Axioms

13 pages, 400 KiB

Open AccessArticle

Impulsive Linearly Implicit Euler Method for the SIR Epidemic Model with Pulse Vaccination Strategy

by Gui-Lai Zhang, Zhi-Yong Zhu, Lei-Ke Chen and Song-Shu Liu

Axioms 2024, 13(12), 854; https://doi.org/10.3390/axioms13120854 - 4 Dec 2024

In this paper, a new numerical scheme, which we call the impulsive linearly implicit Euler method, for the SIR epidemic model with pulse vaccination strategy is constructed based on the linearly implicit Euler method. The sufficient conditions for global attractivity of an infection-free [...] Read more.

In this paper, a new numerical scheme, which we call the impulsive linearly implicit Euler method, for the SIR epidemic model with pulse vaccination strategy is constructed based on the linearly implicit Euler method. The sufficient conditions for global attractivity of an infection-free periodic solution of the impulsive linearly implicit Euler method are obtained. We further show that the limit of the disease-free periodic solution of the impulsive linearly implicit Euler method is the disease-free periodic solution of the exact solution when the step size tends to 0. Finally, two numerical experiments are given to confirm the conclusions. Full article

(This article belongs to the Special Issue Differential Equations and Inverse Problems, 2nd Editon)

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19 pages, 308 KiB

Open AccessArticle

Method for Investigation of Convergence of Formal Series Involved in Asymptotics of Solutions of Second-Order Differential Equations in the Neighborhood of Irregular Singular Points

by Maria Korovina and Ilya Smirnov

Axioms 2024, 13(12), 853; https://doi.org/10.3390/axioms13120853 - 3 Dec 2024

Abstract

The aim of the article is to create a method for studying the asymptotics of solutions to second-order differential equations with irregular singularities. The method allows us to prove the convergence of formal series included in the asymptotics of solutions for a wide [...] Read more.

The aim of the article is to create a method for studying the asymptotics of solutions to second-order differential equations with irregular singularities. The method allows us to prove the convergence of formal series included in the asymptotics of solutions for a wide class of second-order differential equations in the neighborhoods of their irregular singular points, including the point at infinity, which is generally irregular. The article provides a number of applications of the method for studying the asymptotics of solutions to both ordinary differential equations and partial differential equations. Full article

18 pages, 11181 KiB

Open AccessArticle

Topological Properties of the Intersection Curves Between a Torus and Families of Parabolic or Elliptical Cylinders

by Ana Breda, Alexandre Trocado and José Dos Santos

Axioms 2024, 13(12), 852; https://doi.org/10.3390/axioms13120852 - 3 Dec 2024

Abstract

This paper reports the research work carried out with the goal of geometrically and algebraically describing, as well as topologically classifying, the curves resulting from the intersection of a torus with families of parabolic and elliptical cylinders within a purely Euclidean framework. The [...] Read more.

This paper reports the research work carried out with the goal of geometrically and algebraically describing, as well as topologically classifying, the curves resulting from the intersection of a torus with families of parabolic and elliptical cylinders within a purely Euclidean framework. The parabolic cylinders under analysis have generatrices parallel to the axis of the torus, whereas the elliptical cylinders, centered at the same point as the torus, have axes either aligned with or orthogonal to the torus’s axis. For the topological classification of these intersection curves, we consider their number of connected components and self-intersection points. GeoGebra, which was used to create the 3D visual geometric representations of the intersection curves, and Maple, which was used to perform the essential symbolic algebraic calculations, were critical computational tools in the development of this work. Theoretical and computational approaches are interwoven throughout this study, with the computational work serving as the foundation for exploration and providing insights that contributed to the theoretical validation of the results revealed through GeoGebra simulations. Full article

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10 pages, 237 KiB

Open AccessArticle

A Note on Injective g-Frames in Quaternionic Hilbert Spaces

by Jianxia Zhang, Fugen Gao and Guoqing Hong

Axioms 2024, 13(12), 851; https://doi.org/10.3390/axioms13120851 - 3 Dec 2024

Abstract

Motivated by recent advancements in the quantum detection problem employing both discrete and continuous frames, this paper delves into a quantum detection problem utilizing g-frames within the context of quaternionic Hilbert spaces. We offer several equivalent representations of injective g-frames in separable quaternionic [...] Read more.

