertainty. We present evidence of the approach’s applicability exploiting a real case study with the participation of several managers. The approach is assessed through the satisfaction level of each manager. Axioms, Vol. 10, Pages 104: An Intelligent System for Allocating Times to the Main Activities of Managers

Authors: Efrain Solares Liliana Guerrero Alberto Aguilera Juana María Hernández Sandra Rodríguez Víctor De-León-Gómez

Correctly allocating times to the main activities of a manager is a crucial task that directly affects the possibility of success for any company. Decision support based on state-of-the-art methods can lead to better performance in this activity. However, allocating times to managerial activities is not straightforward; the decision support should provide a flexible recommendation so the manager can make a final decision while ensuring robustness. This paper describes and assesses a novel approach where a search for the best distribution of the manager’s time is performed by an intelligent decision support system. The approach consists of eliciting manager preferences to define the value of the manager’s main activities and, by using a portfolio-like optimization based on differential evolution, obtaining the best time allocation. Aiming at applicability in practical scenarios, the approach can deal with many activities, group decisions, cope with imprecision, vagueness, ill-determination, and other types of uncertainty. We present evidence of the approach’s applicability exploiting a real case study with the participation of several managers. The approach is assessed through the satisfaction level of each manager.

]]>
An Intelligent System for Allocating Times to the Main Activities of Managers Efrain Solares Liliana Guerrero Alberto Aguilera Juana María Hernández Sandra Rodríguez Víctor De-León-Gómez doi: 10.3390/axioms10020104 Axioms 2021-05-25 Axioms 2021-05-25 10 2 Article 104 10.3390/axioms10020104 https://www.mdpi.com/2075-1680/10/2/104
Axioms, Vol. 10, Pages 103: On the Existence of Coupled Fractional Jerk Equations with Multi-Point Boundary Conditions https://www.mdpi.com/2075-1680/10/2/103 By coincidence degree theory due to Mawhin, some sufficient conditions for the existence of solution for a class of coupled jerk equations with multi-point conditions are established. The new existence results have not yet been reported before. Novel coupled fractional jerk equations with resonant boundary value conditions are discussed in detail for the first time. Our work is interesting and complements known results. Axioms, Vol. 10, Pages 103: On the Existence of Coupled Fractional Jerk Equations with Multi-Point Boundary Conditions

Authors: Lei Hu Yaozhen Han Shuqin Zhang

By coincidence degree theory due to Mawhin, some sufficient conditions for the existence of solution for a class of coupled jerk equations with multi-point conditions are established. The new existence results have not yet been reported before. Novel coupled fractional jerk equations with resonant boundary value conditions are discussed in detail for the first time. Our work is interesting and complements known results.

]]>
On the Existence of Coupled Fractional Jerk Equations with Multi-Point Boundary Conditions Lei Hu Yaozhen Han Shuqin Zhang doi: 10.3390/axioms10020103 Axioms 2021-05-24 Axioms 2021-05-24 10 2 Article 103 10.3390/axioms10020103 https://www.mdpi.com/2075-1680/10/2/103
Axioms, Vol. 10, Pages 102: Macroscopic and Multi-Scale Models for Multi-Class Vehicular Dynamics with Uneven Space Occupancy: A Case Study https://www.mdpi.com/2075-1680/10/2/102 In this paper, we propose two models describing the dynamics of heavy and light vehicles on a road network, taking into account the interactions between the two classes. The models are tailored for two-lane highways where heavy vehicles cannot overtake. This means that heavy vehicles cannot saturate the whole road space, while light vehicles can. In these conditions, the creeping phenomenon can appear, i.e., one class of vehicles can proceed even if the other class has reached the maximal density. The first model we propose couples two first-order macroscopic LWR models, while the second model couples a second-order microscopic follow-the-leader model with a first-order macroscopic LWR model. Numerical results show that both models are able to catch some second-order (inertial) phenomena such as stop and go waves. Models are calibrated by means of real data measured by fixed sensors placed along the A4 Italian highway Trieste–Venice and its branches, provided by Autovie Venete S.p.A. Axioms, Vol. 10, Pages 102: Macroscopic and Multi-Scale Models for Multi-Class Vehicular Dynamics with Uneven Space Occupancy: A Case Study

Authors: Maya Briani Emiliano Cristiani Paolo Ranut

In this paper, we propose two models describing the dynamics of heavy and light vehicles on a road network, taking into account the interactions between the two classes. The models are tailored for two-lane highways where heavy vehicles cannot overtake. This means that heavy vehicles cannot saturate the whole road space, while light vehicles can. In these conditions, the creeping phenomenon can appear, i.e., one class of vehicles can proceed even if the other class has reached the maximal density. The first model we propose couples two first-order macroscopic LWR models, while the second model couples a second-order microscopic follow-the-leader model with a first-order macroscopic LWR model. Numerical results show that both models are able to catch some second-order (inertial) phenomena such as stop and go waves. Models are calibrated by means of real data measured by fixed sensors placed along the A4 Italian highway Trieste–Venice and its branches, provided by Autovie Venete S.p.A.

]]>
Macroscopic and Multi-Scale Models for Multi-Class Vehicular Dynamics with Uneven Space Occupancy: A Case Study Maya Briani Emiliano Cristiani Paolo Ranut doi: 10.3390/axioms10020102 Axioms 2021-05-24 Axioms 2021-05-24 10 2 Article 102 10.3390/axioms10020102 https://www.mdpi.com/2075-1680/10/2/102
Axioms, Vol. 10, Pages 101: Common Fixed Point Results for Almost ℛg-Geraghty Type Contraction Mappings in b2-Metric Spaces with an Application to Integral Equations https://www.mdpi.com/2075-1680/10/2/101 In this paper, we define almost Rg-Geraghty type contractions and utilize the same to establish some coincidence and common fixed point results in the setting of b2-metric spaces endowed with binary relations. As consequences of our newly proved results, we deduce some coincidence and common fixed point results for almost g-α-η Geraghty type contraction mappings in b2-metric spaces. In addition, we derive some coincidence and common fixed point results in partially ordered b2-metric spaces. Moreover, to show the utility of our main results, we provide an example and an application to non-linear integral equations. Axioms, Vol. 10, Pages 101: Common Fixed Point Results for Almost ℛg-Geraghty Type Contraction Mappings in b2-Metric Spaces with an Application to Integral Equations

In this paper, we define almost Rg-Geraghty type contractions and utilize the same to establish some coincidence and common fixed point results in the setting of b2-metric spaces endowed with binary relations. As consequences of our newly proved results, we deduce some coincidence and common fixed point results for almost g-α-η Geraghty type contraction mappings in b2-metric spaces. In addition, we derive some coincidence and common fixed point results in partially ordered b2-metric spaces. Moreover, to show the utility of our main results, we provide an example and an application to non-linear integral equations.

]]>
Common Fixed Point Results for Almost ℛg-Geraghty Type Contraction Mappings in b2-Metric Spaces with an Application to Integral Equations Samera M. Saleh Salvatore Sessa Waleed M. Alfaqih Fawzia Shaddad doi: 10.3390/axioms10020101 Axioms 2021-05-24 Axioms 2021-05-24 10 2 Article 101 10.3390/axioms10020101 https://www.mdpi.com/2075-1680/10/2/101
Axioms, Vol. 10, Pages 100: Applying Transformer Insulation Using Weibull Extended Distribution Based on Progressive Censoring Scheme https://www.mdpi.com/2075-1680/10/2/100 In this paper, the Weibull extension distribution parameters are estimated under a progressive type-II censoring scheme with random removal. The parameters of the model are estimated using the maximum likelihood method, maximum product spacing, and Bayesian estimation methods. In classical estimation (maximum likelihood method and maximum product spacing), we did use the Newton–Raphson algorithm. The Bayesian estimation is done using the Metropolis–Hastings algorithm based on the square error loss function. The proposed estimation methods are compared using Monte Carlo simulations under a progressive type-II censoring scheme. An empirical study using a real data set of transformer insulation and a simulation study is performed to validate the introduced methods of inference. Based on the result of our study, it can be concluded that the Bayesian method outperforms the maximum likelihood and maximum product-spacing methods for estimating the Weibull extension parameters under a progressive type-II censoring scheme in both simulation and empirical studies. Axioms, Vol. 10, Pages 100: Applying Transformer Insulation Using Weibull Extended Distribution Based on Progressive Censoring Scheme

Authors: Hisham M. Almongy Fatma Y. Alshenawy Ehab M. Almetwally Doaa A. Abdo

In this paper, the Weibull extension distribution parameters are estimated under a progressive type-II censoring scheme with random removal. The parameters of the model are estimated using the maximum likelihood method, maximum product spacing, and Bayesian estimation methods. In classical estimation (maximum likelihood method and maximum product spacing), we did use the Newton–Raphson algorithm. The Bayesian estimation is done using the Metropolis–Hastings algorithm based on the square error loss function. The proposed estimation methods are compared using Monte Carlo simulations under a progressive type-II censoring scheme. An empirical study using a real data set of transformer insulation and a simulation study is performed to validate the introduced methods of inference. Based on the result of our study, it can be concluded that the Bayesian method outperforms the maximum likelihood and maximum product-spacing methods for estimating the Weibull extension parameters under a progressive type-II censoring scheme in both simulation and empirical studies.

