Axioms

14 pages, 249 KiB

Open AccessArticle

Qualitative Outcomes on Monotone Iterative Technique and Quasilinearization Method on Time Scale

by Şahap Çetin, Yalçın Yılmaz and Coşkun Yakar

Axioms 2024, 13(9), 640; https://doi.org/10.3390/axioms13090640 (registering DOI) - 19 Sep 2024

In this paper, a nonlinear dynamic equation with an initial value problem (IVP) on a time scale is considered. First, applying comparison results with a coupled lower solution (LS) and an upper solution (US), we improved the quasilinearization method (QLM) for the IVP. [...] Read more.

In this paper, a nonlinear dynamic equation with an initial value problem (IVP) on a time scale is considered. First, applying comparison results with a coupled lower solution (LS) and an upper solution (US), we improved the quasilinearization method (QLM) for the IVP. Unlike other studies, we consider the LS and US pair of the seventh type instead of the natural type. It was determined that the solutions of the dynamic equation converge uniformly and monotonically to the unique solution of the IVP, and the convergence is quadratic. Moreover, we will use the delta derivative

(Δ γ)

instead of the classical derivative

(d γ)

in the proof because it studies a time scale. In the second part of the paper, we applied the monotone iterative technique (MIT) coupled with the LS and US, which is an effective method, proving a clear analytical representation for the solution of the equation when the relevant functions are monotonically non-decreasing and non-increasing. Then an example is given to illustrate the results obtained. Full article

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

11 pages, 305 KiB

Open AccessArticle

A Framework for I*-Statistical Convergence of Fuzzy Numbers

by Tanushri, Ayaz Ahmad and Ayhan Esi

Axioms 2024, 13(9), 639; https://doi.org/10.3390/axioms13090639 - 18 Sep 2024

Abstract

In this study, we investigate the concept of

I^{*}

-statistical convergence for sequences of fuzzy numbers. We establish several theorems that provide a comprehensive understanding of this notion, including the uniqueness of limits, the relationship between

I^{*}

-statistical convergence and classical [...] Read more.

In this study, we investigate the concept of

I^{*}

-statistical convergence for sequences of fuzzy numbers. We establish several theorems that provide a comprehensive understanding of this notion, including the uniqueness of limits, the relationship between

I^{*}

-statistical convergence and classical convergence, and the algebraic properties of

I^{*}

-statistically convergent sequences. We also introduce the concept of

I^{*}

-statistical pre-Cauchy and

I^{*}

-statistical Cauchy sequences and explore its connection to

I^{*}

-statistical convergence. Our results show that every

I^{*}

-statistically convergent sequence is

I^{*}

-statistically pre-Cauchy, but the converse is not necessarily true. Furthermore, we provide a sufficient condition for an

I^{*}

-statistically pre-Cauchy sequence to be

I^{*}

-statistically convergent, which involves the concept of

I^{*} - l i m i n f

. Full article

(This article belongs to the Special Issue Advances in Fuzzy MCDM, Hybrid Methods, Fuzzy Number Ranking and Their Applications)

17 pages, 311 KiB

Open AccessArticle

Extension of Meir-Keeler-Khan (ψ − α) Type Contraction in Partial Metric Space

by Dimple Singh, Priya Goel, Ramandeep Behl and Iñigo Sarría

Axioms 2024, 13(9), 638; https://doi.org/10.3390/axioms13090638 (registering DOI) - 18 Sep 2024

Abstract

In numerous scientific and engineering domains, fractional-order derivatives and integral operators are frequently used to represent many complex phenomena. They also have numerous practical applications in the area of fixed point iteration. In this article, we introduce the notion of generalized Meir-Keeler-Khan-Rational type [...] Read more.