Motivated by recent advancements in the quantum detection problem employing both discrete and continuous frames, this paper delves into a quantum detection problem utilizing g-frames within the context of quaternionic Hilbert spaces. We offer several equivalent representations of injective g-frames in separable quaternionic Hilbert spaces. By normalizing the trace, we establish a classification for the g-frame injectivity problem. Additionally, we propose a method to derive an injective g-frame by leveraging an injective frame within quaternionic Hilbert spaces. Furthermore, we demonstrate that the injectivity of a g-frame remains intact under a linear isomorphism, while injective g-frames exhibit instability in infinite-dimensional scenarios. Full article

(This article belongs to the Section Geometry and Topology)

16 pages, 4018 KiB

Open AccessArticle

Fractals as Julia Sets for a New Complex Function via a Viscosity Approximation Type Iterative Methods

by Ahmad Almutlg and Iqbal Ahmad

Axioms 2024, 13(12), 850; https://doi.org/10.3390/axioms13120850 - 3 Dec 2024

Abstract

In this article, we examine and investigate various variants of Julia set patterns for complex exponential functions

W (z) = α e^{z^{n}} + \frac{β}{z^{m}} + log c^{t},

and [...] Read more.

In this article, we examine and investigate various variants of Julia set patterns for complex exponential functions

W (z) = α e^{z^{n}} + \frac{β}{z^{m}} + log c^{t},

and

T (z) = α e^{z^{n}} + \frac{β}{z^{m}} + γ

(which are analytic except at

z = 0

) where

n \geq 2,

m, n \in N,

α, β, γ \in C, c \in C ∖ {0}

and

t \in R, t \geq 1,

by employing a viscosity approximation-type iterative method. We employ the proposed iterative method to establish an escape criterion for visualizing Julia sets. We provide graphical illustrations of Julia sets that emphasize their sensitivity to different iteration parameters. We present graphical illustrations of Julia sets; the color, size, and shape of the images change with variations in the iteration parameters. With fixed input parameters, we observe the intriguing behavior of Julia sets for different values of n and m. We hope that the conclusions of this study will inspire researchers with an interest in fractal geometry. Full article

(This article belongs to the Special Issue Fractal Analysis and Mathematical Integration)

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23 pages, 469 KiB

Open AccessArticle

Variational Bayesian Estimation of Quantile Nonlinear Dynamic Latent Variable Models with Possible Nonignorable Missingness

by Mulati Tuerde and Ahmadjan Muhammadhaji

Axioms 2024, 13(12), 849; https://doi.org/10.3390/axioms13120849 - 3 Dec 2024

Abstract

Our study presents an innovative variational Bayesian parameter estimation method for the Quantile Nonlinear Dynamic Latent Variable Model (QNDLVM), particularly when dealing with missing data and nonparametric priors. This method addresses the computational inefficiencies associated with the traditional Markov chain Monte Carlo (MCMC) [...] Read more.

Our study presents an innovative variational Bayesian parameter estimation method for the Quantile Nonlinear Dynamic Latent Variable Model (QNDLVM), particularly when dealing with missing data and nonparametric priors. This method addresses the computational inefficiencies associated with the traditional Markov chain Monte Carlo (MCMC) approach, which struggles with large datasets and high-dimensional parameters due to its prolonged computation times, slow convergence, and substantial memory consumption. By harnessing the deterministic variational Bayesian framework, we convert the complex parameter estimation into a more manageable deterministic optimization problem. This is achieved by leveraging the hierarchical structure of the QNDLVM and the principle of efficiently optimizing approximate posterior distributions within the variational Bayesian framework. We further optimize the evidence lower bound using the coordinate ascent algorithm. To specify propensity scores for missing data manifestations and covariates, we adopt logistic and probit models, respectively, with conditionally conjugate mean field variational Bayes for logistic models. Additionally, we utilize Bayesian local influence to analyze the Ecological Momentary Assessment (EMA) dataset. Our results highlight the variational Bayesian approach’s notable accuracy and its ability to significantly alleviate computational demands, as demonstrated through simulation studies and practical applications. Full article

