Axioms

14 pages, 413 KiB

Open AccessArticle

From the Crossing Numbers of K₅ + P_n and K₅ + C_n to the Crossing Numbers of W_m + S_n and W_m + W_n

by Michal Staš, Jana Fortes and Mária Švecová

Axioms 2024, 13(7), 427; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13070427 (registering DOI) - 25 Jun 2024

Viewed by 74

The crossing number of a graph is a significant measure that indicates the complexity of the graph and the difficulty of visualizing it. In this paper, we examine the crossing numbers of join products involving the complete graph

K_{5}

with discrete graphs, [...] Read more.

The crossing number of a graph is a significant measure that indicates the complexity of the graph and the difficulty of visualizing it. In this paper, we examine the crossing numbers of join products involving the complete graph

K_{5}

with discrete graphs, paths, and cycles. We analyze optimal drawings of

K_{5}

, identify all five non-isomorphic drawings, and address previously hypothesized crossing numbers for

K_{5} + P_{n}

, and

K_{5} + C_{n}

. Through a simplified approach, we first establish

cr (K_{5} + D_{n})

and then extend our method to prove the crossing numbers

cr (K_{5} + P_{n})

and

cr (K_{5} + C_{n})

. These results also lead to new hypotheses for

cr (W_{m} + S_{n})

and

cr (W_{m} + W_{n})

involving wheels and stars. Our findings correct previous inaccuracies in the literature and provide a foundation for future research. Full article

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

► Show Figures

Figure 1

16 pages, 384 KiB

Open AccessArticle

Optimal and Efficient Approximations of Gradients of Functions with Nonindependent Variables

by Matieyendou Lamboni

Axioms 2024, 13(7), 426; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13070426 (registering DOI) - 25 Jun 2024

Viewed by 66

Abstract

Gradients of smooth functions with nonindependent variables are relevant for exploring complex models and for the optimization of the functions subjected to constraints. In this paper, we investigate new and simple approximations and computations of such gradients by making use of independent, central, [...] Read more.

Gradients of smooth functions with nonindependent variables are relevant for exploring complex models and for the optimization of the functions subjected to constraints. In this paper, we investigate new and simple approximations and computations of such gradients by making use of independent, central, and symmetric variables. Such approximations are well suited for applications in which the computations of the gradients are too expansive or impossible. The derived upper bounds of the biases of our approximations do not suffer from the curse of dimensionality for any 2-smooth function, and they theoretically improve the known results. Also, our estimators of such gradients reach the optimal (mean squared error) rates of convergence (i.e.,

O (N^{- 1})

) for the same class of functions. Numerical comparisons based on a test case and a high-dimensional PDE model show the efficiency of our approach. Full article

(This article belongs to the Special Issue Recent Research on Functions with Non-Independent Variables)

► Show Figures

Figure 1

14 pages, 252 KiB

Open AccessArticle

Three Existence Results in the Fixed Point Theory

by Alexander J. Zaslavski

Axioms 2024, 13(7), 425; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13070425 (registering DOI) - 25 Jun 2024

Viewed by 105

Abstract

In the present paper, we obtain three results on the existence of a fixed point for nonexpansive mappings. Two of them are generalizations of the result for F-contraction, while third one is a generalization of a recent result for set-valued contractions. Full article

(This article belongs to the Special Issue Trends in Fixed Point Theory and Fractional Calculus)

31 pages, 13721 KiB

Open AccessArticle

An Enhanced Fuzzy Hybrid of Fireworks and Grey Wolf Metaheuristic Algorithms

by Juan Barraza, Luis Rodríguez, Oscar Castillo, Patricia Melin and Fevrier Valdez

Axioms 2024, 13(7), 424; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13070424 - 24 Jun 2024

Viewed by 157

Abstract

This research work envisages addressing fuzzy adjustment of parameters into a hybrid optimization algorithm for solving mathematical benchmark function problems. The problem of benchmark mathematical functions consists of finding the minimal values. In this study, we considered function optimization. We are presenting an [...] Read more.

