Axioms

25 pages, 11760 KB

Open AccessArticle

Regular, Beating and Dilogarithmic Breathers in Biased Photorefractive Crystals

by Carlos Alberto Betancur-Silvera, Aurea Espinosa-Cerón, Boris A. Malomed and Jorge Fujioka

Axioms 2024, 13(5), 338; https://doi.org/10.3390/axioms13050338 - 20 May 2024

Viewed by 1289

The propagation of light beams in photovoltaic pyroelectric photorefractive crystals is modelled by a specific generalization of the nonlinear Schrödinger equation (GNLSE). We use a variational approximation (VA) to predict the propagation of solitary-wave inputs in the crystals, finding that the VA equations [...] Read more.

The propagation of light beams in photovoltaic pyroelectric photorefractive crystals is modelled by a specific generalization of the nonlinear Schrödinger equation (GNLSE). We use a variational approximation (VA) to predict the propagation of solitary-wave inputs in the crystals, finding that the VA equations involve a dilogarithm special function. The VA predicts that solitons and breathers exist, and the Vakhitov–Kolokolov criterion predicts that the solitons are stable solutions. Direct simulations of the underlying GNLSE corroborates the existence of such stable modes. The numerical solutions produce both regular breathers and ones featuring beats (long-period modulations of fast oscillations). In the latter case, the Fourier transform of amplitude oscillations reveals a nearly discrete spectrum characterizing the beats dynamics. Numerical solutions of another type demonstrate the spontaneous splitting of the input pulse in two or several secondary ones. Full article

(This article belongs to the Special Issue Nonlinear Schrödinger Equations)

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13 pages, 299 KB

Open AccessArticle

High Perturbations of a Fractional Kirchhoff Equation with Critical Nonlinearities

by Shengbin Yu, Lingmei Huang and Jiangbin Chen

Axioms 2024, 13(5), 337; https://doi.org/10.3390/axioms13050337 - 20 May 2024

Viewed by 1169

Abstract

This paper concerns a fractional Kirchhoff equation with critical nonlinearities and a negative nonlocal term. In the case of high perturbations (large values of

α

, i.e., the parameter of a subcritical nonlinearity), existence results are obtained by the concentration compactness principle together [...] Read more.

This paper concerns a fractional Kirchhoff equation with critical nonlinearities and a negative nonlocal term. In the case of high perturbations (large values of

α

, i.e., the parameter of a subcritical nonlinearity), existence results are obtained by the concentration compactness principle together with the mountain pass theorem and cut-off technique. The multiplicity of solutions are further considered with the help of the symmetric mountain pass theorem. Moreover, the nonexistence and asymptotic behavior of positive solutions are also investigated. Full article

(This article belongs to the Section Mathematical Analysis)

16 pages, 310 KB

Open AccessArticle

Blow-Up Analysis of L²-Norm Solutions for an Elliptic Equation with a Varying Nonlocal Term

by Xincai Zhu and Chunxia He

Axioms 2024, 13(5), 336; https://doi.org/10.3390/axioms13050336 - 20 May 2024

Viewed by 1279

Abstract

This paper is devoted to studying a type of elliptic equation that contains a varying nonlocal term. We provide a detailed analysis of the existence, non-existence, and blow-up behavior of

L^{2}

-norm solutions for the related equation when the potential function [...] Read more.

This paper is devoted to studying a type of elliptic equation that contains a varying nonlocal term. We provide a detailed analysis of the existence, non-existence, and blow-up behavior of

L^{2}

-norm solutions for the related equation when the potential function

V (x)

fulfills an appropriate choice. Full article

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

31 pages, 1619 KB

Open AccessArticle

Respiratory Condition Detection Using Audio Analysis and Convolutional Neural Networks Optimized by Modified Metaheuristics

by Nebojsa Bacanin, Luka Jovanovic, Ruxandra Stoean, Catalin Stoean, Miodrag Zivkovic, Milos Antonijevic and Milos Dobrojevic

Axioms 2024, 13(5), 335; https://doi.org/10.3390/axioms13050335 - 18 May 2024

Cited by 18 | Viewed by 2538

Abstract

Respiratory conditions have been a focal point in recent medical studies. Early detection and timely treatment are crucial factors in improving patient outcomes for any medical condition. Traditionally, doctors diagnose respiratory conditions through an investigation process that involves listening to the patient’s lungs. [...] Read more.

