Axioms

58 pages, 639 KiB

Open AccessReview

A Comprehensive Review on the Fejér-Type Inequality Pertaining to Fractional Integral Operators

by Muhammad Tariq, Sotiris K. Ntouyas and Asif Ali Shaikh

Axioms 2023, 12(7), 719; https://doi.org/10.3390/axioms12070719 - 24 Jul 2023

Viewed by 1040

A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented. In the numerous families of convexities, it includes classical convex functions, s-convex functions, quasi-convex functions, strongly convex functions, [...] Read more.

A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented. In the numerous families of convexities, it includes classical convex functions, s-convex functions, quasi-convex functions, strongly convex functions, harmonically convex functions, harmonically quasi-convex functions, quasi-geometrically convex functions, p-convex functions, convexity with respect to strictly monotone function, co-ordinated-convex functions,

(θ, h - m) - p

-convex functions, and h-preinvex functions. Included in the fractional integral operators are Riemann–Liouville fractional integral,

(k - p)

-Riemann–Liouville, k-Riemann–Liouville fractional integral, Riemann–Liouville fractional integrals with respect to another function, the weighted fractional integrals of a function with respect to another function, fractional integral operators with the exponential kernel, Hadamard fractional integral, Raina fractional integral operator, conformable integrals, non-conformable fractional integral, and Katugampola fractional integral. Finally, Fejér-type fractional integral inequalities for invex functions and

(p, q)

-calculus are also included. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inequalities)

14 pages, 294 KiB

Open AccessArticle

Stability of Stochastic Partial Differential Equations

by Allaberen Ashyralyev and Ülker Okur

Axioms 2023, 12(7), 718; https://doi.org/10.3390/axioms12070718 - 24 Jul 2023

Cited by 1 | Viewed by 1041

Abstract

In this paper, we study the stability of the stochastic parabolic differential equation with dependent coefficients. We consider the stability of an abstract Cauchy problem for the solution of certain stochastic parabolic differential equations in a Hilbert space. For the solution of the [...] Read more.

In this paper, we study the stability of the stochastic parabolic differential equation with dependent coefficients. We consider the stability of an abstract Cauchy problem for the solution of certain stochastic parabolic differential equations in a Hilbert space. For the solution of the initial-boundary value problems (IBVPs), we obtain the stability estimates for stochastic parabolic equations with dependent coefficients in specific applications. Full article

(This article belongs to the Topic Numerical Methods for Partial Differential Equations)

24 pages, 830 KiB

Open AccessArticle

Robust Fisher-Regularized Twin Extreme Learning Machine with Capped L₁-Norm for Classification

by Zhenxia Xue and Linchao Cai

Axioms 2023, 12(7), 717; https://doi.org/10.3390/axioms12070717 - 24 Jul 2023

Cited by 2 | Viewed by 1121

Abstract

Twin extreme learning machine (TELM) is a classical and high-efficiency classifier. However, it neglects the statistical knowledge hidden inside the data. In this paper, in order to make full use of statistical information from sample data, we first come up with a Fisher-regularized [...] Read more.

Twin extreme learning machine (TELM) is a classical and high-efficiency classifier. However, it neglects the statistical knowledge hidden inside the data. In this paper, in order to make full use of statistical information from sample data, we first come up with a Fisher-regularized twin extreme learning machine (FTELM) by applying Fisher regularization into TELM learning framework. This strategy not only inherits the advantages of TELM, but also minimizes the within-class divergence of samples. Further, in an effort to further boost the anti-noise ability of FTELM method, we propose a new capped

L_{1}

-norm FTELM (C

L_{1}

-FTELM) by introducing capped

L_{1}

-norm in FTELM to dwindle the influence of abnormal points, and C

L_{1}

-FTELM improves the robust performance of our FTELM. Then, for the proposed FTELM method, we utilize an efficient successive overrelaxation algorithm to solve the corresponding optimization problem. For the proposed C

L_{1}

-FTELM, an iterative method is designed to solve the corresponding optimization based on re-weighted technique. Meanwhile, the convergence and local optimality of C

L_{1}

-FTELM are proved theoretically. Finally, numerical experiments on manual and UCI datasets show that the proposed methods achieve better classification effects than the state-of-the-art methods in most cases, which demonstrates the effectiveness and stability of the proposed methods. Full article

(This article belongs to the Special Issue Application of Machine Learning and Optimization Methods in Engineering Mathematics)

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Four types of data without noise. Full article ">Figure 2
Accuracy for TELM, FTELM, C<math display="inline"><semantics><msub><mi>L</mi><mn>1</mn></msub></semantics></math>-TWSVM, and C<math display="inline"><semantics><msub><mi>L</mi><mn>1</mn></msub></semantics></math>-FTELM on four types of data with 0%, 20%, and 25% noise. Full article ">Figure 3
Accuracy for TELM, FTELM, C<math display="inline"><semantics><msub><mi>L</mi><mn>1</mn></msub></semantics></math>-TWSVM, and C<math display="inline"><semantics><msub><mi>L</mi><mn>1</mn></msub></semantics></math>-FTELM on Circle dataset with noise in different ratios. Full article ">Figure 3 Cont.
Accuracy for TELM, FTELM, C<math display="inline"><semantics><msub><mi>L</mi><mn>1</mn></msub></semantics></math>-TWSVM, and C<math display="inline"><semantics><msub><mi>L</mi><mn>1</mn></msub></semantics></math>-FTELM on Circle dataset with noise in different ratios. Full article ">Figure 4
Accuracies of six algorithms via different noises factors. Full article ">Figure 5
Objective values of C<math display="inline"><semantics><msub><mi>L</mi><mn>1</mn></msub></semantics></math>-FTELM on four datasets. Full article ">Figure 6
Examples of four high-dimensional image datasets. Full article ">

12 pages, 318 KiB

Open AccessArticle

Generalized Bayes Prediction Study Based on Joint Type-II Censoring

by Yahia Abdel-Aty, Mohamed Kayid and Ghadah Alomani

Axioms 2023, 12(7), 716; https://doi.org/10.3390/axioms12070716 - 23 Jul 2023

Cited by 2 | Viewed by 1131

Abstract

In this paper, the problem of predicting future failure times based on a jointly type-II censored sample from k exponential populations is considered. The Bayesian prediction intervals and point predictors were then obtained. Generalized Bayes is a Bayesian study based on a learning [...] Read more.

