Axioms

9 pages, 247 KiB

Open AccessArticle

Mellin Transform of Logarithm and Quotient Function with Reducible Quartic Polynomial in Terms of the Lerch Function

by Robert Reynolds and Allan Stauffer

Axioms 2021, 10(3), 236; https://doi.org/10.3390/axioms10030236 - 21 Sep 2021

Viewed by 1624

A class of definite integrals involving a quotient function with a reducible polynomial, logarithm and nested logarithm functions are derived with a possible connection to contact problems for a wedge. The derivations are expressed in terms of the Lerch function. Special cases are [...] Read more.

A class of definite integrals involving a quotient function with a reducible polynomial, logarithm and nested logarithm functions are derived with a possible connection to contact problems for a wedge. The derivations are expressed in terms of the Lerch function. Special cases are also derived in terms fundamental constants. The majority of the results in this work are new. Full article

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

12 pages, 266 KiB

Open AccessArticle

Integral Characterizations for Uniform Stability with Growth Rates in Banach Spaces

by Rovana Boruga (Toma), Mihail Megan and Daniela Maria-Magdalena Toth

Axioms 2021, 10(3), 235; https://doi.org/10.3390/axioms10030235 - 21 Sep 2021

Cited by 7 | Viewed by 1857

Abstract

The aim of this paper is to present some integral characterizations for the concept of uniform stability with growth rates in Banach spaces. In this sense, we prove necessary and sufficient conditions (of Barbashin and Datko type) for an evolution operator to be [...] Read more.

The aim of this paper is to present some integral characterizations for the concept of uniform stability with growth rates in Banach spaces. In this sense, we prove necessary and sufficient conditions (of Barbashin and Datko type) for an evolution operator to be uniform h- stable. As particular cases of this notion, we obtain four characterizations for uniform exponential stability and two characterizations for uniform polynomial stability. Full article

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

11 pages, 257 KiB

Open AccessArticle

On Solvability Conditions for a Certain Conjugation Problem

by Vladimir Vasilyev and Nikolai Eberlein

Axioms 2021, 10(3), 234; https://doi.org/10.3390/axioms10030234 - 20 Sep 2021

Cited by 2 | Viewed by 1602

Abstract

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general [...] Read more.

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions. Full article

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

14 pages, 298 KiB

Open AccessArticle

On r-Noncommuting Graph of Finite Rings

by Rajat Kanti Nath, Monalisha Sharma, Parama Dutta and Yilun Shang

Axioms 2021, 10(3), 233; https://doi.org/10.3390/axioms10030233 - 19 Sep 2021

Cited by 4 | Viewed by 2066

Abstract

Let R be a finite ring and

r \in R

. The r-noncommuting graph of R, denoted by

Γ_{R}^{r}

, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if [...] Read more.

Let R be a finite ring and

r \in R

. The r-noncommuting graph of R, denoted by

Γ_{R}^{r}

, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if

[x, y] \neq r

and

[x, y] \neq - r

. In this paper, we obtain expressions for vertex degrees and show that

Γ_{R}^{r}

is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that

Γ_{R}^{r}

is a tree, in particular a star graph. It is also shown that

Γ_{R_{1}}^{r}

and

Γ_{R_{2}}^{ψ (r)}

are isomorphic if

R_{1}

and

R_{2}

are two isoclinic rings with isoclinism

(ϕ, ψ)

. Further, we consider the induced subgraph

Δ_{R}^{r}

of

Γ_{R}^{r}

(induced by the non-central elements of R) and obtain results on clique number and diameter of

Δ_{R}^{r}

along with certain characterizations of finite noncommutative rings such that

Δ_{R}^{r}

is n-regular for some positive integer n. As applications of our results, we characterize certain finite noncommutative rings such that their noncommuting graphs are n-regular for

n \leq 6

. Full article

(This article belongs to the Special Issue Latest Trends in Noncommutative Algebra)

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20 pages, 335 KiB

Open AccessArticle

Two Forms of An Inverse Operator to the Generalized Bessel Potential

by Akhmed Dzhabrailov, Yuri Luchko and Elina Shishkina

Axioms 2021, 10(3), 232; https://doi.org/10.3390/axioms10030232 - 18 Sep 2021

Cited by 6 | Viewed by 1778

Abstract

In this paper, we treat a convolution-type operator called the generalized Bessel potential. Our main result is the derivation of two different forms of its inversion. The first inversion is provided in terms of an approximative inverse operator using the method of an [...] Read more.