Respiratory conditions have been a focal point in recent medical studies. Early detection and timely treatment are crucial factors in improving patient outcomes for any medical condition. Traditionally, doctors diagnose respiratory conditions through an investigation process that involves listening to the patient’s lungs. This study explores the potential of combining audio analysis with convolutional neural networks to detect respiratory conditions in patients. Given the significant impact of proper hyperparameter selection on network performance, contemporary optimizers are employed to enhance efficiency. Moreover, a modified algorithm is introduced that is tailored to the specific demands of this study. The proposed approach is validated using a real-world medical dataset and has demonstrated promising results. Two experiments are conducted: the first tasked models with respiratory condition detection when observing mel spectrograms of patients’ breathing patterns, while the second experiment considered the same data format for multiclass classification. Contemporary optimizers are employed to optimize the architecture selection and training parameters of models in both cases. Under identical test conditions, the best models are optimized by the introduced modified metaheuristic, with an accuracy of 0.93 demonstrated for condition detection, and a slightly reduced accuracy of 0.75 for specific condition identification. Full article

(This article belongs to the Special Issue Advances in Parameter-Tuning Techniques for Metaheuristic Algorithms)

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21 pages, 354 KB

Open AccessArticle

Exponential Stability of the Numerical Solution of a Hyperbolic System with Nonlocal Characteristic Velocities

by Rakhmatillo Djuraevich Aloev, Abdumauvlen Suleimanovich Berdyshev, Vasila Alimova and Kymbat Slamovna Bekenayeva

Axioms 2024, 13(5), 334; https://doi.org/10.3390/axioms13050334 - 17 May 2024

Cited by 1 | Viewed by 1171

Abstract

In this paper, we investigate the problem of the exponential stability of a stationary solution for a hyperbolic system with nonlocal characteristic velocities and measurement error. The formulation of the initial boundary value problem of boundary control for the specified hyperbolic system is [...] Read more.

In this paper, we investigate the problem of the exponential stability of a stationary solution for a hyperbolic system with nonlocal characteristic velocities and measurement error. The formulation of the initial boundary value problem of boundary control for the specified hyperbolic system is given. A difference scheme is constructed for the numerical solution of the considered initial boundary value problem. The definition of the exponential stability of the numerical solution in

ℓ^{2}

-norm with respect to a discrete perturbation of the equilibrium state of the initial boundary value difference problem is given. A discrete Lyapunov function for a numerical solution is constructed, and a theorem on the exponential stability of a stationary solution of the initial boundary value difference problem in

ℓ^{2}

-norm with respect to a discrete perturbation is proved. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

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38 pages, 522 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Static Spherically Symmetric Perfect Fluid Solutions in Teleparallel F(T) Gravity

by Alexandre Landry

Axioms 2024, 13(5), 333; https://doi.org/10.3390/axioms13050333 - 17 May 2024

Cited by 8 | Viewed by 1838

Abstract

In this paper, we investigate static spherically symmetric teleparallel

F (T)

gravity containing a perfect isotropic fluid. We first write the field equations and proceed to find new teleparallel

F (T)

solutions for perfect isotropic and linear fluids. By [...] Read more.

In this paper, we investigate static spherically symmetric teleparallel

F (T)

gravity containing a perfect isotropic fluid. We first write the field equations and proceed to find new teleparallel

F (T)

solutions for perfect isotropic and linear fluids. By using a power-law ansatz for the coframe components, we find several classes of new non-trivial teleparallel

F (T)

solutions. We also find a new class of teleparallel

F (T)

solutions for a matter dust fluid. After, we solve the field equations for a non-linear perfect fluid. Once again, there are several new exact teleparallel

F (T)

solutions and also some approximated teleparallel

F (T)

solutions. All these classes of new solutions may be relevant for future cosmological and astrophysical applications. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Mathematical Physics)

12 pages, 291 KB

Open AccessArticle

Eigenvalue of (p,q)-Biharmonic System along the Ricci Flow

by Lixu Yan, Yanlin Li, Apurba Saha, Abimbola Abolarinwa, Suraj Ghosh and Shyamal Kumar Hui

Axioms 2024, 13(5), 332; https://doi.org/10.3390/axioms13050332 - 17 May 2024

Cited by 1 | Viewed by 1313

Abstract

In this paper, we determine the variation formula for the first eigenvalue of

(p, q)

-biharmonic system on a closed Riemannian manifold. Several monotonic quantities are also derived. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)

16 pages, 281 KB

Open AccessArticle

Solving Nonlinear Second-Order ODEs via the Eisenhart Lift and Linearization

by Andronikos Paliathanasis

Axioms 2024, 13(5), 331; https://doi.org/10.3390/axioms13050331 - 16 May 2024

Cited by 3 | Viewed by 1428

Abstract

The linearization of nonlinear differential equations represents a robust approach to solution derivation, typically achieved through Lie symmetry analysis. This study adopts a geometric methodology grounded in the Eisenhart lift, revealing transformative techniques that linearize a set of second-order ordinary differential equations. The [...] Read more.