In this paper, the problem of predicting future failure times based on a jointly type-II censored sample from k exponential populations is considered. The Bayesian prediction intervals and point predictors were then obtained. Generalized Bayes is a Bayesian study based on a learning rate parameter. This study investigated the effects of the learning rate parameters on the prediction results. The loss functions of squared error, Linex, and general entropy were used as point predictors. Monte Carlo simulations were performed to show the effectiveness of the learning rate parameter in improving the results of prediction intervals and point predictors. Full article

(This article belongs to the Special Issue Reliability and Risk of Complex Systems: Modelling, Analysis and Optimization)

21 pages, 619 KiB

Open AccessArticle

Stability Results and Reckoning Fixed Point Approaches by a Faster Iterative Method with an Application

by Hasanen A. Hammad and Doha A. Kattan

Axioms 2023, 12(7), 715; https://doi.org/10.3390/axioms12070715 - 23 Jul 2023

Viewed by 1094

Abstract

In this manuscript, we investigate some convergence and stability results for reckoning fixed points using a faster iterative scheme in a Banach space. Also, weak and strong convergence are discussed for close contraction mappings in a Banach space and for Suzuki generalized nonexpansive [...] Read more.

In this manuscript, we investigate some convergence and stability results for reckoning fixed points using a faster iterative scheme in a Banach space. Also, weak and strong convergence are discussed for close contraction mappings in a Banach space and for Suzuki generalized nonexpansive mapping in a uniformly convex Banach space. Our method opens the door to many expansions in the problems of monotone variational inequalities, image restoration, convex optimization, and split convex feasibility. Moreover, some experimental examples were conducted to gauge the usefulness and efficiency of the technique compared with the iterative methods in the literature. Finally, the proposed approach is applied to solve the nonlinear Volterra integral equation with a delay. Full article

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

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12 pages, 309 KiB

Open AccessArticle

Classes of Harmonic Functions Related to Mittag-Leffler Function

by Abeer A. Al-Dohiman, Basem Aref Frasin, Naci Taşar and Fethiye Müge Sakar

Axioms 2023, 12(7), 714; https://doi.org/10.3390/axioms12070714 - 23 Jul 2023

Cited by 2 | Viewed by 1061

Abstract

The purpose of this paper is to find new inclusion relations of the harmonic class

HF (ϱ, γ)

with the subclasses

S_{HF}^{*},

K_{HF}

and

T N_{HF} (τ)

of harmonic functions by applying the [...] Read more.

The purpose of this paper is to find new inclusion relations of the harmonic class

HF (ϱ, γ)

with the subclasses

S_{HF}^{*},

K_{HF}

and

T N_{HF} (τ)

of harmonic functions by applying the convolution operator

Θ (ℑ)

associated with the Mittag-Leffler function. Further for

ϱ = 0

, several special cases of the main results are also obtained. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

12 pages, 285 KiB

Open AccessArticle

A New Class of Quantile Regression Ratio-Type Estimators for Finite Population Mean in Stratified Random Sampling

by Tuba Koç and Haydar Koç

Axioms 2023, 12(7), 713; https://doi.org/10.3390/axioms12070713 - 22 Jul 2023

Cited by 5 | Viewed by 1496

Abstract

Quantile regression is one of the alternative regression techniques used when the assumptions of classical regression analysis are not met, and it estimates the values of the study variable in various quantiles of the distribution. This study proposes ratio-type estimators of a population [...] Read more.

Quantile regression is one of the alternative regression techniques used when the assumptions of classical regression analysis are not met, and it estimates the values of the study variable in various quantiles of the distribution. This study proposes ratio-type estimators of a population mean using the information on quantile regression for stratified random sampling. The proposed ratio-type estimators are investigated with the help of the mean square error equations. Efficiency comparisons between the proposed estimators and classical estimators are presented in certain conditions. Under these obtained conditions, it is seen that the proposed estimators outperform the classical estimators. In addition, the theoretical results are supported by a real data application. Full article

(This article belongs to the Section Mathematical Analysis)

13 pages, 294 KiB

Open AccessArticle

Generalized Cauchy–Schwarz Inequalities and A-Numerical Radius Applications

by Najla Altwaijry, Kais Feki and Shigeru Furuichi

Axioms 2023, 12(7), 712; https://doi.org/10.3390/axioms12070712 - 22 Jul 2023

Viewed by 1374

Abstract

The purpose of this research paper is to introduce new Cauchy–Schwarz inequalities that are valid in semi-Hilbert spaces, which are generalizations of Hilbert spaces. We demonstrate how these new inequalities can be employed to derive novel A-numerical radius inequalities, where A denotes [...] Read more.

The purpose of this research paper is to introduce new Cauchy–Schwarz inequalities that are valid in semi-Hilbert spaces, which are generalizations of Hilbert spaces. We demonstrate how these new inequalities can be employed to derive novel A-numerical radius inequalities, where A denotes a positive semidefinite operator in a complex Hilbert space. Some of our novel A-numerical radius inequalities expand upon the existing literature on numerical radius inequalities with Hilbert space operators, which are important tools in functional analysis. We use techniques from semi-Hilbert space theory to prove our results and highlight some applications of our findings. Full article

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

19 pages, 10268 KiB

Open AccessArticle

Solving Location Assignment and Order Picker-Routing Problems in Warehouse Management

by Johanna Bolaños-Zuñiga, M. Angélica Salazar-Aguilar and Jania Astrid Saucedo-Martínez

Axioms 2023, 12(7), 711; https://doi.org/10.3390/axioms12070711 - 22 Jul 2023

Cited by 1 | Viewed by 2228

Abstract

One of the critical warehousing processes is the order-picking process. This activity consists of retrieving items from their storage locations to fulfill the demand specified in the pick lists. Therefore, the storage location assignment affects the picking time and, consequently, reduces the operating [...] Read more.