In numerous scientific and engineering domains, fractional-order derivatives and integral operators are frequently used to represent many complex phenomena. They also have numerous practical applications in the area of fixed point iteration. In this article, we introduce the notion of generalized Meir-Keeler-Khan-Rational type

(ψ - α)

-contraction mapping and propose fixed point results in partial metric spaces. Our proposed results extend, unify, and generalize existing findings in the literature. In regards to applicability, we provide evidence for the existence of a solution for the fractional-order differential operator. In addition, the solution of the integral equation and its uniqueness are also discussed. Finally, we conclude that our results are superior and generalized as compared to the existing ones. Full article

(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)

18 pages, 1055 KiB

Open AccessArticle

Mathematical Model for the Control of Red Palm Weevil

by Zuhur Alqahtani, Areej Almuneef and Moustafa El-Shahed

Axioms 2024, 13(9), 637; https://doi.org/10.3390/axioms13090637 - 18 Sep 2024

Abstract

The red palm weevil (Rhynchophorus ferrugineus) is a highly destructive pest, causing severe damage to palm trees and significantly reducing their productivity. This paper aims to develop and analyze a mathematical model that captures the interactions between palm trees, Rhynchophorus ferrugineus [...] Read more.

The red palm weevil (Rhynchophorus ferrugineus) is a highly destructive pest, causing severe damage to palm trees and significantly reducing their productivity. This paper aims to develop and analyze a mathematical model that captures the interactions between palm trees, Rhynchophorus ferrugineus, and entomopathogenic nematodes as a means of integrated control. We identify the equilibrium points of the system and perform a stability analysis to assess the system’s behavior. Additionally, we design a linear quadratic regulator (LQR) to limit the spread of the red palm weevil within a locally linearized framework. The feedback control law, which is both straightforward and immediately implementable, is employed to avoid the need for complex cost calculations, thus simplifying the solution to the optimal control problem. Numerical simulations demonstrate that the proposed control strategy is effective in reducing the number of infected palm trees. The results indicate that increasing the population of entomopathogenic nematodes can significantly decrease the red palm weevil population, offering a promising approach to mitigating this pest’s impact. Full article

(This article belongs to the Topic Mathematical Modeling)

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17 pages, 295 KiB

Open AccessArticle

On Spacelike Hypersurfaces in Generalized Robertson–Walker Spacetimes

by Norah Alessa and Mohammed Guediri

Axioms 2024, 13(9), 636; https://doi.org/10.3390/axioms13090636 - 17 Sep 2024

Abstract

This paper investigates generalized Robertson–Walker (GRW) spacetimes by analyzing Riemannian hypersurfaces within pseudo-Riemannian warped product manifolds of the form

(\bar{M}, \bar{g})

, where

\bar{M} = R \times_{f} M

and [...] Read more.

This paper investigates generalized Robertson–Walker (GRW) spacetimes by analyzing Riemannian hypersurfaces within pseudo-Riemannian warped product manifolds of the form

(\bar{M}, \bar{g})

, where

\bar{M} = R \times_{f} M

and

\bar{g} = ϵ d t^{2} + f^{2} (t) g_{M}

. We focus on the scalar curvature of these hypersurfaces, establishing upper and lower bounds, particularly in the case where

(\bar{M}, \bar{g})

is an Einstein manifold. These bounds facilitate the characterization of slices in GRW spacetimes. In addition, we use the vector field

\partial_{t}

and the so-called support function

θ

to derive generalized Minkowski-type integral formulas for compact Riemannian and spacelike hypersurfaces. These formulas are applied to establish, under certain conditions, results concerning the existence or non-existence of such compact hypersurfaces with scalar curvature, either bounded from above or below. Full article

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

11 pages, 610 KiB

Open AccessArticle

Necessary and Sufficient Criteria for a Four-Weight Weak-Type Maximal Inequality in the Orlicz Class

by Erxin Zhang

Axioms 2024, 13(9), 635; https://doi.org/10.3390/axioms13090635 - 17 Sep 2024

Abstract

Let

Φ_{i} (i = 1, 2)

be two N-functions, f be a

μ

-measurable function, and

ω_{i} (i = 1, 2, 3, 4)

be four weight functions. This study presents necessary and [...] Read more.

