Axioms

24 pages, 4982 KiB

Open AccessArticle

A Computational Approach to a Model for HIV and the Immune System Interaction

by Attaullah, Zeeshan, Muhammad Tufail Khan, Sultan Alyobi, Mansour F. Yassen and Din Prathumwan

Axioms 2022, 11(10), 578; https://doi.org/10.3390/axioms11100578 - 21 Oct 2022

Cited by 5 | Viewed by 2210

Abstract

This study deals with the numerical solution of the human immunodeficiency virus (HIV) infection model, which is a significant problem for global public health. Acquired immunodeficiency syndrome (AIDS) is a communicable disease, and HIV is the causative agent for AIDS, which damages the [...] Read more.

This study deals with the numerical solution of the human immunodeficiency virus (HIV) infection model, which is a significant problem for global public health. Acquired immunodeficiency syndrome (AIDS) is a communicable disease, and HIV is the causative agent for AIDS, which damages the ability of the body to fight against disease and easily usual innocuous infections attack the body. On entering the body, HIV infects a large amount of CD4+ T-cells and disturbs the supply rate of these cells from the thymus. Herein, we consider the model with variable source terms in which the production of these cells is a monotonically decreasing function of viral load. Based on the reproduction number, we describe the stability of free equilibrium. The continuous Galerkin–Petrov method, in particular the cGP(2)-method, is implemented to determine the numerical solutions of the model. The influence of different parameters on the population dynamics of healthy/infected CD4+ T-cells and free HIV particles are examined, and the results are presented graphically. On the other hand, the model is solved using the fourth-order Runge–Kutta method, and briefly, the RK4-method, and the results of the proposed schemes are compared with those obtained from other classical schemes such as the Bessel collocation method (BCM), Laplace Adomian decomposition method (LADM), perturbation iteration algorithm (PIA), modified variational iteration method (MVIM), differential transform method (DTM), and exponential Galerkin method (EGM), numerically. Furthermore, absolute errors relative to the RK4 method are computed to describe the accuracy of the proposed scheme. It is presented that the cGP(2)-method gains accurate results at larger time step sizes in comparison with the results of the aforementioned methods. The numerical and graphical comparison reveals that the proposed scheme yields more accurate results relative to other traditional schemes from the literature. Full article

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

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8 pages, 333 KiB

Open AccessArticle

Stability Analysis of a Patchy Predator–Prey Model with Fear Effect

by Tingting Liu and Lijuan Chen

Axioms 2022, 11(10), 577; https://doi.org/10.3390/axioms11100577 - 21 Oct 2022

Cited by 1 | Viewed by 1538

Abstract

In this paper, a predator–prey model with fear effect and dispersal is proposed. Assume that only the prey migrates at a constant rate between patches and the migration of prey on each patch is faster than the time scale of local predator–prey interaction. [...] Read more.

In this paper, a predator–prey model with fear effect and dispersal is proposed. Assume that only the prey migrates at a constant rate between patches and the migration of prey on each patch is faster than the time scale of local predator–prey interaction. Using two time scales, an aggregation system of total prey density for two patches is constructed. Mathematical analysis shows that there may exist a trivial, a boundary and a unique positive equilibrium point. Under certain conditions, the corresponding unique equilibrium point is global asymptotically stable. The impact of the fear effect on the system is also investigated, i.e., the predator density decreases when the amount of fear effect increases. Moreover, dispersal has a great impact on the persistence of the predator and the prey. Numerical experiments are also presented to verify the feasibility of our conclusion. Full article

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11 pages, 312 KiB

Open AccessArticle

Extended Gevrey Regularity via Weight Matrices

by Nenad Teofanov and Filip Tomić

Axioms 2022, 11(10), 576; https://doi.org/10.3390/axioms11100576 - 21 Oct 2022

Cited by 2 | Viewed by 1597

Abstract

The main aim of this paper is to compare two recent approaches for investigating the interspace between the union of Gevrey spaces

G_{t} (U)

and the space of smooth functions

C^{\infty} (U)

. The first approach in [...] Read more.

The main aim of this paper is to compare two recent approaches for investigating the interspace between the union of Gevrey spaces

G_{t} (U)

and the space of smooth functions

C^{\infty} (U)

. The first approach in the style of Komatsu is based on the properties of two parameter sequences

M_{p} = p^{τ p^{σ}}

,

τ > 0

,

σ > 1

. The other one uses weight matrices defined by certain weight functions. We prove the equivalence of the corresponding spaces in the Beurling case by taking projective limits with respect to matrix parameters, while in the Roumieu case we need to consider a larger space than the one obtained as the inductive limit of extended Gevrey classes. Full article

(This article belongs to the Special Issue Time-Frequency Analysis, Distributions, and Operators)

19 pages, 383 KiB

Open AccessArticle

Stability and Hopf Bifurcation Analysis of a Stage-Structured Predator–Prey Model with Delay

by Xueyong Zhou

Axioms 2022, 11(10), 575; https://doi.org/10.3390/axioms11100575 - 20 Oct 2022

Cited by 1 | Viewed by 1801

Abstract

In this work, a Lotka–Volterra type predator–prey system with time delay and stage structure for the predators is proposed and analyzed. By using the permanence theory for infinite dimensional system, we get that the system is permanent if some conditions are satisfied. The [...] Read more.