One of the critical warehousing processes is the order-picking process. This activity consists of retrieving items from their storage locations to fulfill the demand specified in the pick lists. Therefore, the storage location assignment affects the picking time and, consequently, reduces the operating costs of the warehouse. This work presents two alternative mixed-integer linear models and an adaptive multi-start heuristic (AMH) for solving the integrated storage location and picker-routing problem. The problem considers a warehouse with a general layout and precedence constraints for picking according to the products weight. Experimental work confirms the efficiency of the proposed reformulations since we found out a total of 334 tested instances and optimal solutions for 51 new cases and 62 new feasible solutions. The proposed AMH improved more than 29% of the best-known solutions and required an average execution time of 117 s. Consequently, our proposed algorithm is an attractive decision-making tool to achieve efficiency when solving practical situations in a warehouse. Full article

(This article belongs to the Special Issue Applied Mathematical Modeling and Optimization)

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21 pages, 481 KiB

Open AccessArticle

Constant-Stress Modeling of Log-Normal Data under Progressive Type-I Interval Censoring: Maximum Likelihood and Bayesian Estimation Approaches

by Mohamed Sief, Xinsheng Liu, Mona Hosny and Abd El-Raheem M. Abd El-Raheem

Axioms 2023, 12(7), 710; https://doi.org/10.3390/axioms12070710 - 21 Jul 2023

Cited by 1 | Viewed by 1248

Abstract

This paper discusses inferential approaches for the problem of constant-stress accelerated life testing when the failure data are progressive type-I interval censored. Both frequentist and Bayesian estimations are carried out under the assumption that the log-normal location parameter is nonconstant and follows a [...] Read more.

This paper discusses inferential approaches for the problem of constant-stress accelerated life testing when the failure data are progressive type-I interval censored. Both frequentist and Bayesian estimations are carried out under the assumption that the log-normal location parameter is nonconstant and follows a log-linear life-stress model. The confidence intervals of unknown parameters are also constructed based on asymptotic theory and Bayesian techniques. An analysis of a real data set is combined with a Monte Carlo simulation to provide a thorough assessment of the proposed methods. Full article

(This article belongs to the Special Issue Mathematical and Statistical Analysis Methods with Multidisciplinary Applications)

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18 pages, 853 KiB

Open AccessArticle

Application of Polling Scheduling in Mobile Edge Computing

by Xiong Wang, Zhijun Yang and Hongwei Ding

Axioms 2023, 12(7), 709; https://doi.org/10.3390/axioms12070709 - 21 Jul 2023

Cited by 5 | Viewed by 1534

Abstract

With the Internet of Things (IoT) development, there is an increasing demand for multi-service scheduling for Mobile Edge Computing (MEC). We propose using polling for scheduling in edge computing to accommodate multi-service scheduling methods better. Given the complexity of asymmetric polling systems, we [...] Read more.

With the Internet of Things (IoT) development, there is an increasing demand for multi-service scheduling for Mobile Edge Computing (MEC). We propose using polling for scheduling in edge computing to accommodate multi-service scheduling methods better. Given the complexity of asymmetric polling systems, we have used an information-theoretic approach to analyse the model. Firstly, we propose an asymmetric two-level scheduling approach with priority based on a polling scheduling approach. Secondly, the mathematical model of the system in the continuous time state is established by using the embedded Markov chain theory and the probability-generating function. By solving for the probability-generating function’s first-order partial and second-order partial derivatives, we calculate the exact expressions of the average queue length, the average polling period, and the average delay with an approximate analysis of periodic query way. Finally, we design a simulation experiment to verify that our derived parameters are correct. Our proposed model can better differentiate priorities in MEC scheduling and meet the needs of IoT multi-service scheduling. Full article

(This article belongs to the Special Issue Advances in Mathematical Methods and Applications for High-Performance Computing)

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28 pages, 3020 KiB

Open AccessArticle

An Analytic Solution for 2D Heat Conduction Problems with Space–Time-Dependent Dirichlet Boundary Conditions and Heat Sources

by Heng-Pin Hsu, Jer-Rong Chang, Chih-Yuan Weng and Chun-Jung Huang

Axioms 2023, 12(7), 708; https://doi.org/10.3390/axioms12070708 - 20 Jul 2023

Viewed by 2216

Abstract

This study proposes a closed-form solution for the two-dimensional (2D) transient heat conduction in a rectangular cross-section of an infinite bar with space–time-dependent Dirichlet boundary conditions and heat sources. The main purpose of this study is to eliminate the limitations of the previous [...] Read more.