The linearization of nonlinear differential equations represents a robust approach to solution derivation, typically achieved through Lie symmetry analysis. This study adopts a geometric methodology grounded in the Eisenhart lift, revealing transformative techniques that linearize a set of second-order ordinary differential equations. The research underscores the effectiveness of this geometric approach in the linearization of a class of Newtonian systems that cannot be linearized through symmetry analysis. Full article

(This article belongs to the Special Issue Differential Equations and Its Application)

13 pages, 506 KB

Open AccessArticle

Extremal Bicyclic Graphs with Respect to Permanental Sums and Hosoya Indices

by Tingzeng Wu, Yinggang Bai and Shoujun Xu

Axioms 2024, 13(5), 330; https://doi.org/10.3390/axioms13050330 - 16 May 2024

Viewed by 1109

Abstract

Graph polynomials is one of the important research directions in mathematical chemistry. The coefficients of some graph polynomials, such as matching polynomial and permanental polynomial, are related to structural properties of graphs. The Hosoya index of a graph is the sum of the [...] Read more.

Graph polynomials is one of the important research directions in mathematical chemistry. The coefficients of some graph polynomials, such as matching polynomial and permanental polynomial, are related to structural properties of graphs. The Hosoya index of a graph is the sum of the absolute value of all coefficients for the matching polynomial. And the permanental sum of a graph is the sum of the absolute value of all coefficients of the permanental polynomial. In this paper, we characterize the second to sixth minimal Hosoya indices of all bicyclic graphs. Furthermore, using the results, the second to sixth minimal permanental sums of all bicyclic graphs are also characterized. Full article

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19 pages, 1128 KB

Open AccessArticle

Nonuniform Sampling in L^p-Subspaces Associated with the Multi-Dimensional Special Affine Fourier Transform

by Yingchun Jiang and Jing Yang

Axioms 2024, 13(5), 329; https://doi.org/10.3390/axioms13050329 - 15 May 2024

Cited by 1 | Viewed by 1238

Abstract

In this paper, the sampling and reconstruction problems in function subspaces of

L^{p} (R^{n})

associated with the multi-dimensional special affine Fourier transform (SAFT) are discussed. First, we give the definition of the multi-dimensional SAFT and study its properties including [...] Read more.

In this paper, the sampling and reconstruction problems in function subspaces of

L^{p} (R^{n})

associated with the multi-dimensional special affine Fourier transform (SAFT) are discussed. First, we give the definition of the multi-dimensional SAFT and study its properties including the Parseval’s relation, the canonical convolution theorems and the chirp-modulation periodicity. Then, a kind of function spaces are defined by the canonical convolution in the multi-dimensional SAFT domain, the existence and the properties of the dual basis functions are demonstrated, and the

L^{p}

-stability of the basis functions is established. Finally, based on the nonuniform samples taken on a dense set, we propose an iterative reconstruction algorithm with exponential convergence to recover the signals in a

L^{p}

-subspace associated with the multi-dimensional SAFT, and the validity of the algorithm is demonstrated via simulations. Full article

(This article belongs to the Special Issue Advances in Mathematical Methods in Signal Processing and Its Applications)

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20 pages, 338 KB

Open AccessArticle

Coercive and Noncoercive Mixed Generalized Complementarity Problems

by Ram N. Mohapatra, Bijaya K. Sahu and Gayatri Pany

Axioms 2024, 13(5), 328; https://doi.org/10.3390/axioms13050328 - 15 May 2024

Viewed by 1328

Abstract

Impressed with the very recent developments of noncoercive complementarity problems and the use of recession sets in complementarity problems, here, we discuss mixed generalized complementarity problems in Hausdorff topological vector spaces. We used the Tikhonov regularization procedure, as well as arguments from the [...] Read more.