In this work, a Lotka–Volterra type predator–prey system with time delay and stage structure for the predators is proposed and analyzed. By using the permanence theory for infinite dimensional system, we get that the system is permanent if some conditions are satisfied. The local and global stability of the positive equilibrium is presented. The existence of Hopf bifurcation around the positive equilibrium is observed. Further, by using the normal form theory and center manifold approach, we derive the explicit formulas determining the stability of bifurcating periodic solutions and the direction of Hopf bifurcation. Numerical simulations are carried out by Matlab software to explain the theoretical results. We find that combined time delay and stage structure can affect the dynamical behavior of the system. Full article

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15 pages, 1216 KiB

Open AccessArticle

A Reliable Technique for Solving Fractional Partial Differential Equation

by Azzh Saad Alshehry, Rasool Shah, Nehad Ali Shah and Ioannis Dassios

Axioms 2022, 11(10), 574; https://doi.org/10.3390/axioms11100574 - 20 Oct 2022

Cited by 14 | Viewed by 2497

Abstract

The development of numeric-analytic solutions and the construction of fractional-order mathematical models for practical issues are of the greatest importance in a variety of applied mathematics, physics, and engineering problems. The Laplace residual-power-series method (LRPSM), a new and dependable technique for resolving fractional [...] Read more.

The development of numeric-analytic solutions and the construction of fractional-order mathematical models for practical issues are of the greatest importance in a variety of applied mathematics, physics, and engineering problems. The Laplace residual-power-series method (LRPSM), a new and dependable technique for resolving fractional partial differential equations, is introduced in this study. The residual-power-series method (RPSM), a well-known technique, and the Laplace transform (LT) are elegantly combined in the suggested technique. This innovative approach computes the fractional derivative in the Caputo sense. The proposed method for handling fractional partial differential equations is provided in detail, along with its implementation. The novel approach yields a series solution to fractional partial differential equations. To validate the simplicity, effectiveness, and viability of the suggested technique, the provided model is tested and simulated. A numerical and graphical description of the effects of the fractional order

γ

on approximating the solutions is provided. Comparative results show that the suggested method approximates more precisely than current methods such as the natural homotopy perturbation method. The study showed that the aforementioned method is straightforward, trustworthy, and suitable for analysing non-linear engineering and physical issues. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations)

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17 pages, 331 KiB

Open AccessArticle

Reich-Type and (α, F)-Contractions in Partially Ordered Double-Controlled Metric-Type Spaces with Applications to Non-Linear Fractional Differential Equations and Monotonic Iterative Method

by Muhammad Farhan, Umar Ishtiaq, Muhammad Saeed, Aftab Hussain and Hamed Al Sulami

Axioms 2022, 11(10), 573; https://doi.org/10.3390/axioms11100573 - 20 Oct 2022

Cited by 6 | Viewed by 1684

Abstract

In this manuscript, we defined (α, F)-contractions in the context of double-controlled metric spaces and partially ordered double-controlled metric spaces. We established new fixed-point results and defined the notion of double-controlled metric space on a Reich-type contraction. Our findings are [...] Read more.

In this manuscript, we defined (α, F)-contractions in the context of double-controlled metric spaces and partially ordered double-controlled metric spaces. We established new fixed-point results and defined the notion of double-controlled metric space on a Reich-type contraction. Our findings are generalizations of a few well-known findings in the literature. Some non-trivial examples and certain consequences are also provided to illustrate the significance of the presented results. The existence and uniqueness of the solution of non-linear fractional differential equations and the monotone iterative method are also determined using the fixed-point method. Full article

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

19 pages, 350 KiB

Open AccessArticle

A Novel Approach in Solving Improper Integrals

by Mohammad Abu-Ghuwaleh, Rania Saadeh and Ahmad Qazza

Axioms 2022, 11(10), 572; https://doi.org/10.3390/axioms11100572 - 20 Oct 2022

Cited by 12 | Viewed by 2185

Abstract

To resolve several challenging applications in many scientific domains, general formulas of improper integrals are provided and established for use in this article. The suggested theorems can be considered generators for new improper integrals with precise solutions, without requiring complex computations. New criteria [...] Read more.

To resolve several challenging applications in many scientific domains, general formulas of improper integrals are provided and established for use in this article. The suggested theorems can be considered generators for new improper integrals with precise solutions, without requiring complex computations. New criteria for handling improper integrals are illustrated in tables to simplify the usage and the applications of the obtained outcomes. The results of this research are compared with those obtained by I.S. Gradshteyn and I.M. Ryzhik in the classical table of integrations. Some well-known theorems on improper integrals are considered to be simple cases in the context of our work. Some applications related to finding Green’s function, one-dimensional vibrating string problems, wave motion in elastic solids, and computing Fourier transforms are presented. Full article

(This article belongs to the Special Issue Theory of Functions and Applications)

13 pages, 10352 KiB

Open AccessArticle

Simulation of Marine Debris Path Using Mathematical Model in the Gulf of Thailand

by Jettapol Phiphit, Angkool Wangwongchai and Usa Wannasingha Humphries

Axioms 2022, 11(10), 571; https://doi.org/10.3390/axioms11100571 - 20 Oct 2022

Cited by 2 | Viewed by 2063

Abstract

Marine debris is an important environmental problem that affects aquatic animals, ecosystems, economy, society, and humans. This research aims to simulate the path of marine debris in the Gulf of Thailand using a mathematical model that includes two models: the Oceanic Model (OCM), [...] Read more.