This study proposes a closed-form solution for the two-dimensional (2D) transient heat conduction in a rectangular cross-section of an infinite bar with space–time-dependent Dirichlet boundary conditions and heat sources. The main purpose of this study is to eliminate the limitations of the previous study and add heat sources to the heat conduction system. The restriction of the previous study is that the values of the boundary conditions and initial conditions at the four corners of the rectangular region should be zero. First, the boundary value problem of 2D heat conduction system is transformed into a dimensionless form. Second, the dimensionless temperature function is transformed so that the temperatures at the four endpoints of the boundary of the rectangular region become zero. Dividing the system into two one-dimensional (1D) subsystems and solving them by combining the proposed shifting function method with the eigenfunction expansion theorem, the complete solution in series form is obtained through the superposition of the subsystem solutions. Three examples are studied to illustrate the efficiency and reliability of the method. For convenience, the space–time-dependent functions used in the examples are considered separable in the space–time domain. The linear, parabolic, and sine functions are chosen as the space-dependent functions, and the sine, cosine, and exponential functions are chosen as the time-dependent functions. The solutions in the literature are used to verify the correctness of the solutions derived using the proposed method, and the results are completely consistent. The parameter influence of the time-dependent function of the boundary conditions and heat sources on the temperature variation is also investigated. The time-dependent function includes exponential type and harmonic type. For the exponential time-dependent function, a smaller decay constant of the time-dependent function leads to a greater temperature drop. For the harmonic time-dependent function, a higher frequency of the time-dependent function leads to a more frequent fluctuation of the temperature change. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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The 2D heat conduction in a rectangular cross-section of an infinite bar with space–time-dependent Dirichlet boundary conditions and heat sources: (a) an infinite bar with a rectangular cross-section and heat source; (b) the 2D heat conduction problem in a rectangular region (cross-section). Full article ">Figure 1 Cont.
The 2D heat conduction in a rectangular cross-section of an infinite bar with space–time-dependent Dirichlet boundary conditions and heat sources: (a) an infinite bar with a rectangular cross-section and heat source; (b) the 2D heat conduction problem in a rectangular region (cross-section). Full article ">Figure 2
A 2D heat conduction system after the variable transformation of temperature: (a) dimensionless form of a physical system; (b) transformation results for dimensionless systems. Full article ">Figure 3
The two subsystems of the 2D heat conduction system with space–time-dependent Dirichlet boundary conditions and heat source in the medium: (a) for subsystem A; (b) for subsystem B. Full article ">Figure 4
Temperature variation in the middle of the rectangular region with different decay constants of exponential-type time-dependent functions (case 1 of Example 3): (a) for different decay constants of the boundary conditions; (b) for different decay constants of the heat source. Full article ">Figure 5
Temperature variation in the middle of the rectangular region with different decay constants of harmonic-type time-dependent functions (case 2 of Example 3): (a) for different decay constants of the boundary conditions; (b) for different decay constants of the heat source. Full article ">

22 pages, 6569 KiB

Open AccessArticle

Proposed Shaft Coupling Based on RPRRR Mechanism: Positional Analysis and Consequences

by Stelian Alaci, Ioan Doroftei, Florina-Carmen Ciornei, Ionut-Cristian Romanu, Toma-Marian Ciocirlan and Mariana-Catalina Ciornei

Axioms 2023, 12(7), 707; https://doi.org/10.3390/axioms12070707 - 20 Jul 2023

Viewed by 993

Abstract

This study proposes a solution for the transmission of rotation motion between two shafts with crossed directions. For constructive simplicity, the solutions including the planar pair were preferred and, from the two variants, namely structurally symmetric, revolute–planar–revolute (RPR), or asymmetric RRP, the last [...] Read more.

This study proposes a solution for the transmission of rotation motion between two shafts with crossed directions. For constructive simplicity, the solutions including the planar pair were preferred and, from the two variants, namely structurally symmetric, revolute–planar–revolute (RPR), or asymmetric RRP, the last was selected. The resulting solution, RPRRR, is a non-Denavit–Hartenberg (non-D–H) mechanism. The D–H methodology is laborious since the structure of the equivalent mechanism is more complex than the actual one. For this reason, in the present paper, the kinematic analysis of the mechanism uses geometrical conditions of existence of the planar pair. The system is solved analytically and two main conclusions result: for a set of constructive data and a stipulated position of the driving element, two different assembling positions exist and a rotation motion occurs in the final revolute joint, but in the internal revolute pairs, the motion is oscillatory. The correctness of the theoretical results was corroborated by a CATIA model. The mechanism was also constructed and smooth running was noticed. Two main concerns were considered for the design of the mechanism: avoiding mechanical interference between the elements and estimating the stresses and deformations. Full article

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

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8 pages, 232 KiB

Open AccessArticle

Poisson Bracket Filter for the Effective Lagrangians

by Katalin Gambár and Ferenc Márkus

Axioms 2023, 12(7), 706; https://doi.org/10.3390/axioms12070706 - 20 Jul 2023

Viewed by 995

Abstract

One might think that a Lagrangian function of any form is suitable for a complete description of a process. Indeed, it does not matter in terms of the equations of motion, but it seems that this is not enough. Expressions with Poisson brackets [...] Read more.

One might think that a Lagrangian function of any form is suitable for a complete description of a process. Indeed, it does not matter in terms of the equations of motion, but it seems that this is not enough. Expressions with Poisson brackets are displayed as required fulfillment filters. In the case of the Schrödinger equation for a free particle, we show what we have to be careful about. Full article

(This article belongs to the Section Mathematical Physics)

14 pages, 285 KiB

Open AccessArticle

On Sufficiency Conditions for Some Robust Variational Control Problems

by Tareq Saeed and Savin Treanţă

Axioms 2023, 12(7), 705; https://doi.org/10.3390/axioms12070705 - 20 Jul 2023

Cited by 3 | Viewed by 855

Abstract

We study the sufficient optimality conditions for a class of fractional variational control problems involving data uncertainty in the cost functional. Concretely, by using the parametric technique, we prove the sufficiency of the robust necessary optimality conditions by considering convexity, quasi-convexity, strictly quasi-convexity, [...] Read more.

We study the sufficient optimality conditions for a class of fractional variational control problems involving data uncertainty in the cost functional. Concretely, by using the parametric technique, we prove the sufficiency of the robust necessary optimality conditions by considering convexity, quasi-convexity, strictly quasi-convexity, and/or monotonic quasi-convexity assumptions of the involved functionals. Full article

(This article belongs to the Special Issue New Developments in Analysis of Variational Inequalities and Related Fields)

15 pages, 760 KiB

Open AccessArticle

Geometric Properties of Planar and Spherical Interception Curves

by Yagub N. Aliyev

Axioms 2023, 12(7), 704; https://doi.org/10.3390/axioms12070704 - 20 Jul 2023

Viewed by 1614

Abstract

In this paper, some geometric properties of the plane interception curve defined by a nonlinear ordinary differential equation are discussed. Its parametric representation is used to find the limits of some triangle elements associated with the curve. These limits have some connections with [...] Read more.

