Axioms

18 pages, 1547 KiB

Open AccessArticle

Maneuvering Object Tracking and Movement Parameters Identification by Indirect Observations with Random Delays

by Alexey Bosov

Axioms 2024, 13(10), 668; https://doi.org/10.3390/axioms13100668 (registering DOI) - 26 Sep 2024

The paper presents an approach to solving the problem of unknown motion parameters Bayesian identification for the stochastic dynamic system model with randomly delayed observations. The system identification and the object tracking tasks obtain solutions in the form of recurrent Bayesian relations for [...] Read more.

The paper presents an approach to solving the problem of unknown motion parameters Bayesian identification for the stochastic dynamic system model with randomly delayed observations. The system identification and the object tracking tasks obtain solutions in the form of recurrent Bayesian relations for a posteriori probability density. These relations are not practically applicable due to the computational challenges they present. For practical implementation, we propose a conditionally minimax nonlinear filter that implements the concept of conditionally optimal estimation. The random delays model source is the area of autonomous underwater vehicle control. The paper discusses in detail a computational experiment based on a model that is closely aligned with this practical need. The discussion includes both a description of the filter synthesis features based on the geometric interpretation of the simulated measurements and an impact analysis of the effectiveness of model special factors, such as time delays and model unknown parameters. Furthermore, the paper puts forth a novel approach to the identification problem statement, positing a random jumping change in the motion parameters values. Full article

(This article belongs to the Special Issue Stochastic Modeling and Analysis for Applications and Technologies)

17 pages, 316 KiB

Open AccessArticle

The Bundle of Tensor Densities and Its Covariant Derivatives

by Joan Grandes Umbert and Tom Mestdag

Axioms 2024, 13(10), 667; https://doi.org/10.3390/axioms13100667 - 26 Sep 2024

Abstract

We construct the smooth vector bundle of tensor densities of arbitrary weight in a coordinate-independent way. We prove the general existence of a globally smooth tensor density field, as well as the existence of a globally smooth metric density for a pseudo-Riemannian manifold, [...] Read more.

We construct the smooth vector bundle of tensor densities of arbitrary weight in a coordinate-independent way. We prove the general existence of a globally smooth tensor density field, as well as the existence of a globally smooth metric density for a pseudo-Riemannian manifold, specifically. We study the coordinate description of a covariant derivative over densities, and define a natural extension of affine connections to densities. We provide an equivalent characterization, in the case of a pseudo-Riemannian manifold. Full article

(This article belongs to the Section Mathematical Analysis)

24 pages, 513 KiB

Open AccessArticle

A New Contribution in Fractional Integral Calculus and Inequalities over the Coordinated Fuzzy Codomain

by Zizhao Zhou, Ahmad Aziz Al Ahmadi, Alina Alb Lupas and Khalil Hadi Hakami

Axioms 2024, 13(10), 666; https://doi.org/10.3390/axioms13100666 - 26 Sep 2024

Abstract

The correct derivation of integral inequalities on fuzzy-number-valued mappings depends on applying fractional calculus to fuzzy number analysis. The purpose of this article is to introduce a new class of convex mappings and generalize various previously published results on the fuzzy number and [...] Read more.

The correct derivation of integral inequalities on fuzzy-number-valued mappings depends on applying fractional calculus to fuzzy number analysis. The purpose of this article is to introduce a new class of convex mappings and generalize various previously published results on the fuzzy number and interval-valued mappings via fuzzy-order relations using fuzzy coordinated ỽ-convexity mappings so that the new version of the well-known Hermite–Hadamard (H-H) inequality can be presented in various variants via the fractional integral operators (Riemann–Liouville). Some new product forms of these inequalities for coordinated ỽ-convex fuzzy-number-valued mappings (coordinated ỽ-convex FNVMs) are also discussed. Additionally, we provide several fascinating non-trivial examples and exceptional cases to show that these results are accurate. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

11 pages, 345 KiB

Open AccessArticle

Hamiltonian Formulation for Continuous Systems with Second-Order Derivatives: A Study of Podolsky Generalized Electrodynamics

by Yazen M. Alawaideh, Alina Alb Lupas, Bashar M. Al-khamiseh, Majeed A. Yousif, Pshtiwan Othman Mohammed and Y. S. Hamed

Axioms 2024, 13(10), 665; https://doi.org/10.3390/axioms13100665 - 26 Sep 2024

Abstract

This paper presents an analysis of the Hamiltonian formulation for continuous systems with second-order derivatives derived from Dirac’s theory. This approach offers a unique perspective on the equations of motion compared to the traditional Euler–Lagrange formulation. Focusing on Podolsky’s generalized electrodynamics, the Hamiltonian [...] Read more.