Marine debris is an important environmental problem that affects aquatic animals, ecosystems, economy, society, and humans. This research aims to simulate the path of marine debris in the Gulf of Thailand using a mathematical model that includes two models: the Oceanic Model (OCM), which is based on the Shallow Water Equations (SWE), and the Lagrangian Particle Tracking (LPT) model. The OCM is the partial derivative equation system solved by the finite difference method to satisfy the Arakawa C-grid and the splitting method. The LPT model includes the current velocity, wind velocity at 10 m above sea level, random walk term, and the buoyancy ratio of marine debris with six cases, which are 100:1, 10:1, 1:1, 0:1, 1:10, and 1:100. The current velocity from OCM is applied to the LPT model. This research uses a garbage boat that capsized near Koh Samui on 1 August 2020 as a case study. The simulated current velocity of OCM is compared with Ocean Surface Current Analyses Real-time (OSCAR) data. The Root Mean Square Error (RMSE) of u-velocity is 0.070 m/s, and that of v-velocity is 0.058 m/s. The simulation of the marine debris’s path from the LPT model demonstrates the movement to Koh Samui, Koh Taen, Koh Wang Nai, Koh Wang Nok, Koh Rap, the east coast of Nakorn Si Thammarat province, Phu Quoc Island of Vietnam and the middle of the Gulf of Thailand with the different buoyancy ratios and time durations. Full article

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

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12 pages, 307 KiB

Open AccessArticle

On Symmetrized Stochastic Harmonically Convexity and Hermite–Hadamard Type Inequalities

by Muhammad Amer Latif

Axioms 2022, 11(10), 570; https://doi.org/10.3390/axioms11100570 - 20 Oct 2022

Cited by 3 | Viewed by 1525

Abstract

Throughout this study, the concept of symmetrized harmonically convex stochastic processes will be discussed in further detail. Some certain characterizations for symmetrized harmonically convex stochastic processes are discussed that use Hermite–Hadamard-type inequalities. A Hyers–Ulam-type stability result for harmonically convex stochastic processes is given [...] Read more.

Throughout this study, the concept of symmetrized harmonically convex stochastic processes will be discussed in further detail. Some certain characterizations for symmetrized harmonically convex stochastic processes are discussed that use Hermite–Hadamard-type inequalities. A Hyers–Ulam-type stability result for harmonically convex stochastic processes is given as well. Full article

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

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17 pages, 1369 KiB

Open AccessArticle

Estimation of Truncation Error in Statistical Description of Communication Signals over mm-Wave Channels

by Zvezdan Marjanović, Dejan N. Milić and Goran T. Đorđević

Axioms 2022, 11(10), 569; https://doi.org/10.3390/axioms11100569 - 19 Oct 2022

Cited by 2 | Viewed by 1776

Abstract

This paper presents an illustration of how knowledge from the field of special functions, orthogonal polynomials and numerical series can be applied to solve a very important problem in the field of modern wireless communications. We present the formulas for the probability density [...] Read more.

This paper presents an illustration of how knowledge from the field of special functions, orthogonal polynomials and numerical series can be applied to solve a very important problem in the field of modern wireless communications. We present the formulas for the probability density function (PDF) and cumulative distribution function (CDF) of the composite signal envelope over an

m

m

-Wave channel. The formulas for the PDF and CDF are expressed in the convergent infinity series form. The main contribution of the paper is in estimating the upper bounds for absolute truncation error in evaluating PDF and CDF of the signal envelope. We also derive the formulas for the required number of terms in the summation under the condition of achieving a given accuracy for typical values of channel parameters. In deriving these formulas, we use the alternating series estimation theorem, as well as some properties of orthogonal polynomials in order to derive upper bounds for hypergeometric functions. Based on the newly derived formulas, numerical results are presented and commented upon. The analytical results are verified by Monte Carlo simulations. The results are essential in the designing and performance estimating of the fifth-generation (5G) and beyond wireless networks. Full article

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

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13 pages, 309 KiB

Open AccessArticle

Geometric Properties of Some Generalized Mathieu Power Series inside the Unit Disk

by Živorad Tomovski, Stefan Gerhold, Deepak Bansal and Amit Soni

Axioms 2022, 11(10), 568; https://doi.org/10.3390/axioms11100568 - 19 Oct 2022

Cited by 1 | Viewed by 1519

Abstract

We consider two parametric families of special functions: One is defined by a power series generalizing the classical Mathieu series, and the other one is a generalized Mathieu type power series involving factorials in its coefficients. Using criteria due to Fejér and Ozaki, [...] Read more.

