Mathematics

23 pages, 442 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Estimation and Model Misspecification for Recurrent Event Data with Covariates Under Measurement Errors

by Ravinath Alahakoon, Gideon K. D. Zamba, Xuerong Meggie Wen and Akim Adekpedjou

Mathematics 2025, 13(1), 113; https://doi.org/10.3390/math13010113 - 30 Dec 2024

Viewed by 554

For subject i, we monitor an event that can occur multiple times over a random observation window [0,

τ_{i}

). At each recurrence, p concomitant variables,

x_{i}

, associated to the event recurrence are recorded—a subset (

q \leq p

[...] Read more.

For subject i, we monitor an event that can occur multiple times over a random observation window [0,

τ_{i}

). At each recurrence, p concomitant variables,

x_{i}

, associated to the event recurrence are recorded—a subset (

q \leq p

) of which is measured with errors. To circumvent the problem of bias and consistency associated with parameter estimation in the presence of measurement errors, we propose inference for corrected estimating equations with well-behaved roots under an additive measurement errors model. We show that estimation is essentially unbiased under the corrected profile likelihood for recurrent events, in comparison to biased estimations under a likelihood function that ignores correction. We propose methods for obtaining estimators of error variance and discuss the properties of the estimators. We further investigate the case of misspecified error models and show that the resulting estimators under misspecification converge to a value different from that of the true parameter—thereby providing a basis for bias assessment. We demonstrate the foregoing correction methods on an open-source rhDNase dataset gathered in a clinical setting. Full article

(This article belongs to the Section D1: Probability and Statistics)

► Show Figures

Figure 1

19 pages, 307 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Existence of Bounded Solutions for a Class of Degenerate Fourth-Order Elliptic Equations with Convection Terms

by Salvatore D’Asero

Mathematics 2025, 13(1), 3; https://doi.org/10.3390/math13010003 - 24 Dec 2024

Viewed by 510

Abstract

This paper deals with the existence of bounded and locally Hölder continuous weak solutions of the following nonlinear fourth-order Dirichlet problem: [...] Read more.

This paper deals with the existence of bounded and locally Hölder continuous weak solutions of the following nonlinear fourth-order Dirichlet problem:

\sum_{| α | = 1, 2} {(- 1)}^{| α |} D^{α} [A_{α} (x, u, D^{1} u, D^{2} u) - E_{α} (x) {| u |}^{λ (p_{α} - 1)} sign u] = f

in

Ω

, where the coefficients

A_{α}

satisfy a strengthened degenerate coercivity condition. Full article

(This article belongs to the Special Issue Problems and Methods in Nonlinear Analysis)

33 pages, 522 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Monochromatic Graph Decompositions Inspired by Anti-Ramsey Theory and Parity Constraints

by Yair Caro and Zsolt Tuza

Mathematics 2024, 12(23), 3665; https://doi.org/10.3390/math12233665 - 22 Nov 2024

Cited by 1 | Viewed by 705

Abstract

We open here many new tracks of research in anti-Ramsey Theory, considering edge-coloring problems inspired by rainbow coloring and further by odd colorings and conflict-free colorings. Let G be a graph and

F

any given family of graphs. For every integer [...] Read more.

We open here many new tracks of research in anti-Ramsey Theory, considering edge-coloring problems inspired by rainbow coloring and further by odd colorings and conflict-free colorings. Let G be a graph and

F

any given family of graphs. For every integer

n \geq | G |

, let

f (n, G | F)

denote the smallest integer k such that any edge coloring of

K_{n}

with at least k colors forces a copy of G in which each color class induces a member of

F

. Observe that in anti-Ramsey problems, each color class is a single edge, i.e.,

F = {K_{2}}

. Among the many results introduced in this paper, we mention the following. (1) For every graph G, there exists a constant

c = c (G)

such that in any edge coloring of

K_{n}

with at least

c n

colors there is a copy of G in which every vertex v is incident with an edge whose color appears only once among all edges incident with v. (2) In sharp contrast to the above result we prove that if

F

is the class of all odd graphs (having vertices with odd degrees only) then

f (n, K_{k} | F) = (1 + o (1)) ex (n, K_{⌈ k / 2 ⌉})

, which is quadratic for

k \geq 5

. (3) We exactly determine

f (n, G | F)

for small graphs when

F

belongs to several families representing various odd/even coloring constraints. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

21 pages, 543 KiB

Open AccessEditor’s ChoiceArticle

Brauer Configuration Algebras Induced by Integer Partitions and Their Applications in the Theory of Branched Coverings

by Agustín Moreno Cañadas, José Gregorio Rodríguez-Nieto and Olga Patricia Salazar Díaz

Mathematics 2024, 12(22), 3626; https://doi.org/10.3390/math12223626 - 20 Nov 2024

Cited by 1 | Viewed by 854

Abstract

Brauer configuration algebras are path algebras induced by appropriated multiset systems. Since their structures underlie combinatorial data, the general description of some of their algebraic invariants (e.g., their dimensions or the dimensions of their centers) is a hard problem. Integer partitions and compositions [...] Read more.

Brauer configuration algebras are path algebras induced by appropriated multiset systems. Since their structures underlie combinatorial data, the general description of some of their algebraic invariants (e.g., their dimensions or the dimensions of their centers) is a hard problem. Integer partitions and compositions of a given integer number are examples of multiset systems which can be used to define Brauer configuration algebras. This paper gives formulas for the dimensions of Brauer configuration algebras (and their centers) induced by some integer partitions. As an application of these results, we give examples of Brauer configurations, which can be realized as branch data of suitable branched coverings over different surfaces. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics, 4th Edition)

► Show Figures

Figure 1

19 pages, 1058 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Large Sample Behavior of the Least Trimmed Squares Estimator

by Yijun Zuo

Mathematics 2024, 12(22), 3586; https://doi.org/10.3390/math12223586 - 15 Nov 2024

Cited by 1 | Viewed by 1252

Abstract

The least trimmed squares (LTS) estimator is popular in location, regression, machine learning, and AI literature. Despite the empirical version of least trimmed squares (LTS) being repeatedly studied in the literature, the population version of the LTS has never been introduced and studied. [...] Read more.

The least trimmed squares (LTS) estimator is popular in location, regression, machine learning, and AI literature. Despite the empirical version of least trimmed squares (LTS) being repeatedly studied in the literature, the population version of the LTS has never been introduced and studied. The lack of the population version hinders the study of the large sample properties of the LTS utilizing the empirical process theory. Novel properties of the objective function in both empirical and population settings of the LTS and other properties, are established for the first time in this article. The primary properties of the objective function facilitate the establishment of other original results, including the influence function and Fisher consistency. The strong consistency is established with the help of a generalized Glivenko–Cantelli Theorem over a class of functions for the first time. Differentiability and stochastic equicontinuity promote the establishment of asymptotic normality with a concise and novel approach. Full article

(This article belongs to the Special Issue Advances in High-Dimensional Data Analysis)

► Show Figures

Figure 1

14 pages, 292 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

New Results on Graph Matching from Degree-Preserving Growth

by Péter L. Erdős, Shubha R. Kharel, Tamás Róbert Mezei and Zoltán Toroczkai

Mathematics 2024, 12(22), 3518; https://doi.org/10.3390/math12223518 - 11 Nov 2024

Viewed by 1405

Abstract

The recently introduced model in S. R. Kharel et al.’s study [Degree-preserving network growth. Nature Physics 2022, 18, 100–106] uses matchings to insert new vertices of prescribed degrees into the current graph of an ever-growing graph sequence. The process depends [...] Read more.

The recently introduced model in S. R. Kharel et al.’s study [Degree-preserving network growth. Nature Physics 2022, 18, 100–106] uses matchings to insert new vertices of prescribed degrees into the current graph of an ever-growing graph sequence. The process depends both on the size of the largest available matching, which is the focus of this paper, as well as on the actual choice of the matching. Here, we first show that the question of whether a graphic degree sequence, extended with a new degree

2 δ

, remains graphic is equivalent to the existence of a realization of the original degree sequence with a matching of size

δ

. Secondly, we present lower bounds for the size of the maximum matchings in any realization of the degree sequence. We then study the bounds on the size of maximal matchings in some realizations of the sequence, known as the potential matching number. We also estimate the minimum size of both maximal and maximum matchings, as determined by the degree sequence, independently of graphical realizations. Along this line we answer a question raised by T. Biedl et al.: Tight bounds on maximal and maximum matchings. Discrete Mathematics 2004, 285, 7–15. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

16 pages, 8983 KiB

Open AccessEditor’s ChoiceArticle

A Finite-Time Disturbance Observer for Tracking Control of Nonlinear Systems Subject to Model Uncertainties and Disturbances

by Manh Hung Nguyen and Kyoung Kwan Ahn

Mathematics 2024, 12(22), 3512; https://doi.org/10.3390/math12223512 - 10 Nov 2024

Cited by 1 | Viewed by 1181

Abstract

In this study, a finite-time disturbance observer (FTDOB) with a new structure is originally put forward for the motion tracking problem of a class of nonlinear systems subject to model uncertainties and exogenous disturbances. Compared to existing disturbance estimator designs in the literature, [...] Read more.