Let

Φ_{i} (i = 1, 2)

be two N-functions, f be a

μ

-measurable function, and

ω_{i} (i = 1, 2, 3, 4)

be four weight functions. This study presents necessary and sufficient conditions for weight functions

(ω_{1}, ω_{2}, ω_{3}, ω_{4})

such that the inequality

\int_{{x : M f (x) > λ}} Φ_{1} (λ ω_{1} (x)) ω_{2} (x) d μ (x) \leq c_{1} \int_{X} Φ_{2} (c_{1} | f (x) | ω_{3} (x))

ω_{4} (x) d μ (x)

holds, which extends several established results. Full article

(This article belongs to the Special Issue Theory of Functions and Applications II)

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15 pages, 1006 KiB

Open AccessArticle

Robust State Feedback Control with D-Admissible Assurance for Uncertain Discrete Singular Systems

by Chih-Peng Huang

Axioms 2024, 13(9), 634; https://doi.org/10.3390/axioms13090634 - 17 Sep 2024

Abstract

This study addresses the state feedback control associated with D-admissible assurance for discrete singular systems subjected to parameter uncertainties in both the difference term and system matrices. Firstly, a refined analysis criterion of D-admissible assurance is presented, where the distinct form embraces multiple [...] Read more.

This study addresses the state feedback control associated with D-admissible assurance for discrete singular systems subjected to parameter uncertainties in both the difference term and system matrices. Firstly, a refined analysis criterion of D-admissible assurance is presented, where the distinct form embraces multiple slack matrices and has lessened linear matrix inequalities (LMIs) constraints, which may be beneficial for reducing the conservatism of admissibility analysis. In consequence, by hiring the state feedback control, controller design issues with pole locations, which directly dominate the system performance, are mainly treated. For all the presented criteria can be formulated by the strict LMIs, they are thus suitably solved via current LMI solvers to conduct a state feedback controller with specific poles’ locations of system’s performance requirements. Finally, two numerical examples illustrate that the presented results are efficient and practicable. Full article

(This article belongs to the Special Issue Advances in Dynamical Systems and Control)

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10 pages, 253 KiB

Open AccessArticle

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

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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12 pages, 305 KiB

Open AccessArticle

Extremal Trees for Logarithmic VDB Topological Indices

by Zhenhua Su and Hanyuan Deng

Axioms 2024, 13(9), 632; https://doi.org/10.3390/axioms13090632 - 16 Sep 2024

Abstract

Vertex-degree-based (VDB) topological indices have been applied in the study of molecular structures and chemical properties. At present, the exponential VDB index has been studied extensively. Naturally, we began to consider the logarithmic VDB index

ln T_{f}

. In this paper, we [...] Read more.

Vertex-degree-based (VDB) topological indices have been applied in the study of molecular structures and chemical properties. At present, the exponential VDB index has been studied extensively. Naturally, we began to consider the logarithmic VDB index

ln T_{f}

. In this paper, we first discuss the necessity of a logarithmic VDB index, and then present sufficient conditions so that

P_{n}

and

S_{n}

are the only trees with the smallest and greatest values of

ln T_{f} (T)

. As applications, the minimal and maximal trees of some logarithmic VDB indices are determined. Through our work, we found that the logarithmic VDB index

ln T_{f}

has excellent discriminability, but the relevant results are not completely opposite to the exponential VDB index. The study of logarithmic VDB indices is an interesting but difficult task that requires further resolution. Full article

(This article belongs to the Special Issue Advances in Combinatorial Optimization and Discrete Mathematics with Applications)

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15 pages, 295 KiB

Open AccessFeature PaperArticle

On Closed Forms of Some Trigonometric Series

by Slobodan B. Tričković and Miomir S. Stanković

Axioms 2024, 13(9), 631; https://doi.org/10.3390/axioms13090631 - 14 Sep 2024

Abstract

We have derived alternative closed-form formulas for the trigonometric series over sine or cosine functions when the immediate replacement of the parameter appearing in the denominator with a positive integer gives rise to a singularity. By applying the Choi–Srivastava theorem, we reduce these [...] Read more.

