Mathematics

47 pages, 11591 KB

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 23 | Viewed by 3367

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)

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13 pages, 1088 KB

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 5 | Viewed by 5498

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)

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20 pages, 581 KB

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 3 | Viewed by 3317

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)

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29 pages, 452 KB

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 1487

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

15 pages, 591 KB

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 1852

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)

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24 pages, 2433 KB

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

Cited by 1 | Viewed by 5799

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)

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41 pages, 3974 KB

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 32 | Viewed by 3866

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

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31 pages, 961 KB

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 3 | Viewed by 1951

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)

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27 pages, 1590 KB

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 3 | Viewed by 2764

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)

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13 pages, 273 KB

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 2805

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 KB

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 3 | Viewed by 2556

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 KB

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 6 | Viewed by 2086

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)

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63 pages, 6195 KB

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 3 | Viewed by 2699

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)

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25 pages, 9744 KB

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 20 | Viewed by 3511

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)

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8 pages, 240 KB

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 1891

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)

29 pages, 429 KB

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 24 | Viewed by 9926

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)

11 pages, 250 KB

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

Cited by 3 | Viewed by 2132

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 KB

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 1968

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 KB

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

Cited by 1 | Viewed by 1610

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 KB

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 1589

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 KB

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

Cited by 2 | Viewed by 2274

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)

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18 pages, 1826 KB

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 26 | Viewed by 3529

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)

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19 pages, 1495 KB

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

Cited by 4 | Viewed by 1751

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)

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13 pages, 727 KB

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 14 | Viewed by 2875

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)

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21 pages, 3231 KB

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 1395

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)

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21 pages, 377 KB

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 12 | Viewed by 4094

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)

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29 pages, 10500 KB

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 34 | Viewed by 4262

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

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18 pages, 681 KB

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

Cited by 2 | Viewed by 2030

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)

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21 pages, 507 KB

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 2264

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)

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25 pages, 419 KB

Open AccessEditor’s ChoiceArticle

The Generalized Fox–Wright Function: The Laplace Transform, the Erdélyi–Kober Fractional Integral and Its Role in Fractional Calculus

by Jordanka Paneva-Konovska and Virginia Kiryakova

Mathematics 2024, 12(12), 1918; https://doi.org/10.3390/math12121918 - 20 Jun 2024

Cited by 10 | Viewed by 2889

Abstract

In this paper, we consider and study in detail the generalized Fox–Wright function

{}_{p}{\tilde{Ψ}}_{q}

introduced in our recent work as an extension of the Fox–Wright function

{}_{p}Ψ_{q}

. This special function can be seen as an important case [...] Read more.

In this paper, we consider and study in detail the generalized Fox–Wright function

{}_{p}{\tilde{Ψ}}_{q}

introduced in our recent work as an extension of the Fox–Wright function

{}_{p}Ψ_{q}

. This special function can be seen as an important case of the so-called I-functions of Rathie and

\bar{H}

-functions of Inayat-Hussain, that in turn extend the Fox H-functions and appear to include some Feynman integrals in statistical physics, in polylogarithms, in Riemann Zeta-type functions and in other important mathematical functions. Depending on the parameters,

{}_{p}{\tilde{Ψ}}_{q}

is an entire function or is analytic in an open disc with a final radius. We derive its basic properties, such as its order and type, and its images under the Laplace transform and under classical fractional-order integrals. Particular cases of

{}_{p}{\tilde{Ψ}}_{q}

are specified, including the Mittag-Leffler and Le Roy-type functions and their multi-index analogues and many other special functions of Fractional Calculus. The corresponding results are illustrated. Finally, we emphasize the role of these new generalized hypergeometric functions as eigenfunctions of operators of new Fractional Calculus with specific I-functions as singular kernels. This paper can be considered as a natural supplement to our previous surveys “Going Next after ‘A Guide to Special Functions in Fractional Calculus’: A Discussion Survey”, and “A Guide to Special Functions of Fractional Calculus”, published recently in this journal. Full article

(This article belongs to the Special Issue Fractional Calculus in Natural and Social Sciences)

19 pages, 322 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Existence, Regularity, and Uniqueness of Solutions to Some Noncoercive Nonlinear Elliptic Equations in Unbounded Domains

by Patrizia Di Gironimo

Mathematics 2024, 12(12), 1860; https://doi.org/10.3390/math12121860 - 14 Jun 2024

Cited by 1 | Viewed by 2122

Abstract

In this paper, we study a noncoercive nonlinear elliptic operator with a drift term in an unbounded domain. The singular first-order term grows like

| E (x) | | \nabla u |

, where

E (x)

is a vector [...] Read more.