]]>
Applying Transformer Insulation Using Weibull Extended Distribution Based on Progressive Censoring Scheme Hisham M. Almongy Fatma Y. Alshenawy Ehab M. Almetwally Doaa A. Abdo doi: 10.3390/axioms10020100 Axioms 2021-05-21 Axioms 2021-05-21 10 2 Article 100 10.3390/axioms10020100 https://www.mdpi.com/2075-1680/10/2/100
Axioms, Vol. 10, Pages 99: Perturbation Bounds for Eigenvalues and Determinants of Matrices. A Survey https://www.mdpi.com/2075-1680/10/2/99 The paper is a survey of the recent results of the author on the perturbations of matrices. A part of the results presented in the paper is new. In particular, we suggest a bound for the difference of the determinants of two matrices which refines the well-known Bhatia inequality. We also derive new estimates for the spectral variation of a perturbed matrix with respect to a given one, as well as estimates for the Hausdorff and matching distances between the spectra of two matrices. These estimates are formulated in the terms of the entries of matrices and via so called departure from normality. In appropriate situations they improve the well-known results. We also suggest a bound for the angular sectors containing the spectra of matrices. In addition, we suggest a new bound for the similarity condition numbers of diagonalizable matrices. The paper also contains a generalization of the famous Kahan inequality on perturbations of Hermitian matrices by non-normal matrices. Finally, taking into account that any matrix having more than one eigenvalue is similar to a block-diagonal matrix, we obtain a bound for the condition numbers in the case of non-diagonalizable matrices, and discuss applications of that bound to matrix functions and spectrum perturbations. The main methodology presented in the paper is based on a combined usage of the recent norm estimates for matrix-valued functions with the traditional methods and results. Axioms, Vol. 10, Pages 99: Perturbation Bounds for Eigenvalues and Determinants of Matrices. A Survey

Authors: Michael Gil’

The paper is a survey of the recent results of the author on the perturbations of matrices. A part of the results presented in the paper is new. In particular, we suggest a bound for the difference of the determinants of two matrices which refines the well-known Bhatia inequality. We also derive new estimates for the spectral variation of a perturbed matrix with respect to a given one, as well as estimates for the Hausdorff and matching distances between the spectra of two matrices. These estimates are formulated in the terms of the entries of matrices and via so called departure from normality. In appropriate situations they improve the well-known results. We also suggest a bound for the angular sectors containing the spectra of matrices. In addition, we suggest a new bound for the similarity condition numbers of diagonalizable matrices. The paper also contains a generalization of the famous Kahan inequality on perturbations of Hermitian matrices by non-normal matrices. Finally, taking into account that any matrix having more than one eigenvalue is similar to a block-diagonal matrix, we obtain a bound for the condition numbers in the case of non-diagonalizable matrices, and discuss applications of that bound to matrix functions and spectrum perturbations. The main methodology presented in the paper is based on a combined usage of the recent norm estimates for matrix-valued functions with the traditional methods and results.

]]>
Perturbation Bounds for Eigenvalues and Determinants of Matrices. A Survey Michael Gil’ doi: 10.3390/axioms10020099 Axioms 2021-05-21 Axioms 2021-05-21 10 2 Review 99 10.3390/axioms10020099 https://www.mdpi.com/2075-1680/10/2/99
Axioms, Vol. 10, Pages 98: Optimal Segmentation over a Generalized Customer Distribution https://www.mdpi.com/2075-1680/10/2/98 This paper investigates the impact of consumer preferences on the intensity of competition for companies in a duopoly market. A classical Hotelling’s competition problem will be different if consumers are allowed to distribute non-uniformly. New results in competition intensity are established and conditions for the existence of a subgame perfect Nash equilibrium is identified through a model that considers generic distribution in consumer preferences. A competition strategy is demonstrated to depend on the signs of local change rates of the density function at the endpoints of market segments. Axioms, Vol. 10, Pages 98: Optimal Segmentation over a Generalized Customer Distribution

Authors: Tsung-Yin Ou Yenming J. Chen

This paper investigates the impact of consumer preferences on the intensity of competition for companies in a duopoly market. A classical Hotelling’s competition problem will be different if consumers are allowed to distribute non-uniformly. New results in competition intensity are established and conditions for the existence of a subgame perfect Nash equilibrium is identified through a model that considers generic distribution in consumer preferences. A competition strategy is demonstrated to depend on the signs of local change rates of the density function at the endpoints of market segments.

]]>
Optimal Segmentation over a Generalized Customer Distribution Tsung-Yin Ou Yenming J. Chen doi: 10.3390/axioms10020098 Axioms 2021-05-21 Axioms 2021-05-21 10 2 Article 98 10.3390/axioms10020098 https://www.mdpi.com/2075-1680/10/2/98
Axioms, Vol. 10, Pages 97: Conditional Intuitionistic Fuzzy Mean Value https://www.mdpi.com/2075-1680/10/2/97 The conditional mean value has applications in regression analysis and in financial mathematics, because they are used in it. We can find papers from recent years that use the conditional mean value in fuzzy cases. As the intuitionstic fuzzy sets are an extension of fuzzy sets, we will try to define a conditional mean value for the intuitionistic fuzzy case. The conditional mean value in crisp intuitionistic fuzzy events was first studied by V. Valenčáková in 2009. She used Gödel connectives. Her approach can only be used for special cases of intuitionistic fuzzy events, therefore, we want to define a conditional mean value for all elements of a family of intuitionistic fuzzy events. In this paper, we define the conditional mean value for intuitionistic fuzzy events using Lukasiewicz connectives. We use a Kolmogorov approach and the notions from a classical probability theory for construction. B. Riečan formulated a conditional intuitionistic fuzzy probability for intuitionistic fuzzy events using an intuitionistic fuzzy state in 2012. In classical cases, there exists a connection between the conditional probability and the conditional mean value, therefore we show a connection between the conditional intuitionistic fuzzy probability induced by the intuitionistic fuzzy state and the conditional intuitionistic fuzzy mean value. Axioms, Vol. 10, Pages 97: Conditional Intuitionistic Fuzzy Mean Value

Authors: Katarína Čunderlíková

The conditional mean value has applications in regression analysis and in financial mathematics, because they are used in it. We can find papers from recent years that use the conditional mean value in fuzzy cases. As the intuitionstic fuzzy sets are an extension of fuzzy sets, we will try to define a conditional mean value for the intuitionistic fuzzy case. The conditional mean value in crisp intuitionistic fuzzy events was first studied by V. Valenčáková in 2009. She used Gödel connectives. Her approach can only be used for special cases of intuitionistic fuzzy events, therefore, we want to define a conditional mean value for all elements of a family of intuitionistic fuzzy events. In this paper, we define the conditional mean value for intuitionistic fuzzy events using Lukasiewicz connectives. We use a Kolmogorov approach and the notions from a classical probability theory for construction. B. Riečan formulated a conditional intuitionistic fuzzy probability for intuitionistic fuzzy events using an intuitionistic fuzzy state in 2012. In classical cases, there exists a connection between the conditional probability and the conditional mean value, therefore we show a connection between the conditional intuitionistic fuzzy probability induced by the intuitionistic fuzzy state and the conditional intuitionistic fuzzy mean value.

]]>
Conditional Intuitionistic Fuzzy Mean Value Katarína Čunderlíková doi: 10.3390/axioms10020097 Axioms 2021-05-21 Axioms 2021-05-21 10 2 Article 97 10.3390/axioms10020097 https://www.mdpi.com/2075-1680/10/2/97
Axioms, Vol. 10, Pages 96: Mathematical Approach for System Repair Rate Analysis Used in Maintenance Decision Making https://www.mdpi.com/2075-1680/10/2/96 Reliability, the number of spare parts and repair time have a great impact on system availability. In this paper, we observed a repairable system comprised of several components. The aim was to determine the repair rate by emphasizing its stochastic nature. A model for the statistical analysis of the component repair rate in function of the desired level of availability is presented. Furthermore, based on the presented model, the approach for the calculation of probability density functions of maximal and minimal repair times for a system comprised of observed components was developed as an important measure that unambiguously defines the total annual repair time. The obtained generalized analytical expressions that can be used to predict the total repair time for an observed entity are the main contributions of the manuscript. The outputs of the model can be useful for making decisions in which time interval repair or replacement should be done to maintain the system and component availability. In addition to planning maintenance activities, the presented models could be used for service capacity planning and the dynamic forecasting of system characteristics. Axioms, Vol. 10, Pages 96: Mathematical Approach for System Repair Rate Analysis Used in Maintenance Decision Making

Authors: Nataša Kontrec Stefan Panić Biljana Panić Aleksandar Marković Dejan Stošović

Reliability, the number of spare parts and repair time have a great impact on system availability. In this paper, we observed a repairable system comprised of several components. The aim was to determine the repair rate by emphasizing its stochastic nature. A model for the statistical analysis of the component repair rate in function of the desired level of availability is presented. Furthermore, based on the presented model, the approach for the calculation of probability density functions of maximal and minimal repair times for a system comprised of observed components was developed as an important measure that unambiguously defines the total annual repair time. The obtained generalized analytical expressions that can be used to predict the total repair time for an observed entity are the main contributions of the manuscript. The outputs of the model can be useful for making decisions in which time interval repair or replacement should be done to maintain the system and component availability. In addition to planning maintenance activities, the presented models could be used for service capacity planning and the dynamic forecasting of system characteristics.