In this paper, some geometric properties of the plane interception curve defined by a nonlinear ordinary differential equation are discussed. Its parametric representation is used to find the limits of some triangle elements associated with the curve. These limits have some connections with the lemniscate constants

A, B

and Gauss’s constant G, which are used to compare with the classical pursuit curve. The analogous spherical geometry problem is solved using a spherical curve defined by the Gudermannian function. It is shown that the results agree with the angle-preserving property of Mercator and Stereographic projections. The Mercator and Stereographic projections also reveal the symmetry of this curve with respect to Spherical and Logarithmic Spirals. The geometric properties of the spherical curve are proved in two ways, analytically and using a lemma about spherical angles. A similar lemma for the planar case is also mentioned. The paper shows symmetry/asymmetry between the spherical and planar cases and the derivation of properties of these curves as limiting cases of some plane and spherical geometry results. Full article

(This article belongs to the Special Issue Advances in Mathematics and Its Applications, 2nd Edition)

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Interception curve (blue) on a Plane. Its tangent line (black), the line containing its position vector (black), and the line <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (blue) are also shown. Full article ">Figure 2
Comparison of interception (left, red) and pursuit (right, red) curves. Created using parametrization (5) and Bouguer’s formula. The tangent lines (green), the lines containing the position vectors (thin blue) of the curves, and lines <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (thick blue) are also shown. Full article ">Figure 3
Interception curve (light blue) on a Unit Sphere. Its tangent great circle (dark blue), meridian (red) and equator (black) great circles are also shown. Full article ">Figure 4
Mercator and stereographic projections of the spherical interception curve (blue) and spherical spiral (red). The other colors are explained in the text and in <a href="#app1-axioms-12-00704" class="html-app">Appendix A</a>, <a href="#axioms-12-00704-t0A1" class="html-table">Table A1</a>. Full article ">Figure 5
Lemma 1 for spherical case. The green small circle is used in Lemma 2. Full article ">Figure 6
Proof of Lemma 1. Full article ">Figure 7
Lemma 3 for planar case. Full article ">

21 pages, 2003 KiB

Open AccessArticle

The Zakharov–Shabat Spectral Problem for Complexification and Perturbation of the Korteweg–de Vries Equation

by Tatyana V. Redkina, Arthur R. Zakinyan and Robert G. Zakinyan

Axioms 2023, 12(7), 703; https://doi.org/10.3390/axioms12070703 - 19 Jul 2023

Viewed by 1066

Abstract

In this paper we consider examples of complex expansion (cKdV) and perturbation (pKdV) of the Korteweg–de Vries equation (KdV) and show that these equations have a representation in the form of the zero-curvature equation. In this case, we use the Lie algebra of [...] Read more.

In this paper we consider examples of complex expansion (cKdV) and perturbation (pKdV) of the Korteweg–de Vries equation (KdV) and show that these equations have a representation in the form of the zero-curvature equation. In this case, we use the Lie algebra of 4-dimensional quadratic nilpotent matrices. Moreover, it is shown that the simplest possible matrix representation of this algebra leads to the possibility of constructing a countable number of conservation laws for these equations. Full article

(This article belongs to the Special Issue Applied Nonlinear Dynamical Systems in Mathematical Physics)

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27 pages, 9540 KiB

Open AccessArticle

UAV Path Planning Based on an Improved Chimp Optimization Algorithm

by Qinglong Chen, Qing He and Damin Zhang

Axioms 2023, 12(7), 702; https://doi.org/10.3390/axioms12070702 - 19 Jul 2023

Cited by 9 | Viewed by 1941

Abstract

Path planning is one of the key issues in the research of unmanned aerial vehicle technology. Its purpose is to find the best path between the starting point and the destination. Although there are many research recommendations on UAV path planning in the [...] Read more.

Path planning is one of the key issues in the research of unmanned aerial vehicle technology. Its purpose is to find the best path between the starting point and the destination. Although there are many research recommendations on UAV path planning in the literature, there is a lack of path optimization methods that consider both the complex flight environment and the performance constraints of the UAV itself. We propose an enhanced version of the Chimp Optimization Algorithm (TRS-ChOA) to solve the UAV path planning problem in a 3D environment. Firstly, we combine the differential mutation operator to enhance the search capability of the algorithm and prevent premature convergence. Secondly, we use improved reverse learning to expand the search range of the algorithm, effectively preventing the algorithm from missing high-quality solutions. Finally, we propose a similarity preference weight to prevent individuals from over-assimilation and enhance the algorithm’s ability to escape local optima. Through testing on 13 benchmark functions and 29 CEC2017 complex functions, TRS-ChOA demonstrates superior optimization capability and robustness compared to other algorithms. We apply TRS-ChOA along with five well-known algorithms to solve path planning problems in three 3D environments. The experimental results reveal that TRS-ChOA reduces the average path length/fitness value by 23.4%/65.0%, 8.6%/81.0%, and 16.3%/41.7% compared to other algorithms in the three environments, respectively. This indicates that the flight paths planned by TRS-ChOA are more cost-effective, smoother, and safer. Full article

(This article belongs to the Special Issue Automatic Design of Optimization Algorithms and Their Practical Applications)

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24 pages, 5231 KiB

Open AccessArticle

Optimal Decision for Repairable Products Sale and Warranty under Two-Dimensional Deterioration with Consideration of Production Capacity and Customers’ Heterogeneity

by Ming-Nan Chen and Chih-Chiang Fang

Axioms 2023, 12(7), 701; https://doi.org/10.3390/axioms12070701 - 19 Jul 2023

Cited by 1 | Viewed by 1136

Abstract

An effective warranty policy is not only an obligation for the manufacturer or vendor, but it also enhances the willingness of customers to purchase from them in the future. To earn more customers and increase sales, manufacturers or vendors should be inclined to [...] Read more.

