Axioms

12 pages, 308 KiB

Open AccessArticle

Some New Observations for F-Contractions in Vector-Valued Metric Spaces of Perov’s Type

by Nikola Mirkov, Stojan Radenović and Slobodan Radojević

Axioms 2021, 10(2), 127; https://doi.org/10.3390/axioms10020127 - 21 Jun 2021

Cited by 7 | Viewed by 1908

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 [...] Read more.

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, + \infty)}^{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. Full article

19 pages, 339 KiB

Open AccessArticle

Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in

C^{2}

by Hong Li and Hongyan Xu

Axioms 2021, 10(2), 126; https://doi.org/10.3390/axioms10020126 - 21 Jun 2021

Cited by 3 | Viewed by 2049

Abstract

This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations [...] Read more.

This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations

f {(z + c)}^{2} + 2 α f (z) f (z + c) + f {(z)}^{2} = e^{g (z)},

and the partial differential difference equations

f {(z + c)}^{2} + 2 α f (z + c) \frac{\partial f (z)}{\partial z_{1}} + {(\frac{\partial f (z)}{\partial z_{1}})}^{2} = e^{g (z)},

f {(z + c)}^{2} + 2 α f (z + c) (\frac{\partial f (z)}{\partial z_{1}} + \frac{\partial f (z)}{\partial z_{2}}) + {(\frac{\partial f (z)}{\partial z_{1}} + \frac{\partial f (z)}{\partial z_{2}})}^{2} = e^{g (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. Full article

(This article belongs to the Special Issue Complex Analysis)

11 pages, 286 KiB

Open AccessArticle

Explicit Formulas for Some Infinite ₃F₂(1)-Series

by Kwang-Wu Chen

Axioms 2021, 10(2), 125; https://doi.org/10.3390/axioms10020125 - 19 Jun 2021

Cited by 7 | Viewed by 2285

Abstract

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. Full article

(This article belongs to the Collection Mathematical Analysis and Applications)

12 pages, 700 KiB

Open AccessArticle

An Innovative Grey Approach for Group Multi-Criteria Decision Analysis Based on the Median of Ratings by Using Python

by Dragiša Stanujkić, Darjan Karabašević, Gabrijela Popović, Predrag S. Stanimirović, Florentin Smarandache, Muzafer Saračević, Alptekin Ulutaş and Vasilios N. Katsikis

Axioms 2021, 10(2), 124; https://doi.org/10.3390/axioms10020124 - 19 Jun 2021

Cited by 3 | Viewed by 2936

Abstract

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 [...] Read more.

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. Full article

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

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16 pages, 4368 KiB

Open AccessArticle

Strong Interacting Internal Waves in Rotating Ocean: Novel Fractional Approach

by Pundikala Veeresha, Haci Mehmet Baskonus and Wei Gao

Axioms 2021, 10(2), 123; https://doi.org/10.3390/axioms10020123 - 16 Jun 2021

Cited by 54 | Viewed by 2657

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)

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23 pages, 2988 KiB

Open AccessArticle

New Procedures of a Fractional Order Model of Novel Coronavirus (COVID-19) Outbreak via Wavelets Method

by Maryamsadat Hedayati, Reza Ezzati and Samad Noeiaghdam

Axioms 2021, 10(2), 122; https://doi.org/10.3390/axioms10020122 - 16 Jun 2021

Cited by 24 | Viewed by 2792

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Mathematics of the COVID-19)

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10 pages, 455 KiB

Open AccessArticle

Classical Partition Function for Non-Relativistic Gravity

by Mir Hameeda, Angelo Plastino, Mario Carlos Rocca and Javier Zamora

Axioms 2021, 10(2), 121; https://doi.org/10.3390/axioms10020121 - 16 Jun 2021

Cited by 4 | Viewed by 2429

Abstract

We considered the canonical gravitational partition function Z associated to the classical Boltzmann–Gibbs (BG) distribution

\frac{e^{- β H}}{Z}

. It is popularly thought that it cannot be built up because the integral involved in constructing Z diverges at the origin. Contrariwise, [...] Read more.