]]>
Mathematical Approach for System Repair Rate Analysis Used in Maintenance Decision Making Nataša Kontrec Stefan Panić Biljana Panić Aleksandar Marković Dejan Stošović doi: 10.3390/axioms10020096 Axioms 2021-05-20 Axioms 2021-05-20 10 2 Article 96 10.3390/axioms10020096 https://www.mdpi.com/2075-1680/10/2/96
Axioms, Vol. 10, Pages 95: Comments on the Navier–Stokes Problem https://www.mdpi.com/2075-1680/10/2/95 The aim of this paper is to explain for broad audience the author’s result concerning the Navier–Stokes problem (NSP) in R3 without boundaries. It is proved that the NSP is contradictory in the following sense: if one assumes that the initial data v(x,0)≢0, ∇·v(x,0)=0 and the solution to the NSP exists for all t≥0, then one proves that the solution v(x,t) to the NSP has the property v(x,0)=0. This paradox shows that the NSP is not a correct description of the fluid mechanics problem and the NSP does not have a solution. In the exceptional case, when the data are equal to zero, the solution v(x,t) to the NSP exists for all t≥0 and is equal to zero, v(x,t)≡0. Thus, one of the millennium problems is solved. Axioms, Vol. 10, Pages 95: Comments on the Navier–Stokes Problem

Authors: Alexander Ramm

The aim of this paper is to explain for broad audience the author’s result concerning the Navier–Stokes problem (NSP) in R3 without boundaries. It is proved that the NSP is contradictory in the following sense: if one assumes that the initial data v(x,0)≢0, ∇·v(x,0)=0 and the solution to the NSP exists for all t≥0, then one proves that the solution v(x,t) to the NSP has the property v(x,0)=0. This paradox shows that the NSP is not a correct description of the fluid mechanics problem and the NSP does not have a solution. In the exceptional case, when the data are equal to zero, the solution v(x,t) to the NSP exists for all t≥0 and is equal to zero, v(x,t)≡0. Thus, one of the millennium problems is solved.

]]>
Comments on the Navier–Stokes Problem Alexander Ramm doi: 10.3390/axioms10020095 Axioms 2021-05-20 Axioms 2021-05-20 10 2 Article 95 10.3390/axioms10020095 https://www.mdpi.com/2075-1680/10/2/95
Axioms, Vol. 10, Pages 94: Degenerate Canonical Forms of Ordinary Second-Order Linear Homogeneous Differential Equations https://www.mdpi.com/2075-1680/10/2/94 For each fundamental and widely used ordinary second-order linear homogeneous differential equation of mathematical physics, we derive a family of associated differential equations that share the same “degenerate” canonical form. These equations can be solved easily if the original equation is known to possess analytic solutions, otherwise their properties and the properties of their solutions are de facto known as they are comparable to those already deduced for the fundamental equation. We analyze several particular cases of new families related to some of the famous differential equations applied to physical problems, and the degenerate eigenstates of the radial Schrödinger equation for the hydrogen atom in N dimensions. Axioms, Vol. 10, Pages 94: Degenerate Canonical Forms of Ordinary Second-Order Linear Homogeneous Differential Equations

Authors: Dimitris M. Christodoulou Eric Kehoe Qutaibeh D. Katatbeh

For each fundamental and widely used ordinary second-order linear homogeneous differential equation of mathematical physics, we derive a family of associated differential equations that share the same “degenerate” canonical form. These equations can be solved easily if the original equation is known to possess analytic solutions, otherwise their properties and the properties of their solutions are de facto known as they are comparable to those already deduced for the fundamental equation. We analyze several particular cases of new families related to some of the famous differential equations applied to physical problems, and the degenerate eigenstates of the radial Schrödinger equation for the hydrogen atom in N dimensions.

]]>
Degenerate Canonical Forms of Ordinary Second-Order Linear Homogeneous Differential Equations Dimitris M. Christodoulou Eric Kehoe Qutaibeh D. Katatbeh doi: 10.3390/axioms10020094 Axioms 2021-05-19 Axioms 2021-05-19 10 2 Article 94 10.3390/axioms10020094 https://www.mdpi.com/2075-1680/10/2/94
Axioms, Vol. 10, Pages 93: Transdisciplinary Scientific Strategies for Soft Computing Development: Towards an Era of Data and Business Analytics https://www.mdpi.com/2075-1680/10/2/93 This research is a review and analysis paper that offers a transdisciplinary, methodological, and strategic vision for soft computing development towards a wider favorable impact in data analytics. Strategies are defined, explained, and illustrated by examples. The paper also shows how these strategies are expressed in three dimensions of an ambitious actions plan. They are all integrated into a master strategy called wide knowledge discovery, which offers a way towards the augmented analytics paradigm. Some contributions of this work are defining what kind of mathematical elements should be introduced into soft computing towards a better impact on the area of data analytics, offering orientation towards building new mathematical elements, and defining why and how they can be introduced. Axioms, Vol. 10, Pages 93: Transdisciplinary Scientific Strategies for Soft Computing Development: Towards an Era of Data and Business Analytics

Authors: Rafael A. Espin-Andrade Witold Pedrycz Efrain Solares Laura Cruz-Reyes

This research is a review and analysis paper that offers a transdisciplinary, methodological, and strategic vision for soft computing development towards a wider favorable impact in data analytics. Strategies are defined, explained, and illustrated by examples. The paper also shows how these strategies are expressed in three dimensions of an ambitious actions plan. They are all integrated into a master strategy called wide knowledge discovery, which offers a way towards the augmented analytics paradigm. Some contributions of this work are defining what kind of mathematical elements should be introduced into soft computing towards a better impact on the area of data analytics, offering orientation towards building new mathematical elements, and defining why and how they can be introduced.

]]>
Transdisciplinary Scientific Strategies for Soft Computing Development: Towards an Era of Data and Business Analytics Rafael A. Espin-Andrade Witold Pedrycz Efrain Solares Laura Cruz-Reyes doi: 10.3390/axioms10020093 Axioms 2021-05-18 Axioms 2021-05-18 10 2 Article 93 10.3390/axioms10020093 https://www.mdpi.com/2075-1680/10/2/93
Axioms, Vol. 10, Pages 92: Ranking Road Sections Based on MCDM Model: New Improved Fuzzy SWARA (IMF SWARA) https://www.mdpi.com/2075-1680/10/2/92 Traffic management is a significantly difficult and demanding task. It is necessary to know the main parameters of road networks in order to adequately meet traffic management requirements. Through this paper, an integrated fuzzy model for ranking road sections based on four inputs and four outputs was developed. The goal was to determine the safety degree of the observed road sections by the methodology developed. The greatest contribution of the paper is reflected in the development of the improved fuzzy step-wise weight assessment ratio analysis (IMF SWARA) method and integration with the fuzzy measurement alternatives and ranking according to the compromise solution (fuzzy MARCOS) method. First, the data envelopment analysis (DEA) model was applied, showing that three road sections have a high traffic risk. After that, IMF SWARA was applied to determine the values of the weight coefficients of the criteria, and the fuzzy MARCOS method was used for the final ranking of the sections. The obtained results were verified through a three-phase sensitivity analysis with an emphasis on forming 40 new scenarios in which input values were simulated. The stability of the model was proven in all phases of sensitivity analysis. Axioms, Vol. 10, Pages 92: Ranking Road Sections Based on MCDM Model: New Improved Fuzzy SWARA (IMF SWARA)

Authors: Sabahudin Vrtagić Edis Softić Marko Subotić Željko Stević Milan Dordevic Mirza Ponjavic

Traffic management is a significantly difficult and demanding task. It is necessary to know the main parameters of road networks in order to adequately meet traffic management requirements. Through this paper, an integrated fuzzy model for ranking road sections based on four inputs and four outputs was developed. The goal was to determine the safety degree of the observed road sections by the methodology developed. The greatest contribution of the paper is reflected in the development of the improved fuzzy step-wise weight assessment ratio analysis (IMF SWARA) method and integration with the fuzzy measurement alternatives and ranking according to the compromise solution (fuzzy MARCOS) method. First, the data envelopment analysis (DEA) model was applied, showing that three road sections have a high traffic risk. After that, IMF SWARA was applied to determine the values of the weight coefficients of the criteria, and the fuzzy MARCOS method was used for the final ranking of the sections. The obtained results were verified through a three-phase sensitivity analysis with an emphasis on forming 40 new scenarios in which input values were simulated. The stability of the model was proven in all phases of sensitivity analysis.