In this study, a finite-time disturbance observer (FTDOB) with a new structure is originally put forward for the motion tracking problem of a class of nonlinear systems subject to model uncertainties and exogenous disturbances. Compared to existing disturbance estimator designs in the literature, in which the estimation error only converges to the origin asymptotically under assumptions that the first and/or second derivatives are vanishing, the suggested DOB is able to estimate the disturbance exactly in finite time. Firstly, uncertainties (parametric and unstructured uncertainties), unknown dynamics, and external disturbances in system dynamics are lumped into a generalized disturbance term that is subsequently estimated by the proposed DOB. Based on this, a DOB-based backstepping controller is synthesized to ensure high-accuracy tracking performance under various working conditions. The stability analysis of not only the DOB but also the overall closed-loop system is theoretically confirmed by the Lyapunov stability theory. Finally, the advantages of the proposed FTDOB and the FTDOB-based controller over other DOBs and existing DOB-based controllers are explicitly simultaneously demonstrated by a series of numerical simulations on a second-order mechanical system and comparative experiments on an actual DC motor system. Full article

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

► Show Figures

Figure 1

12 pages, 1496 KiB

Open AccessEditor’s ChoiceArticle

A Long-Memory Model for Multiple Cycles with an Application to the US Stock Market

by Guglielmo Maria Caporale and Luis Alberiko Gil-Alana

Mathematics 2024, 12(22), 3487; https://doi.org/10.3390/math12223487 - 7 Nov 2024

Viewed by 6013

Abstract

This paper proposes a long-memory model that includes multiple cycles in addition to the long-run component. Specifically, instead of a single pole or singularity in the spectrum, it allows for multiple poles and, thus, different cycles with different degrees of persistence. It also [...] Read more.

This paper proposes a long-memory model that includes multiple cycles in addition to the long-run component. Specifically, instead of a single pole or singularity in the spectrum, it allows for multiple poles and, thus, different cycles with different degrees of persistence. It also incorporates non-linear deterministic structures in the form of Chebyshev polynomials in time. Simulations are carried out to analyze the finite sample properties of the proposed test, which is shown to perform well in the case of a relatively large sample with at least 1000 observations. The model is then applied to weekly data on the S&P 500 from 1 January 1970 to 26 October 2023 as an illustration. The estimation results based on the first differenced logged values (i.e., the returns) point to the existence of three cyclical structures in the series, with lengths of approximately one month, one year, and four years, respectively, and to orders of integration in the range (0, 0.20), which implies stationary long memory in all cases. Full article

(This article belongs to the Special Issue Applied Time Series and Artificial Intelligence in Economics and Finance)

► Show Figures

Figure 1

22 pages, 780 KiB

Open AccessEditor’s ChoiceArticle

Adaptive Production Rescheduling System for Managing Unforeseen Disruptions

by Andy J. Figueroa, Raul Poler and Beatriz Andres

Mathematics 2024, 12(22), 3478; https://doi.org/10.3390/math12223478 - 7 Nov 2024

Cited by 2 | Viewed by 1576

Abstract

This work presents a mixed-integer linear programming (MILP) model to solve the production rescheduling problem in a job shop manufacturing system impacted by unexpected events, aiming to minimize production costs and disruptions to the initial schedule. The approach begins by generating an optimal [...] Read more.

This work presents a mixed-integer linear programming (MILP) model to solve the production rescheduling problem in a job shop manufacturing system impacted by unexpected events, aiming to minimize production costs and disruptions to the initial schedule. The approach begins by generating an optimal production plan through batch assignments to machines. When unforeseen events, such as machine breakdowns or raw material shortages, occur, a dynamic rescheduling process is triggered, employing an iterative and reactive algorithm to adapt the plan to the real-time conditions on the shop floor. The results demonstrate that this rescheduling method efficiently adjusts to the new conditions while minimizing deviations from the original schedule, achieving solutions within acceptable computational times. Full article

(This article belongs to the Section E: Applied Mathematics)

► Show Figures

Figure 1

21 pages, 1179 KiB

Open AccessEditor’s ChoiceArticle

The High-Order ADI Difference Method and Extrapolation Method for Solving the Two-Dimensional Nonlinear Parabolic Evolution Equations

by Xin Shen, Xuehua Yang and Haixiang Zhang

Mathematics 2024, 12(22), 3469; https://doi.org/10.3390/math12223469 - 6 Nov 2024

Cited by 6 | Viewed by 942

Abstract

In this paper, the numerical solution for two-dimensional nonlinear parabolic equations is studied using an alternating-direction implicit (ADI) Crank–Nicolson (CN) difference scheme. Firstly, we use the CN format in the time direction, and then use the CN format in the space direction before [...] Read more.

In this paper, the numerical solution for two-dimensional nonlinear parabolic equations is studied using an alternating-direction implicit (ADI) Crank–Nicolson (CN) difference scheme. Firstly, we use the CN format in the time direction, and then use the CN format in the space direction before discretizing the second-order center difference quotient. In addition, we strictly prove that the proposed ADI difference scheme has unique solvability and is unconditionally stable and convergent. The extrapolation method is further applied to improve the numerical solution accuracy. Finally, two numerical examples are given to verify our theoretical results. Full article

► Show Figures

Figure 1

13 pages, 433 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Families of Planar Orbits in Polar Coordinates Compatible with Potentials

by Thomas Kotoulas

Mathematics 2024, 12(21), 3435; https://doi.org/10.3390/math12213435 - 2 Nov 2024

Viewed by 607

Abstract

In light of the planar inverse problem of Newtonian Dynamics, we study the monoparametric family of regular orbits

f (r, θ) = c

in polar coordinates (where c is the parameter varying along the family of orbits), which are generated [...] Read more.

In light of the planar inverse problem of Newtonian Dynamics, we study the monoparametric family of regular orbits

f (r, θ) = c

in polar coordinates (where c is the parameter varying along the family of orbits), which are generated by planar potentials

V = V (r, θ)

. The corresponding family of orbits can be uniquely represented by the “slope function”

γ = \frac{f_{θ}}{f_{r}}

. By using the basic partial differential equation of the planar inverse problem, which combines families of orbits and potentials, we apply a new methodology in order to find specific potentials, e.g.,

V = A (r) + B (θ)

or

V = H (γ)

and one-dimensional potentials, e.g.,

V = A (r)

or

V = G (θ)

. In order to determine such potentials, differential conditions on the family of orbits

f (r, θ)

= c are imposed. If these conditions are fulfilled, then we can find a potential of the above form analytically. For the given families of curves, such as ellipses, parabolas, Bernoulli’s lemniscates, etc., we find potentials that produce them. We present suitable examples for all cases and refer to the case of straight lines. Full article

(This article belongs to the Section E4: Mathematical Physics)

► Show Figures

Figure 1

31 pages, 1871 KiB

Open AccessEditor’s ChoiceArticle

3D Reconstruction of Geometries for Urban Areas Supported by Computer Vision or Procedural Generations

by Hanli Liu, Carlos J. Hellín, Abdelhamid Tayebi, Carlos Delgado and Josefa Gómez

Mathematics 2024, 12(21), 3331; https://doi.org/10.3390/math12213331 - 23 Oct 2024

Cited by 1 | Viewed by 1173

Abstract

This work presents a numerical mesh generation method for 3D urban scenes that could be easily converted into any 3D format, different from most implementations which are limited to specific environments in their applicability. The building models have shaped roofs and faces with [...] Read more.

This work presents a numerical mesh generation method for 3D urban scenes that could be easily converted into any 3D format, different from most implementations which are limited to specific environments in their applicability. The building models have shaped roofs and faces with static colors, combining the buildings with a ground grid. The building generation uses geographic positions and shape names, which can be extracted from OpenStreetMap. Additional steps, like a computer vision method, can be integrated into the generation optionally to improve the quality of the model, although this is highly time-consuming. Its function is to classify unknown roof shapes from satellite images with adequate resolution. The generation can also use custom geographic information. This aspect was tested using information created by procedural processes. The method was validated by results generated for many realistic scenarios with multiple building entities, comparing the results between using computer vision and not. The generated models were attempted to be rendered under Graphics Library Transmission Format and Unity Engine. In future work, a polygon-covering algorithm needs to be completed to process the building footprints more effectively, and a solution is required for the missing height values in OpenStreetMap. Full article

(This article belongs to the Special Issue Object Detection: Algorithms, Computations and Practices)

► Show Figures

Figure 1

30 pages, 10941 KiB

Open AccessEditor’s ChoiceArticle

Closed-Boundary Reflections of Shallow Water Waves as an Open Challenge for Physics-Informed Neural Networks

by Kubilay Timur Demir, Kai Logemann and David S. Greenberg

Mathematics 2024, 12(21), 3315; https://doi.org/10.3390/math12213315 - 22 Oct 2024

Viewed by 1571

Abstract

Physics-informed neural networks (PINNs) have recently emerged as a promising alternative to traditional numerical methods for solving partial differential equations (PDEs) in fluid dynamics. By using PDE-derived loss functions and auto-differentiation, PINNs can recover solutions without requiring costly simulation data, spatial gridding, or [...] Read more.

Physics-informed neural networks (PINNs) have recently emerged as a promising alternative to traditional numerical methods for solving partial differential equations (PDEs) in fluid dynamics. By using PDE-derived loss functions and auto-differentiation, PINNs can recover solutions without requiring costly simulation data, spatial gridding, or time discretization. However, PINNs often exhibit slow or incomplete convergence, depending on the architecture, optimization algorithms, and complexity of the PDEs. To address these difficulties, a variety of novel and repurposed techniques have been introduced to improve convergence. Despite these efforts, their effectiveness is difficult to assess due to the wide range of problems and network architectures. As a novel test case for PINNs, we propose one-dimensional shallow water equations with closed boundaries, where the solutions exhibit repeated boundary wave reflections. After carefully constructing a reference solution, we evaluate the performance of PINNs across different architectures, optimizers, and special training techniques. Despite the simplicity of the problem for classical methods, PINNs only achieve accurate results after prohibitively long training times. While some techniques provide modest improvements in stability and accuracy, this problem remains an open challenge for PINNs, suggesting that it could serve as a valuable testbed for future research on PINN training techniques and optimization strategies. Full article

► Show Figures

Figure 1

30 pages, 10109 KiB

Open AccessEditor’s ChoiceArticle

AI-Powered Approaches for Hypersurface Reconstruction in Multidimensional Spaces

by Kostadin Yotov, Emil Hadzhikolev, Stanka Hadzhikoleva and Mariyan Milev

Mathematics 2024, 12(20), 3285; https://doi.org/10.3390/math12203285 - 19 Oct 2024

Viewed by 1212

Abstract

The present article explores the possibilities of using artificial neural networks to solve problems related to reconstructing complex geometric surfaces in Euclidean and pseudo-Euclidean spaces, examining various approaches and techniques for training the networks. The main focus is on the possibility of training [...] Read more.

