Axioms

20 pages, 465 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation

by Maria Carmela De Bonis and Donatella Occorsio

Axioms 2024, 13(11), 750; https://doi.org/10.3390/axioms13110750 - 30 Oct 2024

Viewed by 544

In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order

α

[...] Read more.

In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order

α

(D^{α} f) (y) = \frac{1}{Γ (m - α)} \int_{0}^{y} {(y - x)}^{m - α - 1} f^{(m)} (x) d x, y > 0

, with

m - 1 < α \leq m, m \in N .

The numerical procedure is based on approximating

f^{(m)}

by the m-th derivative of a Lagrange polynomial, interpolating f at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function f according to the best polynomial approximation error and depending on order

α

. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure. Full article

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

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10 pages, 262 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Greedoids and Violator Spaces

by Yulia Kempner and Vadim E. Levit

Axioms 2024, 13(9), 633; https://doi.org/10.3390/axioms13090633 - 17 Sep 2024

Viewed by 605

Abstract

This research explores the interplay between violator spaces and greedoids—two distinct theoretical frameworks developed independently. Violator spaces were introduced as a generalization of linear programming, while greedoids were designed to characterize combinatorial structures where greedy algorithms yield optimal solutions. These frameworks have, until [...] Read more.

This research explores the interplay between violator spaces and greedoids—two distinct theoretical frameworks developed independently. Violator spaces were introduced as a generalization of linear programming, while greedoids were designed to characterize combinatorial structures where greedy algorithms yield optimal solutions. These frameworks have, until now, existed in isolation. This paper bridges the gap by showing that greedoids can be defined using a modified violator operator. The established connections not only deepen our understanding of these theories but also provide a new characterization of antimatroids. Full article

(This article belongs to the Section Algebra and Number Theory)

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24 pages, 613 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Round-Off Error Suppression by Statistical Averaging

by Andrej Liptaj

Axioms 2024, 13(9), 615; https://doi.org/10.3390/axioms13090615 - 11 Sep 2024

Viewed by 615

Abstract

Regarding round-off errors as random is often a necessary simplification to describe their behavior. Assuming, in addition, the symmetry of their distributions, we show that one can, in unstable (ill-conditioned) computer calculations, suppress their effect by statistical averaging. For this, one slightly perturbs [...] Read more.

Regarding round-off errors as random is often a necessary simplification to describe their behavior. Assuming, in addition, the symmetry of their distributions, we show that one can, in unstable (ill-conditioned) computer calculations, suppress their effect by statistical averaging. For this, one slightly perturbs the argument of

f (x_{0})

many times and averages the resulting function values. In this text, we forward arguments to support the assumed properties of round-off errors and critically evaluate the validity of the averaging approach in several numerical experiments. Full article

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

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17 pages, 304 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

On a Generic Fractional Derivative Associated with the Riemann–Liouville Fractional Integral

by Yuri Luchko

Axioms 2024, 13(9), 604; https://doi.org/10.3390/axioms13090604 - 4 Sep 2024

Cited by 1 | Viewed by 1620

Abstract

In this paper, a generic fractional derivative is defined as a set of the linear operators left-inverse to the Riemann–Liouville fractional integral. Then, the theory of the left-invertible operators developed by Przeworska-Rolewicz is applied to deduce its properties. In particular, we characterize its [...] Read more.

In this paper, a generic fractional derivative is defined as a set of the linear operators left-inverse to the Riemann–Liouville fractional integral. Then, the theory of the left-invertible operators developed by Przeworska-Rolewicz is applied to deduce its properties. In particular, we characterize its domain, null-space, and projector operator; establish the interrelations between its different realizations; and present a generalized fractional Taylor formula involving the generic fractional derivative. Then, we consider the fractional relaxation equation containing the generic fractional derivative, derive a closed-form formula for its unique solution, and study its complete monotonicity. Full article

(This article belongs to the Section Mathematical Analysis)

14 pages, 673 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Fox’s H-Functions: A Gentle Introduction to Astrophysical Thermonuclear Functions

by Hans J. Haubold, Dilip Kumar and Ashik A. Kabeer

Axioms 2024, 13(8), 532; https://doi.org/10.3390/axioms13080532 - 6 Aug 2024

Cited by 1 | Viewed by 803

Abstract

Needed for cosmological and stellar nucleosynthesis, we are studying the closed-form analytic evaluation of thermonuclear reaction rates. In this context, we undertake a comprehensive analysis of three largely distinct velocity distributions, namely the Maxwell–Boltzmann distribution, the pathway distribution, and the Mittag-Leffler distribution. Moreover, [...] Read more.

Needed for cosmological and stellar nucleosynthesis, we are studying the closed-form analytic evaluation of thermonuclear reaction rates. In this context, we undertake a comprehensive analysis of three largely distinct velocity distributions, namely the Maxwell–Boltzmann distribution, the pathway distribution, and the Mittag-Leffler distribution. Moreover, a natural generalization of the Maxwell–Boltzmann velocity distribution is discussed. Furthermore, an explicit evaluation of the reaction rate integral in the high-energy cut-off case is carried out. Generalized special functions of mathematical physics like Meijer’s G-function and Fox’s H-functions and their utilization in mathematical physics are the prime focus of this paper. Full article

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8 pages, 220 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Integrable Couplings and Two-Dimensional Unital Algebras

by Wen-Xiu Ma

Axioms 2024, 13(7), 481; https://doi.org/10.3390/axioms13070481 - 18 Jul 2024

Cited by 15 | Viewed by 733

Abstract

The paper aims to demonstrate that a linear expansion in a unital two-dimensional algebra can generate integrable couplings, proposing a novel approach for their construction. The integrable couplings presented encompass a range of perturbation equations and nonlinear integrable couplings. Their corresponding Lax pairs [...] Read more.

The paper aims to demonstrate that a linear expansion in a unital two-dimensional algebra can generate integrable couplings, proposing a novel approach for their construction. The integrable couplings presented encompass a range of perturbation equations and nonlinear integrable couplings. Their corresponding Lax pairs and hereditary recursion operators are explicitly detailed. Concrete applications to the KdV equation and the AKNS system of nonlinear Schrödinger equations are extensively explored. Full article

21 pages, 747 KiB

Open AccessEditor’s ChoiceArticle

A Reduced-Dimension Weighted Explicit Finite Difference Method Based on the Proper Orthogonal Decomposition Technique for the Space-Fractional Diffusion Equation

by Xuehui Ren and Hong Li

Axioms 2024, 13(7), 461; https://doi.org/10.3390/axioms13070461 - 8 Jul 2024

Cited by 1 | Viewed by 575

Abstract

A kind of reduced-dimension method based on a weighted explicit finite difference scheme and the proper orthogonal decomposition (POD) technique for diffusion equations with Riemann–Liouville fractional derivatives in space are discussed. The constructed approximation method written in matrix form can not only ensure [...] Read more.

A kind of reduced-dimension method based on a weighted explicit finite difference scheme and the proper orthogonal decomposition (POD) technique for diffusion equations with Riemann–Liouville fractional derivatives in space are discussed. The constructed approximation method written in matrix form can not only ensure a sufficient accuracy order but also reduce the degrees of freedom, decrease storage requirements, and accelerate the computation rate. Uniqueness, stabilization, and error estimation are demonstrated by matrix analysis. The procedural steps of the POD algorithm, which reduces dimensionality, are outlined. Numerical simulations to assess the viability and effectiveness of the reduced-dimension weighted explicit finite difference method are given. A comparison between the reduced-dimension method and the classical weighted explicit finite difference scheme is presented, including the error in the

L_{2}

norm, the accuracy order, and the CPU time. Full article

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16 pages, 344 KiB

Open AccessEditor’s ChoiceReview

Monogenity and Power Integral Bases: Recent Developments

by István Gaál

Axioms 2024, 13(7), 429; https://doi.org/10.3390/axioms13070429 - 26 Jun 2024

Cited by 2 | Viewed by 6561

Abstract

Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area. Several of the listed results were presented at a series of online conferences titled [...] Read more.

Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area. Several of the listed results were presented at a series of online conferences titled “Monogenity and Power Integral Bases”. We also give a collection of the most important methods used in several of these papers. A list of open problems for further research is also given. Full article

(This article belongs to the Special Issue Advances in Generalized Hypergeometric Functions, Integral Transforms and Number Theory)

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12 pages, 930 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Constructing Approximations to Bivariate Piecewise-Smooth Functions

by David Levin

Axioms 2024, 13(7), 428; https://doi.org/10.3390/axioms13070428 - 26 Jun 2024

Viewed by 1186

Abstract

This paper demonstrates that the space of piecewise-smooth bivariate functions can be well-approximated by the space of the functions defined by a set of simple (non-linear) operations on smooth uniform tensor product splines. The examples include bivariate functions with jump discontinuities or normal [...] Read more.

This paper demonstrates that the space of piecewise-smooth bivariate functions can be well-approximated by the space of the functions defined by a set of simple (non-linear) operations on smooth uniform tensor product splines. The examples include bivariate functions with jump discontinuities or normal discontinuities across curves, and even across more involved geometries such as a three-corner discontinuity. The provided data may be uniform or non-uniform, and noisy, and the approximation procedure involves non-linear least-squares minimization. Also included is a basic approximation theorem for functions with jump discontinuity across a smooth curve. Full article

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

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16 pages, 284 KiB

Open AccessEditor’s ChoiceArticle

On the Generalized Stabilities of Functional Equations via Isometries

by Muhammad Sarfraz, Jiang Zhou, Yongjin Li and John Michael Rassias

Axioms 2024, 13(6), 403; https://doi.org/10.3390/axioms13060403 - 14 Jun 2024

Cited by 1 | Viewed by 684

Abstract

The main goal of this research article is to investigate the stability of generalized norm-additive functional equations. This study demonstrates that these equations are Hyers-Ulam stable for surjective functions from an arbitrary group G to a real Banach space B using the large [...] Read more.

The main goal of this research article is to investigate the stability of generalized norm-additive functional equations. This study demonstrates that these equations are Hyers-Ulam stable for surjective functions from an arbitrary group G to a real Banach space B using the large perturbation method. Furthermore, hyperstability results are investigated for a generalized Cauchy equation. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization, 2nd Edition)

22 pages, 533 KiB

Open AccessEditor’s ChoiceArticle

Fixed Time Synchronization of Stochastic Takagi–Sugeno Fuzzy Recurrent Neural Networks with Distributed Delay under Feedback and Adaptive Controls

by Yiran Niu, Xiaofeng Xu and Ming Liu

Axioms 2024, 13(6), 391; https://doi.org/10.3390/axioms13060391 - 11 Jun 2024

Cited by 1 | Viewed by 650

Abstract

In this paper, the stochastic Takagi–Sugeno fuzzy recurrent neural networks (STSFRNNS) with distributed delay is established based on the Takagi–Sugeno (TS) model and the fixed time synchronization problem is investigated. In order to synchronize the networks, we design two kinds of controllers: a [...] Read more.

In this paper, the stochastic Takagi–Sugeno fuzzy recurrent neural networks (STSFRNNS) with distributed delay is established based on the Takagi–Sugeno (TS) model and the fixed time synchronization problem is investigated. In order to synchronize the networks, we design two kinds of controllers: a feedback controller and an adaptive controller. Then, we obtain the synchronization criteria in a fixed time by combining the Lyapunov method and the related inequality theory of the stochastic differential equation and calculate the stabilization time for the STSFRNNS. In addition, to verify the authenticity of the theoretical results, we use MATLABR2023A to carry out numerical simulation. Full article

(This article belongs to the Special Issue Recent Advances in Applied Mathematics and Artificial Intelligence)

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21 pages, 370 KiB

Open AccessEditor’s ChoiceArticle

Exploring Clique Transversal Problems for d-degenerate Graphs with Fixed d: From Polynomial-Time Solvability to Parameterized Complexity

by Chuan-Min Lee

Axioms 2024, 13(6), 382; https://doi.org/10.3390/axioms13060382 - 4 Jun 2024

Cited by 1 | Viewed by 706

Abstract

This paper explores the computational challenges of clique transversal problems in d-degenerate graphs, which are commonly encountered across theoretical computer science and various network applications. We examine d-degenerate graphs to highlight their utility in representing sparse structures and assess several variations [...] Read more.

This paper explores the computational challenges of clique transversal problems in d-degenerate graphs, which are commonly encountered across theoretical computer science and various network applications. We examine d-degenerate graphs to highlight their utility in representing sparse structures and assess several variations of clique transversal problems, including the b-fold and

{b}

-clique transversal problems, focusing on their computational complexities for different graph categories. Our analysis identifies that certain instances of these problems are polynomial-time solvable in specific graph classes, such as 1-degenerate or 2-degenerate graphs. However, for d-degenerate graphs where

d \geq 2

, these problems generally escalate to NP-completeness. We also explore the parameterized complexity, pinpointing specific conditions that render these problems fixed-parameter tractable. Full article

(This article belongs to the Special Issue Advances in Graph Theory and Combinatorial Optimization)

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38 pages, 522 KiB

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 1 | Viewed by 725

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)

20 pages, 383 KiB

Open AccessEditor’s ChoiceArticle

Ideals and Filters on Neutrosophic Topologies Generated by Neutrosophic Relations

by Ravi P. Agarwal, Soheyb Milles, Brahim Ziane, Abdelaziz Mennouni and Lemnaouar Zedam

Axioms 2024, 13(5), 292; https://doi.org/10.3390/axioms13050292 - 25 Apr 2024

Cited by 2 | Viewed by 990

Abstract

Recently, Milles and Hammami presented and studied the concept of a neutrosophic topology generated by a neutrosophic relation. As a continuation in the same direction, this paper studies the concepts of neutrosophic ideals and neutrosophic filters on that topology. More precisely, we offer [...] Read more.

Recently, Milles and Hammami presented and studied the concept of a neutrosophic topology generated by a neutrosophic relation. As a continuation in the same direction, this paper studies the concepts of neutrosophic ideals and neutrosophic filters on that topology. More precisely, we offer the lattice structure of neutrosophic open sets of a neutrosophic topology generated via a neutrosophic relation and examine its different characteristics. Furthermore, we enlarge to this lattice structure the notions of ideals (respectively, filters) and characterize them with regard to the lattice operations. We end this work by studying the prime neutrosophic ideal and prime neutrosophic filter as interesting types of neutrosophic ideals and neutrosophic filters. Full article

(This article belongs to the Special Issue Advances in Classical and Applied Mathematics)

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25 pages, 10343 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Jordan-Type Inequalities and Stratification

by Miloš Mićović and Branko Malešević

Axioms 2024, 13(4), 262; https://doi.org/10.3390/axioms13040262 - 14 Apr 2024

Cited by 1 | Viewed by 1566

Abstract

In this paper, two double Jordan-type inequalities are introduced that generalize some previously established inequalities. As a result, some new upper and lower bounds and approximations of the sinc function are obtained. This extension of Jordan’s inequality is enabled by considering the corresponding [...] Read more.

In this paper, two double Jordan-type inequalities are introduced that generalize some previously established inequalities. As a result, some new upper and lower bounds and approximations of the sinc function are obtained. This extension of Jordan’s inequality is enabled by considering the corresponding inequalities through the concept of stratified families of functions. Based on this approach, some optimal approximations of the sinc function are derived by determining the corresponding minimax approximants. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications, 2nd Edition)

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12 pages, 310 KiB

Open AccessEditor’s ChoiceArticle

Personalized Treatment Policies with the Novel Buckley-James Q-Learning Algorithm

by Jeongjin Lee and Jong-Min Kim

Axioms 2024, 13(4), 212; https://doi.org/10.3390/axioms13040212 - 25 Mar 2024

Cited by 2 | Viewed by 1317

Abstract

This research paper presents the Buckley-James Q-learning (BJ-Q) algorithm, a cutting-edge method designed to optimize personalized treatment strategies, especially in the presence of right censoring. We critically assess the algorithm’s effectiveness in improving patient outcomes and its resilience across various scenarios. Central to [...] Read more.