]]>
Ranking Road Sections Based on MCDM Model: New Improved Fuzzy SWARA (IMF SWARA) Sabahudin Vrtagić Edis Softić Marko Subotić Željko Stević Milan Dordevic Mirza Ponjavic doi: 10.3390/axioms10020092 Axioms 2021-05-15 Axioms 2021-05-15 10 2 Article 92 10.3390/axioms10020092 https://www.mdpi.com/2075-1680/10/2/92
Axioms, Vol. 10, Pages 91: The Formal Framework for Collective Systems https://www.mdpi.com/2075-1680/10/2/91 Automated reasoning is becoming crucial for information systems. Building one uniform decision support system has become too complicated. The natural approach is to divide the task and combine the results from different subsystems into one uniform answer. It is the basic idea behind the system approach, where one solution is a composition of multiple subsystems. In this paper, the main emphasis is on establishing the theoretical framework that combines various reasoning methods into a collective system. The system’s formal abstraction uses graph theory and provides a discussion on possible aggregation function definitions. The proposed framework is a tool for building and testing specific approaches rather than the solution itself. Axioms, Vol. 10, Pages 91: The Formal Framework for Collective Systems

Authors: Rafał Palak Krystian Wojtkiewicz

Automated reasoning is becoming crucial for information systems. Building one uniform decision support system has become too complicated. The natural approach is to divide the task and combine the results from different subsystems into one uniform answer. It is the basic idea behind the system approach, where one solution is a composition of multiple subsystems. In this paper, the main emphasis is on establishing the theoretical framework that combines various reasoning methods into a collective system. The system’s formal abstraction uses graph theory and provides a discussion on possible aggregation function definitions. The proposed framework is a tool for building and testing specific approaches rather than the solution itself.

]]>
The Formal Framework for Collective Systems Rafał Palak Krystian Wojtkiewicz doi: 10.3390/axioms10020091 Axioms 2021-05-15 Axioms 2021-05-15 10 2 Article 91 10.3390/axioms10020091 https://www.mdpi.com/2075-1680/10/2/91
Axioms, Vol. 10, Pages 90: Applications Optimal Math Model to Solve Difficult Problems for Businesses Producing and Processing Agricultural Products in Vietnam https://www.mdpi.com/2075-1680/10/2/90 During the months of the COVID-19 epidemic, the number of processing and manufacturing agricultural products enterprises in Vietnam faced many difficulties in the consumption of agricultural products, even when they were unable to sell. Therefore, the enterprises are more and more difficult. Many enterprises are not strong enough to restore production, so it is necessary to find solutions to overcome this difficult period. In this research, the author used the application of modern statistical techniques, along with the Grey method, to predict enterprises’ business results in the future, combined with the model of Super-slacks-based-measure efficiency (Super-SBM) to help businesses select the right partners in a supply chain to achieve their business goals. By our proposed approach, the selected solution (AG6 combined with AG10) should be implemented in the future to upgrade efficiency to help stabilize output and raise productivity; thus, both parties are able not only to improve their product quality but also achieve business goals and sustainable development. In fact, it is necessary to further this study, in combination with these factors and other quantitative models, to give investors a more comprehensive view, helping them to make the right decisions and appropriately develop their businesses and social and economic development. Axioms, Vol. 10, Pages 90: Applications Optimal Math Model to Solve Difficult Problems for Businesses Producing and Processing Agricultural Products in Vietnam

Authors: Han-Khanh Nguyen

During the months of the COVID-19 epidemic, the number of processing and manufacturing agricultural products enterprises in Vietnam faced many difficulties in the consumption of agricultural products, even when they were unable to sell. Therefore, the enterprises are more and more difficult. Many enterprises are not strong enough to restore production, so it is necessary to find solutions to overcome this difficult period. In this research, the author used the application of modern statistical techniques, along with the Grey method, to predict enterprises’ business results in the future, combined with the model of Super-slacks-based-measure efficiency (Super-SBM) to help businesses select the right partners in a supply chain to achieve their business goals. By our proposed approach, the selected solution (AG6 combined with AG10) should be implemented in the future to upgrade efficiency to help stabilize output and raise productivity; thus, both parties are able not only to improve their product quality but also achieve business goals and sustainable development. In fact, it is necessary to further this study, in combination with these factors and other quantitative models, to give investors a more comprehensive view, helping them to make the right decisions and appropriately develop their businesses and social and economic development.

]]>
Applications Optimal Math Model to Solve Difficult Problems for Businesses Producing and Processing Agricultural Products in Vietnam Han-Khanh Nguyen doi: 10.3390/axioms10020090 Axioms 2021-05-14 Axioms 2021-05-14 10 2 Article 90 10.3390/axioms10020090 https://www.mdpi.com/2075-1680/10/2/90
Axioms, Vol. 10, Pages 89: On Factoring Groups into Thin Subsets https://www.mdpi.com/2075-1680/10/2/89 A subset X of a group G is called thin if, for every finite subset F of G, there exists a finite subset H of G such that Fx∩Fy=∅, xF∩yF=∅ for all distinct x,y∈X\H. We prove that every countable topologizable group G can be factorized G=AB into thin subsets A,B. Axioms, Vol. 10, Pages 89: On Factoring Groups into Thin Subsets

Authors: Igor Protasov

A subset X of a group G is called thin if, for every finite subset F of G, there exists a finite subset H of G such that Fx∩Fy=∅, xF∩yF=∅ for all distinct x,y∈X\H. We prove that every countable topologizable group G can be factorized G=AB into thin subsets A,B.

]]>
On Factoring Groups into Thin Subsets Igor Protasov doi: 10.3390/axioms10020089 Axioms 2021-05-14 Axioms 2021-05-14 10 2 Article 89 10.3390/axioms10020089 https://www.mdpi.com/2075-1680/10/2/89
Axioms, Vol. 10, Pages 88: Analyzing Uncertain Dynamical Systems after State-Space Transformations into Cooperative Form: Verification of Control and Fault Diagnosis https://www.mdpi.com/2075-1680/10/2/88 When modeling real-life applications, uncertainty appears in the form of, for example, modeling approximations, measurement errors, or simply physical restrictions. Those uncertainties can either be treated as stochastic or as bounded, with known limits in the form of intervals. The latter is considered in this paper for a real-life application in the form of an electrical circuit. This is reasonable because the electrical circuit is subject to uncertainties, mainly due to circuit element tolerances and variable load conditions. Since conservative worst-case limits for such parameters are commonly known, interval methods can be applied. The aim of this paper is to demonstrate a possible overall handling of the given uncertain system. Firstly, this includes a control and a reliable computation of the states’ interval enclosures. On the one hand, this can be used to predict the system’s behavior, and on the other hand to verify the control numerically. Here, the implemented feedback control is based on linear matrix inequalities (LMIs) and the states are predicted using an interval enclosure technique based on cooperativity. Since the original system is not cooperative, a transformation is performed. Finally, an observer is implemented as a diagnosis tool regarding faulty measurements or component failures. Since adding a state-of-the-art observer would destroy this structure, a cooperativity-preserving method is applied. Overall, this paper combines methods from robust control design and interval-based evaluations, and presents a suitable observer technique to show the applicability of the presented methods. Axioms, Vol. 10, Pages 88: Analyzing Uncertain Dynamical Systems after State-Space Transformations into Cooperative Form: Verification of Control and Fault Diagnosis

Authors: Julia Kersten Andreas Rauh Harald Aschemann

When modeling real-life applications, uncertainty appears in the form of, for example, modeling approximations, measurement errors, or simply physical restrictions. Those uncertainties can either be treated as stochastic or as bounded, with known limits in the form of intervals. The latter is considered in this paper for a real-life application in the form of an electrical circuit. This is reasonable because the electrical circuit is subject to uncertainties, mainly due to circuit element tolerances and variable load conditions. Since conservative worst-case limits for such parameters are commonly known, interval methods can be applied. The aim of this paper is to demonstrate a possible overall handling of the given uncertain system. Firstly, this includes a control and a reliable computation of the states’ interval enclosures. On the one hand, this can be used to predict the system’s behavior, and on the other hand to verify the control numerically. Here, the implemented feedback control is based on linear matrix inequalities (LMIs) and the states are predicted using an interval enclosure technique based on cooperativity. Since the original system is not cooperative, a transformation is performed. Finally, an observer is implemented as a diagnosis tool regarding faulty measurements or component failures. Since adding a state-of-the-art observer would destroy this structure, a cooperativity-preserving method is applied. Overall, this paper combines methods from robust control design and interval-based evaluations, and presents a suitable observer technique to show the applicability of the presented methods.