The present article explores the possibilities of using artificial neural networks to solve problems related to reconstructing complex geometric surfaces in Euclidean and pseudo-Euclidean spaces, examining various approaches and techniques for training the networks. The main focus is on the possibility of training a set of neural networks with information about the available surface points, which can then be used to predict and complete missing parts. A method is proposed for using separate neural networks that reconstruct surfaces in different spatial directions, employing various types of architectures, such as multilayer perceptrons, recursive networks, and feedforward networks. Experimental results show that artificial neural networks can successfully approximate both smooth surfaces and those containing singular points. The article presents the results with the smallest error, showcasing networks of different types, along with a technique for reconstructing geographic relief. A comparison is made between the results achieved by neural networks and those obtained using traditional surface approximation methods such as Bézier curves, k-nearest neighbors, principal component analysis, Markov random fields, conditional random fields, and convolutional neural networks. Full article

(This article belongs to the Special Issue Machine Learning and Evolutionary Algorithms: Theory and Applications)

► Show Figures

Figure 1

29 pages, 466 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Elimination Algorithms for Skew Polynomials with Applications in Cybersecurity

by Raqeeb Rasheed, Ali Safaa Sadiq and Omprakash Kaiwartya

Mathematics 2024, 12(20), 3258; https://doi.org/10.3390/math12203258 - 17 Oct 2024

Viewed by 1556

Abstract

It is evident that skew polynomials offer promising directions for developing cryptographic schemes. This paper focuses on exploring skew polynomials and studying their properties with the aim of exploring their potential applications in fields such as cryptography and combinatorics. We begin by deriving [...] Read more.

It is evident that skew polynomials offer promising directions for developing cryptographic schemes. This paper focuses on exploring skew polynomials and studying their properties with the aim of exploring their potential applications in fields such as cryptography and combinatorics. We begin by deriving the concept of resultants for bivariate skew polynomials. Then, we employ the derived resultant to incrementally eliminate indeterminates in skew polynomial systems, utilising both direct and modular approaches. Finally, we discuss some applications of the derived resultant, including cryptographic schemes (such as Diffie–Hellman) and combinatorial identities (such as Pascal’s identity). We start by considering a bivariate skew polynomial system with two indeterminates; our intention is to isolate and eliminate one of the indeterminates to reduce the system to a simpler form (that is, relying only on one indeterminate in this case). The methodology is composed of two main techniques; in the first technique, we apply our definition of a (bivariate) resultant via a Sylvester-style matrix directly from the polynomials’ coefficients, while the second is based on modular methods where we compute the resultant by using evaluation and interpolation approaches. The idea of this second technique is that instead of computing the resultant directly from the coefficients, we propose to evaluate the polynomials at a set of valid points to compute its corresponding set of partial resultants first; then, we can deduce the original resultant by combining all these partial resultants using an interpolation technique by utilising a theorem we have established. Full article

(This article belongs to the Topic Modeling and Practice for Trustworthy and Secure Systems)

► Show Figures

Figure 1

25 pages, 575 KiB

Open AccessEditor’s ChoiceArticle

Mathematical Analysis of Four Fundamental Epidemiological Models for Monkeypox Disease Outbreaks: On the Pivotal Role of Human–Animal Order Parameters—In Memory of Hermann Haken

by Till D. Frank

Mathematics 2024, 12(20), 3215; https://doi.org/10.3390/math12203215 - 14 Oct 2024

Cited by 1 | Viewed by 1370

Abstract

Four fundamental models that describe the spread of Monkeypox disease are analyzed: the SIR-SIR, SEIR-SIR, SIR-SEIR, and SEIR-SEIR models. They form the basis of most Monkeypox diseases models that are currently discussed in the literature. It is shown that the way the model [...] Read more.

Four fundamental models that describe the spread of Monkeypox disease are analyzed: the SIR-SIR, SEIR-SIR, SIR-SEIR, and SEIR-SEIR models. They form the basis of most Monkeypox diseases models that are currently discussed in the literature. It is shown that the way the model subpopulations are organized in disease outbreaks and evolve relative to each other is determined by the relevant unstable system eigenvectors, also called order parameters. For all models, analytical expressions of the order parameters are derived. Under appropriate conditions these order parameters describe the initial outbreak phases of exponential increase in good approximation. It is shown that all four models exhibit maximally two order parameters and maximally one human–animal order parameter. The human–animal order parameter firmly connects the outbreak dynamics in the animal system with the dynamics in the human system. For the special case of the SIR-SIR model, it is found that the two possible order parameters completely describe the dynamics of infected humans and animals during entire infection waves. Finally, a simulation of a Monkeypox infection wave illustrates that in line with the aforementioned analytical results the leading order parameter explains most of the variance in the infection dynamics. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis of Problems in Ecology, Epidemiology and Oncology)

► Show Figures

Figure 1

13 pages, 283 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Dini’s Theorem for Fuzzy Number-Valued Continuous Functions

by Juan José Font, Sergio Macario and Manuel Sanchis

Mathematics 2024, 12(20), 3209; https://doi.org/10.3390/math12203209 - 13 Oct 2024

Viewed by 1048

Abstract

This work aims to provide several versions of Dini’s theorem for fuzzy number-valued continuous functions defined on a compact set K. In this context, there is a wide variety of possibilities since, unlike the real line, we can consider different topologies and [...] Read more.

This work aims to provide several versions of Dini’s theorem for fuzzy number-valued continuous functions defined on a compact set K. In this context, there is a wide variety of possibilities since, unlike the real line, we can consider different topologies and orders on the set of fuzzy numbers. For example, we will show that the fuzzy Dini’s theorem holds for the usual partial orders and the most commonly used topologies but does not hold for all orders in general. Full article

(This article belongs to the Section D2: Operations Research and Fuzzy Decision Making)

8 pages, 1079 KiB

Open AccessEditor’s ChoiceArticle

Riemannian Manifolds, Closed Geodesic Lines, Topology and Ramsey Theory

by Edward Bormashenko

Mathematics 2024, 12(20), 3206; https://doi.org/10.3390/math12203206 - 13 Oct 2024

Viewed by 1199

Abstract

We applied the Ramsey analysis to the sets of points belonging to Riemannian manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic. This interconnection between the points is mapped into the bi-colored, complete Ramsey graph. The selected points correspond [...] Read more.

We applied the Ramsey analysis to the sets of points belonging to Riemannian manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic. This interconnection between the points is mapped into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. The complete bi-colored graph containing six vertices inevitably contains at least one mono-colored triangle; hence, a mono-colored triangle, built of the green or red links, i.e., non-geodesic or geodesic lines, consequently appears in the graph. We also considered the bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds. Two Riemannian manifolds, namely

(M_{1}, g_{1})

and

(M_{2}, g_{2})

, represented by the Riemann surfaces which intersect along the curve

(M_{1}, g_{1}) \cap (M_{2}, g_{2}) = ℒ

were addressed. Curve

ℒ

does not contain geodesic lines in either of the manifolds

(M_{1}, g_{1})

and

(M_{2}, g_{2})

. Consider six points located on the

ℒ :

{1, \dots 6} \subset ℒ

. The points

{1, \dots 6} \subset ℒ

are connected with two distinguishable kinds of the geodesic lines, namely with the geodesic lines belonging to the Riemannian manifold

(M_{1}, g_{1})

/red links, and, alternatively, with the geodesic lines belonging to the manifold

(M_{2}, g_{2})

/green links. Points

{1, \dots 6} \subset ℒ

form the vertices of the complete graph, connected with two kinds of links. The emerging graph contains at least one closed geodesic line. The extension of the theorem to the Riemann surfaces of various Euler characteristics is presented. Full article

► Show Figures

Figure 1

21 pages, 389 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Constraint Qualifications and Optimality Conditions for Nonsmooth Semidefinite Multiobjective Programming Problems with Mixed Constraints Using Convexificators

by Balendu Bhooshan Upadhyay, Shubham Kumar Singh and Ioan Stancu-Minasian

Mathematics 2024, 12(20), 3202; https://doi.org/10.3390/math12203202 - 12 Oct 2024

Viewed by 976

Abstract

In this article, we investigate a class of non-smooth semidefinite multiobjective programming problems with inequality and equality constraints (in short, NSMPP). We establish the convex separation theorem for the space of symmetric matrices. Employing the properties of the convexificators, we establish Fritz John [...] Read more.

In this article, we investigate a class of non-smooth semidefinite multiobjective programming problems with inequality and equality constraints (in short, NSMPP). We establish the convex separation theorem for the space of symmetric matrices. Employing the properties of the convexificators, we establish Fritz John (in short, FJ)-type necessary optimality conditions for NSMPP. Subsequently, we introduce a generalized version of Abadie constraint qualification (in short, NSMPP-ACQ) for the considered problem, NSMPP. Employing NSMPP-ACQ, we establish strong Karush-Kuhn-Tucker (in short, KKT)-type necessary optimality conditions for NSMPP. Moreover, we establish sufficient optimality conditions for NSMPP under generalized convexity assumptions. In addition to this, we introduce the generalized versions of various other constraint qualifications, namely Kuhn-Tucker constraint qualification (in short, NSMPP-KTCQ), Zangwill constraint qualification (in short, NSMPP-ZCQ), basic constraint qualification (in short, NSMPP-BCQ), and Mangasarian-Fromovitz constraint qualification (in short, NSMPP-MFCQ), for the considered problem NSMPP and derive the interrelationships among them. Several illustrative examples are furnished to demonstrate the significance of the established results. Full article

(This article belongs to the Special Issue Mathematical Optimization and Control: Methods and Applications)

► Show Figures

Figure 1

18 pages, 361 KiB

Open AccessEditor’s ChoiceArticle

A New Class of Braided Block Codes Constructed by Convolutional Interleavers

by Sina Vafi

Mathematics 2024, 12(19), 3127; https://doi.org/10.3390/math12193127 - 6 Oct 2024

Viewed by 1019

Abstract

Parallel Concatenated Block (PCB) codes are conventionally represented as high-rate codes with low error correcting capability. To form a reliable and outstanding code, this paper presents a modification on the structure of PCB codes, which is accomplished by encoding some parity bits of [...] Read more.