This research work envisages addressing fuzzy adjustment of parameters into a hybrid optimization algorithm for solving mathematical benchmark function problems. The problem of benchmark mathematical functions consists of finding the minimal values. In this study, we considered function optimization. We are presenting an enhanced Fuzzy Hybrid Algorithm, which is called Enhanced Fuzzy Hybrid Fireworks and Grey Wolf Metaheuristic Algorithm, and denoted as EF-FWA-GWO. The fuzzy adjustment of parameters is achieved using Fuzzy Inference Systems. For this work, we implemented two variants of the Fuzzy Systems. The first variant utilizes Triangular membership functions, and the second variant employs Gaussian membership functions. Both variants are of a Mamdani Fuzzy Inference Type. The proposed method was applied to 22 mathematical benchmark functions, divided into two parts: the first part consists of 13 functions that can be classified as unimodal and multimodal, and the second part consists of the 9 fixed-dimension multimodal benchmark functions. The proposed method presents better performance with 60 and 90 dimensions, averaging 51% and 58% improvement in the benchmark functions, respectively. And then, a statistical comparison between the conventional hybrid algorithm and the Fuzzy Enhanced Hybrid Algorithm is presented to complement the conclusions of this research. Finally, we also applied the Fuzzy Hybrid Algorithm in a control problem to test its performance in designing a Fuzzy controller for a mobile robot. Full article

(This article belongs to the Special Issue Advances in Mathematical Optimization Algorithms and Its Applications)

► Show Figures

Figure 1

28 pages, 772 KiB

Open AccessArticle

C₀-Semigroups Approach to the Reliability Model Based on Robot-Safety System

by Ehmet Kasim and Aihemaitijiang Yumaier

Axioms 2024, 13(7), 423; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13070423 - 24 Jun 2024

Viewed by 160

Abstract

This paper considers a system with one robot and n safety units (one of which works while the others remain on standby), which is described by an integro-deferential equation. The system can fail in the following three ways: fails with an incident, fails [...] Read more.

This paper considers a system with one robot and n safety units (one of which works while the others remain on standby), which is described by an integro-deferential equation. The system can fail in the following three ways: fails with an incident, fails safely and fails due to the malfunction of the robot. Using the

C_{0} -

semigroups theory of linear operators, we first show that the system has a unique non-negative, time-dependent solution. Then, we obtain the exponential convergence of the time-dependent solution to its steady-state solution. In addition, we study the asymptotic behavior of some time-dependent reliability indices and present a numerical example demonstrating the effects of different parameters on the system. Full article

(This article belongs to the Special Issue Infinite Dynamical System and Differential Equations)

19 pages, 292 KiB

Open AccessArticle

Extensions of Some Statistical Concepts to the Complex Domain

by Arak M. Mathai

Axioms 2024, 13(7), 422; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13070422 - 22 Jun 2024

Viewed by 149

Abstract

This paper deals with the extension of principal component analysis, canonical correlation analysis, the Cramer–Rao inequality, and a few other statistical concepts in the real domain to the corresponding complex domain. Optimizations of Hermitian forms under a linear constraint, a bilinear form under [...] Read more.

This paper deals with the extension of principal component analysis, canonical correlation analysis, the Cramer–Rao inequality, and a few other statistical concepts in the real domain to the corresponding complex domain. Optimizations of Hermitian forms under a linear constraint, a bilinear form under Hermitian-form constraints, and similar maxima/minima problems in the complex domain are discussed. Some vector/matrix differential operators are developed to handle the above types of problems. These operators in the complex domain and the optimization problems in the complex domain are believed to be new and novel. These operators will also be useful in maximum likelihood estimation problems, which will be illustrated in the concluding remarks. Detailed steps are given in the derivations so that the methods are easily accessible to everyone. Full article

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

12 pages, 273 KiB

Open AccessArticle

Symmetric Identities Involving the Extended Degenerate Central Fubini Polynomials Arising from the Fermionic p-Adic Integral on \({\mathbb{Z}_{p}}\)

by Maryam Salem Alatawi, Waseem Ahmad Khan and Ugur Duran

Axioms 2024, 13(7), 421; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13070421 - 22 Jun 2024

Viewed by 169

Abstract

Since the constructions of p-adic q-integrals, these integrals as well as particular cases have been used not only as integral representations of many special functions, polynomials, and numbers, but they also allow for deep examinations of many families of special numbers [...] Read more.