]]>
Analyzing Uncertain Dynamical Systems after State-Space Transformations into Cooperative Form: Verification of Control and Fault Diagnosis Julia Kersten Andreas Rauh Harald Aschemann doi: 10.3390/axioms10020088 Axioms 2021-05-10 Axioms 2021-05-10 10 2 Article 88 10.3390/axioms10020088 https://www.mdpi.com/2075-1680/10/2/88
Axioms, Vol. 10, Pages 87: The Ordered Weighted Average Human Development Index https://www.mdpi.com/2075-1680/10/2/87 The main aim is to propose a new method for estimating the Human Development Index using ordered weighted average. To develop this method, ordered weighted geometric average (OWGA), induced OWGA prioritized OWA (POWA) operator are studied. Using Human Development Index formulation in combination to aggregations operators presented above is proposed the prioritized induced ordered weighted geometric average (PIOWGA) operator. A mathematical application is carried out to estimate the Human Development Index and compare it with the traditional method and other existing methods. Finally, it is noted that decision makers have an influence on the order given in the ranking by its attitude and criterion, and method can capture the subjective information prioritized by them. Axioms, Vol. 10, Pages 87: The Ordered Weighted Average Human Development Index

Authors: Ernesto Leon-Castro Fabio Blanco-Mesa Alma Montserrat Romero-Serrano Marlenne Velázquez-Cazares

The main aim is to propose a new method for estimating the Human Development Index using ordered weighted average. To develop this method, ordered weighted geometric average (OWGA), induced OWGA prioritized OWA (POWA) operator are studied. Using Human Development Index formulation in combination to aggregations operators presented above is proposed the prioritized induced ordered weighted geometric average (PIOWGA) operator. A mathematical application is carried out to estimate the Human Development Index and compare it with the traditional method and other existing methods. Finally, it is noted that decision makers have an influence on the order given in the ranking by its attitude and criterion, and method can capture the subjective information prioritized by them.

]]>
The Ordered Weighted Average Human Development Index Ernesto Leon-Castro Fabio Blanco-Mesa Alma Montserrat Romero-Serrano Marlenne Velázquez-Cazares doi: 10.3390/axioms10020087 Axioms 2021-05-07 Axioms 2021-05-07 10 2 Article 87 10.3390/axioms10020087 https://www.mdpi.com/2075-1680/10/2/87
Axioms, Vol. 10, Pages 86: Properties of λ-Pseudo-Starlike Functions of Complex Order Defined by Subordination https://www.mdpi.com/2075-1680/10/2/86 In this paper, we defined a new class of λ-pseudo-Bazilevič functions of complex order using subordination. Various classes of analytic functions that map unit discs onto a conic domain and some classes of special functions were studied in dual. Some subordination results, inequalities for the initial Taylor–Maclaurin coefficients and the unified solution of the Fekete–Szegő problem for subclasses of analytic functions related to various conic regions, are our main results. Our main results have many applications which are presented in the form of corollaries. Axioms, Vol. 10, Pages 86: Properties of λ-Pseudo-Starlike Functions of Complex Order Defined by Subordination

In this paper, we defined a new class of λ-pseudo-Bazilevič functions of complex order using subordination. Various classes of analytic functions that map unit discs onto a conic domain and some classes of special functions were studied in dual. Some subordination results, inequalities for the initial Taylor–Maclaurin coefficients and the unified solution of the Fekete–Szegő problem for subclasses of analytic functions related to various conic regions, are our main results. Our main results have many applications which are presented in the form of corollaries.

]]>
Properties of λ-Pseudo-Starlike Functions of Complex Order Defined by Subordination Kadhavoor R. Karthikeyan Gangadharan Murugusundaramoorthy Teodor Bulboacă doi: 10.3390/axioms10020086 Axioms 2021-05-07 Axioms 2021-05-07 10 2 Article 86 10.3390/axioms10020086 https://www.mdpi.com/2075-1680/10/2/86
Axioms, Vol. 10, Pages 85: Parrondo’s Paradox for Tent Maps https://www.mdpi.com/2075-1680/10/2/85 In this paper, we study the dynamic Parrondo’s paradox for the well-known family of tent maps. We prove that this paradox is impossible when we consider piecewise linear maps with constant slope. In addition, we analyze the paradox “simple + simple = complex” when a tent map with constant slope and a piecewise linear homeomorphism with two different slopes are considered. Axioms, Vol. 10, Pages 85: Parrondo’s Paradox for Tent Maps

Authors: Jose S. Cánovas

In this paper, we study the dynamic Parrondo’s paradox for the well-known family of tent maps. We prove that this paradox is impossible when we consider piecewise linear maps with constant slope. In addition, we analyze the paradox “simple + simple = complex” when a tent map with constant slope and a piecewise linear homeomorphism with two different slopes are considered.

]]>
Parrondo’s Paradox for Tent Maps Jose S. Cánovas doi: 10.3390/axioms10020085 Axioms 2021-05-06 Axioms 2021-05-06 10 2 Article 85 10.3390/axioms10020085 https://www.mdpi.com/2075-1680/10/2/85
Axioms, Vol. 10, Pages 84: Controllability of Fuzzy Solutions for Neutral Impulsive Functional Differential Equations with Nonlocal Conditions https://www.mdpi.com/2075-1680/10/2/84 In this paper, the controllability of fuzzy solutions for first order nonlocal impulsive neutral functional differential equations is explored using the Banach fixed point theorem. We utilized the concepts of the fuzzy set theory, functional analysis, and the Hausdorff metric. In the conclusion, an illustration is given to bolster the hypothesis. Axioms, Vol. 10, Pages 84: Controllability of Fuzzy Solutions for Neutral Impulsive Functional Differential Equations with Nonlocal Conditions

Authors: Falguni Acharya Vandana Kushawaha Jitendra Panchal Dimplekumar Chalishajar

In this paper, the controllability of fuzzy solutions for first order nonlocal impulsive neutral functional differential equations is explored using the Banach fixed point theorem. We utilized the concepts of the fuzzy set theory, functional analysis, and the Hausdorff metric. In the conclusion, an illustration is given to bolster the hypothesis.

]]>
Controllability of Fuzzy Solutions for Neutral Impulsive Functional Differential Equations with Nonlocal Conditions Falguni Acharya Vandana Kushawaha Jitendra Panchal Dimplekumar Chalishajar doi: 10.3390/axioms10020084 Axioms 2021-05-06 Axioms 2021-05-06 10 2 Article 84 10.3390/axioms10020084 https://www.mdpi.com/2075-1680/10/2/84
Axioms, Vol. 10, Pages 83: About the Normal Projectivity and Injectivity of Krasner Hypermodules https://www.mdpi.com/2075-1680/10/2/83 Inspired by the concepts of projective and injective modules in classical algebraic structure theory, in this paper we initiate the study of the chains of hypermodules over a Krasner hyperring R, endowing first the set HomRn(M,N) of all normal homomorphisms between two R-hypermodules M and N with a structure of R-hypermodule. Then, our study focuses on the concepts of normal injectivity and projectivity of hypermodules over a Krasner hyperring R, characterizing them by the mean of chains of R-hypermodules. Axioms, Vol. 10, Pages 83: About the Normal Projectivity and Injectivity of Krasner Hypermodules

Authors: Hashem Bordbar Irina Cristea

Inspired by the concepts of projective and injective modules in classical algebraic structure theory, in this paper we initiate the study of the chains of hypermodules over a Krasner hyperring R, endowing first the set HomRn(M,N) of all normal homomorphisms between two R-hypermodules M and N with a structure of R-hypermodule. Then, our study focuses on the concepts of normal injectivity and projectivity of hypermodules over a Krasner hyperring R, characterizing them by the mean of chains of R-hypermodules.

]]>
About the Normal Projectivity and Injectivity of Krasner Hypermodules Hashem Bordbar Irina Cristea doi: 10.3390/axioms10020083 Axioms 2021-05-04 Axioms 2021-05-04 10 2 Article 83 10.3390/axioms10020083 https://www.mdpi.com/2075-1680/10/2/83
Axioms, Vol. 10, Pages 82: Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane https://www.mdpi.com/2075-1680/10/2/82 In this paper, we present an explicit formula for the approximate solution of the Cauchy problem for the matrix factorizations of the Helmholtz equation in a bounded domain on the plane. Our formula for an approximate solution also includes the construction of a family of fundamental solutions for the Helmholtz operator on the plane. This family is parameterized by function K(w) which depends on the space dimension. In this paper, based on the results of previous works, the better results can be obtained by choosing the function K(w). Axioms, Vol. 10, Pages 82: Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane

Authors: Davron Aslonqulovich Juraev Samad Noeiaghdam

In this paper, we present an explicit formula for the approximate solution of the Cauchy problem for the matrix factorizations of the Helmholtz equation in a bounded domain on the plane. Our formula for an approximate solution also includes the construction of a family of fundamental solutions for the Helmholtz operator on the plane. This family is parameterized by function K(w) which depends on the space dimension. In this paper, based on the results of previous works, the better results can be obtained by choosing the function K(w).

]]>
Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane Davron Aslonqulovich Juraev Samad Noeiaghdam doi: 10.3390/axioms10020082 Axioms 2021-05-02 Axioms 2021-05-02 10 2 Article 82 10.3390/axioms10020082 https://www.mdpi.com/2075-1680/10/2/82
Axioms, Vol. 10, Pages 81: Certain Unified Integrals Involving a Multivariate Mittag–Leffler Function https://www.mdpi.com/2075-1680/10/2/81 A remarkably large number of unified integrals involving the Mittag–Leffler function have been presented. Here, with the same technique as Choi and Agarwal, we propose the establishment of two generalized integral formulas involving a multivariate generalized Mittag–Leffler function, which are expressed in terms of the generalized Lauricella series due to Srivastava and Daoust. We also present some interesting special cases. Axioms, Vol. 10, Pages 81: Certain Unified Integrals Involving a Multivariate Mittag–Leffler Function

Authors: Shilpi Jain Ravi P. Agarwal Praveen Agarwal Prakash Singh

A remarkably large number of unified integrals involving the Mittag–Leffler function have been presented. Here, with the same technique as Choi and Agarwal, we propose the establishment of two generalized integral formulas involving a multivariate generalized Mittag–Leffler function, which are expressed in terms of the generalized Lauricella series due to Srivastava and Daoust. We also present some interesting special cases.