We consider two parametric families of special functions: One is defined by a power series generalizing the classical Mathieu series, and the other one is a generalized Mathieu type power series involving factorials in its coefficients. Using criteria due to Fejér and Ozaki, we provide sufficient conditions for these functions to be close-to-convex or starlike inside the unit disk, and thus univalent. One of our proofs is assisted by symbolic computation. Full article

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

6 pages, 228 KiB

Open AccessArticle

A Valid Quantization of the Particle in a Box Field Theory, and Well Beyond

by John R. Klauder

Axioms 2022, 11(10), 567; https://doi.org/10.3390/axioms11100567 - 19 Oct 2022

Viewed by 1305

Abstract

The usual particle in a box is turned into a field theory, and its behavior is examined using canonical and affine quantizations. The result leads to a valid affine quantization of the particle in a box field theory, which points toward further valid [...] Read more.

The usual particle in a box is turned into a field theory, and its behavior is examined using canonical and affine quantizations. The result leads to a valid affine quantization of the particle in a box field theory, which points toward further valid quantizations of more realistic field theory models. Full article

(This article belongs to the Special Issue Computational Mathematics and Mathematical Physics)

13 pages, 1354 KiB

Open AccessArticle

The Design of State-Dependent Switching Rules for Second-Order Switched Linear Systems Revisited

by Xiaolan Yuan and Yusheng Zhou

Axioms 2022, 11(10), 566; https://doi.org/10.3390/axioms11100566 - 19 Oct 2022

Cited by 2 | Viewed by 1695

Abstract

This paper focuses on the asymptotic stability of second-order switched linear systems with positive real part conjugate complex roots for each subsystem. Compared with available studies, a more appropriate state-dependent switching rule is designed to stabilize a switched system with the phase trajectories [...] Read more.

This paper focuses on the asymptotic stability of second-order switched linear systems with positive real part conjugate complex roots for each subsystem. Compared with available studies, a more appropriate state-dependent switching rule is designed to stabilize a switched system with the phase trajectories of two subsystems rotating outward in the same direction or the opposite direction. Finally, several numerical examples are used to illustrate the effectiveness and superiority of the proposed method. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems with Applications)

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14 pages, 278 KiB

Open AccessArticle

Development of an Efficient Diagonally Implicit Runge–Kutta–Nyström 5(4) Pair for Special Second Order IVPs

by Musa Ahmed Demba, Norazak Senu, Higinio Ramos and Wiboonsak Watthayu

Axioms 2022, 11(10), 565; https://doi.org/10.3390/axioms11100565 - 18 Oct 2022

Cited by 1 | Viewed by 1632

Abstract

In this work, a new pair of diagonally implicit Runge–Kutta–Nyström methods with four stages is constructed. The proposed method has been derived to solve initial value problems of special second-order ordinary-differential equations. The principal local truncation error of the new method is obtained, [...] Read more.

In this work, a new pair of diagonally implicit Runge–Kutta–Nyström methods with four stages is constructed. The proposed method has been derived to solve initial value problems of special second-order ordinary-differential equations. The principal local truncation error of the new method is obtained, and the main characteristics of the new method are analyzed. Some numerical experiments are performed, which demonstrate the robustness and efficiency of the new embedded pair. Full article

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15 pages, 291 KiB

Open AccessArticle

Weighted Integral Inequalities for Harmonic Convex Functions in Connection with Fejér’s Result

by Muhammad Amer Latif

Axioms 2022, 11(10), 564; https://doi.org/10.3390/axioms11100564 - 18 Oct 2022

Cited by 5 | Viewed by 1303

Abstract

In this study, on the subject of harmonic convex functions, we introduce some new functionals linked with weighted integral inequalities for harmonic convex functions. In addition, certain new inequalities of the Fejér type are discovered. Full article

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

9 pages, 258 KiB

Open AccessArticle

Fixed Point Theory for Multi-Valued Feng–Liu–Subrahmanyan Contractions

by Claudia Luminiţa Mihiţ, Ghiocel Moţ and Adrian Petruşel

Axioms 2022, 11(10), 563; https://doi.org/10.3390/axioms11100563 - 17 Oct 2022

Cited by 4 | Viewed by 1535

Abstract

In this paper, we consider several problems related to the so-called multi-valued Feng–Liu–Subrahmanyan contractions in complete metric spaces. Existence of the fixed points and of the strict fixed points, as well as data dependence and stability properties for the fixed point problem, are [...] Read more.

In this paper, we consider several problems related to the so-called multi-valued Feng–Liu–Subrahmanyan contractions in complete metric spaces. Existence of the fixed points and of the strict fixed points, as well as data dependence and stability properties for the fixed point problem, are discussed. Some results are presented, under appropriate conditions, and some open questions are pointed out. Our results extend recent results given for multi-valued graph contractions and multi-valued Subrahmanyan contractions. Full article

(This article belongs to the Special Issue Special Issue in Honor of the 60th Birthday of Professor Hong-Kun Xu)

16 pages, 312 KiB

Open AccessArticle

A New First Order Expansion Formula with a Reduced Remainder

by Joel Chaskalovic and Hessam Jamshidipour

Axioms 2022, 11(10), 562; https://doi.org/10.3390/axioms11100562 - 17 Oct 2022

Cited by 3 | Viewed by 1386

Abstract

This paper is devoted to a new first order Taylor-like formula, where the corresponding remainder is strongly reduced in comparison with the usual one, which appears in the classical Taylor’s formula. To derive this new formula, we introduce a linear combination of the [...] Read more.