Parallel Concatenated Block (PCB) codes are conventionally represented as high-rate codes with low error correcting capability. To form a reliable and outstanding code, this paper presents a modification on the structure of PCB codes, which is accomplished by encoding some parity bits of one of their component codes. For the newly proposed code, named as the braided code, non-stuff bit-based convolutional interleavers are applied, aiming to minimize the design complexity while ensuring the proper permutations of the original message and selected parity bits. To precisely determine the error correcting capability, a tight bound for the minimum weight of braided code is presented. Additionally, further analyses are provided, which verify iterative decoding performance and the complexity of the constructed code. It is concluded that an outstanding braided code is formed by utilizing a reasonable number of iterations applied at its decoding processes, while maintaining its design complexity at a level similar to other well-known codes. The significant performance of short and long-length-based braided codes is evident in both waterfall and error floor regions. Full article

► Show Figures

Figure 1

47 pages, 11591 KiB

Open AccessEditor’s ChoiceArticle

Spontaneous Formation of Evolutionary Game Strategies for Long-Term Carbon Emission Reduction Based on Low-Carbon Trading Mechanism

by Zhanggen Zhu, Lefeng Cheng and Teng Shen

Mathematics 2024, 12(19), 3109; https://doi.org/10.3390/math12193109 - 4 Oct 2024

Cited by 5 | Viewed by 1011

Abstract

In the context of increasing global efforts to mitigate climate change, effective carbon emission reduction is a pressing issue. Governments and power companies are key stakeholders in implementing low-carbon strategies, but their interactions require careful management to ensure optimal outcomes for both economic [...] Read more.

In the context of increasing global efforts to mitigate climate change, effective carbon emission reduction is a pressing issue. Governments and power companies are key stakeholders in implementing low-carbon strategies, but their interactions require careful management to ensure optimal outcomes for both economic development and environmental protection. This paper addresses this real-world challenge by utilizing evolutionary game theory (EGT) to model the strategic interactions between these stakeholders under a low-carbon trading mechanism. Unlike classical game theory, which assumes complete rationality and perfect information, EGT allows for bounded rationality and learning over time, making it particularly suitable for modeling long-term interactions in complex systems like carbon markets. This study builds an evolutionary game model between the government and power companies to explore how different strategies in carbon emission reduction evolve over time. Using payoff matrices and replicator dynamics equations, we determine the evolutionarily stable equilibrium (ESE) points and analyze their stability through dynamic simulations. The findings show that in the absence of a third-party regulator, neither party achieves an ideal ESE. To address this, a third-party regulatory body is introduced into the model, leading to the formulation of a tripartite evolutionary game. The results highlight the importance of regulatory oversight in achieving stable and optimal low-carbon strategies. This paper offers practical policy recommendations based on the simulation outcomes, providing a robust theoretical framework for government intervention in carbon markets and guiding enterprises towards sustainable practices. Full article

(This article belongs to the Special Issue Artificial Intelligence and Game Theory)

► Show Figures

Figure 1

13 pages, 1088 KiB

Open AccessEditor’s ChoiceArticle

Generalized Kelvin–Voigt Creep Model in Fractal Space–Time

by Eduardo Reyes de Luna, Andriy Kryvko, Juan B. Pascual-Francisco, Ignacio Hernández and Didier Samayoa

Mathematics 2024, 12(19), 3099; https://doi.org/10.3390/math12193099 - 3 Oct 2024

Cited by 1 | Viewed by 1232

Abstract

In this paper, we study the creep phenomena for self-similar models of viscoelastic materials and derive a generalization of the Kelvin–Voigt model in the framework of fractal continuum calculus. Creep compliance for the Kelvin–Voigt model is extended to fractal manifolds through local fractal-continuum [...] Read more.

In this paper, we study the creep phenomena for self-similar models of viscoelastic materials and derive a generalization of the Kelvin–Voigt model in the framework of fractal continuum calculus. Creep compliance for the Kelvin–Voigt model is extended to fractal manifolds through local fractal-continuum differential operators. Generalized fractal creep compliance is obtained, taking into account the intrinsic time

τ

and the fractal dimension of time-scale

β

. The model obtained is validated with experimental data obtained for resin samples with the fractal structure of a Sierpinski carpet and experimental data on rock salt. Comparisons of the model predictions with the experimental data are presented as the curves of slow continuous deformations. Full article

(This article belongs to the Special Issue The Applications of Fractional Calculus in Control Engineering, Dynamical Systems and Signal Processing)

► Show Figures

Figure 1

20 pages, 581 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Mittag–Leffler Fractional Stochastic Integrals and Processes with Applications

by Enrica Pirozzi

Mathematics 2024, 12(19), 3094; https://doi.org/10.3390/math12193094 - 2 Oct 2024

Cited by 2 | Viewed by 2075

Abstract

We study Mittag–Leffler (ML) fractional integrals involved in the solution processes of a system of coupled fractional stochastic differential equations. We introduce the ML fractional stochastic process as a ML fractional stochastic integral with respect to a standard Brownian motion. We provide some [...] Read more.

We study Mittag–Leffler (ML) fractional integrals involved in the solution processes of a system of coupled fractional stochastic differential equations. We introduce the ML fractional stochastic process as a ML fractional stochastic integral with respect to a standard Brownian motion. We provide some representation formulas of solution processes in terms of Mittag–Leffler fractional integrals and processes. Computable expressions of the mean functions and of the covariances of such processes are specifically given. The application in neuronal modeling is provided, and all involved functions and processes are specifically determined. Numerical evaluations are carried out and some results are shown and discussed. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)

► Show Figures

Figure 1

29 pages, 452 KiB

Open AccessEditor’s ChoiceArticle

Some Combined Results from Eneström–Kakeya and Rouché Theorems on the Generalized Schur Stability of Polynomials and the Stability of Quasi-Polynomials-Application to Time-Delay Systems

by Manuel De la Sen

Mathematics 2024, 12(19), 3023; https://doi.org/10.3390/math12193023 - 27 Sep 2024

Viewed by 649

Abstract

This paper derives some generalized Schur-type stability results of polynomials based on several forms and generalizations of the Eneström–Kakeya theorem combined with the Rouché theorem. It is first investigated, under sufficiency-type conditions, the derivation of the eventually generalized Schur stability sufficient conditions which [...] Read more.

This paper derives some generalized Schur-type stability results of polynomials based on several forms and generalizations of the Eneström–Kakeya theorem combined with the Rouché theorem. It is first investigated, under sufficiency-type conditions, the derivation of the eventually generalized Schur stability sufficient conditions which are not necessarily related to the zeros of the polynomial lying in the unit open circle. In a second step, further sufficient conditions were introduced to guarantee that the above generalized Schur stability property persists within either the same above complex nominal stability region or in some larger one. The classical weak and, respectively, strong Schur stability in the closed and, respectively, open complex unit circle centred at zero are particular cases of their corresponding generalized versions. Some of the obtained and proved results are further generalized “ad hoc” for the case of quasi-polynomials whose zeros might be interpreted, in some typical cases, as characteristic zeros of linear continuous-time delayed time-invariant dynamic systems with commensurate constant point delays. Full article

24 pages, 2433 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Generalized Shortest Path Problem: An Innovative Approach for Non-Additive Problems in Conditional Weighted Graphs

by Adrien Durand, Timothé Watteau, Georges Ghazi and Ruxandra Mihaela Botez

Mathematics 2024, 12(19), 2995; https://doi.org/10.3390/math12192995 - 26 Sep 2024

Viewed by 2222

Abstract

The shortest path problem is fundamental in graph theory and has been studied extensively due to its practical importance. Despite this aspect, finding the shortest path between two nodes remains a significant challenge in many applications, as it often becomes complex and time [...] Read more.

The shortest path problem is fundamental in graph theory and has been studied extensively due to its practical importance. Despite this aspect, finding the shortest path between two nodes remains a significant challenge in many applications, as it often becomes complex and time consuming. This complexity becomes even more challenging when constraints make the problem non-additive, thereby increasing the difficulty of finding the optimal path. The objective of this paper is to present a broad perspective on the conventional shortest path problem. It introduces a new method to classify cost functions associated with graphs by defining distinct sets of cost functions. This classification facilitates the exploration of line graphs and an understanding of the upper bounds on the transformation sizes for these types of graphs. Based on these foundations, the paper proposes a practical methodology for solving non-additive shortest path problems. It also provides a proof of optimality and establishes an upper bound on the algorithmic cost of the proposed methodology. This study not only expands the scope of traditional shortest path problems but also highlights their computational complexity and potential solutions. Full article

(This article belongs to the Special Issue Advances in Graph Theory: Algorithms and Applications)

► Show Figures

Figure 1

15 pages, 591 KiB

Open AccessEditor’s ChoiceArticle

Closeness Centrality of Asymmetric Trees and Triangular Numbers

by Nytha Ramanathan, Eduardo Ramirez, Dorothy Suzuki-Burke and Darren A. Narayan

Mathematics 2024, 12(19), 2994; https://doi.org/10.3390/math12192994 - 26 Sep 2024

Cited by 1 | Viewed by 870

Abstract

The combinatorial problem in this paper is motivated by a variant of the famous traveling salesman problem where the salesman must return to the starting point after each delivery. The total length of a delivery route is related to a metric known as [...] Read more.