(This article belongs to the Special Issue Recent Advances in Statistical Modeling and Simulations with Applications, 2nd Edition)

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16 pages, 2141 KiB

Open AccessArticle

An Optimization Model for Production Scheduling in Parallel Machine Systems

by Leting Zu, Wenzhu Liao and Xiaoxia Yang

Axioms 2024, 13(12), 848; https://doi.org/10.3390/axioms13120848 - 2 Dec 2024

Abstract

The efficiency and quality of the manufacturing industry are greatly influenced by production scheduling, which makes it a crucial aspect. A well-designed production scheduling scheme can significantly enhance manufacturing efficiency and reduce enterprise costs. This paper presents a tailored optimization model designed to [...] Read more.

The efficiency and quality of the manufacturing industry are greatly influenced by production scheduling, which makes it a crucial aspect. A well-designed production scheduling scheme can significantly enhance manufacturing efficiency and reduce enterprise costs. This paper presents a tailored optimization model designed to address a more complex production scheduling problem that incorporates parallel machines and preventive maintenance. The proposed solutions aim to achieve a balance between job sequence and machine reliability, considering the minimum maintenance cost rate for determining maintenance cycles of deteriorating machines in real manufacturing scenarios. Furthermore, the objective of minimizing the maximum completion time guides machine assignment and job sequence based on maintenance constraints. The innovation lies in the introduction of a greedy algorithm that utilizes a water injection model to address this NP-hard integrated problem. A pre-distribution model is constructed using the water injection model, and its solution is utilized as input for constructing the production scheduling model, which aids in determining machine assignment and job sequence. This algorithm demonstrates remarkable effectiveness and efficiency, enabling the achievement of an optimal solution. A numerical example is presented to illustrate the computational process, accompanied by an extensive discussion of the results showcasing improved performance. Furthermore, the optimization model developed in this paper can be adapted to tackle the production scheduling problem with modifications tailored for parallel machines. Full article

(This article belongs to the Special Issue Advances in Mathematical Modeling, Analysis and Optimization)

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28 pages, 375 KiB

Open AccessArticle

Functional Differential Equations with an Advanced Neutral Term: New Monotonic Properties of Recursive Nature to Optimize Oscillation Criteria

by Amany Nabih, Wedad Albalawi, Mohammad S. Jazmati, Ali Elrashidi, Hegagi M. Ali and Osama Moaaz

Axioms 2024, 13(12), 847; https://doi.org/10.3390/axioms13120847 - 2 Dec 2024

Abstract

The goal of this study is to derive new conditions that improve the testing of the oscillatory and asymptotic features of fourth-order differential equations with an advanced neutral term. By using Riccati techniques and comparison with lower-order equations, we establish new criteria that [...] Read more.

The goal of this study is to derive new conditions that improve the testing of the oscillatory and asymptotic features of fourth-order differential equations with an advanced neutral term. By using Riccati techniques and comparison with lower-order equations, we establish new criteria that verify the absence of positive solutions and, consequently, the oscillation of all solutions to the investigated equation. Using our results to analyze a few specific instances of the examined equation, we can ultimately clarify the significance of the new inequalities. Our results are an extension of previous results that considered equations with a neutral delay term and also an improvement of previous results that considered only equations with an advanced neutral term. Full article

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

24 pages, 356 KiB

Open AccessArticle

Set-Theoretical Solutions for the Yang–Baxter Equation in GE-Algebras: Applications to Quantum Spin Systems

by Ibrahim Senturk, Tahsin Oner, Abdullah Engin Çalık, Hüseyin Şirin, Metin Bilge and Neelamegarajan Rajesh

Axioms 2024, 13(12), 846; https://doi.org/10.3390/axioms13120846 - 2 Dec 2024

Abstract

This manuscript presents set-theoretical solutions to the Yang–Baxter equation within the framework of GE-algebras by constructing mappings that satisfy the braid condition and exploring the algebraic properties of GE-algebras. Detailed proofs and the use of left and right translation operators are provided to [...] Read more.