We have derived alternative closed-form formulas for the trigonometric series over sine or cosine functions when the immediate replacement of the parameter appearing in the denominator with a positive integer gives rise to a singularity. By applying the Choi–Srivastava theorem, we reduce these trigonometric series to expressions over Hurwitz’s zeta function derivative. Full article

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

11 pages, 257 KiB

Open AccessArticle

Parseval–Goldstein-Type Theorems for Lebedev–Skalskaya Transforms

by Emilio Ramón Negrín, Benito Juan González and Jeetendrasingh Maan

Axioms 2024, 13(9), 630; https://doi.org/10.3390/axioms13090630 - 14 Sep 2024

Abstract

This paper investigates Parseval–Goldstein-type relationships in the framework of Lebedev–Skalskaya transforms. The research also examines the continuity properties of these transforms, along with their adjoint counterparts over weighted Lebesgue spaces. Furthermore, the behavior of Lebedev–Skalskaya transforms and their adjoint transforms in the context [...] Read more.

This paper investigates Parseval–Goldstein-type relationships in the framework of Lebedev–Skalskaya transforms. The research also examines the continuity properties of these transforms, along with their adjoint counterparts over weighted Lebesgue spaces. Furthermore, the behavior of Lebedev–Skalskaya transforms and their adjoint transforms in the context of weighted Lebesgue spaces is analyzed. This study aims to provide deeper insights into the functional properties and applications of these transforms in mathematical analysis. Full article

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

9 pages, 255 KiB

Open AccessArticle

Elephant Random Walk with a Random Step Size and Gradually Increasing Memory and Delays

by Rafik Aguech

Axioms 2024, 13(9), 629; https://doi.org/10.3390/axioms13090629 - 14 Sep 2024

Abstract

The ERW model was introduced twenty years ago to study memory effects in a one-dimensional discrete-time random walk with a complete memory of its past throughout a parameter p between zero and one. Several variations of the ERW model have recently been introduced. [...] Read more.

The ERW model was introduced twenty years ago to study memory effects in a one-dimensional discrete-time random walk with a complete memory of its past throughout a parameter p between zero and one. Several variations of the ERW model have recently been introduced. In this work, we investigate the asymptotic normality of the ERW model with a random step size and gradually increasing memory and delays. In particular, we extend some recent results in this subject. Full article

(This article belongs to the Topic Fractional Calculus, Symmetry Phenomenon and Probability Theory for PDEs, and ODEs)

9 pages, 227 KiB

Open AccessArticle

A Variational Theory for Biunivalent Holomorphic Functions

by Samuel L. Krushkal

Axioms 2024, 13(9), 628; https://doi.org/10.3390/axioms13090628 - 13 Sep 2024

Abstract

Biunivalent holomorphic functions form an interesting class in geometric function theory and are connected with special functions and solutions of complex differential equations. This class has been investigated by many authors, mainly to find the coefficient estimates. The assumption of biunivalence is rigid; [...] Read more.

Biunivalent holomorphic functions form an interesting class in geometric function theory and are connected with special functions and solutions of complex differential equations. This class has been investigated by many authors, mainly to find the coefficient estimates. The assumption of biunivalence is rigid; this rigidity means that, for example, only the initial Taylor coefficients have been estimated. The aim of this paper is to develop a variational technique for biunivalent functions, which provides a power tool for solving the general extremal problems on the classes of such functions. It involves quasiconformal analysis. Full article

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

25 pages, 1082 KiB

Open AccessFeature PaperArticle

On the Existence, Uniqueness and a Numerical Approach to the Solution of Fractional Cauchy–Euler Equation

by Nazim I. Mahmudov, Suzan Cival Buranay and Mtema James Chin

Axioms 2024, 13(9), 627; https://doi.org/10.3390/axioms13090627 - 12 Sep 2024

Abstract

In this research paper, we consider a model of the fractional Cauchy–Euler-type equation, where the fractional derivative operator is the Caputo with order

0 < α < 2

. The problem also constitutes a class of examples of the Cauchy problem of the [...] Read more.