This research paper presents the Buckley-James Q-learning (BJ-Q) algorithm, a cutting-edge method designed to optimize personalized treatment strategies, especially in the presence of right censoring. We critically assess the algorithm’s effectiveness in improving patient outcomes and its resilience across various scenarios. Central to our approach is the innovative use of the survival time to impute the reward in Q-learning, employing the Buckley-James method for enhanced accuracy and reliability. Our findings highlight the significant potential of personalized treatment regimens and introduce the BJ-Q learning algorithm as a viable and promising approach. This work marks a substantial advancement in our comprehension of treatment dynamics and offers valuable insights for augmenting patient care in the ever-evolving clinical landscape. Full article

(This article belongs to the Special Issue New Perspectives in Mathematical Statistics)

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23 pages, 404 KiB

Open AccessEditor’s ChoiceArticle

Optimal Construction for Decoding 2D Convolutional Codes over an Erasure Channel

by Raquel Pinto, Marcos Spreafico and Carlos Vela

Axioms 2024, 13(3), 197; https://doi.org/10.3390/axioms13030197 - 15 Mar 2024

Cited by 1 | Viewed by 1115

Abstract

In general, the problem of building optimal convolutional codes under a certain criteria is hard, especially when size field restrictions are applied. In this paper, we confront the challenge of constructing an optimal 2D convolutional code when communicating over an erasure channel. We [...] Read more.

In general, the problem of building optimal convolutional codes under a certain criteria is hard, especially when size field restrictions are applied. In this paper, we confront the challenge of constructing an optimal 2D convolutional code when communicating over an erasure channel. We propose a general construction method for these codes. Specifically, we provide an optimal construction where the decoding method presented in the bibliography is considered. Full article

(This article belongs to the Special Issue Advances in Linear Algebra with Applications)

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15 pages, 1476 KiB

Open AccessEditor’s ChoiceArticle

A Comparison between Invariant and Equivariant Classical and Quantum Graph Neural Networks

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

Axioms 2024, 13(3), 160; https://doi.org/10.3390/axioms13030160 - 29 Feb 2024

Cited by 3 | Viewed by 1882

Abstract

Machine learning algorithms are heavily relied on to understand the vast amounts of data from high-energy particle collisions at the CERN Large Hadron Collider (LHC). The data from such collision events can naturally be represented with graph structures. Therefore, deep geometric methods, such [...] Read more.

Machine learning algorithms are heavily relied on to understand the vast amounts of data from high-energy particle collisions at the CERN Large Hadron Collider (LHC). The data from such collision events can naturally be represented with graph structures. Therefore, deep geometric methods, such as graph neural networks (GNNs), have been leveraged for various data analysis tasks in high-energy physics. One typical task is jet tagging, where jets are viewed as point clouds with distinct features and edge connections between their constituent particles. The increasing size and complexity of the LHC particle datasets, as well as the computational models used for their analysis, have greatly motivated the development of alternative fast and efficient computational paradigms such as quantum computation. In addition, to enhance the validity and robustness of deep networks, we can leverage the fundamental symmetries present in the data through the use of invariant inputs and equivariant layers. In this paper, we provide a fair and comprehensive comparison of classical graph neural networks (GNNs) and equivariant graph neural networks (EGNNs) and their quantum counterparts: quantum graph neural networks (QGNNs) and equivariant quantum graph neural networks (EQGNN). The four architectures were benchmarked on a binary classification task to classify the parton-level particle initiating the jet. Based on their area under the curve (AUC) scores, the quantum networks were found to outperform the classical networks. However, seeing the computational advantage of quantum networks in practice may have to wait for the further development of quantum technology and its associated application programming interfaces (APIs). Full article

(This article belongs to the Special Issue Computational Aspects of Machine Learning and Quantum Computing)

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20 pages, 381 KiB

Open AccessEditor’s ChoiceArticle

The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving ₁F₂ Hypergeometric Functions That Arise from Them

by Jack C. Straton

Axioms 2024, 13(2), 134; https://doi.org/10.3390/axioms13020134 - 19 Feb 2024

Cited by 1 | Viewed by 1304

Abstract

The Bessel function of the first kind

J_{N} (k x)

is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind

I_{N} (k x)

. The purpose of these expansions in Legendre polynomials was not an [...] Read more.

The Bessel function of the first kind

J_{N} (k x)

is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind

I_{N} (k x)

. The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for

J_{N} (k x)

useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving

_{1} F_{2}

hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes

1 / (2^{i} 3^{j} 5^{k} 7^{l} 11^{m} 13^{n} 17^{o} 19^{p})

multiplying powers of the coefficient k. Full article

(This article belongs to the Special Issue Advances in Generalized Hypergeometric Functions, Integral Transforms and Number Theory)

16 pages, 4276 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Time-Domain Fractional Behaviour Modelling with Rational Non-Singular Kernels

by Jocelyn Sabatier and Christophe Farges

Axioms 2024, 13(2), 99; https://doi.org/10.3390/axioms13020099 - 31 Jan 2024

Cited by 1 | Viewed by 1175

Abstract

This paper proposes a solution to model fractional behaviours with a convolution model involving non-singular kernels and without using fractional calculus. The non-singular kernels considered are rational functions of time. The interest of this class of kernel is demonstrated with a pure power [...] Read more.

This paper proposes a solution to model fractional behaviours with a convolution model involving non-singular kernels and without using fractional calculus. The non-singular kernels considered are rational functions of time. The interest of this class of kernel is demonstrated with a pure power law function that can be approximated in the time domain by a rational function whose pole and zeros are interlaced and linked by geometric laws. The Laplace transform and frequency response of this class of kernel is given and compared with an approximation found in the literature. The comparison reveals less phase oscillation with the solution proposed by the authors. A parameter estimation method is finally proposed to obtain the rational kernel model for general fractional behaviour. An application performed with this estimation method demonstrates the interest in non-singular rational kernels to model fractional behaviours. Another interest is the physical interpretation fractional behaviours that can be implemented with delay distributions. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

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18 pages, 338 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

A Convergence Criterion for a Class of Stationary Inclusions in Hilbert Spaces

by Mircea Sofonea and Domingo A. Tarzia

Axioms 2024, 13(1), 52; https://doi.org/10.3390/axioms13010052 - 15 Jan 2024

Cited by 1 | Viewed by 1215

Abstract

Here, we consider a stationary inclusion in a real Hilbert space X, governed by a set of constraints K, a nonlinear operator A, and an element

f \in X

. Under appropriate assumptions on the data, the inclusion has a [...] Read more.

Here, we consider a stationary inclusion in a real Hilbert space X, governed by a set of constraints K, a nonlinear operator A, and an element

f \in X

. Under appropriate assumptions on the data, the inclusion has a unique solution, denoted by u. We state and prove a covergence criterion, i.e., we provide necessary and sufficient conditions on a sequence

{u_{n}} \subset X

, which guarantee its convergence to the solution u. We then present several applications that provide the continuous dependence of the solution with respect to the data K, A and f on the one hand, and the convergence of an associate penalty problem on the other hand. We use these abstract results in the study of a frictional contact problem with elastic materials that, in a weak formulation, leads to a stationary inclusion for the deformation field. Finally, we apply the abstract penalty method in the analysis of two nonlinear elastic constitutive laws. Full article

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

20 pages, 371 KiB

Open AccessEditor’s ChoiceArticle

Probability Distributions Approximation via Fractional Moments and Maximum Entropy: Theoretical and Computational Aspects

by Pier Luigi Novi Inverardi and Aldo Tagliani

Axioms 2024, 13(1), 28; https://doi.org/10.3390/axioms13010028 - 30 Dec 2023

Cited by 5 | Viewed by 1400

Abstract

In the literature, the use of fractional moments to express the available information in the framework of maximum entropy (MaxEnt) approximation of a distribution F having finite or unbounded positive support, has been essentially considered as a computational tool to improve the performance [...] Read more.