This manuscript presents set-theoretical solutions to the Yang–Baxter equation within the framework of GE-algebras by constructing mappings that satisfy the braid condition and exploring the algebraic properties of GE-algebras. Detailed proofs and the use of left and right translation operators are provided to analyze these algebraic interactions, while an algorithm is introduced to automate the verification process, facilitating broader applications in quantum mechanics and mathematical physics. Additionally, the Yang–Baxter equation is applied to spin transformations in quantum mechanical spin-

\frac{1}{2}

systems, with transformations like rotations and reflections modeled using GE-algebras. A Cayley table is used to represent the algebraic structure of these transformations, and the proposed algorithm ensures that these solutions are consistent with the Yang–Baxter equation, offering new insights into the role of GE-algebras in quantum spin systems. Full article

(This article belongs to the Special Issue Yang-Baxter Equations, Nonassociative Structures and Applications—In Memoriam, Stefan Papadima)

28 pages, 2723 KiB

Open AccessArticle

A Comprehensive Model and Numerical Study of Shear Flow in Compressible Viscous Micropolar Real Gases

by Nelida Črnjarić and Ivan Dražić

Axioms 2024, 13(12), 845; https://doi.org/10.3390/axioms13120845 - 2 Dec 2024

Abstract

Understanding shear flow behavior in compressible, viscous, micropolar real gases is essential for both theoretical advancements and practical engineering applications. This study develops a comprehensive model that integrates micropolar fluid theory with compressible flow dynamics to accurately describe the behavior of real gases [...] Read more.

Understanding shear flow behavior in compressible, viscous, micropolar real gases is essential for both theoretical advancements and practical engineering applications. This study develops a comprehensive model that integrates micropolar fluid theory with compressible flow dynamics to accurately describe the behavior of real gases under shear stress. We formulate the governing equations by incorporating viscosity and micropolar effects and transform the obtained system into the mass Lagrangian coordinates. Two numerical methods, Faedo–Galerkin approximation and finite-difference methods, are used to solve it. These methods are compared using several benchmark examples to assess their accuracy and computational efficiency. Both methods demonstrate good performance, achieving equally precise results in capturing essential flow characteristics. However, the finite-difference method offers advantages in speed, stability, and lower computational demands. This research bridges gaps in existing models and establishes a foundation for further investigations into complex flow phenomena in micropolar real gases. Full article

(This article belongs to the Special Issue Recent Progress in Computational Fluid Dynamics)

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19 pages, 318 KiB

Open AccessArticle

Sharp Second-Order Hankel Determinants Bounds for Alpha-Convex Functions Connected with Modified Sigmoid Functions

by Muhammad Abbas, Reem K. Alhefthi, Daniele Ritelli and Muhammad Arif

Axioms 2024, 13(12), 844; https://doi.org/10.3390/axioms13120844 - 1 Dec 2024

Abstract

The study of the Hankel determinant generated by the Maclaurin series of holomorphic functions belonging to particular classes of normalized univalent functions is one of the most significant problems in geometric function theory. Our goal in this study is first to define a [...] Read more.