In this research paper, we consider a model of the fractional Cauchy–Euler-type equation, where the fractional derivative operator is the Caputo with order

0 < α < 2

. The problem also constitutes a class of examples of the Cauchy problem of the Bagley–Torvik equation with variable coefficients. For proving the existence and uniqueness of the solution of the given problem, the contraction mapping principle is utilized. Furthermore, a numerical method and an algorithm are developed for obtaining the approximate solution. Also, convergence analyses are studied, and simulations on some test problems are given. It is shown that the proposed method and the algorithm are easy to implement on a computer and efficient in computational time and storage. Full article

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

14 pages, 296 KiB

Open AccessArticle

I

-Convergence Sequence Paranormed Spaces of Order (α, β)

by Lian-Ta Su, Ravi Kumar, Sunil K. Sharma, Ajay K. Sharma and Qing-Bo Cai

Axioms 2024, 13(9), 626; https://doi.org/10.3390/axioms13090626 - 12 Sep 2024

Abstract

In this paper, we introduce and rigorously define a novel class of difference sequence spaces, denoted by

w^{I} {(M, ∆_{v}^{u}, r)}_{α}^{β}

, [...] Read more.

In this paper, we introduce and rigorously define a novel class of difference sequence spaces, denoted by

w^{I} {(M, ∆_{v}^{u}, r)}_{α}^{β}

,

w_{0}^{I} {(M, ∆_{v}^{u}, r)}_{α}^{β}, w_{\infty}^{I} {(M, ∆_{v}^{u}, r)}_{α}^{β}

, and

w_{\infty} {(M, ∆_{v}^{u}, r)}_{α}^{β}

. These spaces are constructed through the application of the concept of

I

-convergence of sequences, combined with a Musielak–Orlicz function of order

(α, β)

. The primary focus of our work is to thoroughly investigate the algebraic and topological properties of these defined sequence spaces. We explore their linearity, examine their structure within the framework of paranormed spaces, and analyze various other algebraic characteristics pertinent to these spaces. In addition, we examine the topological nature of these sequence spaces, identifying the conditions under which they exhibit specific topological properties. A significant part of our study is dedicated to examining the inclusion relationships between these sequence spaces, thereby providing a comprehensive understanding of how these spaces are interrelated. Our analysis contributes to the broader field of functional analysis and sequence space theory, offering new insights and potential applications of these advanced mathematical constructs. Full article

(This article belongs to the Special Issue Research on Functional Analysis and Its Applications)

16 pages, 4720 KiB

Open AccessArticle

Dynamics of a New Four-Thirds-Degree Sub-Quadratic Lorenz-like System

by Guiyao Ke, Jun Pan, Feiyu Hu and Haijun Wang

Axioms 2024, 13(9), 625; https://doi.org/10.3390/axioms13090625 - 12 Sep 2024

Abstract

Aiming to explore the subtle connection between the number of nonlinear terms in Lorenz-like systems and hidden attractors, this paper introduces a new simple sub-quadratic four-thirds-degree Lorenz-like system, where

\dot{x} = a (y - x)

, [...] Read more.

Aiming to explore the subtle connection between the number of nonlinear terms in Lorenz-like systems and hidden attractors, this paper introduces a new simple sub-quadratic four-thirds-degree Lorenz-like system, where

\dot{x} = a (y - x)

,

\dot{y} = c x - \sqrt[3]{x} z

,

\dot{z} = - b z + \sqrt[3]{x} y

, and uncovers the following property of these systems: decreasing the powers of the nonlinear terms in a quadratic Lorenz-like system where

\dot{x} = a (y - x)

,

\dot{y} = c x - x z

,

\dot{z} = - b z + x y

, may narrow, or even eliminate the range of the parameter c for hidden attractors, but enlarge it for self-excited attractors. By combining numerical simulation, stability and bifurcation theory, most of the important dynamics of the Lorenz system family are revealed, including self-excited Lorenz-like attractors, Hopf bifurcation and generic pitchfork bifurcation at the origin, singularly degenerate heteroclinic cycles, degenerate pitchfork bifurcation at non-isolated equilibria, invariant algebraic surface, heteroclinic orbits and so on. The obtained results may verify the generalization of the second part of the celebrated Hilbert’s sixteenth problem to some degree, showing that the number and mutual disposition of attractors and repellers may depend on the degree of chaotic multidimensional dynamical systems. Full article

(This article belongs to the Section Mathematical Analysis)

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18 pages, 1135 KiB

Open AccessArticle

Applications of Fuzzy Logic and Probabilistic Neural Networks in E-Service for Malware Detection

by Kristijan Kuk, Aleksandar Stanojević, Petar Čisar, Brankica Popović, Mihailo Jovanović, Zoran Stanković and Olivera Pronić-Rančić

Axioms 2024, 13(9), 624; https://doi.org/10.3390/axioms13090624 - 12 Sep 2024

Abstract

The key point in the process of agent-based management in e-service for malware detection (according to accuracy criteria) is a decision-making process. To determine the optimal e-service for malware detection, two concepts were investigated: Fuzzy Logic (FL) and Probabilistic Neural Networks (PNN). In [...] Read more.