This paper is devoted to a new first order Taylor-like formula, where the corresponding remainder is strongly reduced in comparison with the usual one, which appears in the classical Taylor’s formula. To derive this new formula, we introduce a linear combination of the first derivative of the concerned function, which is computed at

n + 1

equally spaced points between the two points, where the function has to be evaluated. We show that an optimal choice of the weights in the linear combination leads to minimizing the corresponding remainder. Then, we analyze the Lagrange

P_{1}

- interpolation error estimate and the trapezoidal quadrature error, in order to assess the gain of the accuracy we obtain using this new Taylor-like formula. Full article

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

23 pages, 3103 KiB

Open AccessArticle

An Improved Elephant Herding Optimization for Energy-Saving Assembly Job Shop Scheduling Problem with Transportation Times

by Tianhua Jiang, Lu Liu, Huiqi Zhu and Yaping Li

Axioms 2022, 11(10), 561; https://doi.org/10.3390/axioms11100561 - 16 Oct 2022

Cited by 1 | Viewed by 1908

Abstract

The energy-saving scheduling problem (ESSP) has gained increasing attention of researchers in the manufacturing field. However, there is a lack of studies on ESSPs in the assembly job shop environment. In contrast with traditional scheduling problems, the assembly job shop scheduling problem (AJSP) [...] Read more.

The energy-saving scheduling problem (ESSP) has gained increasing attention of researchers in the manufacturing field. However, there is a lack of studies on ESSPs in the assembly job shop environment. In contrast with traditional scheduling problems, the assembly job shop scheduling problem (AJSP) adds the additional consideration of hierarchical precedence constraints between different jobs of each final product. This paper focuses on developing a methodology for an energy-saving assembly job shop scheduling problem with job transportation times. Firstly, a mathematical model is constructed with the objective of minimizing total energy consumption. Secondly, an improved elephant herding optimization (IEHO) is proposed by considering the problem’s characteristics. Finally, thirty-two different instances are designed to verify the performance of the proposed algorithm. Computational results and statistical data demonstrate that the IEHO has advantages over other algorithms in terms of the solving accuracy for the considered problem. Full article

(This article belongs to the Special Issue Computational Optimization and Applications)

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12 pages, 331 KiB

Open AccessArticle

Fekete-Szegö Inequalities for Some Certain Subclass of Analytic Functions Defined with Ruscheweyh Derivative Operator

by Halit Orhan and Luminiţa-Ioana Cotîrlă

Axioms 2022, 11(10), 560; https://doi.org/10.3390/axioms11100560 - 15 Oct 2022

Cited by 3 | Viewed by 1744

Abstract

In our present investigation, we introduce and study some new subclasses of analytic functions associated with Ruscheweyh differential operator

D^{r}

. We obtain a Fekete–Szegö inequality for certain normalized analytic function defined on the open unit disk for which [...] Read more.

In our present investigation, we introduce and study some new subclasses of analytic functions associated with Ruscheweyh differential operator

D^{r}

. We obtain a Fekete–Szegö inequality for certain normalized analytic function defined on the open unit disk for which

{[{(D^{r} l)}^{'} (z)]}^{ϑ} {(\frac{z {(D^{r} l)}^{'} (z)}{(D^{r} l) (z)})}^{1 - ϑ} ≺ e^{z}

(0 \leq ϑ \leq 1)

lies in a starlike region with respect to 1 and symmetric with respect to the real axis. As a special case of this result, the Fekete–Szegö inequality for a class of functions defined through Poisson distribution series is obtained. Full article

(This article belongs to the Special Issue Dedicated to Professor Hari Mohan Srivastava on the Occasion of His 82nd Birthday)

10 pages, 250 KiB

Open AccessArticle

Proving, Refuting, Improving—Looking for a Theorem

by Branislav Boričić

Axioms 2022, 11(10), 559; https://doi.org/10.3390/axioms11100559 - 15 Oct 2022

Viewed by 1376

Abstract

Exploring the proofs and refutations of an abstract statement, conjecture with the aim to give a formal syntactic treatment of its proving–refuting process, we introduce the notion of extrapolation of a possibly unprovable statement having the form if A, then B, and propose [...] Read more.