The study of the Hankel determinant generated by the Maclaurin series of holomorphic functions belonging to particular classes of normalized univalent functions is one of the most significant problems in geometric function theory. Our goal in this study is first to define a family of alpha-convex functions associated with modified sigmoid functions and then to investigate sharp bounds of initial coefficients, Fekete-Szegö inequality, and second-order Hankel determinants. Moreover, we also examine the logarithmic and inverse coefficients of functions within a defined family regarding recent issues. All of the estimations that were found are sharp. Full article

(This article belongs to the Special Issue Recent Advances in Complex Analysis and Related Topics)

29 pages, 11222 KiB

Open AccessArticle

Computational Study on Flow Characteristics of Shocked Light Backward-Triangular Bubbles in Polyatomic Gas

by Salman Saud Alsaeed and Satyvir Singh

Axioms 2024, 13(12), 843; https://doi.org/10.3390/axioms13120843 - 1 Dec 2024

Abstract

This study computationally examined the Richtmyer–Meshkov instability (RMI) evolution in a helium backward-triangular bubble immersed in monatomic argon, diatomic nitrogen, and polyatomic methane under planar shock wave interactions. Using high-fidelity numerical simulations based on the compressible Navier–Fourier equations based on the Boltzmann–Curtiss kinetic [...] Read more.

This study computationally examined the Richtmyer–Meshkov instability (RMI) evolution in a helium backward-triangular bubble immersed in monatomic argon, diatomic nitrogen, and polyatomic methane under planar shock wave interactions. Using high-fidelity numerical simulations based on the compressible Navier–Fourier equations based on the Boltzmann–Curtiss kinetic framework and simulated via a modal discontinuous Galerkin scheme, we analyze the complex interplay of shock-bubble dynamics. Key findings reveal distinct thermal non-equilibrium effects, vorticity generation, enstrophy evolution, kinetic energy dissipation, and interface deformation across gases. Methane, with its molecular complexity and higher viscosity, exhibits the highest levels of vorticity production, enstrophy, and kinetic energy, leading to pronounced Kelvin–Helmholtz instabilities and enhanced mixing. Conversely, argon, due to its simpler atomic structure, shows weaker deformation and mixing. Thermal non-equilibrium effects, quantified by the Rayleigh–Onsager dissipation function, are most significant in methane, indicating delayed energy relaxation and intense turbulence. This study highlights the pivotal role of molecular properties, specific heat ratio, and bulk viscosity in shaping RMI dynamics in polyatomic gases, offering insights on uses such as high-speed aerodynamics, inertial confinement fusion, and supersonic mixing. Full article

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12 pages, 264 KiB

Open AccessArticle

Subclasses of q-Uniformly Starlike Functions Obtained via the q-Carlson–Shaffer Operator

by Qiuxia Hu, Rizwan Salim Badar and Muhammad Ghaffar Khan

Axioms 2024, 13(12), 842; https://doi.org/10.3390/axioms13120842 - 29 Nov 2024

Abstract

This article investigates the applications of the q-Carlson–Shaffer operator on subclasses of q-uniformly starlike functions, introducing the class

S T_{q} (m, c, d, β)

. The study establishes a necessary condition for membership in this class [...] Read more.

This article investigates the applications of the q-Carlson–Shaffer operator on subclasses of q-uniformly starlike functions, introducing the class

S T_{q} (m, c, d, β)

. The study establishes a necessary condition for membership in this class and examines its behavior within conic domains. The article delves into properties such as coefficient bounds, the Fekete–Szegö inequality, and criteria defined via the Hadamard product, providing both necessary and sufficient conditions for these properties. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 3rd Edition)

27 pages, 333 KiB

Open AccessArticle

Fixed-Point Results for Multi-Valued Mappings in Topological Vector Space-Valued Cone Metric Spaces with Applications

by Hala Alzumi and Jamshaid Ahmad

Axioms 2024, 13(12), 841; https://doi.org/10.3390/axioms13120841 - 29 Nov 2024

Abstract

The objective of this research article is to introduce Kikkawa and Suzuki-type contractions in the setting of topological vector space-valued cone metric space with a solid cone and establish some new fixed point results for multi-valued mappings. The problem of finding fixed points [...] Read more.