The key point in the process of agent-based management in e-service for malware detection (according to accuracy criteria) is a decision-making process. To determine the optimal e-service for malware detection, two concepts were investigated: Fuzzy Logic (FL) and Probabilistic Neural Networks (PNN). In this study, three evolutionary variants of fuzzy partitioning, including regular, hierarchical fuzzy partitioning, and k-means, were used to automatically process the design of the fuzzy partition. Also, this study demonstrates the application of a feature selection method to reduce the dimensionality of the data by removing irrelevant features to create fuzzy logic in a dataset. The behaviors of malware are analyzed by fuzzifying relevant features for pattern recognition. The Apriori algorithm was applied to the fuzzified features to find the fuzzy-based rules, and these rules were used for predicting the output of malware detection e-services. Probabilistic neural networks were also used to find the ideal agent-based model for numerous classification problems. The numerical results show that the agent-based management performances trained with the clustering method achieve an accuracy of 100% with the PNN-MCD model. This is followed by the FL model, which classifies on the basis of linguistic variables and achieves an average accuracy of 82%. Full article

(This article belongs to the Special Issue Recent Advances of Computational and Mathematical Applications in Deep Learning)

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27 pages, 401 KiB

Open AccessArticle

Multidimensional Fractional Calculus: Theory and Applications

by Marko Kostić

Axioms 2024, 13(9), 623; https://doi.org/10.3390/axioms13090623 - 12 Sep 2024

Abstract

In this paper, we introduce several new types of partial fractional derivatives in the continuous setting and the discrete setting. We analyze some classes of the abstract fractional differential equations and the abstract fractional difference equations depending on several variables, providing a great [...] Read more.

In this paper, we introduce several new types of partial fractional derivatives in the continuous setting and the discrete setting. We analyze some classes of the abstract fractional differential equations and the abstract fractional difference equations depending on several variables, providing a great number of structural results, useful remarks and illustrative examples. Concerning some specific applications, we would like to mention here our investigation of the fractional partial differential inclusions with Riemann–Liouville and Caputo derivatives. We also establish the complex characterization theorem for the multidimensional vector-valued Laplace transform and provide certain applications. Full article

(This article belongs to the Special Issue Advances in Difference Equations)

13 pages, 464 KiB

Open AccessArticle

A Complete Characterization of Linear Dependence and Independence for All 4-by-4 Metric Matrices

by Ray-Ming Chen

Axioms 2024, 13(9), 622; https://doi.org/10.3390/axioms13090622 - 12 Sep 2024

Abstract

In this article, we study the properties of 4-by-4 metric matrices and characterize their dependence and independence by

M^{4 \times 4} = (M^{4 \times 4} - D M^{4 \times 4}) \cup D M^{4 \times 4}

, where [...] Read more.

In this article, we study the properties of 4-by-4 metric matrices and characterize their dependence and independence by

M^{4 \times 4} = (M^{4 \times 4} - D M^{4 \times 4}) \cup D M^{4 \times 4}

, where

D M^{4 \times 4}

is the set of all dependent metric matrices.

D M^{4 \times 4}

is further characterized by

D M^{4 \times 4} = D M_{1}^{4 \times 4} \cup D M_{2}^{4 \times 4}

, where

D M_{2}^{4 \times 4}

is characterized by

D M_{2}^{4 \times 4} = D M_{21}^{4 \times 4} \cup D M_{22}^{4 \times 4}

. These characterizations provide some insightful findings that go beyond the Euclidean distance or Euclidean distance matrix and link the distance functions to vector spaces, which offers some theoretical and application-related advantages. In the application parts, we show that the metric matrices associated with all Minkowski distance functions over four different points are linearly independent, and that the metric matrices associated with any four concyclic points are also linearly independent. Full article

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