In this paper, we study a noncoercive nonlinear elliptic operator with a drift term in an unbounded domain. The singular first-order term grows like

| E (x) | | \nabla u |

, where

E (x)

is a vector field belonging to a suitable Morrey-type space. Our operator arises as a stationary equation of diffusion–advection problems. We prove existence, regularity, and uniqueness theorems for a Dirichlet problem. To obtain our main results, we use the weak maximum principle and the same a priori estimates. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

20 pages, 337 KB

Open AccessEditor’s ChoiceArticle

Modular Quasi-Pseudo Metrics and the Aggregation Problem

by Maria del Mar Bibiloni-Femenias and Oscar Valero

Mathematics 2024, 12(12), 1826; https://doi.org/10.3390/math12121826 - 12 Jun 2024

Cited by 3 | Viewed by 1563

Abstract

The applicability of the distance aggregation problem has attracted the interest of many authors. Motivated by this fact, in this paper, we face the modular quasi-(pseudo-)metric aggregation problem, which consists of analyzing the properties that a function must have to fuse a collection [...] Read more.

The applicability of the distance aggregation problem has attracted the interest of many authors. Motivated by this fact, in this paper, we face the modular quasi-(pseudo-)metric aggregation problem, which consists of analyzing the properties that a function must have to fuse a collection of modular quasi-(pseudo-)metrics into a single one. In this paper, we characterize such functions as monotone, subadditive and vanishing at zero. Moreover, a description of such functions in terms of triangle triplets is given, and, in addition, the relationship between modular quasi-(pseudo-)metric aggregation functions and modular (pseudo-)metric aggregation functions is discussed. Specifically, we show that the class of modular (quasi-)(pseudo-)metric aggregation functions coincides with that of modular (pseudo-)metric aggregation functions. The characterizations are illustrated with appropriate examples. A few methods to construct modular quasi-(pseudo-)metrics are provided using the exposed theory. By exploring the existence of absorbent and neutral elements of modular quasi-(pseudo-)metric aggregation functions, we find that every modular quasi-pseudo-metric aggregation function with 0 as the neutral element is an Aumann function, is majored by the sum and satisfies the 1-Lipschitz condition. Moreover, a characterization of those modular quasi-(pseudo-)metric aggregation functions that preserve modular quasi-(pseudo-)metrics is also provided. Furthermore, the relationship between modular quasi-(pseudo-)metric aggregation functions and quasi-(pseudo-)metric aggregation functions is studied. Particularly, we have proven that they are the same only when the former functions are finite. Finally, the usefulness of modular quasi-(pseudo-)metric aggregation functions in multi-agent systems is analyzed. Full article

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

17 pages, 356 KB

Open AccessEditor’s ChoiceArticle

Bias-Correction Methods for the Unit Exponential Distribution and Applications

by Hua Xin, Yuhlong Lio, Ya-Yen Fan and Tzong-Ru Tsai

Mathematics 2024, 12(12), 1828; https://doi.org/10.3390/math12121828 - 12 Jun 2024

Cited by 1 | Viewed by 2213

Abstract

The bias of the maximum likelihood estimator can cause a considerable estimation error if the sample size is small. To reduce the bias of the maximum likelihood estimator under the small sample situation, the maximum likelihood and parametric bootstrap bias-correction methods are proposed [...] Read more.

The bias of the maximum likelihood estimator can cause a considerable estimation error if the sample size is small. To reduce the bias of the maximum likelihood estimator under the small sample situation, the maximum likelihood and parametric bootstrap bias-correction methods are proposed in this study to obtain more reliable maximum likelihood estimators of the unit exponential distribution parameters. The procedure to implement the bias-corrected maximum likelihood estimation method is derived analytically, and the steps to obtain the bias-corrected bootstrap estimators are presented. The simulation results show that the proposed maximum likelihood bootstrap bias-correction method can significantly reduce the bias and mean squared error of the maximum likelihood estimators for most of the parameter combinations in the simulation study. A soil moisture data set and a numerical example are used for illustration. Full article

(This article belongs to the Special Issue Fuzzy Applications in Industrial Engineering, 3rd Edition)

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14 pages, 335 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Flexible and Compact MLWE-Based KEM

by Wenqi Liang, Zhaoman Liu, Xuyang Zhao, Yafang Yang and Zhichuang Liang

Mathematics 2024, 12(11), 1769; https://doi.org/10.3390/math12111769 - 6 Jun 2024

Cited by 3 | Viewed by 3064

Abstract

In order to resist the security risks caused by quantum computing, post-quantum cryptography (PQC) has been a research focus. Constructing a key encapsulation mechanism (KEM) based on lattices is one of the promising PQC routines. The algebraically structured learning with errors (LWE) problem [...] Read more.