]]>
Certain Unified Integrals Involving a Multivariate Mittag–Leffler Function Shilpi Jain Ravi P. Agarwal Praveen Agarwal Prakash Singh doi: 10.3390/axioms10020081 Axioms 2021-05-02 Axioms 2021-05-02 10 2 Article 81 10.3390/axioms10020081 https://www.mdpi.com/2075-1680/10/2/81
Axioms, Vol. 10, Pages 80: Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps https://www.mdpi.com/2075-1680/10/2/80 We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and study its properties. Further, we study how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally. Axioms, Vol. 10, Pages 80: Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Authors: Sergey Kryzhevich Viktor Avrutin Nikita Begun Dmitrii Rachinskii Khosro Tajbakhsh

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and study its properties. Further, we study how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

]]>
Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps Sergey Kryzhevich Viktor Avrutin Nikita Begun Dmitrii Rachinskii Khosro Tajbakhsh doi: 10.3390/axioms10020080 Axioms 2021-05-02 Axioms 2021-05-02 10 2 Article 80 10.3390/axioms10020080 https://www.mdpi.com/2075-1680/10/2/80
Axioms, Vol. 10, Pages 79: An Information-Theoretic Framework for Optimal Design: Analysis of Protocols for Estimating Soft Tissue Parameters in Biaxial Experiments https://www.mdpi.com/2075-1680/10/2/79 A new framework for optimal design based on the information-theoretic measures of mutual information, conditional mutual information and their combination is proposed. The framework is tested on the analysis of protocols—a combination of angles along which strain measurements can be acquired—in a biaxial experiment of soft tissues for the estimation of hyperelastic constitutive model parameters. The proposed framework considers the information gain about the parameters from the experiment as the key criterion to be maximised, which can be directly used for optimal design. Information gain is computed through k-nearest neighbour algorithms applied to the joint samples of the parameters and measurements produced by the forward and observation models. For biaxial experiments, the results show that low angles have a relatively low information content compared to high angles. The results also show that a smaller number of angles with suitably chosen combinations can result in higher information gains when compared to a larger number of angles which are poorly combined. Finally, it is shown that the proposed framework is consistent with classical approaches, particularly D-optimal design. Axioms, Vol. 10, Pages 79: An Information-Theoretic Framework for Optimal Design: Analysis of Protocols for Estimating Soft Tissue Parameters in Biaxial Experiments

Authors: Ankush Aggarwal Damiano Lombardi Sanjay Pant

A new framework for optimal design based on the information-theoretic measures of mutual information, conditional mutual information and their combination is proposed. The framework is tested on the analysis of protocols—a combination of angles along which strain measurements can be acquired—in a biaxial experiment of soft tissues for the estimation of hyperelastic constitutive model parameters. The proposed framework considers the information gain about the parameters from the experiment as the key criterion to be maximised, which can be directly used for optimal design. Information gain is computed through k-nearest neighbour algorithms applied to the joint samples of the parameters and measurements produced by the forward and observation models. For biaxial experiments, the results show that low angles have a relatively low information content compared to high angles. The results also show that a smaller number of angles with suitably chosen combinations can result in higher information gains when compared to a larger number of angles which are poorly combined. Finally, it is shown that the proposed framework is consistent with classical approaches, particularly D-optimal design.

]]>
An Information-Theoretic Framework for Optimal Design: Analysis of Protocols for Estimating Soft Tissue Parameters in Biaxial Experiments Ankush Aggarwal Damiano Lombardi Sanjay Pant doi: 10.3390/axioms10020079 Axioms 2021-05-01 Axioms 2021-05-01 10 2 Article 79 10.3390/axioms10020079 https://www.mdpi.com/2075-1680/10/2/79
Axioms, Vol. 10, Pages 78: A Fractional-in-Time Prey–Predator Model with Hunting Cooperation: Qualitative Analysis, Stability and Numerical Approximations https://www.mdpi.com/2075-1680/10/2/78 A prey–predator system with logistic growth of prey and hunting cooperation of predators is studied. The introduction of fractional time derivatives and the related persistent memory strongly characterize the model behavior, as many dynamical systems in the applied sciences are well described by such fractional-order models. Mathematical analysis and numerical simulations are performed to highlight the characteristics of the proposed model. The existence, uniqueness and boundedness of solutions is proved; the stability of the coexistence equilibrium and the occurrence of Hopf bifurcation is investigated. Some numerical approximations of the solution are finally considered; the obtained trajectories confirm the theoretical findings. It is observed that the fractional-order derivative has a stabilizing effect and can be useful to control the coexistence between species. Axioms, Vol. 10, Pages 78: A Fractional-in-Time Prey–Predator Model with Hunting Cooperation: Qualitative Analysis, Stability and Numerical Approximations

Authors: Maria Carfora Isabella Torcicollo

A prey–predator system with logistic growth of prey and hunting cooperation of predators is studied. The introduction of fractional time derivatives and the related persistent memory strongly characterize the model behavior, as many dynamical systems in the applied sciences are well described by such fractional-order models. Mathematical analysis and numerical simulations are performed to highlight the characteristics of the proposed model. The existence, uniqueness and boundedness of solutions is proved; the stability of the coexistence equilibrium and the occurrence of Hopf bifurcation is investigated. Some numerical approximations of the solution are finally considered; the obtained trajectories confirm the theoretical findings. It is observed that the fractional-order derivative has a stabilizing effect and can be useful to control the coexistence between species.

]]>
A Fractional-in-Time Prey–Predator Model with Hunting Cooperation: Qualitative Analysis, Stability and Numerical Approximations Maria Carfora Isabella Torcicollo doi: 10.3390/axioms10020078 Axioms 2021-04-30 Axioms 2021-04-30 10 2 Article 78 10.3390/axioms10020078 https://www.mdpi.com/2075-1680/10/2/78
Axioms, Vol. 10, Pages 77: Trees in Positive Entropy Subshifts https://www.mdpi.com/2075-1680/10/2/77 I give a simple proof for the fact that positive entropy subshifts contain infinite binary trees where branching happens synchronously in each branch, and that the branching times form a set with positive lower asymptotic density. Axioms, Vol. 10, Pages 77: Trees in Positive Entropy Subshifts

Authors: Ville Salo

I give a simple proof for the fact that positive entropy subshifts contain infinite binary trees where branching happens synchronously in each branch, and that the branching times form a set with positive lower asymptotic density.

]]>
Trees in Positive Entropy Subshifts Ville Salo doi: 10.3390/axioms10020077 Axioms 2021-04-29 Axioms 2021-04-29 10 2 Article 77 10.3390/axioms10020077 https://www.mdpi.com/2075-1680/10/2/77
Axioms, Vol. 10, Pages 76: Approximations of an Equilibrium Problem without Prior Knowledge of Lipschitz Constants in Hilbert Spaces with Applications https://www.mdpi.com/2075-1680/10/2/76 The objective of this paper is to introduce an iterative method with the addition of an inertial term to solve equilibrium problems in a real Hilbert space. The proposed iterative scheme is based on the Mann-type iterative scheme and the extragradient method. By imposing certain mild conditions on a bifunction, the corresponding theorem of strong convergence in real Hilbert space is well-established. The proposed method has the advantage of requiring no knowledge of Lipschitz-type constants. The applications of our results to solve particular classes of equilibrium problems is presented. Numerical results are established to validate the proposed method’s efficiency and to compare it to other methods in the literature. Axioms, Vol. 10, Pages 76: Approximations of an Equilibrium Problem without Prior Knowledge of Lipschitz Constants in Hilbert Spaces with Applications

Authors: Chainarong Khanpanuk Nuttapol Pakkaranang Nopparat Wairojjana Nattawut Pholasa

The objective of this paper is to introduce an iterative method with the addition of an inertial term to solve equilibrium problems in a real Hilbert space. The proposed iterative scheme is based on the Mann-type iterative scheme and the extragradient method. By imposing certain mild conditions on a bifunction, the corresponding theorem of strong convergence in real Hilbert space is well-established. The proposed method has the advantage of requiring no knowledge of Lipschitz-type constants. The applications of our results to solve particular classes of equilibrium problems is presented. Numerical results are established to validate the proposed method’s efficiency and to compare it to other methods in the literature.