Exploring the proofs and refutations of an abstract statement, conjecture with the aim to give a formal syntactic treatment of its proving–refuting process, we introduce the notion of extrapolation of a possibly unprovable statement having the form if A, then B, and propose a procedure that should result in the new statement if $A^{'}$ , then $B^{'}$ , which is similar to the starting one, but provable. We think that this procedure, based on the extrapolation method, can be considered a basic methodological tool applicable to prove–refute–improve any conjecture. This new notion, extrapolation, presents a dual counterpart of the well-known interpolation introduced in traditional logic sixty-five years ago. Full article

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

15 pages, 327 KiB

Open AccessArticle

Fourier Transform of the Orthogonal Polynomials on the Unit Ball and Continuous Hahn Polynomials

by Esra Güldoğan Lekesiz, Rabia Aktaş and Iván Area

Axioms 2022, 11(10), 558; https://doi.org/10.3390/axioms11100558 - 14 Oct 2022

Cited by 2 | Viewed by 1756

Abstract

Some systems of univariate orthogonal polynomials can be mapped into other families by the Fourier transform. The most-studied example is related to the Hermite functions, which are eigenfunctions of the Fourier transform. For the multivariate case, by using the Fourier transform and Parseval’s [...] Read more.

Some systems of univariate orthogonal polynomials can be mapped into other families by the Fourier transform. The most-studied example is related to the Hermite functions, which are eigenfunctions of the Fourier transform. For the multivariate case, by using the Fourier transform and Parseval’s identity, very recently, some examples of orthogonal systems of this type have been introduced and orthogonality relations have been discussed. In the present paper, this method is applied for multivariate orthogonal polynomials on the unit ball. The Fourier transform of these orthogonal polynomials on the unit ball is obtained. By Parseval’s identity, a new family of multivariate orthogonal functions is introduced. The results are expressed in terms of the continuous Hahn polynomials. Full article

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

23 pages, 524 KiB

Open AccessArticle

Plane Section Curves on Surfaces of NCP Functions

by Shun-Wei Li, Yu-Lin Chang and Jein-Shan Chen

Axioms 2022, 11(10), 557; https://doi.org/10.3390/axioms11100557 - 14 Oct 2022

Cited by 1 | Viewed by 1456

Abstract

The goal of this paper is to investigate the curves intersected by a vertical plane with the surfaces based on certain NCP functions. The convexity and differentiability of these curves are studied as well. In most cases, the inflection points of the curves [...] Read more.

The goal of this paper is to investigate the curves intersected by a vertical plane with the surfaces based on certain NCP functions. The convexity and differentiability of these curves are studied as well. In most cases, the inflection points of the curves cannot be expressed exactly. Therefore, we instead estimate the interval where the curves are convex under this situation. Then, with the help of differentiability and convexity, we obtain the local minimum or maximum of the curves accordingly. The study of these curves is very useful to binary quadratic programming. Full article

(This article belongs to the Special Issue Special Issue in Honor of the 60th Birthday of Professor Hong-Kun Xu)

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13 pages, 2550 KiB

Open AccessArticle

A Shortcut Method to Solve for a 1D Heat Conduction Model under Complicated Boundary Conditions

by Ting Wei, Yuezan Tao, Honglei Ren and Fei Lin

Axioms 2022, 11(10), 556; https://doi.org/10.3390/axioms11100556 - 14 Oct 2022

Cited by 4 | Viewed by 2104

Abstract

The function of boundary temperature variation with time, f(t) is generally defined according to measured data. For f(t), which has a complicated expression, a corresponding one-dimensional heat conduction model was constructed under the first type of boundary [...] Read more.

The function of boundary temperature variation with time, f(t) is generally defined according to measured data. For f(t), which has a complicated expression, a corresponding one-dimensional heat conduction model was constructed under the first type of boundary conditions (Dirichlet conditions) in a semi-infinite domain. By taking advantage of the Fourier transform properties, a theoretical solution was given for the model, under the condition that f(t) does not directly participate in the transformation process. The solution consists of the product of erfc(t) and f(0) and the convolution of erfc(t) and the derivative of f(t). The piecewise linear interpolation equation of f(t), based on the measured data of temperature, was substituted into the theoretical solution, thus quickly solving the model and deriving a corresponding analytical solution. Based on the analytical solution under the linear decay function boundary condition, the inflection point method and curve fitting method for calculating the thermal diffusivity were introduced and exemplified, and the variation laws of the appearance moment of the inflection point were discussed. The obtained results show that the values of thermal diffusivity calculated by the two methods are basically consistent, and that the inflection point values rise with the increasing values of the initial temperature variation of the boundary, the decrease in boundary temperature velocity, and the distance from the boundary, respectively. Full article

(This article belongs to the Special Issue Computational Heat Transfer and Fluid Dynamics)

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12 pages, 286 KiB

Open AccessReview

On the Method of Differential Invariants for Solving Higher Order Ordinary Differential Equations

by Winter Sinkala and Molahlehi Charles Kakuli

Axioms 2022, 11(10), 555; https://doi.org/10.3390/axioms11100555 - 14 Oct 2022

Cited by 1 | Viewed by 1539

Abstract

There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an nth-order ODE that admits an r-parameter Lie group

(3 \leq r \leq n)

, there is a powerful method of [...] Read more.