In order to resist the security risks caused by quantum computing, post-quantum cryptography (PQC) has been a research focus. Constructing a key encapsulation mechanism (KEM) based on lattices is one of the promising PQC routines. The algebraically structured learning with errors (LWE) problem over power-of-two cyclotomics has been one of the most widely used hardness assumptions for lattice-based cryptographic schemes. However, power-of-two cyclotomic rings may be exploited in the inflexibility of selecting parameters. Recently, trinomial cyclotomic rings of the form

Z_{q} [x] / (x^{n} - x^{n / 2} + 1)

, where

n = 2^{k} 3^{l}

,

k \geq 1, l \geq 0

, have received widespread attention due to their flexible parameter selection. In this paper, we propose Tyber, a variant scheme of the NIST-standardized KEM candidate Kyber over trinomial cyclotomic rings. We provide three parameter sets, aiming at the quantum security of 128, 192, and 256 bits (actually achieving 129, 197, and 276 bits) with matching and negligible error probabilities. When compared to Kyber, our Tyber exhibits stronger quantum security, by 22, 31, and 44 bits, than Kyber for three security levels. Full article

(This article belongs to the Special Issue New Advances in Cryptographic Theory and Application)

17 pages, 3355 KB

Open AccessEditor’s ChoiceArticle

Towards Analysis of Multivariate Time Series Using Topological Data Analysis

by Jingyi Zheng, Ziqin Feng and Arne D. Ekstrom

Mathematics 2024, 12(11), 1727; https://doi.org/10.3390/math12111727 - 1 Jun 2024

Cited by 6 | Viewed by 5311

Abstract

Topological data analysis (TDA) has proven to be a potent approach for extracting intricate topological structures from complex and high-dimensional data. In this paper, we propose a TDA-based processing pipeline for analyzing multi-channel scalp EEG data. The pipeline starts with extracting both frequency [...] Read more.

Topological data analysis (TDA) has proven to be a potent approach for extracting intricate topological structures from complex and high-dimensional data. In this paper, we propose a TDA-based processing pipeline for analyzing multi-channel scalp EEG data. The pipeline starts with extracting both frequency and temporal information from the signals via the Hilbert–Huang Transform. The sequences of instantaneous frequency and instantaneous amplitude across all electrode channels are treated as approximations of curves in the high-dimensional space. TDA features, which represent the local topological structure of the curves, are further extracted and used in the classification models. Three sets of scalp EEG data, including one collected in a lab and two Brain–computer Interface (BCI) competition data, were used to validate the proposed methods, and compare with other state-of-art TDA methods. The proposed TDA-based approach shows superior performance and outperform the winner of the BCI competition. Besides BCI, the proposed method can also be applied to spatial and temporal data in other domains such as computer vision, remote sensing, and medical imaging. Full article

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12 pages, 288 KB

Open AccessEditor’s ChoiceArticle

Solution to Several Split Quaternion Matrix Equations

by Xin Liu, Tong Shi and Yang Zhang

Mathematics 2024, 12(11), 1707; https://doi.org/10.3390/math12111707 - 30 May 2024

Cited by 2 | Viewed by 2037

Abstract

Split quaternions have various applications in mathematics, computer graphics, robotics, physics, and so on. In this paper, two useful, real representations of a split quaternion matrix are proposed. Based on this, we derive their fundamental properties. Then, via the real representation method, we [...] Read more.