Since the constructions of p-adic q-integrals, these integrals as well as particular cases have been used not only as integral representations of many special functions, polynomials, and numbers, but they also allow for deep examinations of many families of special numbers and polynomials, such as central Fubini, Bernoulli, central Bell, and Changhee numbers and polynomials. One of the key applications of these integrals is for obtaining the symmetric identities of certain special polynomials. In this study, we focus on a novel generalization of degenerate central Fubini polynomials. First, we introduce two variable degenerate w-torsion central Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. Through this representation, we investigate several symmetric identities for these polynomials using special p-adic integral techniques. Also, using series manipulation methods, we obtain an identity of symmetry for the two variable degenerate w-torsion central Fubini polynomials. Finally, we provide a representation of the degenerate differential operator on the two variable degenerate w-torsion central Fubini polynomials related to the degenerate central factorial polynomials of the second kind. Full article

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

16 pages, 582 KiB

Open AccessArticle

Generalization of the Distance Fibonacci Sequences

by Nur Şeyma Yilmaz, Andrej Włoch and Engin Özkan

Axioms 2024, 13(7), 420; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13070420 - 21 Jun 2024

Viewed by 200

Abstract

In this study, we introduced a generalization of distance Fibonacci sequences and investigate some of its basic properties. We then proposed a generalization of distance Fibonacci sequences for negative integers and investigated some basic properties. Additionally, we explored the construction of matrix generators [...] Read more.

In this study, we introduced a generalization of distance Fibonacci sequences and investigate some of its basic properties. We then proposed a generalization of distance Fibonacci sequences for negative integers and investigated some basic properties. Additionally, we explored the construction of matrix generators for these sequences and offered a graphical representation to clarify their structure. Furthermore, we demonstrated how these generalizations can be applied to obtain the Padovan and Narayana sequences for specific parameter values. Full article

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

32 pages, 1415 KiB

Open AccessArticle

Enhanced Kepler Optimization Method for Nonlinear Multi-Dimensional Optimal Power Flow

by Mohammed H. Alqahtani, Sulaiman Z. Almutairi, Abdullah M. Shaheen and Ahmed R. Ginidi

Axioms 2024, 13(7), 419; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13070419 - 21 Jun 2024

Viewed by 177

Abstract

Multi-Dimensional Optimal Power Flow (MDOPF) is a fundamental task in power systems engineering aimed at optimizing the operation of electrical networks while considering various constraints such as power generation, transmission, and distribution. The mathematical model of MDOPF involves formulating it as a non-linear, [...] Read more.

Multi-Dimensional Optimal Power Flow (MDOPF) is a fundamental task in power systems engineering aimed at optimizing the operation of electrical networks while considering various constraints such as power generation, transmission, and distribution. The mathematical model of MDOPF involves formulating it as a non-linear, non-convex optimization problem aimed at minimizing specific objective functions while adhering to equality and inequality constraints. The objective function typically includes terms representing the Fuel Cost (FC), Entire Network Losses (ENL), and Entire Emissions (EE), while the constraints encompass power balance equations, generator operating limits, and network constraints, such as line flow limits and voltage limits. This paper presents an innovative Improved Kepler Optimization Technique (IKOT) for solving MDOPF problems. The IKOT builds upon the traditional KOT and incorporates enhanced local escaping mechanisms to overcome local optima traps and improve convergence speed. The mathematical model of the IKOT algorithm involves defining a population of candidate solutions (individuals) represented as vectors in a high-dimensional search space. Each individual corresponds to a potential solution to the MDOPF problem, and the algorithm iteratively refines these solutions to converge towards the optimal solution. The key innovation of the IKOT lies in its enhanced local escaping mechanisms, which enable it to explore the search space more effectively and avoid premature convergence to suboptimal solutions. Experimental results on standard IEEE test systems demonstrate the effectiveness of the proposed IKOT in solving MDOPF problems. The proposed IKOT obtained the FC, EE, and ENL of USD 41,666.963/h, 1.039 Ton/h, and 9.087 MW, respectively, in comparison with the KOT, which achieved USD 41,677.349/h, 1.048 Ton/h, 11.277 MW, respectively. In comparison to the base scenario, the IKOT achieved a reduction percentage of 18.85%, 58.89%, and 64.13%, respectively, for the three scenarios. The IKOT consistently outperformed the original KOT and other state-of-the-art metaheuristic optimization algorithms in terms of solution quality, convergence speed, and robustness. Full article