In the literature, the use of fractional moments to express the available information in the framework of maximum entropy (MaxEnt) approximation of a distribution F having finite or unbounded positive support, has been essentially considered as a computational tool to improve the performance of the analogous procedure based on integer moments. No attention has been paid to two formal aspects concerning fractional moments, such as conditions for the existence of the maximum entropy approximation based on them or convergence in entropy of this approximation to F. This paper aims to fill this gap by providing proofs of these two fundamental results. In fact, convergence in entropy can be involved in the optimal selection of the order of fractional moments for accelerating the convergence of the MaxEnt approximation to F, to clarify the entailment relationships of this type of convergence with other types of convergence useful in statistical applications, and to preserve some important prior features of the underlying F distribution. Full article

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

28 pages, 1776 KiB

Open AccessEditor’s ChoiceArticle

On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations

by Nikolay K. Vitanov

Axioms 2023, 12(12), 1106; https://doi.org/10.3390/axioms12121106 - 8 Dec 2023

Viewed by 4462

Abstract

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea [...] Read more.

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg–de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave–solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons. Full article

(This article belongs to the Topic Mathematical Modeling)

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12 pages, 311 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Ramsey Chains in Linear Forests

by Gary Chartrand, Ritabrato Chatterjee and Ping Zhang

Axioms 2023, 12(11), 1019; https://doi.org/10.3390/axioms12111019 - 29 Oct 2023

Viewed by 1163

Abstract

Every red–blue coloring of the edges of a graph G results in a sequence

G_{1}

,

G_{2}

, …,

G_{ℓ}

of pairwise edge-disjoint monochromatic subgraphs

G_{i}

(

1 \leq i \leq ℓ

) of size i, such [...] Read more.

Every red–blue coloring of the edges of a graph G results in a sequence

G_{1}

,

G_{2}

, …,

G_{ℓ}

of pairwise edge-disjoint monochromatic subgraphs

G_{i}

(

1 \leq i \leq ℓ

) of size i, such that

G_{i}

is isomorphic to a subgraph of

G_{i + 1}

for

1 \leq i \leq ℓ - 1

. Such a sequence is called a Ramsey chain in G, and

A R_{c} (G)

is the maximum length of a Ramsey chain in G, with respect to a red–blue coloring c. The Ramsey index

A R (G)

of G is the minimum value of

A R_{c} (G)

among all the red–blue colorings c of G. If G has size m, then

(\binom{k + 1}{2}) \leq m < (\binom{k + 2}{2})

for some positive integer k. It has been shown that there are infinite classes S of graphs, such that for every graph G of size m in S,

A R (G) = k

if and only if

(\binom{k + 1}{2}) \leq m < (\binom{k + 2}{2})

. Two of these classes are the matchings

m K_{2}

and paths

P_{m + 1}

of size m. These are both subclasses of linear forests (a forest of which each of the components is a path). It is shown that if F is any linear forest of size m with

(\binom{k + 1}{2}) < m < (\binom{k + 2}{2})

, then

A R (F) = k

. Furthermore, if F is a linear forest of size

(\binom{k + 1}{2})

, where

k \geq 4

, that has at most

(\binom{k - 1}{2})

components, then

A R (F) = k

, while for each integer t with

(\binom{k - 1}{2}) < t < (\binom{k + 1}{2})

there is a linear forest F of size

(\binom{k + 1}{2})

with t components, such that

A R (F) = k - 1

. Full article

(This article belongs to the Section Algebra and Number Theory)

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12 pages, 6794 KiB

Open AccessEditor’s ChoiceCommunication

A Comparative Study of the Explicit Finite Difference Method and Physics-Informed Neural Networks for Solving the Burgers’ Equation

by Svetislav Savović, Miloš Ivanović and Rui Min

Axioms 2023, 12(10), 982; https://doi.org/10.3390/axioms12100982 - 18 Oct 2023

Cited by 10 | Viewed by 2458

Abstract

The Burgers’ equation is solved using the explicit finite difference method (EFDM) and physics-informed neural networks (PINN). We compare our numerical results, obtained using the EFDM and PINN for three test problems with various initial conditions and Dirichlet boundary conditions, with the analytical [...] Read more.

The Burgers’ equation is solved using the explicit finite difference method (EFDM) and physics-informed neural networks (PINN). We compare our numerical results, obtained using the EFDM and PINN for three test problems with various initial conditions and Dirichlet boundary conditions, with the analytical solutions, and, while both approaches yield very good agreement, the EFDM results are more closely aligned with the analytical solutions. Since there is good agreement between all of the numerical findings from the EFDM, PINN, and analytical solutions, both approaches are competitive and deserving of recommendation. The conclusions that are provided are significant for simulating a variety of nonlinear physical phenomena, such as those that occur in flood waves in rivers, chromatography, gas dynamics, and traffic flow. Additionally, the concepts of the solution techniques used in this study may be applied to the development of numerical models for this class of nonlinear partial differential equations by present and future model developers of a wide range of diverse nonlinear physical processes. Full article

(This article belongs to the Special Issue Axioms and Methods for Handling Differential Equations and Inverse Problems)

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23 pages, 9565 KiB

Open AccessEditor’s ChoiceArticle

Color Image Recovery Using Generalized Matrix Completion over Higher-Order Finite Dimensional Algebra

by Liang Liao, Zhuang Guo, Qi Gao, Yan Wang, Fajun Yu, Qifeng Zhao, Stephen John Maybank, Zhoufeng Liu, Chunlei Li and Lun Li

Axioms 2023, 12(10), 954; https://doi.org/10.3390/axioms12100954 - 10 Oct 2023

Cited by 55 | Viewed by 1718

Abstract

To improve the accuracy of color image completion with missing entries, we present a recovery method based on generalized higher-order scalars. We extend the traditional second-order matrix model to a more comprehensive higher-order matrix equivalent, called the “t-matrix” model, which incorporates a pixel [...] Read more.

To improve the accuracy of color image completion with missing entries, we present a recovery method based on generalized higher-order scalars. We extend the traditional second-order matrix model to a more comprehensive higher-order matrix equivalent, called the “t-matrix” model, which incorporates a pixel neighborhood expansion strategy to characterize the local pixel constraints. This “t-matrix” model is then used to extend some commonly used matrix and tensor completion algorithms to their higher-order versions. We perform extensive experiments on various algorithms using simulated data and publicly available images. The results show that our generalized matrix completion model and the corresponding algorithm compare favorably with their lower-order tensor and conventional matrix counterparts. Full article

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18 pages, 346 KiB

Open AccessEditor’s ChoiceArticle

Positive Solutions for a System of Hadamard Fractional Boundary Value Problems on an Infinite Interval

by Alexandru Tudorache and Rodica Luca

Axioms 2023, 12(8), 793; https://doi.org/10.3390/axioms12080793 - 16 Aug 2023

Cited by 5 | Viewed by 1308

Abstract

Our investigation is devoted to examining the existence, uniqueness, and multiplicity of positive solutions for a system of Hadamard fractional differential equations. This system is defined on an infinite interval and is subject to coupled nonlocal boundary conditions. These boundary conditions encompass both [...] Read more.

Our investigation is devoted to examining the existence, uniqueness, and multiplicity of positive solutions for a system of Hadamard fractional differential equations. This system is defined on an infinite interval and is subject to coupled nonlocal boundary conditions. These boundary conditions encompass both Hadamard fractional derivatives and Riemann–Stieltjes integrals, and the nonlinearities within the system are non-negative functions that may not be bounded. To establish the main results, we rely on the utilization of mathematical theorems such as the Schauder fixed-point theorem, the Banach contraction mapping principle, and the Avery–Peterson fixed-point theorem. Full article

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

13 pages, 353 KiB

Open AccessEditor’s ChoiceArticle

Solvability, Approximation and Stability of Periodic Boundary Value Problem for a Nonlinear Hadamard Fractional Differential Equation with

p

-Laplacian

by Kaihong Zhao

Axioms 2023, 12(8), 733; https://doi.org/10.3390/axioms12080733 - 27 Jul 2023

Cited by 16 | Viewed by 1201

Abstract

The fractional order

p

-Laplacian differential equation model is a powerful tool for describing turbulent problems in porous viscoelastic media. The study of such models helps to reveal the dynamic behavior of turbulence. Therefore, this article is mainly concerned with the periodic boundary [...] Read more.