In this paper, we treat a convolution-type operator called the generalized Bessel potential. Our main result is the derivation of two different forms of its inversion. The first inversion is provided in terms of an approximative inverse operator using the method of an improving multiplier. The second one employs the regularization technique for the divergent integrals in the form of the appropriate segments of the Taylor–Delsarte series. Full article

5 pages, 229 KiB

Open AccessArticle

Fixed Point Results for Frum-Ketkov Type Contractions in b-Metric Spaces

by Cristian Chifu, Erdal Karapınar and Gabriela Petrusel

Axioms 2021, 10(3), 231; https://doi.org/10.3390/axioms10030231 - 18 Sep 2021

Cited by 2 | Viewed by 1388

Abstract

The purpose of this paper is to present some fixed point results for Frum-Ketkov type operators in complete b-metric spaces. Full article

(This article belongs to the Special Issue Advances in Nonlinear Analysis and Related Fixed Point Problems)

21 pages, 1181 KiB

Open AccessArticle

The Approximate and Analytic Solutions of the Time-Fractional Intermediate Diffusion Wave Equation Associated with the Fokker–Planck Operator and Applications

by Entsar A. Abdel-Rehim

Axioms 2021, 10(3), 230; https://doi.org/10.3390/axioms10030230 - 17 Sep 2021

Cited by 4 | Viewed by 1916

Abstract

In this paper, the time-fractional wave equation associated with the space-fractional Fokker–Planck operator and with the time-fractional-damped term is studied. The concept of the Green function is implemented to drive the analytic solution of the three-term time-fractional equation. The explicit expressions for the [...] Read more.

In this paper, the time-fractional wave equation associated with the space-fractional Fokker–Planck operator and with the time-fractional-damped term is studied. The concept of the Green function is implemented to drive the analytic solution of the three-term time-fractional equation. The explicit expressions for the Green function

G_{3} (t)

of the three-term time-fractional wave equation with constant coefficients is also studied for two physical and biological models. The explicit analytic solutions, for the two studied models, are expressed in terms of the Weber, hypergeometric, exponential, and Mittag–Leffler functions. The relation to the diffusion equation is given. The asymptotic behaviors of the Mittag–Leffler function, the hypergeometric function

1 F 1

, and the exponential functions are compared numerically. The Grünwald–Letnikov scheme is used to derive the approximate difference schemes of the Caputo time-fractional operator and the Feller–Riesz space-fractional operator. The explicit difference scheme is numerically studied, and the simulations of the approximate solutions are plotted for different values of the fractional orders. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

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16 pages, 314 KiB

Open AccessArticle

Statistical Riemann and Lebesgue Integrable Sequence of Functions with Korovkin-Type Approximation Theorems

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

Axioms 2021, 10(3), 229; https://doi.org/10.3390/axioms10030229 - 16 Sep 2021

Cited by 9 | Viewed by 1923

Abstract

In this work we introduce and investigate the ideas of statistical Riemann integrability, statistical Riemann summability, statistical Lebesgue integrability and statistical Lebesgue summability via deferred weighted mean. We first establish some fundamental limit theorems connecting these beautiful and potentially useful notions. Furthermore, based [...] Read more.

In this work we introduce and investigate the ideas of statistical Riemann integrability, statistical Riemann summability, statistical Lebesgue integrability and statistical Lebesgue summability via deferred weighted mean. We first establish some fundamental limit theorems connecting these beautiful and potentially useful notions. Furthermore, based upon our proposed techniques, we establish the Korovkin-type approximation theorems with algebraic test functions. Finally, we present two illustrative examples under the consideration of positive linear operators in association with the Bernstein polynomials to exhibit the effectiveness of our findings. Full article

(This article belongs to the Special Issue Mathematical Tools and Techniques Applicable to Probability Theory and Statistics)

24 pages, 906 KiB

Open AccessArticle

Mathematical Modeling and Forecasting of COVID-19 in Saudi Arabia under Fractal-Fractional Derivative in Caputo Sense with Power-Law

by Mdi Begum Jeelani, Abeer S. Alnahdi, Mohammed S. Abdo, Mansour A. Abdulwasaa, Kamal Shah and Hanan A. Wahash

Axioms 2021, 10(3), 228; https://doi.org/10.3390/axioms10030228 - 15 Sep 2021

Cited by 19 | Viewed by 2725

Abstract

This manuscript is devoted to investigating a fractional-order mathematical model of COVID-19. The corresponding derivative is taken in Caputo sense with power-law of fractional order

μ

and fractal dimension

χ

. We give some detailed analysis on the existence and uniqueness of the [...] Read more.