There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an nth-order ODE that admits an r-parameter Lie group

(3 \leq r \leq n)

, there is a powerful method of Lie symmetry analysis by which the ODE is reduced to an

(n - r)

th-order ODE plus r quadratures provided that the Lie algebra formed by the infinitesimal generators of the group is solvable. It would seem this method is not widely appreciated and/or used as it is not mentioned in many related articles centred around integration of higher order ODEs. In the interest of mainstreaming the method, we describe the method in detail and provide four illustrative examples. We use the case of a third-order ODE that admits a three-dimensional solvable Lie algebra to present the gist of the integration algorithm. Full article

(This article belongs to the Special Issue Higher Order Differential Equations)

9 pages, 258 KiB

Open AccessArticle

On Entire Function Solutions to Fermat Delay-Differential Equations

by Xue-Ying Zhang, Ze-Kun Xu and Wei-Ran Lü

Axioms 2022, 11(10), 554; https://doi.org/10.3390/axioms11100554 - 14 Oct 2022

Viewed by 1491

Abstract

This paper concerns the existence and precise expression form of entire solutions to a certain type of delay-differential equation. The significance of our results lie in that we generalize and supplement the related results obtained recently. Full article

(This article belongs to the Special Issue Differential Equations and Genetic Algorithms)

22 pages, 1472 KiB

Open AccessArticle

Statistical Analysis of Alpha Power Exponential Parameters Using Progressive First-Failure Censoring with Applications

by Mazen Nassar, Refah Alotaibi and Ahmed Elshahhat

Axioms 2022, 11(10), 553; https://doi.org/10.3390/axioms11100553 - 13 Oct 2022

Cited by 2 | Viewed by 1745

Abstract

This paper is an endeavor to investigate some estimation problems of the unknown parameters and some reliability measures of the alpha power exponential distribution in the presence of progressive first-failure censored data. In this regard, the classical and Bayesian approaches are considered to [...] Read more.

This paper is an endeavor to investigate some estimation problems of the unknown parameters and some reliability measures of the alpha power exponential distribution in the presence of progressive first-failure censored data. In this regard, the classical and Bayesian approaches are considered to acquire the point and interval estimates of the different quantities. The maximum likelihood approach is proposed to obtain the estimates of the unknown parameters, reliability, and hazard rate functions. The approximate confidence intervals are also considered. The Bayes estimates are obtained by considering both symmetric and asymmetric loss functions. The Bayes estimates and the associated highest posterior density credible intervals are given by applying the Monte Carlo Markov Chain technique. Due to the complexity of the given estimators which cannot be compared theoretically, a simulation study is implemented to compare the performance of the different procedures. In addition, diverse optimality criteria are employed to pick the best progressive censoring plans. Two engineering applications are considered to illustrate the applicability of the offered estimators. The numerical outcomes showed that the Bayes estimates based on symmetric or asymmetric loss functions perform better than other estimates in terms of minimum root mean square errors and interval lengths. Full article

(This article belongs to the Special Issue Computational Statistics & Data Analysis)

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13 pages, 295 KiB

Open AccessArticle

Diffusion Effect in Quantum Hydrodynamics

by Moise Bonilla-Licea, Dieter Schuch and Moises Bonilla Estrada

Axioms 2022, 11(10), 552; https://doi.org/10.3390/axioms11100552 - 13 Oct 2022

Cited by 2 | Viewed by 1728

Abstract

In this paper, we introduce (at least formally) a diffusion effect that is based on an axiom postulated by Werner Heisenberg in the early days of quantum mechanics. His statement was that—in quantum mechanics—kinematical quantities such as velocity must be treated as complex [...] Read more.

In this paper, we introduce (at least formally) a diffusion effect that is based on an axiom postulated by Werner Heisenberg in the early days of quantum mechanics. His statement was that—in quantum mechanics—kinematical quantities such as velocity must be treated as complex matrices. In the hydrodynamic formulation of quantum mechanics according to Madelung, the complex Schrödinger equation is rewritten in terms of two real equations—a continuity equation and a modified Hamilton–Jacobi equation. Considering seriously Heisenberg’s axiom, the velocity occurring in the continuity equation should be replaced by a complex one, thus introducing a diffusion term with an imaginary diffusion coefficient. Therefore, in quantum mechanics, there should be a diffusion effect showing up in the dynamics. This is the case in the time evolution of the free-motion wave packet under time reversal. The maximum returns to the initial position; however, the width of the wave packet does not shrink to its initial width. This effect is obvious but—as far as we know—it is not mentioned in any textbook. In our paper, we discuss this effect in detail and show the connection with a complex version of quantum hydrodynamics. Full article

(This article belongs to the Special Issue Advances in Quantum Theory and Quantum Computing)

13 pages, 1658 KiB

Open AccessFeature PaperArticle

A Multilevel Fuzzy Transform Method for High Resolution Image Compression

by Ferdinando Di Martino and Salvatore Sessa

Axioms 2022, 11(10), 551; https://doi.org/10.3390/axioms11100551 - 13 Oct 2022

Cited by 4 | Viewed by 1768

Abstract

The Multilevel Fuzzy Transform technique (MF-tr) is a hierarchical image compression method based on Fuzzy Transform, which is successfully used to compress images and manage the information loss of the reconstructed image. Unlike other lossy image compression methods, it ensures that the quality [...] Read more.