Impressed with the very recent developments of noncoercive complementarity problems and the use of recession sets in complementarity problems, here, we discuss mixed generalized complementarity problems in Hausdorff topological vector spaces. We used the Tikhonov regularization procedure, as well as arguments from the recession analysis, to establish the existence of solutions for mixed generalized complementarity problems without coercivity assumptions in Banach spaces. Full article

(This article belongs to the Special Issue Large-Scale Optimization and Variational Inequalities: Theory and Applications)

5 pages, 179 KB

Open AccessEditorial

Mathematical Methods in Applied Sciences

by Nuno R. O. Bastos and Touria Karite

Axioms 2024, 13(5), 327; https://doi.org/10.3390/axioms13050327 - 15 May 2024

Viewed by 1399

Abstract

In this editorial, we introduce “Mathematical Methods in Applied Sciences”, a Special Issue of Axioms comprising 17 articles [...] Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

20 pages, 280 KB

Open AccessArticle

Tractability of Multivariate Approximation Problem on Euler and Wiener Integrated Processes

by Jie Zhang

Axioms 2024, 13(5), 326; https://doi.org/10.3390/axioms13050326 - 15 May 2024

Viewed by 930

Abstract

This paper examines the tractability of multivariate approximation problems under the normalized error criterion for a zero-mean Gaussian measure in an average-case setting. The Gaussian measure is associated with a covariance kernel, which is represented by the tensor product of one-dimensional kernels corresponding [...] Read more.

This paper examines the tractability of multivariate approximation problems under the normalized error criterion for a zero-mean Gaussian measure in an average-case setting. The Gaussian measure is associated with a covariance kernel, which is represented by the tensor product of one-dimensional kernels corresponding to Euler and Wiener integrated processes with non-negative and nondecreasing smoothness parameters

{r_{d}}_{d \in N}

. We give matching sufficient and necessary conditions for various concepts of tractability in terms of the asymptotic properties of the regularity parameters, except for (s, 0)-WT. Full article

(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics, 2nd Edition)

20 pages, 319 KB

Open AccessArticle

Multiplicity of Solutions for the Noncooperative Kirchhoff-Type Variable Exponent Elliptic System with Nonlinear Boundary Conditions

by Yiying Mao and Yang Yang

Axioms 2024, 13(5), 325; https://doi.org/10.3390/axioms13050325 - 14 May 2024

Viewed by 1043

Abstract

Considering the solutions of a class of noncooperative Kirchhoff-type

p (x)

-Laplacian elliptic systems with nonlinear boundary conditions, we derive a sequence of solutions utilizing both the variational method and limit index theory under certain underlying assumptions. The novelty of this [...] Read more.

Considering the solutions of a class of noncooperative Kirchhoff-type

p (x)

-Laplacian elliptic systems with nonlinear boundary conditions, we derive a sequence of solutions utilizing both the variational method and limit index theory under certain underlying assumptions. The novelty of this study is that we verify the

{(P S)}_{c}^{*}

condition using another method, diverging from the approaches cited in the previous literature. Full article

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

20 pages, 1388 KB

Open AccessArticle

A Vector Representation of Multicomplex Numbers and Its Application to Radio Frequency Signals

by Daniele Borio

Axioms 2024, 13(5), 324; https://doi.org/10.3390/axioms13050324 - 14 May 2024

Cited by 1 | Viewed by 1523

Abstract

Hypercomplex numbers, which are multi-dimensional extensions of complex numbers, have been proven beneficial in the development of advanced signal processing algorithms, including multi-dimensional filter design, linear regression and classification. We focus on multicomplex numbers, sets of hypercomplex numbers with commutative products, and introduce [...] Read more.

Hypercomplex numbers, which are multi-dimensional extensions of complex numbers, have been proven beneficial in the development of advanced signal processing algorithms, including multi-dimensional filter design, linear regression and classification. We focus on multicomplex numbers, sets of hypercomplex numbers with commutative products, and introduce a vector representation allowing one to isolate the hyperbolic real and imaginary parts of a multicomplex number. The orthogonal decomposition of a multicomplex number is also discussed, and its connection with Hadamard matrices is highlighted. Finally, a multicomplex polar representation is provided. These properties are used to extend the standard complex baseband signal representation to the multi-dimensional case. It is shown that a set of

2^{n}

Radio Frequency (RF) signals can be represented as the real part of a single multicomplex signal modulated by several frequencies. The signal RFs are related through a Hadamard matrix to the modulating frequencies adopted in the multicomplex baseband representation. Moreover, an orthogonal decomposition is provided for the obtained multicomplex baseband signal as a function of the complex baseband representations of the input RF signals. Full article

(This article belongs to the Special Issue Advances in Mathematical Methods in Signal Processing and Its Applications)

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14 pages, 684 KB

Open AccessArticle

Quantum Vision Transformers for Quark–Gluon Classification

by Marçal Comajoan Cara, Gopal Ramesh Dahale, Zhongtian Dong, Roy T. Forestano, Sergei Gleyzer, Daniel Justice, Kyoungchul Kong, Tom Magorsch, Konstantin T. Matchev, Katia Matcheva and Eyup B. Unlu

Axioms 2024, 13(5), 323; https://doi.org/10.3390/axioms13050323 - 13 May 2024

Viewed by 3713

Abstract

We introduce a hybrid quantum-classical vision transformer architecture, notable for its integration of variational quantum circuits within both the attention mechanism and the multi-layer perceptrons. The research addresses the critical challenge of computational efficiency and resource constraints in analyzing data from the upcoming [...] Read more.