An effective warranty policy is not only an obligation for the manufacturer or vendor, but it also enhances the willingness of customers to purchase from them in the future. To earn more customers and increase sales, manufacturers or vendors should be inclined to prolong the service life of their products as an effort to gain more customers. Nevertheless, manufacturers or vendors will not provide a boundless warranty in order to dominate the market, since the related warranty costs will eventually exceed the profits in the end. Therefore, it is a question of weighing the advantage of extending the warranty term in order to earn the trust of new customers against the investment. In addition, since deterioration depends on both time and usage, the deterioration estimation for durable products may be incorrect when considering only one factor. For such problems, a two-dimensional deterioration model is suitable, and the failure times are drawn from a non-homogeneous Poisson process (NHPP). Moreover, customers’ heterogeneity, manufacturers’ production capacity, and preventive maintenance services are also considered in this study. A mathematical model with the corresponding solution algorithm is proposed to assist manufacturers in making systematic decisions about pricing, production, and warranty. Finally, managerial implications are also provided for refining related decision-making. Full article

(This article belongs to the Special Issue Reliability and Risk of Complex Systems: Modelling, Analysis and Optimization)

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14 pages, 327 KiB

Open AccessArticle

Convergence of Collocation Methods for One Class of Impulsive Delay Differential Equations

by Zhiwei Wang, Guilai Zhang and Yang Sun

Axioms 2023, 12(7), 700; https://doi.org/10.3390/axioms12070700 - 19 Jul 2023

Cited by 2 | Viewed by 948

Abstract

This paper is concerned with collocation methods for one class of impulsive delay differential equations (IDDEs). Some results for the convergence, global superconvergence and local superconvergence of collocation methods are given. We choose a suitable piecewise continuous collocation space to obtain high-order numerical [...] Read more.

This paper is concerned with collocation methods for one class of impulsive delay differential equations (IDDEs). Some results for the convergence, global superconvergence and local superconvergence of collocation methods are given. We choose a suitable piecewise continuous collocation space to obtain high-order numerical methods. Some illustrative examples are given to verify the theoretical results. Full article

(This article belongs to the Special Issue Differential Equations and Inverse Problems)

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17 pages, 1227 KiB

Open AccessReview

Topological Comparison of Some Dimension Reduction Methods Using Persistent Homology on EEG Data

by Eddy Kwessi

Axioms 2023, 12(7), 699; https://doi.org/10.3390/axioms12070699 - 18 Jul 2023

Cited by 1 | Viewed by 1271

Abstract

In this paper, we explore how to use topological tools to compare dimension reduction methods. We first make a brief overview of some of the methods often used in dimension reduction such as isometric feature mapping, Laplacian Eigenmaps, fast independent component analysis, kernel [...] Read more.

In this paper, we explore how to use topological tools to compare dimension reduction methods. We first make a brief overview of some of the methods often used in dimension reduction such as isometric feature mapping, Laplacian Eigenmaps, fast independent component analysis, kernel ridge regression, and t-distributed stochastic neighbor embedding. We then give a brief overview of some of the topological notions used in topological data analysis, such as barcodes, persistent homology, and Wasserstein distance. Theoretically, when these methods are applied on a data set, they can be interpreted differently. From EEG data embedded into a manifold of high dimension, we discuss these methods and we compare them across persistent homologies of dimensions 0, 1, and 2, that is, across connected components, tunnels and holes, shells around voids, or cavities. We find that from three dimension clouds of points, it is not clear how distinct from each other the methods are, but Wasserstein and Bottleneck distances, topological tests of hypothesis, and various methods show that the methods qualitatively and significantly differ across homologies. We can infer from this analysis that topological persistent homologies do change dramatically at seizure, a finding already obtained in previous analyses. This suggests that looking at changes in homology landscapes could be a predictor of seizure. Full article

(This article belongs to the Special Issue Recent Advances in Special Functions and Applications)

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22 pages, 352 KiB

Open AccessArticle

N-Widths of Multivariate Sobolev Spaces with Common Smoothness in Probabilistic and Average Settings in the S_q Norm

by Yuqi Liu, Xuehua Li and Huan Li

Axioms 2023, 12(7), 698; https://doi.org/10.3390/axioms12070698 - 17 Jul 2023

Cited by 1 | Viewed by 1041

Abstract

In this article, we give the sharp bounds of probabilistic Kolmogorov

(N, δ)

-widths and probabilistic linear

(N, δ)

-widths of the multivariate Sobolev space

W_{2}^{A}

with common smoothness on a

S_{q}

norm equipped with the Gaussian measure [...] Read more.

In this article, we give the sharp bounds of probabilistic Kolmogorov

(N, δ)

-widths and probabilistic linear

(N, δ)

-widths of the multivariate Sobolev space

W_{2}^{A}

with common smoothness on a

S_{q}

norm equipped with the Gaussian measure

μ

, where

A \subset R^{d}

is a finite set. And we obtain the sharp bounds of average width from the results of the probabilistic widths. These results develop the theory of approximation of functions and play important roles in the research of related approximation algorithms for Sobolev spaces. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

26 pages, 1211 KiB

Open AccessArticle

A Novel Weld-Seam Defect Detection Algorithm Based on the S-YOLO Model

by Yi Zhang and Qingjian Ni

Axioms 2023, 12(7), 697; https://doi.org/10.3390/axioms12070697 - 17 Jul 2023

Cited by 9 | Viewed by 2737

Abstract

Detecting small targets and handling target occlusion and overlap are critical challenges in weld defect detection. In this paper, we propose the S-YOLO model, a novel weld defect detection method based on the YOLOv8-nano model and several mathematical techniques, specifically tailored to address [...] Read more.