The combinatorial problem in this paper is motivated by a variant of the famous traveling salesman problem where the salesman must return to the starting point after each delivery. The total length of a delivery route is related to a metric known as closeness centrality. The closeness centrality of a vertex v in a graph G was defined in 1950 by Bavelas to be

C C (v) = \frac{| V (G) | - 1}{S D (v)}

, where

S D (v)

is the sum of the distances from v to each of the other vertices (which is one-half of the total distance in the delivery route). We provide a real-world example involving the Metro Atlanta Rapid Transit Authority rail network and identify stations whose

S D

values are nearly identical, meaning they have a similar proximity to other stations in the network. We then consider theoretical aspects involving asymmetric trees. For integer values of k, we considered the asymmetric tree with paths of lengths

k, 2 k, \dots, n k

that are incident to a center vertex. We investigated trees with different values of k, and for

k = 1

and

k = 2

, we established necessary and sufficient conditions for the existence of two vertices with identical

S D

values, which has a surprising connection to the triangular numbers. Additionally, we investigated asymmetric trees with paths incident to two vertices and found a sufficient condition for vertices to have equal

S D

values. This leads to new combinatorial proofs of identities arising from Pascal’s triangle. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 2nd Edition)

► Show Figures

Figure 1

41 pages, 3974 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Adaptation of an Eddy Current Model for Characterizing Subsurface Defects in CFRP Plates Using FEM Analysis Based on Energy Functional

by Mario Versaci, Filippo Laganà, Francesco Carlo Morabito, Annunziata Palumbo and Giovanni Angiulli

Mathematics 2024, 12(18), 2854; https://doi.org/10.3390/math12182854 - 13 Sep 2024

Cited by 13 | Viewed by 1212

Abstract

In this work, a known Eddy Current (EC) model is adapted to characterize subsurface defects in carbon fiber-reinforced polymer (CFRP) plates intended for the civil aerospace industry. The considered defects include delaminations, microcracks, porosity, fiber breakage, and the simultaneous presence of these defects. [...] Read more.

In this work, a known Eddy Current (EC) model is adapted to characterize subsurface defects in carbon fiber-reinforced polymer (CFRP) plates intended for the civil aerospace industry. The considered defects include delaminations, microcracks, porosity, fiber breakage, and the simultaneous presence of these defects. Each defect is modeled as an additive variation in the material’s electrical conductivity tensor, allowing for a detailed mathematical representation of the defect’s influence on the CFRP’s electromagnetic behavior. The additivity of the variations in the conductivity tensor is justified by the assumption that the defects are not visible to the naked eye, implying that the material does not require non-destructive testing. The adapted EC model admits a unique and stable solution by verifying that all analytical steps are satisfied. To reconstruct 2D maps of the magnetic flux density amplitude, a FEM formulation is adopted, based on the energy functional because it ensures a stable and consistent numerical formulation given its coercivity. Moreover, the numerical approach allows precise and reliable numerical solutions, enhancing the capability to detect and quantify defects. The numerical results show that the obtained 2D maps are entirely superimposable on those highlighting the distribution of mechanical stress states known in the literature, offering a clear advantage in terms of detection costs. This approach provides an effective and economical solution for the non-destructive inspection of CFRP, ensuring accurate and timely defect diagnosis for maintaining structural integrity. Full article

► Show Figures

Figure 1

27 pages, 1590 KiB

Open AccessEditor’s ChoiceArticle

Sojourn Time Analysis of a Single-Server Queue with Single- and Batch-Service Customers

by Yusei Koyama, Ayane Nakamura and Tuan Phung-Duc

Mathematics 2024, 12(18), 2820; https://doi.org/10.3390/math12182820 - 11 Sep 2024

Cited by 1 | Viewed by 1136

Abstract

There are various types of sharing economy services, such as ride-sharing and shared-taxi rides. Motivated by these services, we consider a single-server queue in which customers probabilistically select the type of service, that is, the single service or batch service, or other services [...] Read more.

There are various types of sharing economy services, such as ride-sharing and shared-taxi rides. Motivated by these services, we consider a single-server queue in which customers probabilistically select the type of service, that is, the single service or batch service, or other services (e.g., train). In the proposed model, which is denoted by the M+M(K)/M/1 queue, we assume that the arrival process of all the customers follows a Poisson distribution, the batch size is constant, and the common service time (for the single- and batch-service customers) follows an exponential distribution. In this model, the derivation of the sojourn time distribution is challenging because the sojourn time of a batch-service customer is not determined upon arrival but depends on single customers who arrive later. This results in a two-dimensional recursion, which is not generally solvable, but we made it possible by utilizing a special structure of our model. We present an analysis using a quasi-birth-and-death process, deriving the exact and approximated sojourn time distributions (for the single-service customers, batch-service customers, and all the customers). Through numerical experiments, we demonstrate that the approximated sojourn time distribution is sufficiently accurate compared to the exact sojourn time distributions. We also present a reasonable approximation for the distribution of the total number of customers in the system, which would be challenging with a direct-conventional method. Furthermore, we presented an accurate approximation method for a more general model where the service time of single-service customers and that of batch-service customers follow two distinct distributions, based on our original model. Full article

(This article belongs to the Special Issue Advances in Queueing Theory and Applications)

► Show Figures

Figure 1

31 pages, 961 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Optimal Epidemic Control with Nonmedical and Medical Interventions

by Alexandra Smirnova, Mona Baroonian and Xiaojing Ye

Mathematics 2024, 12(18), 2811; https://doi.org/10.3390/math12182811 - 11 Sep 2024

Cited by 1 | Viewed by 952

Abstract

In this study, we investigate different epidemic control scenarios through theoretical analysis and numerical simulations. To account for two important types of control at the early ascending stage of an outbreak, nonmedical interventions, and medical treatments, a compartmental model is considered with the [...] Read more.

In this study, we investigate different epidemic control scenarios through theoretical analysis and numerical simulations. To account for two important types of control at the early ascending stage of an outbreak, nonmedical interventions, and medical treatments, a compartmental model is considered with the first control aimed at lowering the disease transmission rate through behavioral changes and the second control set to lower the period of infectiousness by means of antiviral medications and other forms of medical care. In all experiments, the implementation of control strategies reduces the daily cumulative number of cases and successfully “flattens the curve”. The reduction in the cumulative cases is achieved by eliminating or delaying new cases. This delay is incredibly valuable, as it provides public health organizations with more time to advance antiviral treatments and devise alternative preventive measures. The main theoretical result of the paper, Theorem 1, concludes that the two optimal control functions may be increasing initially. However, beyond a certain point, both controls decline (possibly causing the number of newly infected people to grow). The numerical simulations conducted by the authors confirm theoretical findings, which indicates that, ideally, around the time that early interventions become less effective, the control strategy must be upgraded through the addition of new and improved tools, such as vaccines, therapeutics, testing, air ventilation, and others, in order to successfully battle the virus going forward. Full article

(This article belongs to the Special Issue Mathematical Methods and Models in Epidemiology)

► Show Figures

Figure 1

13 pages, 273 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Some Fractional Integral and Derivative Formulas Revisited

by Juan Luis González-Santander and Francesco Mainardi

Mathematics 2024, 12(17), 2786; https://doi.org/10.3390/math12172786 - 9 Sep 2024

Cited by 2 | Viewed by 1016

Abstract

In the most common literature about fractional calculus, we find that

{}_{a}D_{t}^{α} f (t) = {}_{a}I_{t}^{- α} f (t)

is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the [...] Read more.

In the most common literature about fractional calculus, we find that

{}_{a}D_{t}^{α} f (t) = {}_{a}I_{t}^{- α} f (t)

is assumed implicitly in the tables of fractional integrals and derivatives. However, this is not straightforward from the definitions of

{}_{a}I_{t}^{α} f (t)

and

{}_{a}D_{t}^{α} f (t)

. In this sense, we prove that

{}_{0}D_{t} f (t) = {}_{0}I_{t}^{- α} f (t)

is true for

f (t) = t^{ν - 1} \log t

, and

f (t) = e^{λ t}

, despite the fact that these derivations are highly non-trivial. Moreover, the corresponding formulas for

{}_{- \infty}D_{t}^{α} {|t|}^{- δ}

and

{}_{- \infty}I_{t}^{α} {|t|}^{- δ}

found in the literature are incorrect; thus, we derive the correct ones, proving in turn that

{}_{- \infty}D_{t}^{α} {|t|}^{- δ} = {}_{- \infty}I_{t}^{- α} {|t|}^{- δ}

holds true. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

19 pages, 311 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Pricing a Defaultable Zero-Coupon Bond under Imperfect Information and Regime Switching

by Ashwaq Ali Zarban, David Colwell and Donna Mary Salopek

Mathematics 2024, 12(17), 2740; https://doi.org/10.3390/math12172740 - 2 Sep 2024

Cited by 2 | Viewed by 1396

Abstract

We propose a pricing formula for a defaultable zero-coupon bond with imperfect information under a regime switching model using a structural form of credit risk modelling. This paper provides explicit representations of risky debt under regime switching with a constant interest rate and [...] Read more.

We propose a pricing formula for a defaultable zero-coupon bond with imperfect information under a regime switching model using a structural form of credit risk modelling. This paper provides explicit representations of risky debt under regime switching with a constant interest rate and risky debt under regime switching with a regime switching interest rate. While the value of the firm’s equity is observed continuously, we assume that the total value of the firm is only observed at discrete times, such as the dates of the release of the firm’s annual reports, or quarterly reports. This uncertainty about the true value of the firm results in credit spreads that do not approach zero as the debt approaches maturity, which is a problem with many structural models. The firm’s value is typically decomposed into its equity and debt; however, we consider the asset–to–equity ratio, an accounting ratio used to examine a firm’s financial well-being. The parameters in our model are regime switching, where the regime can be thought of as the state of the economy. A Markov chain with a constant transition rate matrix produces the regime switching. Full article

(This article belongs to the Special Issue Stochastic Analysis and Applications in Financial Mathematics)

25 pages, 2056 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Zhang Neuro-PID Control for Generalized Bi-Variable Function Projective Synchronization of Nonautonomous Nonlinear Systems with Various Perturbations

by Meichun Huang and Yunong Zhang

Mathematics 2024, 12(17), 2715; https://doi.org/10.3390/math12172715 - 30 Aug 2024

Cited by 1 | Viewed by 757

Abstract

Nonautonomous nonlinear (NN) systems have broad application prospects and significant research value in nonlinear science. In this paper, a new synchronization type—namely, generalized bi-variable function projective synchronization (GBVFPS)—is proposed. The scaling function matrix of GBVFPS is not one-variable but bi-variable. This indicates that [...] Read more.