This paper presents an analysis of the Hamiltonian formulation for continuous systems with second-order derivatives derived from Dirac’s theory. This approach offers a unique perspective on the equations of motion compared to the traditional Euler–Lagrange formulation. Focusing on Podolsky’s generalized electrodynamics, the Hamiltonian and corresponding equations of motion are derived. The findings demonstrate that both Hamiltonian and Euler–Lagrange formulations yield equivalent results. This study highlights the Hamiltonian approach as a valuable alternative for understanding the dynamics of second-order systems, validated through a specific application within generalized electrodynamics. The novelty of the research lies in developing advanced theoretical models through Hamiltonian formalism for continuous systems with second-order derivatives. The research employs an alternative method to the Euler–Lagrange formulas by applying Dirac’s theory to study the generalized Podolsky electrodynamics, contributing to a better understanding of complex continuous systems. Full article

(This article belongs to the Special Issue Mathematical Models and Simulations, 2nd edition)

24 pages, 607 KiB

Open AccessArticle

Bivariate Length-Biased Exponential Distribution under Progressive Type-II Censoring: Incorporating Random Removal and Applications to Industrial and Computer Science Data

by Aisha Fayomi, Ehab M. Almetwally and Maha E. Qura

Axioms 2024, 13(10), 664; https://doi.org/10.3390/axioms13100664 (registering DOI) - 26 Sep 2024

Abstract

In this paper, we address the analysis of bivariate lifetime data from a length-biased exponential distribution observed under Type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial distribution. We derive the likelihood [...] Read more.

In this paper, we address the analysis of bivariate lifetime data from a length-biased exponential distribution observed under Type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial distribution. We derive the likelihood function for the progressive Type II censoring scheme with random removals and apply it to the bivariate length-biased exponential distribution. The parameters of the proposed model are estimated using both likelihood and Bayesian methods for point and interval estimators, including asymptotic confidence intervals and bootstrap confidence intervals. We also employ different loss functions to construct Bayesian estimators. Additionally, a simulation study is conducted to compare the performance of censoring schemes. The effectiveness of the proposed methodology is demonstrated through the analysis of two real datasets from the industrial and computer science domains, providing valuable insights for illustrative purposes. Full article

(This article belongs to the Special Issue Applications of Bayesian Methods in Statistical Analysis)

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10 pages, 2308 KiB

Open AccessArticle

Exact Solutions to Fractional Schrödinger–Hirota Equation Using Auxiliary Equation Method

by Guangyuan Tian and Xianji Meng

Axioms 2024, 13(10), 663; https://doi.org/10.3390/axioms13100663 - 26 Sep 2024

Viewed by 120

Abstract

In this paper, we consider the fractional Schrödinger–Hirota (FSH) equation in the sense of a conformable fractional derivative. Through a traveling wave transformation, we change the FSH equation to an ordinary differential equation. We obtain several exact solutions through the auxiliary equation method, [...] Read more.

In this paper, we consider the fractional Schrödinger–Hirota (FSH) equation in the sense of a conformable fractional derivative. Through a traveling wave transformation, we change the FSH equation to an ordinary differential equation. We obtain several exact solutions through the auxiliary equation method, including soliton, exponential and periodic solutions, which are useful to analyze the behaviors of the FSH equation. We show that the auxiliary equation method improves the speed of the discovery of exact solutions. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

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16 pages, 348 KiB

Open AccessArticle

A Bimodal Extension of the Beta-Binomial Distribution with Applications

by Jimmy Reyes, Josu Najera-Zuloaga, Dae-Jin Lee, Jaime Arrué and Yuri A. Iriarte

Axioms 2024, 13(10), 662; https://doi.org/10.3390/axioms13100662 - 25 Sep 2024

Viewed by 143

Abstract

In this paper, we propose an alternative distribution to model count data exhibiting uni/bimodality. It arises as a weighted version of the beta-binomial distribution, which is defined by a parametric weight function that admits up to two modes for the resulting probability mass [...] Read more.