Detecting small targets and handling target occlusion and overlap are critical challenges in weld defect detection. In this paper, we propose the S-YOLO model, a novel weld defect detection method based on the YOLOv8-nano model and several mathematical techniques, specifically tailored to address these issues. Our approach includes several key contributions. Firstly, we introduce omni-dimensional dynamic convolution, which is sensitive to small targets, for improved feature extraction. Secondly, the NAM attention mechanism enhances feature representation in the region of interest. NAM computes the channel-wise and spatial-wise attention weights by matrix multiplications and element-wise operations, and then applies them to the feature maps. Additionally, we replace the SPPF module with a context augmentation module to improve feature map resolution and quality. To minimize information loss, we utilize Carafe upsampling instead of the conventional upsampling operations. Furthermore, we use a loss function that combines IoU, binary cross-entropy, and focal loss to improve bounding box regression and object classification. We use stochastic gradient descent (SGD) with momentum and weight decay to update the parameters of our model. Through rigorous experimental validation, our S-YOLO model demonstrates outstanding accuracy and efficiency in weld defect detection. It effectively tackles the challenges of small target detection, target occlusion, and target overlap. Notably, the proposed model achieves an impressive 8.9% improvement in mean Average Precision (mAP) compared to the native model. Full article

(This article belongs to the Special Issue Mathematical Modeling of Signal Processing and Analysis in Light of Deep Learning)

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16 pages, 5989 KiB

Open AccessArticle

Simulation Analysis of a Novel Digital Pump with Direct Recycling of Hydraulic Energy

by Daling Yue, Xiukun Zuo, Zengguang Liu, Yinshui Liu, Liejiang Wei, Jisu Sun and Hongfei Gao

Axioms 2023, 12(7), 696; https://doi.org/10.3390/axioms12070696 - 17 Jul 2023

Viewed by 1401

Abstract

There is a permanent and strong need for energy recovery to improve the efficiency of the hydraulic system in the field of the construction machinery. In addition, the digital pump will become powerful and versatile by employing different configurations and intelligent control of [...] Read more.

There is a permanent and strong need for energy recovery to improve the efficiency of the hydraulic system in the field of the construction machinery. In addition, the digital pump will become powerful and versatile by employing different configurations and intelligent control of the flow distribution valves. Considering this case, we have proposed a novel digital pump in which every plunger is equipped with two flow distribution valves. By controlling these two valves, external hydraulic energy can be directly reused without other components. Based on the structure and working principle of the digital pump, the mathematical model is established and three working modes are detailed. To verify the feasibility and correctness of control methods, a performance simulation testing platform including a digital pump, load module, hydraulic energy to be recovered, and controller module was developed in AMESim R15 software. The pressure, flow rate, and torque simulations of the digital pump in three working modes were carried out. The simulation results have shown that the digital pump not only can be used as an ordinary pump but also has the function of recovery and immediate reutilization of another hydraulic energy. Meanwhile, the corresponding variable displacement control strategy is effective and the positive torque required to drive the digital pump can be reduced, which verified the energy-saving of this scheme. The ideas and contents in this paper can offer significant references for energy conservation technology of various engineering machineries and the intensive study of digital hydraulics. Full article

(This article belongs to the Special Issue Advances of Mathematical Modeling in Fluid Mechanics)

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18 pages, 4559 KiB

Open AccessArticle

Stabilization Control for a Class of Fractional-Order HIV-1 Infection Model with Time Delays

by Zitong Li and Zhe Zhang

Axioms 2023, 12(7), 695; https://doi.org/10.3390/axioms12070695 - 17 Jul 2023

Cited by 2 | Viewed by 1093

Abstract

In this study, we investigated a novel asymptotic stabilization control method for a fractional-order HIV-1 infection model. First, we constructed a mathematical model of the fractional-order HIV-1 infection using the state-space equations of Caputo fractional calculus. Subsequently, a new control strategy was designed [...] Read more.

In this study, we investigated a novel asymptotic stabilization control method for a fractional-order HIV-1 infection model. First, we constructed a mathematical model of the fractional-order HIV-1 infection using the state-space equations of Caputo fractional calculus. Subsequently, a new control strategy was designed for the fractional-order HIV-1 infection model, and the corresponding asymptotic stabilization criterion was proposed by combining a novel vector Lyapunov function with the M-matrix method. Additionally, we incorporated a time delay, which was generated by the interaction between different variables in the actual system, into the fractional-order HIV-1 infection model, forming a system with a time delay. Based on the vector Lyapunov function associated with the M-matrix measure and Razumikhin interpretation, a control strategy was developed for the fractional-order HIV-1 infection model with a time delay. Finally, we show the results of two numerical simulations of the fractional-order HIV-1 infection model, with and without time delay, to illustrate the accuracy, usefulness, and universality of the proposed measure in our paper. Full article

(This article belongs to the Special Issue Advances in Fractional Order Information Measures and Applications)

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22 pages, 3547 KiB

Open AccessArticle

Exact and Approximate Solutions for Linear and Nonlinear Partial Differential Equations via Laplace Residual Power Series Method

by Haneen Khresat, Ahmad El-Ajou, Shrideh Al-Omari, Sharifah E. Alhazmi and Moa’ath N. Oqielat

Axioms 2023, 12(7), 694; https://doi.org/10.3390/axioms12070694 - 17 Jul 2023

Cited by 6 | Viewed by 1866

Abstract

The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize the Laplace residual power series method to generate analytic solutions to various kinds [...] Read more.