We considered the canonical gravitational partition function Z associated to the classical Boltzmann–Gibbs (BG) distribution

\frac{e^{- β H}}{Z}

. 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. Full article

(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)

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23 pages, 2473 KiB

Open AccessArticle

A Two-Stage Closed-Loop Supply Chain Pricing Decision: Cross-Channel Recycling and Channel Preference

by Wenjun Pan and Miao Lin

Axioms 2021, 10(2), 120; https://doi.org/10.3390/axioms10020120 - 15 Jun 2021

Cited by 6 | Viewed by 2516

Abstract

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 [...] Read more.

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. Full article

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29 pages, 1335 KiB

Open AccessArticle

A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

by Marcoen J. T. F. Cabbolet

Axioms 2021, 10(2), 119; https://doi.org/10.3390/axioms10020119 - 14 Jun 2021

Cited by 1 | Viewed by 2576

Abstract

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 [...] Read more.

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. Full article

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27 pages, 396 KiB

Open AccessFeature PaperArticle

Vector Fields and Differential Forms on the Orbit Space of a Proper Action

by Larry Bates, Richard Cushman and Jędrzej Śniatycki

Axioms 2021, 10(2), 118; https://doi.org/10.3390/axioms10020118 - 10 Jun 2021

Cited by 1 | Viewed by 2386 | Correction

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Applications of Differential Geometry II)

8 pages, 270 KiB

Open AccessArticle

About Cogredient and Contragredient Linear Differential Equations

by Vasily Gorelov

Axioms 2021, 10(2), 117; https://doi.org/10.3390/axioms10020117 - 10 Jun 2021

Cited by 1 | Viewed by 2071

Abstract

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. Full article

(This article belongs to the Special Issue Recent Developments in Methods, Techniques, and Approaches to Study the Qualitative Properties of Differential Equations)

10 pages, 272 KiB

Open AccessArticle

A Note on the Computation of the Modular Inverse for Cryptography

by Michele Bufalo, Daniele Bufalo and Giuseppe Orlando

Axioms 2021, 10(2), 116; https://doi.org/10.3390/axioms10020116 - 9 Jun 2021

Cited by 7 | Viewed by 4134

Abstract

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 [...] Read more.

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

{(z_{j})}_{j \geq 0}

, where

z_{j} = z_{j - 1} + a β_{j} - n

,

a < 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} = ⌊ i m ⌋ + 1

with

m = n / a

. The computational cost of such an index i is

O (a)

, which is less than

O (n ln n)

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 (ln n)

(extended Euclidean algorithm). Therefore, the above procedure is more convenient when

a < < n

(e.g.,

a < ln n

). Full article

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12 pages, 775 KiB

Open AccessFeature PaperArticle

Analytic Representation of Maxwell—Boltzmann and Tsallis Thermonuclear Functions with Depleted Tail

by Dilip Kumar and Hans J. Haubold

Axioms 2021, 10(2), 115; https://doi.org/10.3390/axioms10020115 - 7 Jun 2021

Cited by 2 | Viewed by 2413

Abstract

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 [...] Read more.

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. Full article

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19 pages, 2021 KiB

Open AccessArticle

Numerical Solution of an Interval-Based Uncertain SIR (Susceptible–Infected–Recovered) Epidemic Model by Homotopy Analysis Method

by Emmanuel A. Bakare, Snehashish Chakraverty and Radovan Potucek

Axioms 2021, 10(2), 114; https://doi.org/10.3390/axioms10020114 - 6 Jun 2021

Viewed by 2204

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Numerical Analysis and Computational Mathematics)

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37 pages, 2921 KiB

Open AccessArticle

A Decision-Making Approach Based on New Aggregation Operators under Fermatean Fuzzy Linguistic Information Environment

by Rajkumar Verma

Axioms 2021, 10(2), 113; https://doi.org/10.3390/axioms10020113 - 4 Jun 2021

Cited by 14 | Viewed by 2543

Abstract

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 [...] Read more.