The Multilevel Fuzzy Transform technique (MF-tr) is a hierarchical image compression method based on Fuzzy Transform, which is successfully used to compress images and manage the information loss of the reconstructed image. Unlike other lossy image compression methods, it ensures that the quality of the reconstructed image is not lower than a prefixed threshold. However, this method is not suitable for compressing massive images due to the high processing times and memory usage. In this paper, we propose a variation of MF-tr for the compression of massive images. The image is divided into tiles, each of which is individually compressed using MF-tr; thereafter, the image is reconstructed by merging the decompressed tiles. Comparative tests performed on remote sensing images show that the proposed method provides better performance than MF-tr in terms of compression rate and CPU time. Moreover, comparison tests show that our method reconstructs the image with CPU times that are at least two times less than those obtained using the MF-tr algorithm. Full article

(This article belongs to the Special Issue Fuzzy Transforms and Their Applications II)

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11 pages, 261 KiB

Open AccessArticle

Price Risk Strategy Analysis for Budget Hotels in the Post-Pandemic Era

by I-Fei Chen, Pi-Ying Kuo and Ruey-Chyn Tsaur

Axioms 2022, 11(10), 550; https://doi.org/10.3390/axioms11100550 - 13 Oct 2022

Viewed by 2081

Abstract

The supply chain of the tourism industry, including air transportation, travel agencies, souvenirs, and hotel services, is almost at a breaking point, causing a rise in unemployment with huge losses during the COVID-19 pandemic period. In order to overcome these losses, we propose [...] Read more.

The supply chain of the tourism industry, including air transportation, travel agencies, souvenirs, and hotel services, is almost at a breaking point, causing a rise in unemployment with huge losses during the COVID-19 pandemic period. In order to overcome these losses, we propose that luxury hotels should consider offering budget hotels at a lower cost but with satisfactory accommodation in order to create some turn-arounds in the post-pandemic era. However, budget hotels that branch off from luxury hotels cannot post the same room rates because there are some uncertain factors that affect the traveler experience when staying in budget hotels. In this study, we define four types of risk factors for the self-selection of the consumer model, and then find that the optimal room price appears to be independent of the performance risk for the service quality, brand image, and shuttle buses, but is dependent on physical risk in terms of priority number risk, the financial risk of refund rates, and the privacy risk of investment in the system. Finally, we discuss how government subsidies can encourage branched budget hotels by describing three sensitivity scenarios. The results show that subsidies that go towards staff training and higher-frequency shuttle buses will cause consumers to book more stays in budget hotels and, thereby, contribute to a higher profit. By lobbying the policy on government subsidies, budget hotels that branch off from luxury hotels are a profitable business model for a reduction in the huge losses occurred during the period of the spread of COVID-19. Full article

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

23 pages, 4350 KiB

Open AccessArticle

MEM and MEM4PP: New Tools Supporting the Parallel Generation of Critical Metrics in the Evaluation of Statistical Models

by Daniel Homocianu and Cristina Tîrnăucă

Axioms 2022, 11(10), 549; https://doi.org/10.3390/axioms11100549 - 12 Oct 2022

Cited by 3 | Viewed by 1905

Abstract

This paper describes MEM and MEM4PP as new Stata tools and commands. They support the automatic reporting and selection of the best regression and classification models by adding supplemental performance metrics based on statistical post-estimation and custom computation. In particular, MEM provides helpful [...] Read more.

This paper describes MEM and MEM4PP as new Stata tools and commands. They support the automatic reporting and selection of the best regression and classification models by adding supplemental performance metrics based on statistical post-estimation and custom computation. In particular, MEM provides helpful metrics, such as the maximum acceptable variance inflation factor (maxAcceptVIF) together with the maximum computed variance inflation factor (maxComputVIF) for ordinary least squares (OLS) regression, the maximum absolute value of the correlation coefficient in the predictors’ correlation matrix (maxAbsVPMCC), the area under the curve of receiving operator characteristics (AUC-ROC), p and chi-squared of the goodness-of-fit (GOF) test for logit and probit, and also the maximum probability thresholds (maxProbNlogPenultThrsh and maxProbNlogLastThrsh) from Zlotnik and Abraira risk-prediction nomograms (nomolog) for logistic regressions. This new tool also performs the automatic identification of the list of variables if run after most regression commands. After simple successive invocations of MEM (in a .do file acting as a batch file), the collectible results are produced in the console or exported to specially designated files (one .csv for all models in a batch). MEM4PP is MEM’s version for parallel processing. It starts from the same batch (the same .do file with its path provided as a parameter) and triggers different instances of Stata to parallelly generate the same results (one .csv for each model in a batch). The paper also includes some examples using real-world data from the World Values Survey (the evidence between 1981 and 2020, version number 1.6). They help us understand how MEM and MEM4PP support the testing of predictor independence, reverse causality checks, the best model selection starting from such metrics, and, ultimately, the replication of all these steps. Full article

(This article belongs to the Special Issue Soft Computing with Applications to Decision Making and Data Mining)

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Axioms, Volume 11, Issue 10 (October 2022) – 90 articles

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