The fractional order

p

-Laplacian differential equation model is a powerful tool for describing turbulent problems in porous viscoelastic media. The study of such models helps to reveal the dynamic behavior of turbulence. Therefore, this article is mainly concerned with the periodic boundary value problem (BVP) for a class of nonlinear Hadamard fractional differential equation with

p

-Laplacian operator. By virtue of an important fixed point theorem on a complete metric space with two distances, we study the solvability and approximation of this BVP. Based on nonlinear analysis methods, we further discuss the generalized Ulam-Hyers (GUH) stability of this problem. Eventually, we supply two example and simulations to verify the correctness and availability of our main results. Compared to many previous studies, our approach enables the solution of the system to exist in metric space rather than normed space. In summary, we obtain some sufficient conditions for the existence, uniqueness, and stability of solutions in the metric space. Full article

(This article belongs to the Special Issue Differential Equations in Applied Mathematics)

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25 pages, 3239 KiB

Open AccessEditor’s ChoiceArticle

Fuzzy Logic and Decision Making Applied to Customer Service Optimization

by Gabriel Marín Díaz and Ramón Alberto Carrasco González

Axioms 2023, 12(5), 448; https://doi.org/10.3390/axioms12050448 - 30 Apr 2023

Cited by 8 | Viewed by 2897

Abstract

In the literature, the Information Technology Infrastructure Library (ITIL) methodology recommends determining the priority of incident resolution based on the impact and urgency of interactions. The RFID model, based on the parameters of Recency, Frequency, Importance and Duration in the resolution of incidents, [...] Read more.

In the literature, the Information Technology Infrastructure Library (ITIL) methodology recommends determining the priority of incident resolution based on the impact and urgency of interactions. The RFID model, based on the parameters of Recency, Frequency, Importance and Duration in the resolution of incidents, provides an individual assessment and a clustering of customers based on these factors. We can improve the traditional concept of waiting queues for customer service management by using a procedure that adds to the evaluation provided by RFID such additional factors as Impact, Urgency and Emotional character of each interaction. If we also include aspects such as Waiting Time and Contact Center Workload, we have a procedure that allows prioritizing interactions between the customer and the Contact Center dynamically and in real time. In this paper we propose to apply a model of unification of heterogeneous information in 2-tuple linguistic evaluations, to obtain a global evaluation of each interaction by applying the Analytic Hierarchy Process (AHP), and in this way be able to have a dynamic process of prioritization of interactions. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Its Applications in Decision Making)

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21 pages, 1852 KiB

Open AccessEditor’s ChoiceArticle

An Analytic Solution for 2D Heat Conduction Problems with General Dirichlet Boundary Conditions

by Heng-Pin Hsu, Te-Wen Tu and Jer-Rong Chang

Axioms 2023, 12(5), 416; https://doi.org/10.3390/axioms12050416 - 24 Apr 2023

Cited by 3 | Viewed by 4106

Abstract

This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case [...] Read more.

This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case of space–time dependence. First, the physical system is decomposed into two one-dimensional subsystems, each of which can be solved by combining the proposed shifting function method with the eigenfunction expansion theorem. Therefore, through the superposition of the solutions of the two subsystems, the complete solution in the form of series can be obtained. Two numerical examples are used to investigate the analytic solution of the 2D heat conduction problems with space–time-dependent boundary conditions. The considered space–time-dependent functions are separable in the space–time domain for convenience. The space-dependent function is specified as a sine function and/or a parabolic function, and the time-dependent function is specified as an exponential function and/or a cosine function. In order to verify the correctness of the proposed method, the case of the space-dependent sinusoidal function and time-dependent exponential function is studied, and the consistency between the derived solution and the literature solution is verified. The parameter influence of the time-dependent function of the boundary conditions on the temperature variation is also investigated, and the time-dependent function includes harmonic type and exponential type. Full article

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

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22 pages, 4652 KiB

Open AccessEditor’s ChoiceArticle

A Coupled PDE-ODE Model for Nonlinear Transient Heat Transfer with Convection Heating at the Boundary: Numerical Solution by Implicit Time Discretization and Sequential Decoupling

by Stefan M. Filipov, Jordan Hristov, Ana Avdzhieva and István Faragó

Axioms 2023, 12(4), 323; https://doi.org/10.3390/axioms12040323 - 24 Mar 2023

Cited by 1 | Viewed by 2446

Abstract

This article considers heat transfer in a solid body with temperature-dependent thermal conductivity that is in contact with a tank filled with liquid. The liquid in the tank is heated by hot liquid entering the tank through a pipe. Liquid at a lower [...] Read more.

This article considers heat transfer in a solid body with temperature-dependent thermal conductivity that is in contact with a tank filled with liquid. The liquid in the tank is heated by hot liquid entering the tank through a pipe. Liquid at a lower temperature leaves the tank through another pipe. We propose a one-dimensional mathematical model that consists of a nonlinear PDE for the temperature along the solid body, coupled to a linear ODE for the temperature in the tank, the boundary and the initial conditions. All equations are converted into a dimensionless form reducing the input parameters to three dimensionless numbers and a dimensionless function. A steady-state analysis is performed. To solve the transient problem, a nontrivial numerical approach is proposed whereby the differential equations are first discretized in time. This reduces the problem to a sequence of nonlinear two-point boundary value problems (TPBVP) and a sequence of linear algebraic equations coupled to it. We show that knowing the temperature in the system at time level n − 1 allows us to decouple the TPBVP and the corresponding algebraic equation at time level n. Thus, starting from the initial conditions, the equations are decoupled and solved sequentially. The TPBVPs are solved by FDM with the Newtonian method. Full article

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

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14 pages, 361 KiB

Open AccessEditor’s ChoiceArticle

Renyi Entropy of the Residual Lifetime of a Reliability System at the System Level

by Mhamed Mesfioui, Mohamed Kayid and Mansour Shrahili

Axioms 2023, 12(4), 320; https://doi.org/10.3390/axioms12040320 - 23 Mar 2023

Cited by 7 | Viewed by 1364

Abstract

The measurement of uncertainty across the lifetimes of engineering systems has drawn more attention in recent years. It is a helpful metric for assessing how predictable a system’s lifetime is. In these circumstances, Renyi entropy, a Shannon entropy extension, is particularly appealing. In [...] Read more.

The measurement of uncertainty across the lifetimes of engineering systems has drawn more attention in recent years. It is a helpful metric for assessing how predictable a system’s lifetime is. In these circumstances, Renyi entropy, a Shannon entropy extension, is particularly appealing. In this paper, we develop the system signature to give an explicit formula for the Renyi entropy of the residual lifetime of a coherent system when all system components have lived to a time t. In addition, several findings are studied for the aforementioned entropy, including the bounds and order characteristics. It is possible to compare the residual lifespan predictability of two coherent systems with known signatures using the findings of this study. Full article

(This article belongs to the Special Issue Information Theory in Economics, Finance, and Management)

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20 pages, 4619 KiB

Open AccessEditor’s ChoiceArticle

Analysis of

ℝ = P [Y < X < Z]

Using Ranked Set Sampling for a Generalized Inverse Exponential Model

by Amal S. Hassan, Najwan Alsadat, Mohammed Elgarhy, Christophe Chesneau and Heba F. Nagy

Axioms 2023, 12(3), 302; https://doi.org/10.3390/axioms12030302 - 15 Mar 2023

Cited by 10 | Viewed by 1826

Abstract

In many real-world situations, systems frequently fail due to demanding operating conditions. In particular, when systems reach their lowest, highest, or both extremes operating conditions, they usually fail to accomplish their intended functions. This study considers estimating the stress–strength reliability, for a component [...] Read more.