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. Full article

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

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9 pages, 308 KiB

Open AccessArticle

On a Class of Isoperimetric Constrained Controlled Optimization Problems

by Savin Treanţă

Axioms 2021, 10(2), 112; https://doi.org/10.3390/axioms10020112 - 3 Jun 2021

Cited by 10 | Viewed by 2081

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)

10 pages, 273 KiB

Open AccessArticle

B-Fredholm Spectra of Drazin Invertible Operators and Applications

by Elvis Aponte, Jhixon Macías, José Sanabria and José Soto

Axioms 2021, 10(2), 111; https://doi.org/10.3390/axioms10020111 - 2 Jun 2021

Cited by 8 | Viewed by 3059

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Collection Mathematical Analysis and Applications)

7 pages, 235 KiB

Open AccessArticle

Nonlocal Problem for a Third-Order Equation with Multiple Characteristics with General Boundary Conditions

by Abdukomil Risbekovich Khashimov and Dana Smetanová

Axioms 2021, 10(2), 110; https://doi.org/10.3390/axioms10020110 - 2 Jun 2021

Cited by 1 | Viewed by 2093

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Boundary-Value and Spectral Problems)

18 pages, 368 KiB

Open AccessArticle

A Self-Adaptive Algorithm for the Common Solution of the Split Minimization Problem and the Fixed Point Problem

by Nattakarn Kaewyong and Kanokwan Sitthithakerngkiet

Axioms 2021, 10(2), 109; https://doi.org/10.3390/axioms10020109 - 30 May 2021

Cited by 2 | Viewed by 2338

Abstract

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. [...] Read more.

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. Full article

(This article belongs to the Collection Mathematical Analysis and Applications)

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21 pages, 945 KiB

Open AccessFeature PaperArticle

Relaxation Limit of the Aggregation Equation with Pointy Potential

by Benoît Fabrèges, Frédéric Lagoutière, Sébastien Tran Tien and Nicolas Vauchelet

Axioms 2021, 10(2), 108; https://doi.org/10.3390/axioms10020108 - 28 May 2021

Cited by 2 | Viewed by 2633

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Differential Models, Numerical Simulations and Applications)

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23 pages, 345 KiB

Open AccessArticle

Spectral Transformations and Associated Linear Functionals of the First Kind

by Juan Carlos García-Ardila and Francisco Marcellán

Axioms 2021, 10(2), 107; https://doi.org/10.3390/axioms10020107 - 28 May 2021

Cited by 1 | Viewed by 2509

Abstract

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)

{(P_{n})}_{n \geq 0}

. For a canonical Christoffel transformation [...] Read more.

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)

{(P_{n})}_{n \geq 0}

. For a canonical Christoffel transformation

\tilde{u} = (x - c) u

with SMOP

{({\tilde{P}}_{n})}_{n \geq 0}

, we are interested to study the relation between

\tilde{u}

and

\tilde{u^{(1)}},

where

u^{(1)}

is the linear functional for the associated orthogonal polynomials of the first kind

{(P_{n}^{(1)})}_{n \geq 0}

, and

\tilde{u^{(1)}} = (x - c) u^{(1)}

is its Christoffel transformation. This problem is also studied for canonical Geronimus transformations. Full article

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications)

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16 pages, 2772 KiB

Open AccessArticle

Monitoring and Recognizing Enterprise Public Opinion from High-Risk Users Based on User Portrait and Random Forest Algorithm

by Tinggui Chen, Xiaohua Yin, Lijuan Peng, Jingtao Rong, Jianjun Yang and Guodong Cong

Axioms 2021, 10(2), 106; https://doi.org/10.3390/axioms10020106 - 27 May 2021

Cited by 46 | Viewed by 5784

Abstract

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, [...] Read more.

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. Full article

(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)

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14 pages, 257 KiB

Open AccessFeature PaperArticle

Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

by Lokesh Singh and Dhirendra Bahuguna

Axioms 2021, 10(2), 105; https://doi.org/10.3390/axioms10020105 - 25 May 2021

Cited by 1 | Viewed by 2185

Abstract

In this article, we construct a

C^{1}

stable invariant manifold for the delay differential equation

x^{'} = A x (t) + L x_{t} + f (t, x_{t})

assuming the

ρ

-nonuniform exponential dichotomy for [...] Read more.