In this paper, we propose an alternative distribution to model count data exhibiting uni/bimodality. It arises as a weighted version of the beta-binomial distribution, which is defined by a parametric weight function that admits up to two modes for the resulting probability mass function. Like the baseline beta-binomial distribution, the proposed distribution performs well in modeling overdispersed binomial data. Structural properties of the new distribution are studied. Raw moments are derived, which are used to describe the dispersion behavior relative to the mean and the skewness behavior. Parameter estimation is carried out using the maximum likelihood method. A simulation study is conducted in order to illustrate the behavior of the estimators. Finally, two applications illustrating the usefulness of the proposal are presented. Full article

(This article belongs to the Special Issue Advances in the Theory and Applications of Statistical Distributions)

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15 pages, 6269 KiB

Open AccessArticle

On the Torsional Energy of Deformed Curves and Knots

by Svetozar R. Rančić, Ljubica S. Velimirović and Marija S. Najdanović

Axioms 2024, 13(10), 661; https://doi.org/10.3390/axioms13100661 - 25 Sep 2024

Viewed by 161

Abstract

This paper deals with the study of torsional energy (total squared torsion) at infinitesimal bending of curves and knots in three dimensional Euclidean space. During bending, the curve is subject to change, and its properties are changed. The effect that deformation has on [...] Read more.

This paper deals with the study of torsional energy (total squared torsion) at infinitesimal bending of curves and knots in three dimensional Euclidean space. During bending, the curve is subject to change, and its properties are changed. The effect that deformation has on the curve is measured by variations. Here, we observe the infinitesimal bending of the second order and variations of the first and the second order that occur in this occasion. The subjects of study are curves and knots, in particular torus knots. We analyze various examples both analytically and graphically, using our own calculation and visualization software tool. Full article

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

21 pages, 510 KiB

Open AccessArticle

Capital Asset Pricing Model and Ordered Weighted Average Operator for Selecting Investment Portfolios

by Cristhian R. Uzeta-Obregon, Tanya S. Garcia-Gastelum, Pavel A. Alvarez, Cristhian Mellado-Cid, Fabio Blanco-Mesa and Ernesto Leon-Castro

Axioms 2024, 13(10), 660; https://doi.org/10.3390/axioms13100660 - 25 Sep 2024

Viewed by 228

Abstract

The main objective of this article is to present the formulation of a Capital Asset Pricing Model ordered weighted average CAPM_OWA and its extensions, called CAPM-induced OWA (CAPM_IOWA), CAPM Bonferroni OWA (CAPM_Bon−IOWA), and CAPM Bonferroni-induced OWA [...] Read more.

The main objective of this article is to present the formulation of a Capital Asset Pricing Model ordered weighted average CAPM_OWA and its extensions, called CAPM-induced OWA (CAPM_IOWA), CAPM Bonferroni OWA (CAPM_Bon−IOWA), and CAPM Bonferroni-induced OWA CAPM_Bon−IOWA. A step-by-step process for applying this new proposal in a real case of formulating investment portfolios is generated. These methods show several scenarios, considering the attitude, preferences, and relationship of each argument, when underestimation or overestimation of the information by the decision maker may influence the decision-making process regarding portfolio investments. Finally, the complexity of the method and the incorporation of soft information into the modeling process lead to generating a greater number of scenarios and reflect the attitudes and preferences of decision makers. Full article

(This article belongs to the Special Issue Fuzzy Sets, Simulation and Their Applications)

14 pages, 476 KiB

Open AccessArticle

Calibrating and Visualizing Some Bootstrap Confidence Regions

by Welagedara Arachchilage Dhanushka M. Welagedara and David J. Olive

Axioms 2024, 13(10), 659; https://doi.org/10.3390/axioms13100659 - 25 Sep 2024

Viewed by 192

Abstract

When the bootstrap sample size is moderate, bootstrap confidence regions tend to have undercoverage. Improving the coverage is known as calibrating the confidence region. Consider testing

H_{0} : θ = θ_{0}

versus

H_{1} : θ \neq θ_{0}

. We [...] Read more.

When the bootstrap sample size is moderate, bootstrap confidence regions tend to have undercoverage. Improving the coverage is known as calibrating the confidence region. Consider testing

H_{0} : θ = θ_{0}

versus

H_{1} : θ \neq θ_{0}

. We reject

H_{0}

only if

θ_{0}

is not contained in a large-sample 95% confidence region. If the confidence region has 3% undercoverage for the data set sample size, then the type I error is 8% instead of the nominal 5%. Hence, calibrating confidence regions is also useful for testing hypotheses. Several bootstrap confidence regions are also prediction regions for a future value of a bootstrap statistic. A new bootstrap confidence region uses a simple prediction region calibration technique to improve the coverage. The DD plot for visualizing prediction regions can also be used to visualize some bootstrap confidence regions. Full article

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

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10 pages, 363 KiB

Open AccessArticle

Two-Matchings with Respect to the General Sum-Connectivity Index of Trees

by Roberto Cruz, Mateo Lopez and Juan Rada

Axioms 2024, 13(10), 658; https://doi.org/10.3390/axioms13100658 - 24 Sep 2024

Viewed by 286

Abstract

A vertex-degree-based topological index

φ

associates a real number to a graph G which is invariant under graph isomorphism. It is defined in terms of the degrees of the vertices of G and plays an important role in chemical graph theory, especially in [...] Read more.