]]>
Approximations of an Equilibrium Problem without Prior Knowledge of Lipschitz Constants in Hilbert Spaces with Applications Chainarong Khanpanuk Nuttapol Pakkaranang Nopparat Wairojjana Nattawut Pholasa doi: 10.3390/axioms10020076 Axioms 2021-04-27 Axioms 2021-04-27 10 2 Article 76 10.3390/axioms10020076 https://www.mdpi.com/2075-1680/10/2/76
Axioms, Vol. 10, Pages 75: A Comparison of a Priori Estimates of the Solutions of a Linear Fractional System with Distributed Delays and Application to the Stability Analysis https://www.mdpi.com/2075-1680/10/2/75 In this article, we consider a retarded linear fractional differential system with distributed delays and Caputo type derivatives of incommensurate orders. For this system, several a priori estimates for the solutions, applying the two traditional approaches—by the use of the Gronwall’s inequality and by the use of integral representations of the solutions are obtained. As application of the obtained estimates, different sufficient conditions which guaranty finite-time stability of the solutions are established. A comparison of the obtained different conditions in respect to the used estimates and norms is made. Axioms, Vol. 10, Pages 75: A Comparison of a Priori Estimates of the Solutions of a Linear Fractional System with Distributed Delays and Application to the Stability Analysis

Authors: Hristo Kiskinov Magdalena Veselinova Ekaterina Madamlieva Andrey Zahariev

In this article, we consider a retarded linear fractional differential system with distributed delays and Caputo type derivatives of incommensurate orders. For this system, several a priori estimates for the solutions, applying the two traditional approaches—by the use of the Gronwall’s inequality and by the use of integral representations of the solutions are obtained. As application of the obtained estimates, different sufficient conditions which guaranty finite-time stability of the solutions are established. A comparison of the obtained different conditions in respect to the used estimates and norms is made.

]]>
A Comparison of a Priori Estimates of the Solutions of a Linear Fractional System with Distributed Delays and Application to the Stability Analysis Hristo Kiskinov Magdalena Veselinova Ekaterina Madamlieva Andrey Zahariev doi: 10.3390/axioms10020075 Axioms 2021-04-27 Axioms 2021-04-27 10 2 Article 75 10.3390/axioms10020075 https://www.mdpi.com/2075-1680/10/2/75
Axioms, Vol. 10, Pages 74: Singular Integral Neumann Boundary Conditions for Semilinear Elliptic PDEs https://www.mdpi.com/2075-1680/10/2/74 In this article, we discuss semilinear elliptic partial differential equations with singular integral Neumann boundary conditions. Such boundary value problems occur in applications as mathematical models of nonlocal interaction between interior points and boundary points. Particularly, we are interested in the uniqueness of solutions to such problems. For the sublinear and subcritical case, we calculate, on the one hand, illustrative, rather explicit solutions in the one-dimensional case. On the other hand, we prove in the general case the existence and—via the strong solution of an integro-PDE with a kind of fractional divergence as a lower order term—uniqueness up to a constant. Axioms, Vol. 10, Pages 74: Singular Integral Neumann Boundary Conditions for Semilinear Elliptic PDEs

Authors: Praveen Agarwal Jochen Merker Gregor Schuldt

In this article, we discuss semilinear elliptic partial differential equations with singular integral Neumann boundary conditions. Such boundary value problems occur in applications as mathematical models of nonlocal interaction between interior points and boundary points. Particularly, we are interested in the uniqueness of solutions to such problems. For the sublinear and subcritical case, we calculate, on the one hand, illustrative, rather explicit solutions in the one-dimensional case. On the other hand, we prove in the general case the existence and—via the strong solution of an integro-PDE with a kind of fractional divergence as a lower order term—uniqueness up to a constant.

]]>
Singular Integral Neumann Boundary Conditions for Semilinear Elliptic PDEs Praveen Agarwal Jochen Merker Gregor Schuldt doi: 10.3390/axioms10020074 Axioms 2021-04-24 Axioms 2021-04-24 10 2 Article 74 10.3390/axioms10020074 https://www.mdpi.com/2075-1680/10/2/74
Axioms, Vol. 10, Pages 73: Coupled Optimal Results with an Application to Nonlinear Integral Equations https://www.mdpi.com/2075-1680/10/2/73 In the present work, we consider the best proximal problem related to a coupled mapping, which we define using control functions and weak inequalities. As a consequence, we obtain some results on coupled fixed points. Our results generalize some recent results in the literature. Also, as an application of the results obtained, we present the solution to a system of a coupled Fredholm nonlinear integral equation. Our work is supported by several illustrations. Axioms, Vol. 10, Pages 73: Coupled Optimal Results with an Application to Nonlinear Integral Equations

Authors: Pulak Konar Sumit Chandok Shrutinil Dutta Manuel De la Sen

In the present work, we consider the best proximal problem related to a coupled mapping, which we define using control functions and weak inequalities. As a consequence, we obtain some results on coupled fixed points. Our results generalize some recent results in the literature. Also, as an application of the results obtained, we present the solution to a system of a coupled Fredholm nonlinear integral equation. Our work is supported by several illustrations.

]]>
Coupled Optimal Results with an Application to Nonlinear Integral Equations Pulak Konar Sumit Chandok Shrutinil Dutta Manuel De la Sen doi: 10.3390/axioms10020073 Axioms 2021-04-22 Axioms 2021-04-22 10 2 Article 73 10.3390/axioms10020073 https://www.mdpi.com/2075-1680/10/2/73
Axioms, Vol. 10, Pages 72: Fixed Points Results for Various Types of Tricyclic Contractions https://www.mdpi.com/2075-1680/10/2/72 In this paper, we introduce four new types of contractions called in this order Kanan-S-type tricyclic contraction, Chattergea-S-type tricyclic contraction, Riech-S-type tricyclic contraction, Cirić-S-type tricyclic contraction, and we prove the existence and uniqueness for a fixed point for each situation. Axioms, Vol. 10, Pages 72: Fixed Points Results for Various Types of Tricyclic Contractions

Authors: Mustapha Sabiri Abdelhafid Bassou Jamal Mouline Taoufik Sabar

In this paper, we introduce four new types of contractions called in this order Kanan-S-type tricyclic contraction, Chattergea-S-type tricyclic contraction, Riech-S-type tricyclic contraction, Cirić-S-type tricyclic contraction, and we prove the existence and uniqueness for a fixed point for each situation.

]]>
Fixed Points Results for Various Types of Tricyclic Contractions Mustapha Sabiri Abdelhafid Bassou Jamal Mouline Taoufik Sabar doi: 10.3390/axioms10020072 Axioms 2021-04-20 Axioms 2021-04-20 10 2 Article 72 10.3390/axioms10020072 https://www.mdpi.com/2075-1680/10/2/72
Axioms, Vol. 10, Pages 71: On Convex F-Contraction in b-Metric Spaces https://www.mdpi.com/2075-1680/10/2/71 In this paper, we introduce a notion of convex F-contraction and establish some fixed point results for such contractions in b-metric spaces. Moreover, we give a supportive example to show that our convex F-contraction is quite different from the F-contraction used in the existing literature since our convex F-contraction does not necessarily contain the continuous mapping but the F-contraction contains such mapping. In addition, via some facts, we claim that our results indeed generalize and improve some previous results in the literature. Axioms, Vol. 10, Pages 71: On Convex F-Contraction in b-Metric Spaces

Authors: Huaping Huang Zoran D. Mitrović Kastriot Zoto Stojan Radenović

In this paper, we introduce a notion of convex F-contraction and establish some fixed point results for such contractions in b-metric spaces. Moreover, we give a supportive example to show that our convex F-contraction is quite different from the F-contraction used in the existing literature since our convex F-contraction does not necessarily contain the continuous mapping but the F-contraction contains such mapping. In addition, via some facts, we claim that our results indeed generalize and improve some previous results in the literature.

]]>
On Convex F-Contraction in b-Metric Spaces Huaping Huang Zoran D. Mitrović Kastriot Zoto Stojan Radenović doi: 10.3390/axioms10020071 Axioms 2021-04-19 Axioms 2021-04-19 10 2 Article 71 10.3390/axioms10020071 https://www.mdpi.com/2075-1680/10/2/71
Axioms, Vol. 10, Pages 70: q-Binomial Convolution and Transformations of q-Appell Polynomials https://www.mdpi.com/2075-1680/10/2/70 In this paper, binomial convolution in the frame of quantum calculus is studied for the set Aq of q-Appell sequences. It has been shown that the set Aq of q-Appell sequences forms an Abelian group under the operation of binomial convolution. Several properties for this Abelian group structure Aq have been studied. A new definition of the q-Appell polynomials associated with a random variable is proposed. Scale transformation as well as transformation based on expectation with respect to a random variable is used to present the determinantal form of q-Appell sequences. Axioms, Vol. 10, Pages 70: q-Binomial Convolution and Transformations of q-Appell Polynomials

Authors: Alaa Mohammed Obad Asif Khan Kottakkaran Sooppy Nisar Ahmed Morsy

In this paper, binomial convolution in the frame of quantum calculus is studied for the set Aq of q-Appell sequences. It has been shown that the set Aq of q-Appell sequences forms an Abelian group under the operation of binomial convolution. Several properties for this Abelian group structure Aq have been studied. A new definition of the q-Appell polynomials associated with a random variable is proposed. Scale transformation as well as transformation based on expectation with respect to a random variable is used to present the determinantal form of q-Appell sequences.