This manuscript is devoted to investigating a fractional-order mathematical model of COVID-19. The corresponding derivative is taken in Caputo sense with power-law of fractional order

μ

and fractal dimension

χ

. We give some detailed analysis on the existence and uniqueness of the solution to the proposed problem. Furthermore, some results regarding basic reproduction number and stability are given. For the proposed theoretical analysis, we use fixed point theory while for numerical analysis fractional Adams–Bashforth iterative techniques are utilized. Using our numerical scheme is verified by using some real values of the parameters to plot the approximate solution to the considered model. Graphical presentations corresponding to different values of fractional order and fractal dimensions are given. Moreover, we provide some information regarding the real data of Saudi Arabia from 1 March 2020 till 22 April 2021, then calculated the fatality rates by utilizing the SPSS, Eviews and Expert Modeler procedure. We also built forecasts of infection for the period 23 April 2021 to 30 May 2021, with 95% confidence. Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

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18 pages, 2540 KiB

Open AccessArticle

Chaotic Dynamics by Some Quadratic Jerk Systems

by Mei Liu, Bo Sang, Ning Wang and Irfan Ahmad

Axioms 2021, 10(3), 227; https://doi.org/10.3390/axioms10030227 - 14 Sep 2021

Cited by 15 | Viewed by 2640

Abstract

This paper is about the dynamical evolution of a family of chaotic jerk systems, which have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden attractors are [...] Read more.

This paper is about the dynamical evolution of a family of chaotic jerk systems, which have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden attractors are explored. The self-exited chaotic attractors are found via a supercritical Hopf bifurcation and period-doubling cascades to chaos. The hidden chaotic attractors (related to a subcritical Hopf bifurcation, and with a unique stable equilibrium) are also found via period-doubling cascades to chaos. A circuit implementation is presented for the hidden chaotic attractor. The methods used in this paper will help understand and predict the chaotic dynamics of quadratic jerk systems. Full article

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

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13 pages, 287 KiB

Open AccessArticle

Closed-Form Solutions of Linear Ordinary Differential Equations with General Boundary Conditions

by Efthimios Providas, Stefanos Zaoutsos and Ioannis Faraslis

Axioms 2021, 10(3), 226; https://doi.org/10.3390/axioms10030226 - 14 Sep 2021

Cited by 1 | Viewed by 1964

Abstract

This paper deals with the solution of boundary value problems for ordinary differential equations with general boundary conditions. We obtain closed-form solutions in a symbolic form of problems with the general n-th order differential operator, as well as the composition of linear [...] Read more.

This paper deals with the solution of boundary value problems for ordinary differential equations with general boundary conditions. We obtain closed-form solutions in a symbolic form of problems with the general n-th order differential operator, as well as the composition of linear operators. The method is based on the theory of the extensions of linear operators in Banach spaces. Full article

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

11 pages, 253 KiB

Open AccessArticle

An Expository Lecture of María Jesús Chasco on Some Applications of Fubini’s Theorem

by Alberto Castejón, María Jesús Chasco, Eusebio Corbacho and Virgilio Rodríguez de Miguel

Axioms 2021, 10(3), 225; https://doi.org/10.3390/axioms10030225 - 14 Sep 2021

Viewed by 1671

Abstract

The usefulness of Fubini’s theorem as a measurement instrument is clearly understood from its multiple applications in Analysis, Convex Geometry, Statistics or Number Theory. This article is an expository paper based on a master class given by the second author at the University [...] Read more.

The usefulness of Fubini’s theorem as a measurement instrument is clearly understood from its multiple applications in Analysis, Convex Geometry, Statistics or Number Theory. This article is an expository paper based on a master class given by the second author at the University of Vigo and is devoted to presenting some Applications of Fubini’s theorem. In the first part, we present Brunn–Minkowski’s and Isoperimetric inequalities. The second part is devoted to the estimations of volumes of sections of balls in

R^{n}

. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

23 pages, 5686 KiB

Open AccessArticle

Error Compensation of Strapdown Inertial Navigation System for the Boom-Type Roadheader under Complex Vibration

by Yang Shen, Pengjiang Wang, Weixiong Zheng, Xiaodong Ji, Hai Jiang and Miao Wu

Axioms 2021, 10(3), 224; https://doi.org/10.3390/axioms10030224 - 14 Sep 2021

Cited by 9 | Viewed by 2765

Abstract

The strapdown inertial navigation system can provide the navigation information for the boom-type roadheader in the unmanned roadway tunneling working face of the coal mine. However, the complex vibration caused by the cutting process of the boom-type roadheader may result in significant errors [...] Read more.