The objective of this research article is to introduce Kikkawa and Suzuki-type contractions in the setting of topological vector space-valued cone metric space with a solid cone and establish some new fixed point results for multi-valued mappings. The problem of finding fixed points for multi-valued mappings satisfying locally contractive conditions on a closed ball is also addressed. Our findings generalize a number of well-established results in the literature. To highlight the uniqueness of our key finding, we present an example. As a demonstration of the applicability of our principal theorem, we prove a result in homotopy theory. Full article

(This article belongs to the Special Issue Advances in Fixed Point Theory with Applications)

33 pages, 3247 KiB

Open AccessArticle

Optimization of General Power-Sum Connectivity Index in Uni-Cyclic Graphs, Bi-Cyclic Graphs and Trees by Means of Operations

by Muhammad Yasin Khan, Gohar Ali and Ioan-Lucian Popa

Axioms 2024, 13(12), 840; https://doi.org/10.3390/axioms13120840 - 28 Nov 2024

Abstract

The field of indices has been explored and advanced by various researchers for different purposes. One purpose is the optimization of indices in various problems. In this work, the general power-sum connectivity index is considered. The general power-sum connectivity index was investigated for [...] Read more.

The field of indices has been explored and advanced by various researchers for different purposes. One purpose is the optimization of indices in various problems. In this work, the general power-sum connectivity index is considered. The general power-sum connectivity index was investigated for k-generalized quasi-trees where optimal graphs were found. Further, in this work, we extend the idea of optimization to families of graphs, including uni-cyclic graphs, bi-cyclic graphs and trees. The optimization is carried out by means of operations named as Operation A, B, C and D. The first two operations increase the value of the general power-sum connectivity index, while the last two work opposite to Operations A and B. These operations are explained by means of diagrams, where one can easily obtain their working procedures. Full article

(This article belongs to the Special Issue Graph Theory and Combinatorics: Theory and Applications)

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21 pages, 377 KiB

Open AccessFeature PaperArticle

On Schur Forms for Matrices with Simple Eigenvalues

by Mihail Mihaylov Konstantinov and Petko Hristov Petkov

Axioms 2024, 13(12), 839; https://doi.org/10.3390/axioms13120839 - 28 Nov 2024

Abstract

In this paper, we consider various aspects of the Schur problem for a square complex matrix A, namely the similarity unitary transformation of A into upper triangular form containing the eigenvalues of A on its diagonal. Since the profound work of I. [...] Read more.

In this paper, we consider various aspects of the Schur problem for a square complex matrix A, namely the similarity unitary transformation of A into upper triangular form containing the eigenvalues of A on its diagonal. Since the profound work of I. Schur published in 1909, this has become a fundamental issue in the theory and applications of matrices. Nevertheless, certain details concerning the Schur problem need further clarification, especially in connection with the perturbation analysis of the Schur decomposition relative to perturbations in the matrix A. We consider both canonical and condensed Schur forms. Special attention is paid to matrices with simple eigenvalues. Some new concepts, such as quasi-Schur forms and diagonally spectral matrices, are also introduced and studied. Full article

19 pages, 2188 KiB

Open AccessArticle

Simultaneous Method for Solving Certain Systems of Matrix Equations with Two Unknowns

by Predrag S. Stanimirović, Miroslav Ćirić, Spyridon D. Mourtas, Gradimir V. Milovanović and Milena J. Petrović

Axioms 2024, 13(12), 838; https://doi.org/10.3390/axioms13120838 - 28 Nov 2024

Abstract

Quantitative bisimulations between weighted finite automata are defined as solutions of certain systems of matrix-vector inequalities and equations. In the context of fuzzy automata and max-plus automata, testing the existence of bisimulations and their computing are performed through a sequence of matrices that [...] Read more.