We introduce a hybrid quantum-classical vision transformer architecture, notable for its integration of variational quantum circuits within both the attention mechanism and the multi-layer perceptrons. The research addresses the critical challenge of computational efficiency and resource constraints in analyzing data from the upcoming High Luminosity Large Hadron Collider, presenting the architecture as a potential solution. In particular, we evaluate our method by applying the model to multi-detector jet images from CMS Open Data. The goal is to distinguish quark-initiated from gluon-initiated jets. We successfully train the quantum model and evaluate it via numerical simulations. Using this approach, we achieve classification performance almost on par with the one obtained with the completely classical architecture, considering a similar number of parameters. Full article

(This article belongs to the Section Mathematical Analysis)

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17 pages, 804 KB

Open AccessArticle

A C⁰ Nonconforming Virtual Element Method for the Kirchhoff Plate Obstacle Problem

by Bangmin Wu and Jiali Qiu

Axioms 2024, 13(5), 322; https://doi.org/10.3390/axioms13050322 - 13 May 2024

Viewed by 1228

Abstract

This paper investigates a novel

C^{0}

nonconforming virtual element method (VEM) for solving the Kirchhoff plate obstacle problem, which is described by a fourth-order variational inequality (VI) of the first kind. In our study, we distinguish our approach by introducing new internal [...] Read more.

This paper investigates a novel

C^{0}

nonconforming virtual element method (VEM) for solving the Kirchhoff plate obstacle problem, which is described by a fourth-order variational inequality (VI) of the first kind. In our study, we distinguish our approach by introducing new internal degrees of freedom to the traditional lowest-order

C^{0}

nonconforming VEM, which originally lacked such degrees. This addition not only facilitates error estimation but also enhances its intuitiveness. Importantly, our novel

C^{0}

nonconforming VEM naturally satisfies the constraints of the obstacle problem. We then establish an a priori error estimate for our novel

C^{0}

nonconforming VEM, with the result indicating that the lowest order of our method achieves optimal convergence. Finally, we present a numerical example to validate the theoretical result. Full article

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16 pages, 1807 KB

Open AccessArticle

Quasi-Configurations Derived by Special Arrangements of Lines

by Stefano Innamorati

Axioms 2024, 13(5), 321; https://doi.org/10.3390/axioms13050321 - 11 May 2024

Cited by 1 | Viewed by 1365

Abstract

A quasi-configuration is a point–line incidence structure in which each point is incident with at least three lines and each line is incident with at least three points. We investigate derived quasi-configurations that arise both by duality and intersecting lines of three special [...] Read more.

A quasi-configuration is a point–line incidence structure in which each point is incident with at least three lines and each line is incident with at least three points. We investigate derived quasi-configurations that arise both by duality and intersecting lines of three special arrangements of lines. Sets with few intersection numbers are provided. Full article

(This article belongs to the Special Issue Theory of Curves and Knots with Applications)

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13 pages, 286 KB

Open AccessArticle

Stability of the Borell–Brascamp–Lieb Inequality for Multiple Power Concave Functions

by Meng Qin, Zhuohua Zhang, Rui Luo, Mengjie Ren and Denghui Wu

Axioms 2024, 13(5), 320; https://doi.org/10.3390/axioms13050320 - 11 May 2024

Cited by 1 | Viewed by 1319

Abstract

In this paper, we prove the stability of the Brunn–Minkowski inequality for multiple convex bodies in terms of the concept of relative asymmetry. Using these stability results and the relationship of the compact support of functions, we establish the stability of the Borell–Brascamp–Lieb [...] Read more.

In this paper, we prove the stability of the Brunn–Minkowski inequality for multiple convex bodies in terms of the concept of relative asymmetry. Using these stability results and the relationship of the compact support of functions, we establish the stability of the Borell–Brascamp–Lieb inequality for multiple power concave functions via relative asymmetry. Full article

(This article belongs to the Special Issue Advances in Convex Geometry and Analysis)

16 pages, 573 KB

Open AccessArticle

Strategic Behavior and Optimal Inventory Level in a Make-to-Stock Queueing System with Retrial Customers

by Yuejiao Wang and Chenguang Cai

Axioms 2024, 13(5), 319; https://doi.org/10.3390/axioms13050319 - 11 May 2024

Cited by 1 | Viewed by 1200

Abstract

In this article, we consider a make-to-stock queueing system with retrial customers. Upon their arrival, customers make a decision to either join the system or not based on a reward–cost function. If customers join the retrial queue, they become repeat customers. Each repeat [...] Read more.