The strapdown inertial navigation system can provide the navigation information for the boom-type roadheader in the unmanned roadway tunneling working face of the coal mine. However, the complex vibration caused by the cutting process of the boom-type roadheader may result in significant errors of its attitude and position measured by the strapdown inertial navigation system. Thus, an error compensation method based on the vibration characteristics of the roadheader is proposed in this paper. In order to further analyze the angular and linear vibration of the fuselage, as the main vibration sources of the roadheader, the dynamic model of the roadheader is formulated based on the cutting load. Following that, multiple sub-samples compensation algorithms for the coning and sculling errors are constructed. Simulation experiments were carried out under different subsample compensation algorithms, different coal and rock characteristics, and different types of roadheader. The experimental results show that the proposed error compensation algorithm can eliminate the effect of the angular and linear vibration on the measurement accuracy. The coning and sculling error of the strapdown inertial navigation system can reduce at least 52.21% and 42.89%, respectively. Finally, a strapdown inertial navigation error compensation accuracy experiment system is built, and the validity and superiority of the method proposed in this paper are verified through calculation and analysis of the data collected on the actual tunneling work face. Full article

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13 pages, 278 KiB

Open AccessArticle

On New Generalizations of Hermite-Hadamard Type Inequalities via Atangana-Baleanu Fractional Integral Operators

by Erhan Set, Ahmet Ocak Akdemir, Ali Karaoǧlan, Thabet Abdeljawad and Wasfi Shatanawi

Axioms 2021, 10(3), 223; https://doi.org/10.3390/axioms10030223 - 12 Sep 2021

Cited by 7 | Viewed by 1864

Abstract

Fractional operators are one of the frequently used tools to obtain new generalizations of clasical inequalities in recent years and many new fractional operators are defined in the literature. This development in the field of fractional analysis has led to a new orientation [...] Read more.

Fractional operators are one of the frequently used tools to obtain new generalizations of clasical inequalities in recent years and many new fractional operators are defined in the literature. This development in the field of fractional analysis has led to a new orientation in various branches of mathematics and in many of the applied sciences. Thanks to this orientation, it has brought a whole new dimension to the field of inequality theory as well as many other disciplines. In this study, a new lemma has been proved for the fractional integral operator defined by Atangana and Baleanu. Later with the help of this lemma and known inequalities such as Young, Jensen, Hölder, new generalizations of Hermite-Hadamard inequality are obtained. Many reduced corollaries about the main findings are presented for classical integrals. Full article

(This article belongs to the Special Issue Differential Equations: Theories, Methods and Modern Applications)

12 pages, 315 KiB

Open AccessArticle

Periodic Third-Order Problems with a Parameter

by Feliz Minhós and Nuno Oliveira

Axioms 2021, 10(3), 222; https://doi.org/10.3390/axioms10030222 - 11 Sep 2021

Cited by 3 | Viewed by 1926

Abstract

This work concerns with the solvability of third-order periodic fully problems with a weighted parameter, where the nonlinearity must verify only a local monotone condition and no periodic, coercivity or super or sublinearity restrictions are assumed, as usual in the literature. The arguments [...] Read more.

This work concerns with the solvability of third-order periodic fully problems with a weighted parameter, where the nonlinearity must verify only a local monotone condition and no periodic, coercivity or super or sublinearity restrictions are assumed, as usual in the literature. The arguments are based on a new type of lower and upper solutions method, not necessarily well ordered. A Nagumo growth condition and Leray–Schauder’s topological degree theory are the existence tools. Only the existence of solution is studied here and it will remain open the discussion on the non-existence and the multiplicity of solutions. Last section contains a nonlinear third-order differential model for periodic catatonic phenomena, depending on biological and/or chemical parameters. Full article

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

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5 pages, 216 KiB

Open AccessArticle

Strictly Convex Banach Algebras

by David Yost

Axioms 2021, 10(3), 221; https://doi.org/10.3390/axioms10030221 - 11 Sep 2021

Viewed by 2198

Abstract

We discuss two facets of the interaction between geometry and algebra in Banach algebras. In the class of unital Banach algebras, there is essentially one known example which is also strictly convex as a Banach space. We recall this example, which is finite-dimensional, [...] Read more.

We discuss two facets of the interaction between geometry and algebra in Banach algebras. In the class of unital Banach algebras, there is essentially one known example which is also strictly convex as a Banach space. We recall this example, which is finite-dimensional, and consider the open question of generalising it to infinite dimensions. In

C^{*}

-algebras, we exhibit one striking example of the tighter relationship that exists between algebra and geometry there. Full article

(This article belongs to the Collection Topological Groups)

9 pages, 246 KiB

Open AccessArticle

Inequalities on General L_p-Mixed Chord Integral Difference

by Hongying Xiao, Weidong Wang and Zhaofeng Li

Axioms 2021, 10(3), 220; https://doi.org/10.3390/axioms10030220 - 10 Sep 2021

Viewed by 1472

Abstract

In this article, we introduce the concept of general

L_{p}

-mixed chord integral difference of star bodies. Further, we establish the Brunn–Minkowski type, Aleksandrov–Fenchel type and cyclic inequalities for the