Quantitative bisimulations between weighted finite automata are defined as solutions of certain systems of matrix-vector inequalities and equations. In the context of fuzzy automata and max-plus automata, testing the existence of bisimulations and their computing are performed through a sequence of matrices that is built member by member, whereby the next member of the sequence is obtained by solving a particular system of linear matrix-vector inequalities and equations in which the previously computed member appears. By modifying the systems that define bisimulations, systems of matrix-vector inequalities and equations with k unknowns are obtained. Solutions of such systems, in the case of existence, witness to the existence of a certain type of partial equivalence, where it is not required that the word functions computed by two WFAs match on all input words, but only on all input words whose lengths do not exceed k. Solutions of these new systems represent finite sequences of matrices which, in the context of fuzzy automata and max-plus automata, are also computed sequentially, member by member. Here we deal with those systems in the context of WFAs over the field of real numbers and propose a different approach, where all members of the sequence are computed simultaneously. More precisely, we apply a simultaneous approach in solving the corresponding systems of matrix-vector equations with two unknowns. Zeroing neural network (ZNN) neuro-dynamical systems for approximating solutions of heterotypic bisimulations are proposed. Numerical simulations are performed for various random initial states and comparison with the Matlab, linear programming solver linprog, and the pseudoinverse solution generated by the standard function pinv is given. Full article

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

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16 pages, 307 KiB

Open AccessArticle

Horadam–Lucas Cubes

by Elif Tan, Luka Podrug and Vesna Iršič Chenoweth

Axioms 2024, 13(12), 837; https://doi.org/10.3390/axioms13120837 - 28 Nov 2024

Abstract

In this paper, we introduce a novel class of graphs referred to as the Horadam–Lucas cubes. This class extends the concept of Lucas cubes and retains numerous desirable properties associated with them. Horadam–Lucas cubes can also be viewed as a companion graph family [...] Read more.

In this paper, we introduce a novel class of graphs referred to as the Horadam–Lucas cubes. This class extends the concept of Lucas cubes and retains numerous desirable properties associated with them. Horadam–Lucas cubes can also be viewed as a companion graph family of the Horadam cubes, in a similar way the Lucas cubes relate to Fibonacci cubes or the Lucas-run graphs relate to Fibonacci-run graphs. As special cases, they also give rise to new graph families, such as Pell–Lucas cubes and Jacobsthal–Lucas cubes. We derive the several metric and enumerative properties of these cubes, including their diameter, periphery, radius, fundamental decomposition, number of edges, cube polynomials, and generating function of the cube polynomials. Full article

(This article belongs to the Special Issue Recent Developments in Graph Theory)

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23 pages, 1345 KiB

Open AccessArticle

Fuzzy Decision Tree Based on Fuzzy Rough Sets and Z-Number Rules

by Boya Zhu, Jingqian Wang and Xiaohong Zhang

Axioms 2024, 13(12), 836; https://doi.org/10.3390/axioms13120836 - 28 Nov 2024

Abstract

The decision tree algorithm is widely used in various classification problems due to its ease of implementation and strong interpretability. However, information in the real world often has uncertainty and partial reliability, which poses challenges for classification tasks. To address this issue, this [...] Read more.

The decision tree algorithm is widely used in various classification problems due to its ease of implementation and strong interpretability. However, information in the real world often has uncertainty and partial reliability, which poses challenges for classification tasks. To address this issue, this paper proposes a fuzzy decision tree based on fuzzy rough sets and Z-numbers, aimed at enhancing the decision tree’s ability to handle fuzzy and uncertain information. In the aspect of rule extraction, we combine the fuzzy rough set model to propose a fuzzy confidence based on lower approximation as a metric for attribute selection, effectively addressing the role of imprecise knowledge in classification. In terms of the tree structure, the concept of Z-numbers is introduced, specifically focusing on the fuzzy constraint reliability B, making the information representation more aligned with human evaluation habits, as well as using Z-number rules to replace traditional fuzzy rules in constructing the fuzzy decision tree. Furthermore, as generating Z-numbers still presents certain challenges, this paper also establishes a method for reasonably generating Z-numbers in situations with limited information, utilizing the generated fuzzy constraint reliability B to adjust fuzzy numbers A. Finally, the proposed decision tree algorithm is experimentally compared with other classifiers, and the results indicate that this algorithm demonstrates higher classification accuracy and a more concise tree structure when handling datasets containing fuzzy and uncertain factors. This research enriches the existing research on fuzzy decision trees and shows greater potential in solving practical problems. Full article

(This article belongs to the Special Issue Advances in Fuzzy Theory and Decision-Making Theory)

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