In this article, we consider a make-to-stock queueing system with retrial customers. Upon their arrival, customers make a decision to either join the system or not based on a reward–cost function. If customers join the retrial queue, they become repeat customers. Each repeat customer repeats their demand after an exponential amount of time until they have been successfully served. We explore the equilibrium strategies of customers in both the almost observable and unobservable cases. Furthermore, we also analyze the expected costs of the entire system based on the customers’ behavior in these two cases. Additionally, we determine the optimal inventory levels in both cases through numerical experiments. Full article

(This article belongs to the Section Mathematical Analysis)

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15 pages, 267 KB

Open AccessArticle

Quasi-Contraction Maps in Subordinate Semimetric Spaces

by Areej Alharbi, Hamed Alsulami and Maha Noorwali

Axioms 2024, 13(5), 318; https://doi.org/10.3390/axioms13050318 - 10 May 2024

Viewed by 1007

Abstract

Throughout this study, we discuss the subordinate Pompeiu–Hausdorff metric (SPHM) in subordinate semimetric spaces. Moreover, we present a well-behaved quasi-contraction (WBQC) to solve quasi-contraction (QC) problems in subordinate semimetric spaces under some local constraints. Furthermore, we provide examples to support our conclusion. Full article

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

24 pages, 398 KB

Open AccessArticle

A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments

by Yue-Wu Li and Feng Qi

Axioms 2024, 13(5), 317; https://doi.org/10.3390/axioms13050317 - 10 May 2024

Cited by 15 | Viewed by 4045

Abstract

In this paper, the authors briefly review some closed-form formulas of the Gauss hypergeometric function at specific arguments, alternatively prove four of these formulas, newly extend a closed-form formula of the Gauss hypergeometric function at some specific arguments, successfully apply a special case [...] Read more.

In this paper, the authors briefly review some closed-form formulas of the Gauss hypergeometric function at specific arguments, alternatively prove four of these formulas, newly extend a closed-form formula of the Gauss hypergeometric function at some specific arguments, successfully apply a special case of the newly extended closed-form formula to derive an alternative form for the Maclaurin power series expansion of the Wilf function, and discover two novel increasing rational approximations to a quarter of the circular constant. Full article

23 pages, 353 KB

Open AccessArticle

Convergence Results for History-Dependent Variational Inequalities

by Mircea Sofonea and Domingo A. Tarzia

Axioms 2024, 13(5), 316; https://doi.org/10.3390/axioms13050316 - 10 May 2024

Cited by 3 | Viewed by 1100

Abstract

We consider a history-dependent variational inequality in a real Hilbert space, for which we recall an existence and uniqueness result. We associate this inequality with a gap function, together with two additional problems: a nonlinear equation and a minimization problem. Then, we prove [...] Read more.

We consider a history-dependent variational inequality in a real Hilbert space, for which we recall an existence and uniqueness result. We associate this inequality with a gap function, together with two additional problems: a nonlinear equation and a minimization problem. Then, we prove that solving these problems is equivalent to solving the original history-dependent variational inequality. Next, we state and prove a convergence criterion, i.e., we provide necessary and sufficient conditions which guarantee the convergence of a sequence of functions to the solution of the considered inequality. Based on the equivalence above, we deduce various consequences that present some interest on their own, and, moreover, we obtain convergence results for the two additional problems considered. Finally, we apply our abstract results to the study of an inequality problem in solid mechanics. It concerns the study of a viscoelastic constitutive law with long memory and unilateral constraints, for which we deduce a convergence result and provide the corresponding mechanical interpretations. Full article

(This article belongs to the Section Hilbert’s Sixth Problem)

20 pages, 334 KB

Open AccessArticle

Parameter Estimation in Spatial Autoregressive Models with Missing Data and Measurement Errors

by Tengjun Li, Zhikang Zhang and Yunquan Song

Axioms 2024, 13(5), 315; https://doi.org/10.3390/axioms13050315 - 10 May 2024

Viewed by 2350

Abstract

This study addresses the problem of parameter estimation in spatial autoregressive models with missing data and measurement errors in covariates. Specifically, a corrected likelihood estimation approach is employed to rectify the bias in the log-maximum likelihood function induced by measurement errors. Additionally, a [...] Read more.