(This article belongs to the Special Issue Advances in Mathematical Methods in Optimal Control and Applications)

15 pages, 1485 KiB

Open AccessArticle

Analysis of Fat Big Data Using Factor Models and Penalization Techniques: A Monte Carlo Simulation and Application

by Faridoon Khan and Olayan Albalawi

Axioms 2024, 13(7), 418; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13070418 - 21 Jun 2024

Viewed by 227

Abstract

This article assesses the predictive accuracy of factor models utilizing Partial·Least·Squares (PLS) and Principal·Component·Analysis (PCA) in comparison to autometrics and penalization techniques. The simulation exercise examines three types of scenarios by introducing the issues of multicollinearity, heteroscedasticity, and autocorrelation. The number of predictors [...] Read more.

This article assesses the predictive accuracy of factor models utilizing Partial·Least·Squares (PLS) and Principal·Component·Analysis (PCA) in comparison to autometrics and penalization techniques. The simulation exercise examines three types of scenarios by introducing the issues of multicollinearity, heteroscedasticity, and autocorrelation. The number of predictors and sample size are adjusted to observe the effects. The accuracy of the models is evaluated by calculating the Root·Mean·Square·Error (RMSE) and the Mean·Absolute·Error (MAE). In the presence of severe multicollinearity, the factor approach utilizing (PLS demonstrates exceptional performance in comparison. Autometrics achieves the lowest RMSE and MAE values across all levels of heteroscedasticity. Autometrics provides better forecasts with low and moderate autocorrelation. However, Elastic·Smoothly·Clipped·Absolute·Deviation (E-SCAD) forecasts well with severe autocorrelation. In addition to the simulation, we employ a popular Pakistani macroeconomic dataset for empirical research. The dataset contains 79 monthly variables from January 2013 to December 2020. The competing approaches perform differently compared to the simulation datasets, although “The PLS factor approach outperforms its competing approaches in forecasting, with lower RMSE and MAE”. It is more probable that the actual dataset exhibits a high degree of multicollinearity. Full article

(This article belongs to the Special Issue Applications of Statistical and Mathematical Models)

► Show Figures

Figure 1

33 pages, 492 KiB

Open AccessArticle

Some New Estimations of Left and Right Interval Fractional Pachpatte’s Type Integral Inequalities via Rectangle Plane

by Azzh Saad Alshehry, Loredana Ciurdariu, Yaser Saber and Amal F. Soliman

Axioms 2024, 13(7), 417; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13070417 - 21 Jun 2024

Viewed by 211

Abstract

Inequalities involving fractional operators have been an active area of research, which is crucial in establishing bounds, estimates, and stability conditions for solutions to fractional integrals. In this paper, we initially presented a new class that is known as coordinated left and right [...] Read more.

Inequalities involving fractional operators have been an active area of research, which is crucial in establishing bounds, estimates, and stability conditions for solutions to fractional integrals. In this paper, we initially presented a new class that is known as coordinated left and right

ℏ

-pre-invex interval-valued mappings (

C

·

L

·

R

-

ℏ

-pre-invex

Ι

·

V

-

M

), as well classical convex and nonconvex are also obtained. This newly defined class enabled us to derive novel inequalities, such as Hermite–Hadamard and Pachpatte’s type inequalities. Furthermore, the obtained results allowed us to recapture several special cases of known results for different parameter choices, which can be applications of the main results. Finally, we discussed the validity of the main outcomes. Full article

(This article belongs to the Section Mathematical Analysis)

Journal Menu

Journal Browser

Axioms, Volume 13, Issue 7 (July 2024) – 11 articles

Further Information

Guidelines

MDPI Initiatives

Follow MDPI