L_{p}

-mixed chord integral difference. Full article

(This article belongs to the Special Issue Dedicated to Professor Ji-Huan He on the Occasion of His 55th Birthday)

16 pages, 1881 KiB

Open AccessArticle

A Family of the r-Associated Stirling Numbers of the Second Kind and Generalized Bernoulli Polynomials

by Paolo Emilio Ricci, Rekha Srivastava and Pierpaolo Natalini

Axioms 2021, 10(3), 219; https://doi.org/10.3390/axioms10030219 - 9 Sep 2021

Cited by 5 | Viewed by 2938

Abstract

In this article, we derive representation formulas for a class of r-associated Stirling numbers of the second kind and examine their connections with a class of generalized Bernoulli polynomials. Herein, we use the Blissard umbral approach and the familiar Bell polynomials. Links [...] Read more.

In this article, we derive representation formulas for a class of r-associated Stirling numbers of the second kind and examine their connections with a class of generalized Bernoulli polynomials. Herein, we use the Blissard umbral approach and the familiar Bell polynomials. Links with available literature on this subject are also pointed out. The extension to the bivariate case is discussed. Full article

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

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12 pages, 326 KiB

Open AccessArticle

Calculations on Matrix Transformations Involving an Infinite Tridiagonal Matrix

by Ali Fares, Ali Ayad and Bruno de Malafosse

Axioms 2021, 10(3), 218; https://doi.org/10.3390/axioms10030218 - 8 Sep 2021

Viewed by 1641

Abstract

Given any sequence

z = {(z_{n})}_{n \geq 1}

of positive real numbers and any set E of complex sequences, we write

E_{z}

for the set of all sequences

y = {(y_{n})}_{n \geq 1}

such that [...] Read more.

Given any sequence

z = {(z_{n})}_{n \geq 1}

of positive real numbers and any set E of complex sequences, we write

E_{z}

for the set of all sequences

y = {(y_{n})}_{n \geq 1}

such that

y / z = {(y_{n} / z_{n})}_{n \geq 1} \in E

; in particular,

s_{z}^{0}

denotes the set of all sequences y such that

y / z

tends to zero. Here, we consider the infinite tridiagonal matrix

\tilde{B (r, s, t)}

, obtained from the triangle

B (r, s, t)

, by deleting its first row. Then we determine the sets of all positive sequences

a = {(a_{n})}_{n \geq 1}

such that

{(E_{a})}_{\tilde{B (r, s, t)}} \subset E_{a}

, where

E = ℓ_{\infty}

,

c_{0}

, or c. These results extend some recent results. Full article

(This article belongs to the Special Issue Operator Theory and Its Applications)

6 pages, 243 KiB

Open AccessArticle

Normed Spaces Which Are Not Mackey Groups

by Saak Gabriyelyan

Axioms 2021, 10(3), 217; https://doi.org/10.3390/axioms10030217 - 8 Sep 2021

Cited by 2 | Viewed by 1401

Abstract

It is well known that every normed (even quasibarrelled) space is a Mackey space. However, in the more general realm of locally quasi-convex abelian groups an analogous result does not hold. We give the first examples of normed spaces which are not Mackey [...] Read more.

It is well known that every normed (even quasibarrelled) space is a Mackey space. However, in the more general realm of locally quasi-convex abelian groups an analogous result does not hold. We give the first examples of normed spaces which are not Mackey groups. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

23 pages, 2374 KiB

Open AccessFeature PaperArticle

A Dynamic Model of Multiple Time-Delay Interactions between the Virus-Infected Cells and Body’s Immune System with Autoimmune Diseases

by Hoang Pham

Axioms 2021, 10(3), 216; https://doi.org/10.3390/axioms10030216 - 7 Sep 2021

Cited by 6 | Viewed by 2179

Abstract

The immune system is a complex interconnected network consisting of many parts including organs, tissues, cells, molecules and proteins that work together to protect the body from illness when germs enter the body. An autoimmune disease is a disease in which the body’s [...] Read more.

The immune system is a complex interconnected network consisting of many parts including organs, tissues, cells, molecules and proteins that work together to protect the body from illness when germs enter the body. An autoimmune disease is a disease in which the body’s immune system attacks healthy cells. It is known that when the immune system is working properly, it can clearly recognize and kill the abnormal cells and virus-infected cells. But when it doesn’t work properly, the human body will not be able to recognize the virus-infected cells and, therefore, it can attack the body’s healthy cells when there is no invader or does not stop an attack after the invader has been killed, resulting in autoimmune disease.; This paper presents a mathematical modeling of the virus-infected development in the body’s immune system considering the multiple time-delay interactions between the immune cells and virus-infected cells with autoimmune disease. The proposed model aims to determine the dynamic progression of virus-infected cell growth in the immune system. The patterns of how the virus-infected cells spread and the development of the body’s immune cells with respect to time delays will be derived in the form of a system of delay partial differential equations. The model can be used to determine whether the virus-infected free state can be reached or not as time progresses. It also can be used to predict the number of the body’s immune cells at any given time. Several numerical examples are discussed to illustrate the proposed model. The model can provide a real understanding of the transmission dynamics and other significant factors of the virus-infected disease and the body’s immune system subject to the time delay, including approaches to reduce the growth rate of virus-infected cell and the autoimmune disease as well as to enhance the immune effector cells. Full article