This study addresses the problem of parameter estimation in spatial autoregressive models with missing data and measurement errors in covariates. Specifically, a corrected likelihood estimation approach is employed to rectify the bias in the log-maximum likelihood function induced by measurement errors. Additionally, a combination of inverse probability weighting (IPW) and mean imputation is utilized to mitigate the bias caused by missing data. Under several mild conditions, it is demonstrated that the proposed estimators are consistent and possess oracle properties. The efficacy of the proposed parameter estimation process is assessed through Monte Carlo simulation studies. Finally, the applicability of the proposed method is further substantiated using the Boston Housing Dataset. Full article

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

15 pages, 266 KB

Open AccessArticle

Some Results on Zinbiel Algebras and Rota–Baxter Operators

by Jizhong Gao, Junna Ni and Jianhua Yu

Axioms 2024, 13(5), 314; https://doi.org/10.3390/axioms13050314 - 10 May 2024

Viewed by 1303

Abstract

Rota–Baxter operators (RBOs) play a substantial role in many subfields of mathematics, especially in mathematical physics. In the article, RBOs on Zinbiel algebras (ZAs) and their sub-adjacent algebras are first investigated. Moreover, all the RBOs on two and three-dimensional ZAs are presented. Finally, [...] Read more.

Rota–Baxter operators (RBOs) play a substantial role in many subfields of mathematics, especially in mathematical physics. In the article, RBOs on Zinbiel algebras (ZAs) and their sub-adjacent algebras are first investigated. Moreover, all the RBOs on two and three-dimensional ZAs are presented. Finally, ZAs are also realized in low dimensions of the RBOs of commutative associative algebras. It was found that not all ZAs can be attained in this way. Full article

24 pages, 20089 KB

Open AccessArticle

Basic Computational Algorithms for Representing an Aircraft Flight (Calculation of 3D Displacement and Displaying)

by Adan Ramirez-Lopez

Axioms 2024, 13(5), 313; https://doi.org/10.3390/axioms13050313 - 10 May 2024

Cited by 2 | Viewed by 2752

Abstract

This manuscript describes the computational process to calculate an airplane path and display it in a 2D and 3D coordinate system on a computer screen. The airplane movement is calculated as a function of its dynamic’s conditions according to physical and logical theory. [...] Read more.

This manuscript describes the computational process to calculate an airplane path and display it in a 2D and 3D coordinate system on a computer screen. The airplane movement is calculated as a function of its dynamic’s conditions according to physical and logical theory. Here, the flight is divided into maneuvers and the aircraft conditions are defined as boundary conditions. Then the aircraft position is calculated using nested loops, which execute the calculation procedure at every step time (Δt). The calculation of the aircraft displacement is obtained as a function of the aircraft speed and heading angles. The simulator was created using the C++ programming language, and each part of the algorithm was compiled independently to reduce the source code, allow easy modification, and improve the programming efficiency. Aerial navigation involves very complex phenomena to be considered for an appropriate representation; moreover, in this manuscript, the influence of the mathematical approach to properly represent the aircraft flight is described in detail. The flight simulator was successfully tested by simulating some basic theoretical flights with different maneuvers, which include stationary position, running along the way, take off, and some movements in the airspace. The maximum aircraft speed tested was 120 km/h, the maximum maneuver time was 12 min, and the space for simulation was assumed to be without obstacles. Here, the geometrical description of path and speed is analyzed according to the symmetric and asymmetric results. Finally, an analysis was conducted to evaluate the approach of the numerical methods used; after that, it was possible to confirm that precision increased as the step time was reduced. According to this analysis, no more than 500 steps are required for a good approach in the calculation of the aircraft displacement. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)

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20 pages, 347 KB

Open AccessArticle

Canonical Metrics on Twisted Quiver Bundles over a Class of Non-Compact Gauduchon Manifold

by Shi-Fan Cai, Sudhakar Kumar Chaubey, Xin Xu, Pan Zhang and Zhi-Heng Zhang

Axioms 2024, 13(5), 312; https://doi.org/10.3390/axioms13050312 - 9 May 2024

Viewed by 1450

Abstract

The aim of this paper is to prove a theorem for holomorphic twisted quiver bundles over a special non-compact Gauduchon manifold, connecting the existence of

(σ, τ)

-Hermite–Yang–Mills metric in differential geometry and the analytic

(σ, τ)

[...] Read more.