Nonautonomous nonlinear (NN) systems have broad application prospects and significant research value in nonlinear science. In this paper, a new synchronization type—namely, generalized bi-variable function projective synchronization (GBVFPS)—is proposed. The scaling function matrix of GBVFPS is not one-variable but bi-variable. This indicates that the GBVFPS can be transformed into various synchronization types such as projective synchronization (PS), modified PS, function PS, modified function PS, and generalized function PS. In order to achieve the GBVFPS in two different NN systems with various perturbations, by designing a novel Zhang neuro-PID controller, an effective and anti-perturbation GBVFPS control method is proposed. Rigorous theoretical analyses are presented to prove the convergence performance and anti-perturbation ability of the GBVFPS control method, especially its ability to suppress six different perturbations. Besides, the effectiveness, superiority, and anti-perturbation ability of the proposed GBVFPS control method are further substantiated through two representative numerical simulations, including the synchronization of two NN chaotic systems and the synchronization of two four-dimensional vehicular inverted pendulum systems. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

► Show Figures

Figure 1

63 pages, 6195 KiB

Open AccessEditor’s ChoiceArticle

Matching and Rewriting Rules in Object-Oriented Databases

by Giacomo Bergami, Oliver Robert Fox and Graham Morgan

Mathematics 2024, 12(17), 2677; https://doi.org/10.3390/math12172677 - 28 Aug 2024

Cited by 1 | Viewed by 1140

Abstract

Graph query languages such as Cypher are widely adopted to match and retrieve data in a graph representation, due to their ability to retrieve and transform information. Even though the most natural way to match and transform information is through rewriting rules, those [...] Read more.

Graph query languages such as Cypher are widely adopted to match and retrieve data in a graph representation, due to their ability to retrieve and transform information. Even though the most natural way to match and transform information is through rewriting rules, those are scarcely or partially adopted in graph query languages. Their inability to do so has a major impact on the subsequent way the information is structured, as it might then appear more natural to provide major constraints over the data representation to fix the way the information should be represented. On the other hand, recent works are starting to move towards the opposite direction, as the provision of a truly general semistructured model (GSM) allows to both represent all the available data formats (Network-Based, Relational, and Semistructured) as well as support a holistic query language expressing all major queries in such languages. In this paper, we show that the usage of GSM enables the definition of a general rewriting mechanism which can be expressed in current graph query languages only at the cost of adhering the query to the specificity of the underlying data representation. We formalise the proposed query language in terms declarative graph rewriting mechanisms described as a set of production rules

L \to R

while both providing restriction to the characterisation of L, and extending it to support structural graph nesting operations, useful to aggregate similar information around an entry-point of interest. We further achieve our declarative requirements by determining the order in which the data should be rewritten and multiple rules should be applied while ensuring the application of such updates on the GSM database is persisted in subsequent rewriting calls. We discuss how GSM, by fully supporting index-based data representation, allows for a better physical model implementation leveraging the benefits of columnar database storage. Preliminary benchmarks show the scalability of this proposed implementation in comparison with state-of-the-art implementations. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

► Show Figures

Figure 1

25 pages, 9744 KiB

Open AccessEditor’s ChoiceArticle

An Improved Spider-Wasp Optimizer for Obstacle Avoidance Path Planning in Mobile Robots

by Yujie Gao, Zhichun Li, Haorui Wang, Yupeng Hu, Haoze Jiang, Xintong Jiang and Dong Chen

Mathematics 2024, 12(17), 2604; https://doi.org/10.3390/math12172604 - 23 Aug 2024

Cited by 4 | Viewed by 1619

Abstract

The widespread application of mobile robots holds significant importance for advancing social intelligence. However, as the complexity of the environment increases, existing Obstacle Avoidance Path Planning (OAPP) methods tend to fall into local optimal paths, compromising reliability and practicality. Therefore, based on the [...] Read more.

The widespread application of mobile robots holds significant importance for advancing social intelligence. However, as the complexity of the environment increases, existing Obstacle Avoidance Path Planning (OAPP) methods tend to fall into local optimal paths, compromising reliability and practicality. Therefore, based on the Spider-Wasp Optimizer (SWO), this paper proposes an improved OAPP method called the LMBSWO to address these challenges. Firstly, the learning strategy is introduced to enhance the diversity of the algorithm population, thereby improving its global optimization performance. Secondly, the dual-median-point guidance strategy is incorporated to enhance the algorithm’s exploitation capability and increase its path searchability. Lastly, a better guidance strategy is introduced to enhance the algorithm’s ability to escape local optimal paths. Subsequently, the LMBSWO is employed for OAPP in five different map environments. The experimental results show that the LMBSWO achieves an advantage in collision-free path length, with 100% probability, across five maps of different complexity, while obtaining 80% fault tolerance across different maps, compared to nine existing novel OAPP methods with efficient performance. The LMBSWO ranks first in the trade-off between planning time and path length. With these results, the LMBSWO can be considered as a robust OAPP method with efficient solving performance, along with high robustness. Full article

(This article belongs to the Section E: Applied Mathematics)

► Show Figures

Figure 1

29 pages, 429 KiB

Open AccessFeature PaperEditor’s ChoiceReview

Review about the Permutation Approach in Hypothesis Testing

by Stefano Bonnini, Getnet Melak Assegie and Kamila Trzcinska

Mathematics 2024, 12(17), 2617; https://doi.org/10.3390/math12172617 - 23 Aug 2024

Cited by 3 | Viewed by 1738

Abstract

Today, permutation tests represent a powerful and increasingly widespread tool of statistical inference for hypothesis-testing problems. To the best of our knowledge, a review of the application of permutation tests for complex data in practical data analysis for hypothesis testing is missing. In [...] Read more.

Today, permutation tests represent a powerful and increasingly widespread tool of statistical inference for hypothesis-testing problems. To the best of our knowledge, a review of the application of permutation tests for complex data in practical data analysis for hypothesis testing is missing. In particular, it is essential to review the application of permutation tests in two-sample or multi-sample problems and in regression analysis. The aim of this paper is to consider the main scientific contributions on the subject of permutation methods for hypothesis testing in the mentioned fields. Notes on their use to address the problem of missing data and, in particular, right-censored data, will also be included. This review also tries to highlight the limits and advantages of the works cited with a critical eye and also to provide practical indications to researchers and practitioners who need to identify flexible and distribution-free solutions for the most disparate hypothesis-testing problems. Full article

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

8 pages, 240 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Revisiting a Classic Identity That Implies the Rogers–Ramanujan Identities III

by Hei-Chi Chan

Mathematics 2024, 12(17), 2611; https://doi.org/10.3390/math12172611 - 23 Aug 2024

Viewed by 863

Abstract

This is the third installment in a series of papers on a one-parameter extension of the Rogers–Ramanujan identities (this extension was discovered independently by Rogers and Ramanujan). In this paper, we report a new proof of this identity. Our key ingredient is the [...] Read more.

This is the third installment in a series of papers on a one-parameter extension of the Rogers–Ramanujan identities (this extension was discovered independently by Rogers and Ramanujan). In this paper, we report a new proof of this identity. Our key ingredient is the Bridge Lemma, an identity that connects the both sides of the one-parameter refinement, which differ significantly in terms of their complexity. Full article

(This article belongs to the Special Issue Recent Advances on Ramanujan Theories in Mathematics and Physics)

11 pages, 250 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Explicit Parameterizations of Ortho-Symplectic Matrices in

R

⁴

by Clementina D. Mladenova and Ivaïlo M. Mladenov

Mathematics 2024, 12(16), 2439; https://doi.org/10.3390/math12162439 - 6 Aug 2024

Viewed by 786

Abstract

Starting from the very first principles we derive explicit parameterizations of the ortho-symplectic matrices in the real four-dimensional Euclidean space. These matrices depend on a set of four real parameters which splits naturally as a union of the real line and the three-dimensional [...] Read more.

Starting from the very first principles we derive explicit parameterizations of the ortho-symplectic matrices in the real four-dimensional Euclidean space. These matrices depend on a set of four real parameters which splits naturally as a union of the real line and the three-dimensional space. It turns out that each of these sets is associated with a separate Lie algebra which after exponentiations generates Lie groups that commute between themselves. Besides, by making use of the Cayley and Fedorov maps, we have arrived at alternative realizations of the ortho-symplectic matrices in four dimensions. Finally, relying on the fundamental structure results in Lie group theory we have derived one more explicit parameterization of these matrices which suggests that the obtained earlier results can be viewed as a universal method for building the representations of the unitary groups in arbitrary dimension. Full article

(This article belongs to the Section B: Geometry and Topology)

28 pages, 463 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Hyperpolyadic Structures

by Steven Duplij

Mathematics 2024, 12(15), 2378; https://doi.org/10.3390/math12152378 - 30 Jul 2024

Viewed by 1149

Abstract

We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras

R

,

C

,

H

,

O

without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of [...] Read more.