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

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27 pages, 25974 KiB

Open AccessArticle

Image Encryption and Decryption System through a Hybrid Approach Using the Jigsaw Transform and Langton’s Ant Applied to Retinal Fundus Images

by Andrés Romero-Arellano, Ernesto Moya-Albor, Jorge Brieva, Ivan Cruz-Aceves, Juan Gabriel Avina-Cervantes, Martha Alicia Hernandez-Gonzalez and Luis Miguel Lopez-Montero

Axioms 2021, 10(3), 215; https://doi.org/10.3390/axioms10030215 - 7 Sep 2021

Cited by 10 | Viewed by 4819

Abstract

In this work, a new medical image encryption/decryption algorithm was proposed. It is based on three main parts: the Jigsaw transform, Langton’s ant, and a novel way to add deterministic noise. The Jigsaw transform was used to hide visual information effectively, whereas Langton’s [...] Read more.

In this work, a new medical image encryption/decryption algorithm was proposed. It is based on three main parts: the Jigsaw transform, Langton’s ant, and a novel way to add deterministic noise. The Jigsaw transform was used to hide visual information effectively, whereas Langton’s ant and the deterministic noise algorithm give a reliable and secure approach. As a case study, the proposal was applied to high-resolution retinal fundus images, where a zero mean square error was obtained between the original and decrypted image. The method performance has been proven through several testing methods, such as statistical analysis (histograms and correlation distributions), entropy computation, keyspace assessment, robustness to differential attack, and key sensitivity analysis, showing in each one a high security level. In addition, the method was compared against other works showing a competitive performance and highlighting with a large keyspace (>1

\times 10^{1, 134, 190.38})

. Besides, the method has demonstrated adequate handling of high-resolution images, obtaining entropy values between 7.999988 and 7.999989, an average Number of Pixel Change Rate (NPCR) of

99.5796 % \pm 0.000674

, and a mean Uniform Average Change Intensity (UACI) of

33.4469 % \pm 0.00229

. In addition, when there is a small change in the key, the method does not give additional information to decrypt the image. Full article

(This article belongs to the Special Issue Mathematics Behind Machine Learning)

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22 pages, 4426 KiB

Open AccessArticle

Location of Urban Logistics Spaces (ULS) for Two-Echelon Distribution Systems

by José Ruiz-Meza, Karen Meza-Peralta, Jairo R. Montoya-Torres and Jesus Gonzalez-Feliu

Axioms 2021, 10(3), 214; https://doi.org/10.3390/axioms10030214 - 7 Sep 2021

Cited by 8 | Viewed by 2618

Abstract

The main concern in city logistics is the need to optimize the movement of goods in urban contexts, and to minimize the multiple costs inherent in logistics operations. Inspired by an application in a medium-sized city in Latin America, this paper develops a [...] Read more.

The main concern in city logistics is the need to optimize the movement of goods in urban contexts, and to minimize the multiple costs inherent in logistics operations. Inspired by an application in a medium-sized city in Latin America, this paper develops a bi-objective mixed linear integer programming (MILP) model to locate different types of urban logistics spaces (ULS) for the configuration of a two-echelon urban distribution system. The objective functions seek to minimize the costs associated with distance traveled and relocation, in addition to the costs of violation of time windows. This model considers heterogeneous transport, speed assignment, and time windows. For experimental evaluation, two operational scenarios are considered, and Pareto frontiers are obtained to identify the efficient non-dominated solutions to select the most feasible ones from such a set. A case study of a distribution company of goods for supermarkets in the city of Barranquilla, Colombia, is also used to validate the proposed model. These solutions allow decision-makers to define the configuration of ULS networks for urban product delivery. Full article

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19 pages, 280 KiB

Open AccessArticle

De Moivre’s and Euler Formulas for Matrices of Hybrid Numbers

by Mücahit Akbıyık, Seda Yamaç Akbıyık, Emel Karaca and Fatih Yılmaz

Axioms 2021, 10(3), 213; https://doi.org/10.3390/axioms10030213 - 6 Sep 2021

Cited by 7 | Viewed by 3240

Abstract

It is known that the hybrid numbers are generalizations of complex, hyperbolic and dual numbers. Recently, they have attracted the attention of many scientists. At this paper, we provide the Euler’s and De Moivre’s formulas for the

4 \times 4

matrices associated with [...] Read more.