]]>
q-Binomial Convolution and Transformations of q-Appell Polynomials Alaa Mohammed Obad Asif Khan Kottakkaran Sooppy Nisar Ahmed Morsy doi: 10.3390/axioms10020070 Axioms 2021-04-19 Axioms 2021-04-19 10 2 Article 70 10.3390/axioms10020070 https://www.mdpi.com/2075-1680/10/2/70
Axioms, Vol. 10, Pages 69: The χ-Hessian Quotient for Riemannian Metrics https://www.mdpi.com/2075-1680/10/2/69 Pseudo-Riemannian geometry and Hilbert–Schmidt norms are two important fields of research in applied mathematics. One of the main goals of this paper will be to find a link between these two research fields. In this respect, in the present paper, we will introduce and analyze two important quantities in pseudo-Riemannian geometry, namely the H-distorsion and, respectively, the Hessian χ-quotient. This second quantity will be investigated using the Frobenius (Hilbert–Schmidt) norm. Some important examples will be also given, which will prove the validity of the developed theory along the paper. Axioms, Vol. 10, Pages 69: The χ-Hessian Quotient for Riemannian Metrics

Authors: Pişcoran Laurian-Ioan Akram Ali Barbu Cătălin Ali H. Alkhaldi

Pseudo-Riemannian geometry and Hilbert–Schmidt norms are two important fields of research in applied mathematics. One of the main goals of this paper will be to find a link between these two research fields. In this respect, in the present paper, we will introduce and analyze two important quantities in pseudo-Riemannian geometry, namely the H-distorsion and, respectively, the Hessian χ-quotient. This second quantity will be investigated using the Frobenius (Hilbert–Schmidt) norm. Some important examples will be also given, which will prove the validity of the developed theory along the paper.

]]>
The χ-Hessian Quotient for Riemannian Metrics Pişcoran Laurian-Ioan Akram Ali Barbu Cătălin Ali H. Alkhaldi doi: 10.3390/axioms10020069 Axioms 2021-04-19 Axioms 2021-04-19 10 2 Article 69 10.3390/axioms10020069 https://www.mdpi.com/2075-1680/10/2/69
Axioms, Vol. 10, Pages 68: A Quadratic Mean Field Games Model for the Langevin Equation https://www.mdpi.com/2075-1680/10/2/68 We consider a Mean Field Games model where the dynamics of the agents is given by a controlled Langevin equation and the cost is quadratic. An appropriate change of variables transforms the Mean Field Games system into a system of two coupled kinetic Fokker–Planck equations. We prove an existence result for the latter system, obtaining consequently existence of a solution for the Mean Field Games system. Axioms, Vol. 10, Pages 68: A Quadratic Mean Field Games Model for the Langevin Equation

Authors: Fabio Camilli

We consider a Mean Field Games model where the dynamics of the agents is given by a controlled Langevin equation and the cost is quadratic. An appropriate change of variables transforms the Mean Field Games system into a system of two coupled kinetic Fokker–Planck equations. We prove an existence result for the latter system, obtaining consequently existence of a solution for the Mean Field Games system.

]]>
A Quadratic Mean Field Games Model for the Langevin Equation Fabio Camilli doi: 10.3390/axioms10020068 Axioms 2021-04-19 Axioms 2021-04-19 10 2 Article 68 10.3390/axioms10020068 https://www.mdpi.com/2075-1680/10/2/68
Axioms, Vol. 10, Pages 67: Mann-Type Inertial Subgradient Extragradient Rules for Variational Inequalities and Common Fixed Points of Nonexpansive and Quasi-Nonexpansive Mappings https://www.mdpi.com/2075-1680/10/2/67 Suppose that in a real Hilbert space H, the variational inequality problem with Lipschitzian and pseudomonotone mapping A and the common fixed-point problem of a finite family of nonexpansive mappings and a quasi-nonexpansive mapping with a demiclosedness property are represented by the notations VIP and CFPP, respectively. In this article, we suggest two Mann-type inertial subgradient extragradient iterations for finding a common solution of the VIP and CFPP. Our iterative schemes require only calculating one projection onto the feasible set for every iteration, and the strong convergence theorems are established without the assumption of sequentially weak continuity for A. Finally, in order to support the applicability and implementability of our algorithms, we make use of our main results to solve the VIP and CFPP in two illustrating examples. Axioms, Vol. 10, Pages 67: Mann-Type Inertial Subgradient Extragradient Rules for Variational Inequalities and Common Fixed Points of Nonexpansive and Quasi-Nonexpansive Mappings

Authors: Lu-Chuan Ceng Jen-Chih Yao

Suppose that in a real Hilbert space H, the variational inequality problem with Lipschitzian and pseudomonotone mapping A and the common fixed-point problem of a finite family of nonexpansive mappings and a quasi-nonexpansive mapping with a demiclosedness property are represented by the notations VIP and CFPP, respectively. In this article, we suggest two Mann-type inertial subgradient extragradient iterations for finding a common solution of the VIP and CFPP. Our iterative schemes require only calculating one projection onto the feasible set for every iteration, and the strong convergence theorems are established without the assumption of sequentially weak continuity for A. Finally, in order to support the applicability and implementability of our algorithms, we make use of our main results to solve the VIP and CFPP in two illustrating examples.

]]>
Mann-Type Inertial Subgradient Extragradient Rules for Variational Inequalities and Common Fixed Points of Nonexpansive and Quasi-Nonexpansive Mappings Lu-Chuan Ceng Jen-Chih Yao doi: 10.3390/axioms10020067 Axioms 2021-04-19 Axioms 2021-04-19 10 2 Article 67 10.3390/axioms10020067 https://www.mdpi.com/2075-1680/10/2/67
Axioms, Vol. 10, Pages 66: An Analytic and Numerical Investigation of a Differential Game https://www.mdpi.com/2075-1680/10/2/66 In this paper we present an appropriate singular, zero-sum, linear-quadratic differential game. One of the main features of this game is that the weight matrix of the minimizer’s control cost in the cost functional is singular. Due to this singularity, the game cannot be solved either by applying the Isaacs MinMax principle, or the Bellman–Isaacs equation approach. As an application, we introduced an interception differential game with an appropriate regularized cost functional and developed an appropriate dual representation. By developing the variational derivatives of this regularized cost functional, we apply Popov’s approximation method and show how the numerical results coincide with the dual representation. Axioms, Vol. 10, Pages 66: An Analytic and Numerical Investigation of a Differential Game

Authors: Aviv Gibali Oleg Kelis

In this paper we present an appropriate singular, zero-sum, linear-quadratic differential game. One of the main features of this game is that the weight matrix of the minimizer’s control cost in the cost functional is singular. Due to this singularity, the game cannot be solved either by applying the Isaacs MinMax principle, or the Bellman–Isaacs equation approach. As an application, we introduced an interception differential game with an appropriate regularized cost functional and developed an appropriate dual representation. By developing the variational derivatives of this regularized cost functional, we apply Popov’s approximation method and show how the numerical results coincide with the dual representation.

]]>
An Analytic and Numerical Investigation of a Differential Game Aviv Gibali Oleg Kelis doi: 10.3390/axioms10020066 Axioms 2021-04-17 Axioms 2021-04-17 10 2 Article 66 10.3390/axioms10020066 https://www.mdpi.com/2075-1680/10/2/66
Axioms, Vol. 10, Pages 65: Fractional Coupled Hybrid Sturm–Liouville Differential Equation with Multi-Point Boundary Coupled Hybrid Condition https://www.mdpi.com/2075-1680/10/2/65 The Sturm–Liouville differential equation is an important tool for physics, applied mathematics, and other fields of engineering and science and has wide applications in quantum mechanics, classical mechanics, and wave phenomena. In this paper, we investigate the coupled hybrid version of the Sturm–Liouville differential equation. Indeed, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with multi-point boundary coupled hybrid condition. Furthermore, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with an integral boundary coupled hybrid condition. We give an application and some examples to illustrate our results. Axioms, Vol. 10, Pages 65: Fractional Coupled Hybrid Sturm–Liouville Differential Equation with Multi-Point Boundary Coupled Hybrid Condition

Authors: Mohadeseh Paknazar Manuel De La Sen

The Sturm–Liouville differential equation is an important tool for physics, applied mathematics, and other fields of engineering and science and has wide applications in quantum mechanics, classical mechanics, and wave phenomena. In this paper, we investigate the coupled hybrid version of the Sturm–Liouville differential equation. Indeed, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with multi-point boundary coupled hybrid condition. Furthermore, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with an integral boundary coupled hybrid condition. We give an application and some examples to illustrate our results.

]]>
Fractional Coupled Hybrid Sturm–Liouville Differential Equation with Multi-Point Boundary Coupled Hybrid Condition Mohadeseh Paknazar Manuel De La Sen doi: 10.3390/axioms10020065 Axioms 2021-04-16 Axioms 2021-04-16 10 2 Article 65 10.3390/axioms10020065 https://www.mdpi.com/2075-1680/10/2/65