We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras

R

,

C

,

H

,

O

without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element, we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley–Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, which is proportional to the intermediate arities, and which are not isomorphic to those obtained by the previous constructions. Second, a new polyadic product of vectors in any vector space is defined, which is consistent with the polyadization procedure using vectorization. Endowed with this introduced product, the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process (we call it the “imaginary tower”), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call “half-quaternions” and “half-octonions”. The latter are not the subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced “half-quaternion” norm, we obtain the ternary analog of the sum of two squares identity. We show that the ternary division algebra of imaginary “half-octonions” is unitless and totally associative. Full article

(This article belongs to the Section A: Algebra and Logic)

20 pages, 352 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

The Role of the Table of Games in the Discrete Thermostatted Kinetic Theory

by Carlo Bianca

Mathematics 2024, 12(15), 2356; https://doi.org/10.3390/math12152356 - 28 Jul 2024

Viewed by 1015

Abstract

This paper is concerned with the mathematical modeling of complex living systems whose element microscopic state contains variables which can attain discrete values. Specifically, the main mathematical frameworks of the discrete thermostatted kinetic theory for active particles are reviewed and generalized. In the [...] Read more.

This paper is concerned with the mathematical modeling of complex living systems whose element microscopic state contains variables which can attain discrete values. Specifically, the main mathematical frameworks of the discrete thermostatted kinetic theory for active particles are reviewed and generalized. In the generalized thermostatted frameworks, which are based on nonlinear ordinary or partial differential equations, the elements of the system are viewed as active particles that are able to perform certain strategies modeled by introducing a functional-state variable called activity. Interactions, which are responsible of the evolution of the system, are modeled using the fundamentals of stochastic game theory and may be influenced by the action of an external force field coupled to a Gaussian-type thermostat. In particular, the interaction domain is modeled by introducing a weighted function and different non-homogeneous discrete frameworks are proposed and coupled with a specific thermostat. Two recent models derived within this approach are reviewed and refer to vehicular and pedestrian dynamics. Future research perspectives are discussed in the whole paper from theoretical and modeling viewpoints. Full article

(This article belongs to the Special Issue Mathematical and Numerical Analysis of Nonlinear Evolution Equations: Advances and Perspectives, 2nd Edition)

8 pages, 255 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Radially Symmetric Positive Solutions of the Dirichlet Problem for the p-Laplace Equation

by Bo Yang

Mathematics 2024, 12(15), 2351; https://doi.org/10.3390/math12152351 - 27 Jul 2024

Viewed by 891

Abstract

We consider the p-Laplace boundary value problem with the Dirichlet boundary condition. A new lower estimate for positive solutions of the problem is obtained. As an application of this new lower estimate, some sufficient conditions for the existence and nonexistence of positive [...] Read more.

We consider the p-Laplace boundary value problem with the Dirichlet boundary condition. A new lower estimate for positive solutions of the problem is obtained. As an application of this new lower estimate, some sufficient conditions for the existence and nonexistence of positive solutions for the p-Laplace problem are obtained. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations and Their Applications)

23 pages, 743 KiB

Open AccessEditor’s ChoiceArticle

Ship Selection and Inspection Scheduling in Inland Waterway Transport

by Xizi Qiao, Ying Yang, King-Wah Pang, Yong Jin and Shuaian Wang

Mathematics 2024, 12(15), 2327; https://doi.org/10.3390/math12152327 - 25 Jul 2024

Viewed by 818

Abstract

Inland waterway transport is considered a critical component of sustainable maritime transportation and is subject to strict legal regulations on fuel quality. However, crew members often prefer cheaper, inferior fuels for economic reasons, making government inspections crucial. To address this issue, we formulate [...] Read more.

Inland waterway transport is considered a critical component of sustainable maritime transportation and is subject to strict legal regulations on fuel quality. However, crew members often prefer cheaper, inferior fuels for economic reasons, making government inspections crucial. To address this issue, we formulate the ship selection and inspection scheduling problem into an integer programming model under a multi-inspector and multi-location scenario, alongside a more compact symmetry-eliminated model. The two models are developed based on ship itinerary information and inspection resources, aiming to maximize the total weight of the inspected ships. Driven by the unique property of the problem, a customized heuristic algorithm is also designed to solve the problem. Numerical experiments are conducted using the ships sailing on the Yangtze River as a case study. The results show that, from the perspective of the computation time, the compact model is 102.07 times faster than the original model. Compared with the optimal objectives value, the gap of the solution provided by our heuristic algorithm is 0.37% on average. Meanwhile, our algorithm is 877.19 times faster than the original model, demonstrating the outstanding performance of the proposed algorithm in solving efficiency. Full article

(This article belongs to the Special Issue Data-Driven Decision Making: Models, Methods and Applications, 2nd Edition)

► Show Figures

Figure 1

18 pages, 1826 KiB

Open AccessEditor’s ChoiceArticle

Learning a Context-Aware Environmental Residual Correlation Filter via Deep Convolution Features for Visual Object Tracking

by Sachin Sakthi Kuppusami Sakthivel, Sathishkumar Moorthy, Sathiyamoorthi Arthanari, Jae Hoon Jeong and Young Hoon Joo

Mathematics 2024, 12(14), 2279; https://doi.org/10.3390/math12142279 - 21 Jul 2024

Cited by 11 | Viewed by 1209

Abstract

Visual tracking has become widespread in swarm robots for intelligent video surveillance, navigation, and autonomous vehicles due to the development of machine learning algorithms. Discriminative correlation filter (DCF)-based trackers have gained increasing attention owing to their efficiency. This study proposes “context-aware environmental residual [...] Read more.

Visual tracking has become widespread in swarm robots for intelligent video surveillance, navigation, and autonomous vehicles due to the development of machine learning algorithms. Discriminative correlation filter (DCF)-based trackers have gained increasing attention owing to their efficiency. This study proposes “context-aware environmental residual correlation filter tracking via deep convolution features (CAERDCF)” to enhance the performance of the tracker under ambiguous environmental changes. The objective is to address the challenges posed by intensive environment variations that confound DCF-based trackers, resulting in undesirable tracking drift. We present a selective spatial regularizer in the DCF to suppress boundary effects and use the target’s context information to improve tracking performance. Specifically, a regularization term comprehends the environmental residual among video sequences, enhancing the filter’s discrimination and robustness in unpredictable tracking conditions. Additionally, we propose an efficient method for acquiring environmental data using the current observation without additional computation. A multi-feature integration method is also introduced to enhance the target’s presence by combining multiple metrics. We demonstrate the efficiency and feasibility of our proposed CAERDCF approach by comparing it with existing methods using the OTB2015, TempleColor128, UAV123, LASOT, and GOT10K benchmark datasets. Specifically, our method increased the precision score by 12.9% in OTB2015 and 16.1% in TempleColor128 compared to BACF. Full article

(This article belongs to the Special Issue Advanced Computational Intelligence)

► Show Figures

Figure 1

19 pages, 1495 KiB

Open AccessEditor’s ChoiceArticle

Innovative Methods of Constructing Strict and Strong Fuzzy Negations, Fuzzy Implications and New Classes of Copulas

by Panagiotis Georgiou Mangenakis and Basil Papadopoulos

Mathematics 2024, 12(14), 2254; https://doi.org/10.3390/math12142254 - 19 Jul 2024

Viewed by 1002

Abstract

This paper presents new classes of strong fuzzy negations, fuzzy implications and Copulas. It begins by presenting two theorems with function classes involving the construction of strong fuzzy negations. These classes are based on a well-known equilibrium point theorem. After that, a construction [...] Read more.

This paper presents new classes of strong fuzzy negations, fuzzy implications and Copulas. It begins by presenting two theorems with function classes involving the construction of strong fuzzy negations. These classes are based on a well-known equilibrium point theorem. After that, a construction of fuzzy implication is presented, which is not based on any negation. Finally, moving on to the area concerning copulas, we present proof about the third property of copulas. To conclude, we will present two original constructions of copulas. All the above constructions are motivated by a specific formula. For some specific conditions of the variables x, y and other conditions for the function f(x), the formula presented produces strict and strong fuzzy negations, fuzzy implications and copulas. Full article

(This article belongs to the Special Issue Advances and Applications of Soft Computing)

► Show Figures

Figure 1

13 pages, 727 KiB

Open AccessEditor’s ChoiceArticle

Differential Transform Method and Neural Network for Solving Variational Calculus Problems

by Rafał Brociek and Mariusz Pleszczyński

Mathematics 2024, 12(14), 2182; https://doi.org/10.3390/math12142182 - 11 Jul 2024

Cited by 2 | Viewed by 1386

Abstract

The history of variational calculus dates back to the late 17th century when Johann Bernoulli presented his famous problem concerning the brachistochrone curve. Since then, variational calculus has developed intensively as many problems in physics and engineering are described by equations from this [...] Read more.

The history of variational calculus dates back to the late 17th century when Johann Bernoulli presented his famous problem concerning the brachistochrone curve. Since then, variational calculus has developed intensively as many problems in physics and engineering are described by equations from this branch of mathematical analysis. This paper presents two non-classical, distinct methods for solving such problems. The first method is based on the differential transform method (DTM), which seeks an analytical solution in the form of a certain functional series. The second method, on the other hand, is based on the physics-informed neural network (PINN), where artificial intelligence in the form of a neural network is used to solve the differential equation. In addition to describing both methods, this paper also presents numerical examples along with a comparison of the obtained results.Comparingthe two methods, DTM produced marginally more accurate results than PINNs. While PINNs exhibited slightly higher errors, their performance remained commendable. The key strengths of neural networks are their adaptability and ease of implementation. Both approaches discussed in the article are effective for addressing the examined problems. Full article

(This article belongs to the Special Issue Applications of Symbolic and Soft Computations in Applied Sciences)

► Show Figures

Figure 1

21 pages, 3231 KiB

Open AccessEditor’s ChoiceArticle

Designing Decentralized Multi-Variable Robust Controllers: A Multi-Objective Approach Considering Nearly Optimal Solutions

by Alberto Pajares, Xavier Blasco, Juan Manuel Herrero, Javier Sanchis and Raúl Simarro

Mathematics 2024, 12(13), 2124; https://doi.org/10.3390/math12132124 - 6 Jul 2024

Cited by 1 | Viewed by 771

Abstract

This article presents a new methodology for designing a robust, decentralized control structure that considers stochastic parametric uncertainty and uses a multi-objective approach. This design tunes the loop pairing and controller to be implemented. The proposed approach obtains the optimal and nearly optimal [...] Read more.