It is known that the hybrid numbers are generalizations of complex, hyperbolic and dual numbers. Recently, they have attracted the attention of many scientists. At this paper, we provide the Euler’s and De Moivre’s formulas for the

4 \times 4

matrices associated with hybrid numbers by using trigonometric identities. Also, we give the roots of the matrices of hybrid numbers. Moreover, we give some illustrative examples to support the main formulas. Full article

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

7 pages, 247 KiB

Open AccessArticle

(ρ,η,μ)-Interpolative Kannan Contractions I

by Yaé Ulrich Gaba, Hassen Aydi and Nabil Mlaiki

Axioms 2021, 10(3), 212; https://doi.org/10.3390/axioms10030212 - 3 Sep 2021

Cited by 7 | Viewed by 2001

Abstract

We point out a vital error in the paper of Gaba et al. (2019), showing that a (

ρ

,

η

,

μ

) interpolative Kannan contraction in a complete metric space need not have a fixed point. Then we give an appropriate [...] Read more.

We point out a vital error in the paper of Gaba et al. (2019), showing that a (

ρ

,

η

,

μ

) interpolative Kannan contraction in a complete metric space need not have a fixed point. Then we give an appropriate restriction on a (

ρ

,

η

,

μ

)-interpolative Kannan contraction that guarantees the existence of a fixed point and provide an equivalent formulation. Moreover, we show that this formulation can be extended to the interpolative Reich-Rus-Ćirić type contraction. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics II)

14 pages, 407 KiB

Open AccessArticle

Numerical Algorithms for Computing an Arbitrary Singular Value of a Tensor Sum

by Asuka Ohashi and Tomohiro Sogabe

Axioms 2021, 10(3), 211; https://doi.org/10.3390/axioms10030211 - 31 Aug 2021

Cited by 1 | Viewed by 1688

Abstract

We consider computing an arbitrary singular value of a tensor sum: [...] Read more.

We consider computing an arbitrary singular value of a tensor sum:

T : = I_{n} \otimes I_{m} \otimes A + I_{n} \otimes B \otimes I_{ℓ} + C \otimes I_{m} \otimes I_{ℓ} \in R^{ℓ m n \times ℓ m n},

where

A \in R^{ℓ \times ℓ}

,

B \in R^{m \times m}

,

C \in R^{n \times n}

. We focus on the shift-and-invert Lanczos method, which solves a shift-and-invert eigenvalue problem of

{(T^{T} T - {\tilde{σ}}^{2} I_{ℓ m n})}^{- 1}

, where

\tilde{σ}

is set to a scalar value close to the desired singular value. The desired singular value is computed by the maximum eigenvalue of the eigenvalue problem. This shift-and-invert Lanczos method needs to solve large-scale linear systems with the coefficient matrix

T^{T} T - {\tilde{σ}}^{2} I_{ℓ m n}

. The preconditioned conjugate gradient (PCG) method is applied since the direct methods cannot be applied due to the nonzero structure of the coefficient matrix. However, it is difficult in terms of memory requirements to simply implement the shift-and-invert Lanczos and the PCG methods since the size of T grows rapidly by the sizes of A, B, and C. In this paper, we present the following two techniques: (1) efficient implementations of the shift-and-invert Lanczos method for the eigenvalue problem of

T^{T} T

and the PCG method for

T^{T} T - {\tilde{σ}}^{2} I_{ℓ m n}

using three-dimensional arrays (third-order tensors) and the n-mode products, and (2) preconditioning matrices of the PCG method based on the eigenvalue and the Schur decomposition of T. Finally, we show the effectiveness of the proposed methods through numerical experiments. Full article

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

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18 pages, 824 KiB

Open AccessArticle

Qualitative and Quantitative Analyses of COVID-19 Dynamics

by Taye Samuel Faniran, Leontine Nkague Nkamba and Thomas Timothee Manga

Axioms 2021, 10(3), 210; https://doi.org/10.3390/axioms10030210 - 31 Aug 2021

Cited by 3 | Viewed by 2272

Abstract

COVID-19 is a highly contagious disease which has spread across the world. A deterministic model that considers an important component of individuals with vertically transmitted underlying diseases (high-risk susceptible individuals), rather than the general public, is formulated in this paper. We also consider [...] Read more.