The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize the Laplace residual power series method to generate analytic solutions to various kinds of partial differential equations. Then, by resorting to the above-mentioned technique, we derive certain solutions to different types of linear and nonlinear partial differential equations, including wave equations, nonhomogeneous space telegraph equations, water wave partial differential equations, Klein–Gordon partial differential equations, Fisher equations, and a few others. Moreover, we numerically examine several results by investing some graphs and tables and comparing our results with the exact solutions of some nominated differential equations to display the new approach’s reliability, capability, and efficiency. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics)

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17 pages, 3818 KiB

Open AccessArticle

Modified Wild Horse Optimizer for Constrained System Reliability Optimization

by Anuj Kumar, Sangeeta Pant, Manoj K. Singh, Shshank Chaube, Mangey Ram and Akshay Kumar

Axioms 2023, 12(7), 693; https://doi.org/10.3390/axioms12070693 - 16 Jul 2023

Cited by 9 | Viewed by 1452

Abstract

The last few decades have witnessed advancements in intelligent metaheuristic approaches and system reliability optimization. The huge progress in metaheuristic approaches can be viewed as the main motivator behind further refinement in the system reliability optimization process. Researchers have intensively studied system reliability [...] Read more.

The last few decades have witnessed advancements in intelligent metaheuristic approaches and system reliability optimization. The huge progress in metaheuristic approaches can be viewed as the main motivator behind further refinement in the system reliability optimization process. Researchers have intensively studied system reliability optimization problems (SROPs) to obtain the optimal system design with several constraints in order to optimize the overall system reliability. This article proposes a modified wild horse optimizer (MWHO) for SROPs and investigates the reliability allocation of two complex SROPs, namely, complex bridge system (CBS) and life support system in space capsule (LSSSC), with the help of the same process. The effectiveness of this framework based on MWHO is demonstrated by comparing the results obtained with the results available in the literature. The proposed MWHO algorithm shows better efficiency, as it provides superior solutions to SROPs. Full article

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20 pages, 330 KiB

Open AccessArticle

Interval-Valued Topology on Soft Sets

by Sadi Bayramov, Çiğdem Gündüz Aras and Ljubiša D. R. Kočinac

Axioms 2023, 12(7), 692; https://doi.org/10.3390/axioms12070692 - 16 Jul 2023

Cited by 1 | Viewed by 1062

Abstract

In this paper, we study the concept of interval-valued fuzzy set on the family

SS (X, E)

of all soft sets over X with the set of parameters E and examine its basic properties. Later, we define the concept of interval-valued fuzzy [...] Read more.

In this paper, we study the concept of interval-valued fuzzy set on the family

SS (X, E)

of all soft sets over X with the set of parameters E and examine its basic properties. Later, we define the concept of interval-valued fuzzy topology (cotopology)

τ

on

SS (X, E)

. We obtain that each interval-valued fuzzy topology is a descending family of soft topologies. In addition, we study some topological structures such as interval-valued fuzzy neighborhood system of a soft point, base and subbase of

τ

and investigate some relationships among them. Finally, we give some concepts such as direct sum, open mapping and continuous mapping and consider connections between them. A few examples support the presented results. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

26 pages, 405 KiB

Open AccessArticle

Some New Bullen-Type Inequalities Obtained via Fractional Integral Operators

by Asfand Fahad, Saad Ihsaan Butt, Bahtiyar Bayraktar, Mehran Anwar and Yuanheng Wang

Axioms 2023, 12(7), 691; https://doi.org/10.3390/axioms12070691 - 16 Jul 2023

Cited by 8 | Viewed by 1753

Abstract

In this paper, we establish a new auxiliary identity of the Bullen type for twice-differentiable functions in terms of fractional integral operators. Based on this new identity, some generalized Bullen-type inequalities are obtained by employing convexity properties. Concrete examples are given to illustrate [...] Read more.

In this paper, we establish a new auxiliary identity of the Bullen type for twice-differentiable functions in terms of fractional integral operators. Based on this new identity, some generalized Bullen-type inequalities are obtained by employing convexity properties. Concrete examples are given to illustrate the results, and the correctness is confirmed by graphical analysis. An analysis is provided on the estimations of bounds. According to calculations, improved Hölder and power mean inequalities give better upper-bound results than classical inequalities. Lastly, some applications to quadrature rules, modified Bessel functions and digamma functions are provided as well. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

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21 pages, 10003 KiB

Open AccessArticle

Analysis of WE Parameters of Life Using Adaptive-Progressively Type-II Hybrid Censored Mechanical Equipment Data

by Ahmed Elshahhat, Ehab M. Almetwally, Sanku Dey and Heba S. Mohammed

Axioms 2023, 12(7), 690; https://doi.org/10.3390/axioms12070690 - 16 Jul 2023

Cited by 2 | Viewed by 1138

Abstract

A new two-parameter weighted-exponential (WE) distribution, as a beneficial competitor model to other lifetime distributions, namely: generalized exponential, gamma, and Weibull distributions, is studied in the presence of adaptive progressive Type-II hybrid data. Thus, based on different frequentist and Bayesian estimation methods, we [...] Read more.

A new two-parameter weighted-exponential (WE) distribution, as a beneficial competitor model to other lifetime distributions, namely: generalized exponential, gamma, and Weibull distributions, is studied in the presence of adaptive progressive Type-II hybrid data. Thus, based on different frequentist and Bayesian estimation methods, we study the inferential problem of the WE parameters as well as related reliability indices, including survival and failure functions. In frequentist setups, besides the standard likelihood-based estimation, the product of spacing (PS) approach is also taken into account for estimating all unknown parameters of life. Making use of the delta method and the observed Fisher information of the frequentist estimators, approximated asymptotic confidence intervals for all unknown parameters are acquired. In Bayes methodology, from the squared-error loss with independent gamma density priors, the point and interval estimates of the unknown parameters are offered using both joint likelihood and the product of spacings functions. Because a closed solution to the Bayes estimators is not accessible, the Metropolis–Hastings sampler is presented to approximate the Bayes estimates and also to create their associated highest interval posterior density estimates. To figure out the effectiveness of the developed approaches, extensive Monte Carlo experiments are implemented. To highlight the applicability of the offered methodologies in practice, one real-life data set consisting of 30 failure times of repairable mechanical equipment is analyzed. This application demonstrated that the offered WE model provides a better fit compared to the other eight lifetime models. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)

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Axioms, Volume 12, Issue 7 (July 2023) – 109 articles

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