This article presents a new methodology for designing a robust, decentralized control structure that considers stochastic parametric uncertainty and uses a multi-objective approach. This design tunes the loop pairing and controller to be implemented. The proposed approach obtains the optimal and nearly optimal controllers relevant to the nominal scenario. Once obtained, the robustness of these solutions is analyzed. This methodology is compared with a traditional approach for selecting the most robust control pairings. The traditional approach obtains lightly robust controllers, i.e., the most robust controllers with an acceptable performance for the nominal scenario, and it obtains trade-offs between robustness and nominal performance. However, the traditional approach has a high computational cost because it is necessary to consider uncertainty in the optimization stage. The proposed approach mathematically guarantees the acquisition of at least one neighbor controller for each existing lightly robust controller. Therefore, this approach obtains solutions similar to lightly robust solutions with a significantly lower computational cost. Furthermore, the proposed approach provides the designer with more diversity and interesting solutions that are not lightly robust. The different approaches are compared using an example of a multi-variable process with two alternative control structures. The results show the usefulness of the proposed methodology. Full article

(This article belongs to the Special Issue Advanced Applications Based on Nonlinear Optimal and Robust Control)

► Show Figures

Figure 1

21 pages, 377 KiB

Open AccessEditor’s ChoiceArticle

Joint Statistical Inference for the Area under the ROC Curve and Youden Index under a Density Ratio Model

by Siyan Liu, Qinglong Tian, Yukun Liu and Pengfei Li

Mathematics 2024, 12(13), 2118; https://doi.org/10.3390/math12132118 - 5 Jul 2024

Cited by 4 | Viewed by 1585

Abstract

The receiver operating characteristic (ROC) curve is a valuable statistical tool in medical research. It assesses a biomarker’s ability to distinguish between diseased and healthy individuals. The area under the ROC curve (

A U C

) and the Youden index (J [...] Read more.

The receiver operating characteristic (ROC) curve is a valuable statistical tool in medical research. It assesses a biomarker’s ability to distinguish between diseased and healthy individuals. The area under the ROC curve (

A U C

) and the Youden index (J) are common summary indices used to evaluate a biomarker’s diagnostic accuracy. Simultaneously examining

A U C

and J offers a more comprehensive understanding of the ROC curve’s characteristics. In this paper, we utilize a semiparametric density ratio model to link the distributions of a biomarker for healthy and diseased individuals. Under this model, we establish the joint asymptotic normality of the maximum empirical likelihood estimator of

(A U C, J)

and construct an asymptotically valid confidence region for

(A U C, J)

. Furthermore, we propose a new test to determine whether a biomarker simultaneously exceeds prespecified target values of

A U C_{0}

and

J_{0}

with the null hypothesis

H_{0} : A U C \leq A U C_{0}

or

J \leq J_{0}

against the alternative hypothesis

H_{a} : A U C > A U C_{0}

and

J > J_{0}

. Simulation studies and a real data example on Duchenne Muscular Dystrophy are used to demonstrate the effectiveness of our proposed method and highlight its advantages over existing methods. Full article

(This article belongs to the Special Issue Statistical Analysis and Data Science for Complex Data)

► Show Figures

Figure 1

29 pages, 10500 KiB

Open AccessEditor’s ChoiceArticle

Trajectory Optimization for Adaptive Deformed Wheels to Overcome Steps Using an Improved Hybrid Genetic Algorithm and an Adaptive Particle Swarm Optimization

by Yanjie Liu, Yanlong Wei, Chao Wang and Heng Wu

Mathematics 2024, 12(13), 2077; https://doi.org/10.3390/math12132077 - 2 Jul 2024

Cited by 18 | Viewed by 1530

Abstract

Two-wheeled mobile robots with deformed wheels face low stability when climbing steps, and their success rate in overcoming steps is affected by the trajectory. To address these challenges, we propose an improved hybrid genetic and adaptive particle swarm optimization (HGAPSO) algorithm to optimize [...] Read more.

Two-wheeled mobile robots with deformed wheels face low stability when climbing steps, and their success rate in overcoming steps is affected by the trajectory. To address these challenges, we propose an improved hybrid genetic and adaptive particle swarm optimization (HGAPSO) algorithm to optimize the deformed wheels’ trajectory for overcoming steps. HGAPSO optimizes the maximum and minimum values of the inertial weight and learning factors of the adaptive particle swarm algorithm utilizing the region-wide search capabilities of the genetic algorithm, which substantially improves the convergence speed and adaptability. Furthermore, the analysis of the motion of the deformed wheel overcoming the steps and the examination of the potential interference during the operation are used to construct a wheel’s center-of-mass route based on fifth-order Bézier curves. Comparative simulation experiments of the trajectories optimized using different optimization algorithms under the same working conditions are designed to demonstrate the efficacy of the proposed HGAPSO algorithm in optimizing the trajectory of the deformed wheel overcoming the step. Simulation experiments were conducted using the HGAPSO algorithm to optimize the trajectories of deformation wheels for overcoming steps of various sizes. These optimized trajectories were then compared to unoptimized ones. The results showed that the HGAPSO-optimized trajectories significantly improved the success rate and stability of the mobile robot in overcoming steps. Full article

► Show Figures

Figure 1

18 pages, 681 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

On the Problems of

CF

-Connected Graphs for K_l,m,n

by Michal Staš and Mária Timková

Mathematics 2024, 12(13), 2068; https://doi.org/10.3390/math12132068 - 1 Jul 2024

Viewed by 875

Abstract

A connected graph, G, is Crossing Free-connected (

CF

-connected) if there is a path between every pair of vertices with no crossing on its edges for each optimal drawing of G. We conjecture that a complete tripartite graph, [...] Read more.

A connected graph, G, is Crossing Free-connected (

CF

-connected) if there is a path between every pair of vertices with no crossing on its edges for each optimal drawing of G. We conjecture that a complete tripartite graph,

K_{l, m, n}

, is

CF

-connected if and only if it does not contain any of the following as a subgraph:

K_{1, 2, 7}

,

K_{1, 3, 5}

,

K_{1, 4, 4}

,

K_{2, 2, 5}

,

K_{3, 3, 3}

. We examine the idea that

K_{1, 2, 7}

,

K_{1, 3, 5}

,

K_{1, 4, 4}

, and

K_{2, 2, 5}

are the first non-

CF

-connected complete tripartite graphs. The

CF

-connectedness of

K_{l, m, n}

with

l, m, n \geq 3

is dependent on the knowledge of crossing numbers of

K_{3, 3, n}

. In this paper, we prove various results that support this conjecture. Full article

(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory, 3rd Edition)

► Show Figures

Figure 1

39 pages, 12486 KiB

Open AccessEditor’s ChoiceArticle

Parameter Prediction with Novel Enhanced Wagner Hagras Interval Type-3 Takagi–Sugeno–Kang Fuzzy System with Type-1 Non-Singleton Inputs

by Gerardo Armando Hernández Castorena, Gerardo Maximiliano Méndez, Ismael López-Juárez, María Aracelia Alcorta García, Dulce Citlalli Martinez-Peon and Pascual Noradino Montes-Dorantes

Mathematics 2024, 12(13), 1976; https://doi.org/10.3390/math12131976 - 26 Jun 2024

Cited by 5 | Viewed by 1433

Abstract

This paper presents the novel enhanced Wagner–Hagras interval type-3 Takagi–Sugeno–Kang fuzzy logic system with type-1 non-singleton inputs (EWH IT3 TSK NSFLS-1) that uses the backpropagation (BP) algorithm to train the antecedent and consequent parameters. The proposed methodology dynamically changes the parameters of only [...] Read more.

This paper presents the novel enhanced Wagner–Hagras interval type-3 Takagi–Sugeno–Kang fuzzy logic system with type-1 non-singleton inputs (EWH IT3 TSK NSFLS-1) that uses the backpropagation (BP) algorithm to train the antecedent and consequent parameters. The proposed methodology dynamically changes the parameters of only the alpha-0 level, minimizing some criterion functions as the current information becomes available for each alpha-k level. The novel fuzzy system was applied in two industrial processes and several fuzzy models were used to make comparisons. The experiments demonstrated that the proposed fuzzy system has a superior ability to predict the critical variables of the tested processes with lower prediction errors than those produced by the benchmark fuzzy systems. Full article

► Show Figures

Figure 1

21 pages, 507 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

A Note on the Convergence of Multigrid Methods for the Riesz–Space Equation and an Application to Image Deblurring

by Danyal Ahmad, Marco Donatelli, Mariarosa Mazza, Stefano Serra-Capizzano and Ken Trotti

Mathematics 2024, 12(12), 1916; https://doi.org/10.3390/math12121916 - 20 Jun 2024

Cited by 1 | Viewed by 1125

Abstract

In recent decades, a remarkable amount of research has been carried out regarding fast solvers for large linear systems resulting from various discretizations of fractional differential equations (FDEs). In the current work, we focus on multigrid methods for a Riesz–Space FDE whose theoretical [...] Read more.

In recent decades, a remarkable amount of research has been carried out regarding fast solvers for large linear systems resulting from various discretizations of fractional differential equations (FDEs). In the current work, we focus on multigrid methods for a Riesz–Space FDE whose theoretical convergence analysis of such multigrid methods is currently limited in the relevant literature to the two-grid method. Here we provide a detailed theoretical convergence study in the multilevel setting. Moreover, we discuss its use combined with a band approximation and we compare the result with both

τ

and circulant preconditionings. The numerical tests include 2D problems as well as the extension to the case of a Riesz–FDE with variable coefficients. Finally, we investigate the use of a Riesz–Space FDE in a variational model for image deblurring, comparing the performance of specific preconditioning strategies. Full article

(This article belongs to the Special Issue Mathematical Methods for Image Processing and Understanding)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Editor’s Choice Articles

Further Information

Guidelines

MDPI Initiatives

Follow MDPI