COVID-19 is a highly contagious disease which has spread across the world. A deterministic model that considers an important component of individuals with vertically transmitted underlying diseases (high-risk susceptible individuals), rather than the general public, is formulated in this paper. We also consider key parameters that are concerned with the disease. An epidemiological threshold,

R_{0}

, is computed using next-generation matrix approach. This is used to establish the existence and global stability of equilibria. We identify the most sensitive parameters which effectively contribute to change the disease dynamics with the help of sensitivity analysis. Our results reveal that increasing contact tracing of the exposed individuals who are tested for COVID-19 and hospitalizing them, largely has a negative impact on

R_{0}

. Results further reveal that transmission rate between low-risk/high-risk susceptible individuals and symptomatic infectious individuals

β

and incubation rate of the exposed individuals

σ

have positive impact on

R_{0}

. Numerical simulations show that there are fewer high-risk susceptible individuals than the general public when

R_{0} < 1

. This may be due to the fact that high-risk susceptible individuals may prove a bit more difficult to control than the low-risk susceptible individuals as a result of inherited underlying diseases present in them. We thus conclude that high level of tracing and hospitalizing the exposed individuals, as well as adherence to standard precautions and wearing appropriate Personal Protective Equipment (PPE) while handling emergency cases, are needed to flatten the epidemic curve. Full article

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13 pages, 306 KiB

Open AccessArticle

θ*-Weak Contractions and Discontinuity at the Fixed Point with Applications to Matrix and Integral Equations

by Atiya Perveen, Waleed M. Alfaqih, Salvatore Sessa and Mohammad Imdad

Axioms 2021, 10(3), 209; https://doi.org/10.3390/axioms10030209 - 31 Aug 2021

Cited by 3 | Viewed by 1829

Abstract

In this paper, the notion of

θ^{*}

-weak contraction is introduced, which is utilized to prove some fixed point results. These results are helpful to give a positive response to certain open question raised by Kannan and Rhoades on the existence of [...] Read more.

In this paper, the notion of

θ^{*}

-weak contraction is introduced, which is utilized to prove some fixed point results. These results are helpful to give a positive response to certain open question raised by Kannan and Rhoades on the existence of contractive definition which does not force the mapping to be continuous at the fixed point. Some illustrative examples are also given to support our results. As applications of our result, we investigate the existence and uniqueness of a solution of non-linear matrix equations and integral equations of Volterra type as well. Full article

(This article belongs to the Special Issue Advances in Nonlinear Analysis and Related Fixed Point Problems)

15 pages, 942 KiB

Open AccessArticle

Selection of Logistics Service Provider for the E-Commerce Companies in Pakistan Based on Integrated GRA-TOPSIS Approach

by Muhammad Hamza Naseem, Jiaqi Yang and Ziquan Xiang

Axioms 2021, 10(3), 208; https://doi.org/10.3390/axioms10030208 - 30 Aug 2021

Cited by 8 | Viewed by 3004

Abstract

Recently, the demand for third-party logistics providers has become extremely relevant and the key subject for businesses to enhance their service quality and minimize logistics costs. The key success factor for an e-commerce business is product delivery, and the third-party logistics service provider [...] Read more.

Recently, the demand for third-party logistics providers has become extremely relevant and the key subject for businesses to enhance their service quality and minimize logistics costs. The key success factor for an e-commerce business is product delivery, and the third-party logistics service provider is responsible for that. Each 3PLP has its own business characteristics, meaning it is important to select the most suitable logistics provider for the e-commerce business. This study uses a combination of grey relational analysis (GRA) and the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method, assisting decision makers in choosing the best logistics service provider for their e-business. A case study of an e-commerce company based in Faisalabad, Pakistan, was selected to demonstrate the steps of the proposed methods. In this process, seven criteria of logistics suppliers were considered, and then the best alternatives among four logistics provider companies were selected using the proposed method. Full article

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

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15 pages, 266 KiB

Open AccessArticle

A Parametric Type of Cauchy Polynomials with Higher Level

by Takao Komatsu

Axioms 2021, 10(3), 207; https://doi.org/10.3390/axioms10030207 - 30 Aug 2021

Viewed by 1603

Abstract

There are many kinds of generalizations of Cauchy numbers and polynomials. Recently, a parametric type of the Bernoulli numbers with level 3 was introduced and studied as a kind of generalization of Bernoulli polynomials. A parametric type of Cauchy numbers with level 3 [...] Read more.

There are many kinds of generalizations of Cauchy numbers and polynomials. Recently, a parametric type of the Bernoulli numbers with level 3 was introduced and studied as a kind of generalization of Bernoulli polynomials. A parametric type of Cauchy numbers with level 3 is its analogue. In this paper, as an analogue of a parametric type of Bernoulli polynomials with level 3 and its extension, we introduce a parametric type of Cauchy polynomials with a higher level. We present their characteristic and combinatorial properties. By using recursions, we show some determinant expressions. Full article

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

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