In many real-world situations, systems frequently fail due to demanding operating conditions. In particular, when systems reach their lowest, highest, or both extremes operating conditions, they usually fail to accomplish their intended functions. This study considers estimating the stress–strength reliability, for a component with a strength (

X

) that is independent of the opposing lower bound stress (

Y

) and upper bound stress (

Z

). We assumed that the strength and stress random variables followed a generalized inverse exponential distribution with different shape parameters. Under ranked set sampling (RSS) and simple random sampling (SRS) designs, we obtained four reliability estimators using the maximum likelihood method. The first and second reliability estimators were deduced when the sample data of the strength and stress distributions used the sample design (RSS/SRS). The third reliability estimator was determined when the sample data for

Y

and

Z

were received from the RSS and the sample data for

X

were taken from the SRS. The fourth reliability estimator was derived when the sample data of

Y

and

Z

were selected from the SRS, while the sample data of

X

were taken from the RSS. The accuracy of the suggested estimators was compared using a comprehensive computer simulation. Lastly, three real data sets were used to determine the reliability. Full article

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

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20 pages, 534 KiB

Open AccessEditor’s ChoiceArticle

An Inertial Subgradient Extragradient Method for Approximating Solutions to Equilibrium Problems in Hadamard Manifolds

by Olawale Kazeem Oyewole and Simeon Reich

Axioms 2023, 12(3), 256; https://doi.org/10.3390/axioms12030256 - 1 Mar 2023

Cited by 6 | Viewed by 1577

Abstract

In this work, we are concerned with the iterative approximation of solutions to equilibrium problems in the framework of Hadamard manifolds. We introduce a subgradient extragradient type method with a self-adaptive step size. The use of a step size which is allowed to [...] Read more.

In this work, we are concerned with the iterative approximation of solutions to equilibrium problems in the framework of Hadamard manifolds. We introduce a subgradient extragradient type method with a self-adaptive step size. The use of a step size which is allowed to increase per iteration is to avoid the dependence of our method on the Lipschitz constant of the underlying operator as has been the case in recent articles in this direction. In general, operators satisfying weak monotonicity conditions seem to be more applicable in practice. By using inertial and viscosity techniques, we establish a convergence result for solving a pseudomonotone equilibrium problem under some appropriate conditions. As applications, we use our method to solve some theoretical optimization problems. Finally, we present some numerical illustrations in order to demonstrate the quantitative efficacy and superiority of our proposed method over a previous method present in the literature. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

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10 pages, 307 KiB

Open AccessEditor’s ChoiceArticle

Initial Coefficients Estimates and Fekete–Szegö Inequality Problem for a General Subclass of Bi-Univalent Functions Defined by Subordination

by Mohamed Illafe, Feras Yousef, Maisarah Haji Mohd and Shamani Supramaniam

Axioms 2023, 12(3), 235; https://doi.org/10.3390/axioms12030235 - 23 Feb 2023

Cited by 18 | Viewed by 1650

Abstract

In the present work, we aim to introduce and investigate a novel comprehensive subclass of normalized analytic bi-univalent functions involving Gegenbauer polynomials and the zero-truncated Poisson distribution. For functions in the aforementioned class, we find upper estimates of the second and third Taylor–Maclaurin [...] Read more.

In the present work, we aim to introduce and investigate a novel comprehensive subclass of normalized analytic bi-univalent functions involving Gegenbauer polynomials and the zero-truncated Poisson distribution. For functions in the aforementioned class, we find upper estimates of the second and third Taylor–Maclaurin coefficients, and then we solve the Fekete–Szegö functional problem. Moreover, by setting the values of the parameters included in our main results, we obtain several links to some of the earlier known findings. Full article

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

16 pages, 356 KiB

Open AccessEditor’s ChoiceArticle

Boundary Value Problems for Fractional Differential Equations of Caputo Type and Ulam Type Stability: Basic Concepts and Study

by Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan

Axioms 2023, 12(3), 226; https://doi.org/10.3390/axioms12030226 - 21 Feb 2023

Cited by 7 | Viewed by 2398

Abstract

Boundary value problems are very applicable problems for different types of differential equations and stability of solutions, which are an important qualitative question in the theory of differential equations. There are various types of stability, one of which is the so called Ulam-type [...] Read more.

Boundary value problems are very applicable problems for different types of differential equations and stability of solutions, which are an important qualitative question in the theory of differential equations. There are various types of stability, one of which is the so called Ulam-type stability, and it is a special type of data dependence of solutions of differential equations. For boundary value problems, this type of stability requires some additional understanding, and, in connection with this, we discuss the Ulam-Hyers stability for different types of differential equations, such as ordinary differential equations and generalized proportional Caputo fractional differential equations. To propose an appropriate idea of Ulam-type stability, we consider a boundary condition with a parameter, and the value of the parameter depends on the chosen arbitrary solution of the corresponding differential inequality. Several examples are given to illustrate the theoretical considerations. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

14 pages, 286 KiB

Open AccessEditor’s ChoiceArticle

A New Advanced Class of Convex Functions with Related Results

by Muhammad Adil Khan, Adnan, Tareq Saeed and Eze R. Nwaeze

Axioms 2023, 12(2), 195; https://doi.org/10.3390/axioms12020195 - 13 Feb 2023

Cited by 6 | Viewed by 1985

Abstract

It is the purpose of this paper to propose a novel class of convex functions associated with strong

η

-convexity. A relationship between the newly defined function and an earlier generalized class of convex functions is hereby established. To point out the importance [...] Read more.

It is the purpose of this paper to propose a novel class of convex functions associated with strong

η

-convexity. A relationship between the newly defined function and an earlier generalized class of convex functions is hereby established. To point out the importance of the new class of functions, some examples are presented. Additionally, the famous Hermite–Hadamard inequality is derived for this generalized family of convex functions. Furthermore, some inequalities and results for strong

η

-convex function are also derived. We anticipate that this new class of convex functions will open the research door to more investigations in this direction. Full article

(This article belongs to the Section Mathematical Analysis)

15 pages, 269 KiB

Open AccessEditor’s ChoiceArticle

On Intuitionistic Fuzzy Temporal Topological Structures

by Krassimir Atanassov

Axioms 2023, 12(2), 182; https://doi.org/10.3390/axioms12020182 - 9 Feb 2023

Cited by 6 | Viewed by 1405

Abstract

In the present paper, four different intuitionistic fuzzy temporal topological structures are introduced, and some of their properties are discussed. These topological structures are based on the intuitionistic fuzzy topological operators and on the temporal intuitionistic fuzzy topological operators, which exist in intuitionistic [...] Read more.

In the present paper, four different intuitionistic fuzzy temporal topological structures are introduced, and some of their properties are discussed. These topological structures are based on the intuitionistic fuzzy topological operators and on the temporal intuitionistic fuzzy topological operators, which exist in intuitionistic fuzzy sets theory. The new structures are direct extensions of the IFTSs and will be the basis for introducing of a new type of topological structures. Full article

(This article belongs to the Special Issue Fuzzy Logic and Application in Multi-Criteria Decision-Making (MCDM))

20 pages, 2236 KiB

Open AccessEditor’s ChoiceArticle

Two Novel Models for Traffic Sign Detection Based on YOLOv5s

by Wei Bai, Jingyi Zhao, Chenxu Dai, Haiyang Zhang, Li Zhao, Zhanlin Ji and Ivan Ganchev

Axioms 2023, 12(2), 160; https://doi.org/10.3390/axioms12020160 - 3 Feb 2023

Cited by 28 | Viewed by 3392

Abstract

Object detection and image recognition are some of the most significant and challenging branches in the field of computer vision. The prosperous development of unmanned driving technology has made the detection and recognition of traffic signs crucial. Affected by diverse factors such as [...] Read more.