A vertex-degree-based topological index

φ

associates a real number to a graph G which is invariant under graph isomorphism. It is defined in terms of the degrees of the vertices of G and plays an important role in chemical graph theory, especially in QSPR/QSAR investigations. A subset of k edges in G with no common vertices is called a k-matching of G, and the number of such subsets is denoted by

m (G, k)

. Recently, this number was naturally extended to weighted graphs, where the weight function is induced by the topological index

φ

. This number was denoted by

m_{k} (G, φ)

and called the k-matchings of G with respect to the topological index

φ

. It turns out that

m_{1} (G, φ) = φ (G),

and so for

k \geq 2,

the k-matching numbers

m_{k} (G, φ)

can be viewed as kth order topological indices which involve both the topological index

φ

and the k-matching numbers. In this work, we solve the extremal value problem for the number of 2-matchings with respect to general sum-connectivity indices

{SC}_{α}

, over the set

T_{n}

of trees with n vertices, when

α

is a real number in the interval

[- 1, 0) .

Full article

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

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24 pages, 288 KiB

Open AccessArticle

Some New Algebraic Method Developments in the Characterization of Matrix Equalities

by Yongge Tian

Axioms 2024, 13(10), 657; https://doi.org/10.3390/axioms13100657 - 24 Sep 2024

Viewed by 285

Abstract

Algebraic expressions and equalities can be constructed arbitrarily in a given algebraic framework according to the operational rules provided, and thus it is a prominent and necessary task in mathematics and applications to construct, classify, and characterize various simple general algebraic expressions and [...] Read more.

Algebraic expressions and equalities can be constructed arbitrarily in a given algebraic framework according to the operational rules provided, and thus it is a prominent and necessary task in mathematics and applications to construct, classify, and characterize various simple general algebraic expressions and equalities. As an update to this prominent topic in matrix algebra, this article reviews and improves upon the well-known block matrix methodology and matrix rank methodology in the construction and characterization of matrix equalities. We present a collection of fundamental and useful formulas for calculating the ranks of a wide range of block matrices and then derive from these rank formulas various valuable consequences. In particular, we present several groups of equivalent conditions in the characterizations of the Hermitian matrix, the skew-Hermitian matrix, the normal matrix, etc. Full article

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

17 pages, 456 KiB

Open AccessArticle

Elliptic Quaternion Matrices: Theory and Algorithms

by Hidayet Hüda Kösal, Emre Kişi, Mahmut Akyiğit and Beyza Çelik

Axioms 2024, 13(10), 656; https://doi.org/10.3390/axioms13100656 - 24 Sep 2024

Viewed by 168

Abstract

In this study, we obtained results for the computation of eigen-pairs, singular value decomposition, pseudoinverse, and the least squares problem for elliptic quaternion matrices. Moreover, we established algorithms based on these results and provided illustrative numerical experiments to substantiate the accuracy of our [...] Read more.

In this study, we obtained results for the computation of eigen-pairs, singular value decomposition, pseudoinverse, and the least squares problem for elliptic quaternion matrices. Moreover, we established algorithms based on these results and provided illustrative numerical experiments to substantiate the accuracy of our conclusions. In the experiments, it was observed that the p-value in the algebra of elliptic quaternions directly affects the performance of the problem under consideration. Selecting the optimal p-value for problem-solving and the elliptic behavior of many physical systems make this number system advantageous in applied sciences. Full article

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

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15 pages, 275 KiB

Open AccessArticle

γ-Dual Codes over Finite Commutative Chain Rings

by Hai Q. Dinh, Hiep L. Thi and Roengchai Tansuchat

Axioms 2024, 13(10), 655; https://doi.org/10.3390/axioms13100655 - 24 Sep 2024

Viewed by 197

Abstract

In this article, the notion of

γ

-dual codes over finite chain rings is introduced as an extension of dual codes over finite chain rings. Various characteristics and properties of

γ

-dual codes over finite chain rings are explored. We provide both necessary [...] Read more.

In this article, the notion of

γ

-dual codes over finite chain rings is introduced as an extension of dual codes over finite chain rings. Various characteristics and properties of

γ

-dual codes over finite chain rings are explored. We provide both necessary and sufficient conditions for the existence of

γ

-self-dual codes over finite chain rings. Additionally, we investigate the

γ

-dual of skew

ϕ

-

α

-constacyclic codes over finite chain rings. Full article

(This article belongs to the Special Issue Advances in Computation and Mathematics in Electrical and Electronic Engineering)

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Axioms, Volume 13, Issue 10 (October 2024) – 14 articles

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