In this article, we construct a

C^{1}

stable invariant manifold for the delay differential equation

x^{'} = A x (t) + L x_{t} + f (t, x_{t})

assuming the

ρ

-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the

C^{1}

perturbation,

f (t, x_{t})

, 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, x_{t})

. Full article

(This article belongs to the Special Issue Nonautonomous and Random Dynamical Systems)

17 pages, 340 KiB

Open AccessArticle

An Intelligent System for Allocating Times to the Main Activities of Managers

by Efrain Solares, Liliana Guerrero, Alberto Aguilera, Juana María Hernández, Sandra Rodríguez and Víctor De-León-Gómez

Axioms 2021, 10(2), 104; https://doi.org/10.3390/axioms10020104 - 25 May 2021

Viewed by 2163

Abstract

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 [...] Read more.

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. Full article

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

13 pages, 254 KiB

Open AccessArticle

On the Existence of Coupled Fractional Jerk Equations with Multi-Point Boundary Conditions

by Lei Hu, Yaozhen Han and Shuqin Zhang

Axioms 2021, 10(2), 103; https://doi.org/10.3390/axioms10020103 - 24 May 2021

Cited by 2 | Viewed by 1937

Abstract

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 [...] Read more.

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. Full article

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

21 pages, 3223 KiB

Open AccessArticle

Macroscopic and Multi-Scale Models for Multi-Class Vehicular Dynamics with Uneven Space Occupancy: A Case Study

by Maya Briani, Emiliano Cristiani and Paolo Ranut

Axioms 2021, 10(2), 102; https://doi.org/10.3390/axioms10020102 - 24 May 2021

Cited by 2 | Viewed by 2857

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Differential Models, Numerical Simulations and Applications)

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19 pages, 307 KiB

Open AccessArticle

Common Fixed Point Results for Almost ℛ_g-Geraghty Type Contraction Mappings in b₂-Metric Spaces with an Application to Integral Equations

by Samera M. Saleh, Salvatore Sessa, Waleed M. Alfaqih and Fawzia Shaddad

Axioms 2021, 10(2), 101; https://doi.org/10.3390/axioms10020101 - 24 May 2021

Cited by 4 | Viewed by 2153

Abstract

In this paper, we define almost

R_{g}

-Geraghty type contractions and utilize the same to establish some coincidence and common fixed point results in the setting of

b_{2}

-metric spaces endowed with binary relations. As consequences of our newly proved results, [...] Read more.

In this paper, we define almost

R_{g}

-Geraghty type contractions and utilize the same to establish some coincidence and common fixed point results in the setting of

b_{2}

-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

b_{2}

-metric spaces. In addition, we derive some coincidence and common fixed point results in partially ordered

b_{2}

-metric spaces. Moreover, to show the utility of our main results, we provide an example and an application to non-linear integral equations. Full article

(This article belongs to the Special Issue Theory and Application of Fixed Point)

35 pages, 357 KiB

Open AccessFeature PaperReview

Perturbation Bounds for Eigenvalues and Determinants of Matrices. A Survey

by Michael Gil’

Axioms 2021, 10(2), 99; https://doi.org/10.3390/axioms10020099 - 21 May 2021

Viewed by 2441

Abstract

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 [...] Read more.

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. Full article

8 pages, 275 KiB

Open AccessArticle

Optimal Segmentation over a Generalized Customer Distribution

by Tsung-Yin Ou and Yenming J. Chen

Axioms 2021, 10(2), 98; https://doi.org/10.3390/axioms10020098 - 21 May 2021

Viewed by 1953

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Decision Analysis with Optimization Technique)

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8 pages, 252 KiB

Open AccessArticle

Conditional Intuitionistic Fuzzy Mean Value

by Katarína Čunderlíková

Axioms 2021, 10(2), 97; https://doi.org/10.3390/axioms10020097 - 21 May 2021

Cited by 1 | Viewed by 1804

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Applications)

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