Object detection and image recognition are some of the most significant and challenging branches in the field of computer vision. The prosperous development of unmanned driving technology has made the detection and recognition of traffic signs crucial. Affected by diverse factors such as light, the presence of small objects, and complicated backgrounds, the results of traditional traffic sign detection technology are not satisfactory. To solve this problem, this paper proposes two novel traffic sign detection models, called YOLOv5-DH and YOLOv5-TDHSA, based on the YOLOv5s model with the following improvements (YOLOv5-DH uses only the second improvement): (1) replacing the last layer of the ‘Conv + Batch Normalization + SiLU’ (CBS) structure in the YOLOv5s backbone with a transformer self-attention module (T in the YOLOv5-TDHSA’s name), and also adding a similar module to the last layer of its neck, so that the image information can be used more comprehensively, (2) replacing the YOLOv5s coupled head with a decoupled head (DH in both models’ names) so as to increase the detection accuracy and speed up the convergence, and (3) adding a small-object detection layer (S in the YOLOv5-TDHSA’s name) and an adaptive anchor (A in the YOLOv5-TDHSA’s name) to the YOLOv5s neck to improve the detection of small objects. Based on experiments conducted on two public datasets, it is demonstrated that both proposed models perform better than the original YOLOv5s model and three other state-of-the-art models (Faster R-CNN, YOLOv4-Tiny, and YOLOv5n) in terms of the mean accuracy (mAP) and F1 score, achieving mAP values of 77.9% and 83.4% and F1 score values of 0.767 and 0.811 on the TT100K dataset, and mAP values of 68.1% and 69.8% and F1 score values of 0.71 and 0.72 on the CCTSDB2021 dataset, respectively, for YOLOv5-DH and YOLOv5-TDHSA. This was achieved, however, at the expense of both proposed models having a bigger size, greater number of parameters, and slower processing speed than YOLOv5s, YOLOv4-Tiny and YOLOv5n, surpassing only Faster R-CNN in this regard. The results also confirmed that the incorporation of the T and SA improvements into YOLOv5s leads to further enhancement, represented by the YOLOv5-TDHSA model, which is superior to the other proposed model, YOLOv5-DH, which avails of only one YOLOv5s improvement (i.e., DH). Full article

(This article belongs to the Special Issue Various Deep Learning Algorithms in Computational Intelligence)

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15 pages, 326 KiB

Open AccessEditor’s ChoiceArticle

Third-Order Differential Subordinations Using Fractional Integral of Gaussian Hypergeometric Function

by Georgia Irina Oros, Gheorghe Oros and Lavinia Florina Preluca

Axioms 2023, 12(2), 133; https://doi.org/10.3390/axioms12020133 - 28 Jan 2023

Cited by 6 | Viewed by 1404

Abstract

Sanford S. Miller and Petru T. Mocanu’s theory of second-order differential subordinations was extended for the case of third-order differential subordinations by José A. Antonino and Sanford S. Miller in 2011. In this paper, new results are proved regarding third-order differential subordinations that [...] Read more.

Sanford S. Miller and Petru T. Mocanu’s theory of second-order differential subordinations was extended for the case of third-order differential subordinations by José A. Antonino and Sanford S. Miller in 2011. In this paper, new results are proved regarding third-order differential subordinations that extend the ones involving the classical second-order differential subordination theory. A method for finding a dominant of a third-order differential subordination is provided when the behavior of the function is not known on the boundary of the unit disc. Additionally, a new method for obtaining the best dominant of a third-order differential subordination is presented. This newly proposed method essentially consists of finding the univalent solution for the differential equation that corresponds to the differential subordination considered in the investigation; previous results involving third-order differential subordinations have been obtained mainly by investigating specific classes of admissible functions. The fractional integral of the Gaussian hypergeometric function, previously associated with the theory of fuzzy differential subordination, is used in this paper to obtain an interesting third-order differential subordination by involving a specific convex function. The best dominant is also provided, and the example presented proves the importance of the theoretical results involving the fractional integral of the Gaussian hypergeometric function. Full article

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

23 pages, 363 KiB

Open AccessEditor’s ChoiceArticle

Boundary-Value Problem for Nonlinear Fractional Differential Equations of Variable Order with Finite Delay via Kuratowski Measure of Noncompactness

by Benoumran Telli, Mohammed Said Souid and Ivanka Stamova

Axioms 2023, 12(1), 80; https://doi.org/10.3390/axioms12010080 - 12 Jan 2023

Cited by 9 | Viewed by 1850

Abstract

This paper is devoted to boundary-value problems for Riemann–Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is first studied using a Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness. Secondly, the Ulam–Hyers stability criteria are examined. [...] Read more.

This paper is devoted to boundary-value problems for Riemann–Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is first studied using a Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness. Secondly, the Ulam–Hyers stability criteria are examined. All of the results in this study are established with the help of generalized intervals and piecewise constant functions. We convert the Riemann–Liouville fractional variable-order problem to equivalent standard Riemann–Liouville problems of fractional-constant orders. Finally, two examples are constructed to illustrate the validity of the observed results. Full article

(This article belongs to the Special Issue Mathematical Models and Simulations)

10 pages, 247 KiB

Open AccessEditor’s ChoiceArticle

Fixed Point Theorems for Generalized Classes of Operators

by Cristiana Ionescu

Axioms 2023, 12(1), 69; https://doi.org/10.3390/axioms12010069 - 9 Jan 2023

Cited by 2 | Viewed by 1932

Abstract

In this work, we consider weakly generalized operators, which extend the Geraghty mappings that are studied with regard to the existence and uniqueness of their fixed points, in the setting offered by strong b-metric spaces. Classic results are obtained as corollaries. An [...] Read more.

In this work, we consider weakly generalized operators, which extend the Geraghty mappings that are studied with regard to the existence and uniqueness of their fixed points, in the setting offered by strong b-metric spaces. Classic results are obtained as corollaries. An example is provided to support these outcomes. Full article

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

10 pages, 369 KiB

Open AccessEditor’s ChoiceArticle

New Classes of Degenerate Unified Polynomials

by Daniel Bedoya, Clemente Cesarano, Stiven Díaz and William Ramírez

Axioms 2023, 12(1), 21; https://doi.org/10.3390/axioms12010021 - 25 Dec 2022

Cited by 12 | Viewed by 1646

Abstract

In this paper, we introduce a class of new classes of degenerate unified polynomials and we show some algebraic and differential properties. This class includes the Appell-type classical polynomials and their most relevant generalizations. Most of the results are proved by using generating [...] Read more.

In this paper, we introduce a class of new classes of degenerate unified polynomials and we show some algebraic and differential properties. This class includes the Appell-type classical polynomials and their most relevant generalizations. Most of the results are proved by using generating function methods and we illustrate our results with some examples. Full article

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

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24 pages, 371 KiB

Open AccessEditor’s ChoiceArticle

On Fuzzy Implications Derived from General Overlap Functions and Their Relation to Other Classes

by Jocivania Pinheiro, Helida Santos, Graçaliz P. Dimuro, Benjamin Bedregal, Regivan H. N. Santiago, Javier Fernandez and Humberto Bustince

Axioms 2023, 12(1), 17; https://doi.org/10.3390/axioms12010017 - 24 Dec 2022

Cited by 5 | Viewed by 1828

Abstract

There are distinct techniques to generate fuzzy implication functions. Despite most of them using the combination of associative aggregators and fuzzy negations, other connectives such as (general) overlap/grouping functions may be a better strategy. Since these possibly non-associative operators have been successfully used [...] Read more.

There are distinct techniques to generate fuzzy implication functions. Despite most of them using the combination of associative aggregators and fuzzy negations, other connectives such as (general) overlap/grouping functions may be a better strategy. Since these possibly non-associative operators have been successfully used in many applications, such as decision making, classification and image processing, the idea of this work is to continue previous studies related to fuzzy implication functions derived from general overlap functions. In order to obtain a more general and flexible context, we extend the class of implications derived by fuzzy negations and t-norms, replacing the latter by general overlap functions, obtaining the so-called

(GO, N)

-implication functions. We also investigate their properties, the aggregation of

(GO, N)

-implication functions, their characterization and the intersections with other classes of fuzzy implication functions. Full article

(This article belongs to the Section Logic)

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16 pages, 663 KiB

Open AccessEditor’s ChoiceArticle

The Use of a Log-Normal Prior for the Student t-Distribution

by Se Yoon Lee

Axioms 2022, 11(9), 462; https://doi.org/10.3390/axioms11090462 - 8 Sep 2022

Cited by 7 | Viewed by 3714

Abstract

It is typically difficult to estimate the number of degrees of freedom due to the leptokurtic nature of the Student t-distribution. Particularly in studies with small sample sizes, special care is needed concerning prior choice in order to ensure that the analysis [...] Read more.