The aim of this paper is to prove a theorem for holomorphic twisted quiver bundles over a special non-compact Gauduchon manifold, connecting the existence of

(σ, τ)

-Hermite–Yang–Mills metric in differential geometry and the analytic

(σ, τ)

-stability in algebraic geometry. The proof of the theorem relies on the flow method and the Uhlenbeck–Yau’s continuity method. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Mathematical Physics)

18 pages, 1228 KB

Open AccessArticle

Study of the Six-Compartment Nonlinear COVID-19 Model with the Homotopy Perturbation Method

by Muhammad Rafiullah, Muhammad Asif, Dure Jabeen and Mahmoud A. Ibrahim

Axioms 2024, 13(5), 311; https://doi.org/10.3390/axioms13050311 - 9 May 2024

Viewed by 1758

Abstract

The current study aims to utilize the homotopy perturbation method (HPM) to solve nonlinear dynamical models, with a particular focus on models related to predicting and controlling pandemics, such as the SIR model. Specifically, we apply this method to solve a six-compartment model [...] Read more.

The current study aims to utilize the homotopy perturbation method (HPM) to solve nonlinear dynamical models, with a particular focus on models related to predicting and controlling pandemics, such as the SIR model. Specifically, we apply this method to solve a six-compartment model for the novel coronavirus (COVID-19), which includes susceptible, exposed, asymptomatic infected, symptomatic infected, and recovered individuals, and the concentration of COVID-19 in the environment is indicated by

S (t)

,

E (t)

,

A (t)

,

I (t)

,

R (t)

, and

B (t)

, respectively. We present the series solution of this model by varying the controlling parameters and representing them graphically. Additionally, we verify the accuracy of the series solution (up to the

(n - 1) t h

-degree polynomial) that satisfies both the initial conditions and the model, with all coefficients correct at 18 decimal places. Furthermore, we have compared our results with the Runge–Kutta fourth-order method. Based on our findings, we conclude that the homotopy perturbation method is a promising approach to solve nonlinear dynamical models, particularly those associated with pandemics. This method provides valuable insight into how the control of various parameters can affect the model. We suggest that future studies can expand on our work by exploring additional models and assessing the applicability of other analytical methods. Full article

(This article belongs to the Special Issue Dynamical Systems: Theory and Applications in Mathematical Biology)

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3 pages, 171 KB

Open AccessEditorial

Recent Advances in Fractional Calculus

by Péter Kórus and Juan Eduardo Nápoles Valdés

Axioms 2024, 13(5), 310; https://doi.org/10.3390/axioms13050310 - 8 May 2024

Cited by 1 | Viewed by 1804

Abstract

This Special Issue of the scientific journal Axioms, entitled “Recent Advances in Fractional Calculus”, is dedicated to one of the most dynamic areas of mathematical sciences today [...] Full article

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

17 pages, 966 KB

Open AccessArticle

Study on SEAI Model of COVID-19 Based on Asymptomatic Infection

by Lidong Huang, Yue Xia and Wenjie Qin

Axioms 2024, 13(5), 309; https://doi.org/10.3390/axioms13050309 - 8 May 2024

Cited by 3 | Viewed by 1762

Abstract

In this paper, an SEAI epidemic model with asymptomatic infection is studied under the background of mass transmission of COVID-19. First, we use the next-generation matrix method to obtain the basic reproductive number

R_{0}

and calculate the equilibrium point. Secondly, when [...] Read more.

In this paper, an SEAI epidemic model with asymptomatic infection is studied under the background of mass transmission of COVID-19. First, we use the next-generation matrix method to obtain the basic reproductive number

R_{0}

and calculate the equilibrium point. Secondly, when

R_{0} < 1

, the local asymptotic stability of the disease-free equilibrium is proved by Hurwitz criterion, and the global asymptotic stability of the disease-free equilibrium is proved by constructing the Lyapunov function. When

R_{0} > 1

, the system has a unique endemic equilibrium point and is locally asymptotically stable, and it is also proved that the system is uniformly persistent. Then, the application of optimal control theory is carried out, and the expression of the optimal control solution is obtained. Finally, in order to verify the correctness of the theory, the stability of the equilibrium point is numerically simulated and the sensitivity of the parameters of

R_{0}

is analyzed. We also simulated the comparison of the number of asymptomatic infected people and symptomatic infected people before and after adopting the optimal control strategy. This shows that the infection of asymptomatic people cannot be underestimated in the spread of COVID-19 virus, and an isolation strategy should be adopted to control the spread speed of the disease. Full article

(This article belongs to the Special Issue Recent Advances in Mathematical Modeling of COVID-19 and Other Infectious Diseases)

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Axioms, Volume 13, Issue 5 (May 2024) – 58 articles

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