It is typically difficult to estimate the number of degrees of freedom due to the leptokurtic nature of the Student t-distribution. Particularly in studies with small sample sizes, special care is needed concerning prior choice in order to ensure that the analysis is not overly dominated by any prior distribution. In this article, popular priors used in the existing literature are examined by characterizing their distributional properties on an effective support where it is desirable to concentrate on most of the prior probability mass. Additionally, we suggest a log-normal prior as a viable prior option. We show that the Bayesian estimator based on a log-normal prior compares favorably to other Bayesian estimators based on the priors previously proposed via simulation studies and financial applications. Full article

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

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29 pages, 8785 KiB

Open AccessEditor’s ChoiceArticle

Design of Type-3 Fuzzy Systems and Ensemble Neural Networks for COVID-19 Time Series Prediction Using a Firefly Algorithm

by Patricia Melin, Daniela Sánchez, Juan R. Castro and Oscar Castillo

Axioms 2022, 11(8), 410; https://doi.org/10.3390/axioms11080410 - 17 Aug 2022

Cited by 29 | Viewed by 2537

Abstract

In this work, information on COVID-19 confirmed cases is utilized as a dataset to perform time series predictions. We propose the design of ensemble neural networks (ENNs) and type-3 fuzzy inference systems (FISs) for predicting COVID-19 data. The answers for each ENN module [...] Read more.

In this work, information on COVID-19 confirmed cases is utilized as a dataset to perform time series predictions. We propose the design of ensemble neural networks (ENNs) and type-3 fuzzy inference systems (FISs) for predicting COVID-19 data. The answers for each ENN module are combined using weights provided by the type-3 FIS, in which the ENN is also designed using the firefly algorithm (FA) optimization technique. The proposed method, called ENNT3FL-FA, is applied to the COVID-19 data for confirmed cases from 12 countries. The COVID-19 data have shown to be a complex time series due to unstable behavior in certain periods of time. For example, it is unknown when a new wave will exist and how it will affect each country due to the increase in cases due to many factors. The proposed method seeks mainly to find the number of modules of the ENN and the best possible parameters, such as lower scale and lower lag of the type-3 FIS. Each module of the ENN produces an individual prediction. Each prediction error is an input for the type-3 FIS; moreover, outputs provide a weight for each prediction, and then the final prediction can be calculated. The type-3 fuzzy weighted average (FWA) integration method is compared with the type-2 FWA to verify its ability to predict future confirmed cases by using two data periods. The achieved results show how the proposed method allows better results for the real prediction of 20 future days for most of the countries used in this study, especially when the number of data points increases. In countries such as Germany, India, Italy, Mexico, Poland, Spain, the United Kingdom, and the United States of America, on average, the proposed ENNT3FL-FA achieves a better performance for the prediction of future days for both data points. The proposed method proves to be more stable with complex time series to predict future information such as the one utilized in this study. Intelligence techniques and their combination in the proposed method are recommended for time series with many data points. Full article

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

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24 pages, 485 KiB

Open AccessEditor’s ChoiceArticle

A Theoretical Dynamical Noninteracting Model for General Manipulation Systems Using Axiomatic Geometric Structures

by Paolo Mercorelli

Axioms 2022, 11(7), 309; https://doi.org/10.3390/axioms11070309 - 25 Jun 2022

Cited by 4 | Viewed by 1784

Abstract

This paper presents a new theoretical approach to the study of robotics manipulators dynamics. It is based on the well-known geometric approach to system dynamics, according to which some axiomatic definitions of geometric structures concerning invariant subspaces are used. In such a framework, [...] Read more.

This paper presents a new theoretical approach to the study of robotics manipulators dynamics. It is based on the well-known geometric approach to system dynamics, according to which some axiomatic definitions of geometric structures concerning invariant subspaces are used. In such a framework, certain typical problems in robotics are mathematically formalised and analysed in axiomatic form. The outcomes are sufficiently general that it is possible to discuss the structural properties of robotic manipulation. A generalized theoretical linear model is used, and a thorough analysis is made. The noninteracting nature of this model is also proven through a specific theorem. Full article

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

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18 pages, 320 KiB

Open AccessEditor’s ChoiceArticle

Numerical Processes for Approximating Solutions of Nonlinear Equations

by Samundra Regmi, Ioannis K. Argyros, Santhosh George and Christopher I. Argyros

Axioms 2022, 11(7), 307; https://doi.org/10.3390/axioms11070307 - 24 Jun 2022

Cited by 5 | Viewed by 1829

Abstract

In this article, we present generalized conditions of three-step iterative schemes for solving nonlinear equations. The convergence order is shown using Taylor series, but the existence of high-order derivatives is assumed. However, only the first derivative appears on these schemes. Therefore, the hypotheses [...] Read more.

In this article, we present generalized conditions of three-step iterative schemes for solving nonlinear equations. The convergence order is shown using Taylor series, but the existence of high-order derivatives is assumed. However, only the first derivative appears on these schemes. Therefore, the hypotheses limit the utilization of the schemes to operators that are at least nine times differentiable, although the schemes may converge. To the best of our knowledge, no semi-local convergence has been given in the setting of a Banach space. Our goal is to extend the applicability of these schemes in both the local and semi-local convergence cases. Moreover, we use our idea of recurrent functions and conditions only on the derivative or divided differences of order one that appear in these schemes. This idea can be applied to extend other high convergence multipoint and multistep schemes. Numerical applications where the convergence criteria are tested complement this article. Full article

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

9 pages, 291 KiB

Open AccessEditor’s ChoiceArticle

Some Cardinal and Geometric Properties of the Space of Permutation Degree

by Ljubiša D. R. Kočinac, Farkhod G. Mukhamadiev and Anvar K. Sadullaev

Axioms 2022, 11(6), 290; https://doi.org/10.3390/axioms11060290 - 14 Jun 2022

Cited by 12 | Viewed by 2044

Abstract

This paper is devoted to the investigation of cardinal invariants such as the hereditary density, hereditary weak density, and hereditary Lindelöf number. The relation between the spread and the extent of the space

{SP}^{2} (R, τ (A))

[...] Read more.

This paper is devoted to the investigation of cardinal invariants such as the hereditary density, hereditary weak density, and hereditary Lindelöf number. The relation between the spread and the extent of the space

{SP}^{2} (R, τ (A))

of permutation degree of the Hattori space is discussed. In particular, it is shown that the space

{SP}^{2} (R, τ_{S})

contains a closed discrete subset of cardinality

c

. Moreover, it is shown that the functor

{SP}_{G}^{n}

preserves the homotopy and the retraction of topological spaces. In addition, we prove that if the spaces X and Y are homotopically equivalent, then the spaces

{SP}_{G}^{n} X

and

{SP}_{G}^{n} Y

are also homotopically equivalent. As a result, it has been proved that the functor

{SP}_{G}^{n}

is a covariant homotopy functor. Full article

(This article belongs to the Special Issue Advances in General Topology and Its Application)

17 pages, 1386 KiB

Open AccessFeature PaperEditor’s ChoiceReview

Continuous-Stage Runge–Kutta Approximation to Differential Problems

by Pierluigi Amodio, Luigi Brugnano and Felice Iavernaro

Axioms 2022, 11(5), 192; https://doi.org/10.3390/axioms11050192 - 21 Apr 2022

Cited by 9 | Viewed by 2325

Abstract

In recent years, the efficient numerical solution of Hamiltonian problems has led to the definition of a class of energy-conserving Runge–Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Such methods admit an interesting interpretation in terms of continuous-stage Runge–Kutta methods. In [...] Read more.

In recent years, the efficient numerical solution of Hamiltonian problems has led to the definition of a class of energy-conserving Runge–Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Such methods admit an interesting interpretation in terms of continuous-stage Runge–Kutta methods. In this review paper, we recall this aspect and extend it to higher-order differential problems. Full article

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