Split quaternions have various applications in mathematics, computer graphics, robotics, physics, and so on. In this paper, two useful, real representations of a split quaternion matrix are proposed. Based on this, we derive their fundamental properties. Then, via the real representation method, we obtain the necessary and sufficient conditions for the existence of solutions to two split quaternion matrix equations. In addition, two experimental examples are provided to show their feasibility. Full article

27 pages, 1929 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Bayesian Learning in an Affine GARCH Model with Application to Portfolio Optimization

by Marcos Escobar-Anel, Max Speck and Rudi Zagst

Mathematics 2024, 12(11), 1611; https://doi.org/10.3390/math12111611 - 21 May 2024

Cited by 3 | Viewed by 2703

Abstract

This paper develops a methodology to accommodate uncertainty in a GARCH model with the goal of improving portfolio decisions via Bayesian learning. Given the abundant evidence of uncertainty in estimating expected returns, we focus our analyses on the single parameter driving expected returns. [...] Read more.

This paper develops a methodology to accommodate uncertainty in a GARCH model with the goal of improving portfolio decisions via Bayesian learning. Given the abundant evidence of uncertainty in estimating expected returns, we focus our analyses on the single parameter driving expected returns. After deriving an Uncertainty-Implied GARCH (UI-GARCH) model, we investigate how learning about uncertainty affects investments in a dynamic portfolio optimization problem. We consider an investor with constant relative risk aversion (CRRA) utility who wants to maximize her expected utility from terminal wealth under an Affine GARCH(1,1) model. The corresponding stock evolution, and therefore, the wealth process, is treated as a Bayesian information model that learns about the expected return with each period. We explore the one- and two-period cases, demonstrating a significant impact of uncertainty on optimal allocation and wealth-equivalent losses, particularly in the case of a small sample size or large standard errors in the parameter estimation. These analyses are conducted under well-documented parametric choices. The methodology can be adapted to other GARCH models and applications beyond portfolio optimization. Full article

(This article belongs to the Special Issue Statistics and Mathematics in Economics and Finance: Theory, Methods and Applications)

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25 pages, 17243 KB

Open AccessEditor’s ChoiceArticle

RBF-Assisted Hybrid Neural Network for Solving Partial Differential Equations

by Ying Li, Wei Gao and Shihui Ying

Mathematics 2024, 12(11), 1617; https://doi.org/10.3390/math12111617 - 21 May 2024

Cited by 7 | Viewed by 4198

Abstract

In scientific computing, neural networks have been widely used to solve partial differential equations (PDEs). In this paper, we propose a novel RBF-assisted hybrid neural network for approximating solutions to PDEs. Inspired by the tendency of physics-informed neural networks (PINNs) to become local [...] Read more.

In scientific computing, neural networks have been widely used to solve partial differential equations (PDEs). In this paper, we propose a novel RBF-assisted hybrid neural network for approximating solutions to PDEs. Inspired by the tendency of physics-informed neural networks (PINNs) to become local approximations after training, the proposed method utilizes a radial basis function (RBF) to provide the normalization and localization properties to the input data. The objective of this strategy is to assist the network in solving PDEs more effectively. During the RBF-assisted processing part, the method selects the center points and collocation points separately to effectively manage data size and computational complexity. Subsequently, the RBF processed data are put into the network for predicting the solutions to PDEs. Finally, a series of experiments are conducted to evaluate the novel method. The numerical results confirm that the proposed method can accelerate the convergence speed of the loss function and improve predictive accuracy. Full article

(This article belongs to the Special Issue Mathematical Modeling and Numerical Simulation in Engineering, 2nd Edition)

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14 pages, 297 KB

Open AccessEditor’s ChoiceArticle

Markov Chains and Kinetic Theory: A Possible Application to Socio-Economic Problems

by Bruno Carbonaro and Marco Menale

Mathematics 2024, 12(10), 1571; https://doi.org/10.3390/math12101571 - 17 May 2024

Cited by 5 | Viewed by 2659

Abstract

A very important class of models widely used nowadays to describe and predict, at least in stochastic terms, the behavior of many-particle systems (where the word “particle” is not meant in the purely mechanical sense: particles can be cells of a living tissue, [...] Read more.

A very important class of models widely used nowadays to describe and predict, at least in stochastic terms, the behavior of many-particle systems (where the word “particle” is not meant in the purely mechanical sense: particles can be cells of a living tissue, or cars in a traffic flow, or even members of an animal or human population) is the Kinetic Theory for Active Particles, i.e., a scheme of possible generalizations and re-interpretations of the Boltzmann equation. Now, though in the literature on the subject this point is systematically disregarded, this scheme is based on Markov Chains, which are special stochastic processes with important properties they share with many natural processes. This circumstance is here carefully discussed not only to suggest the different ways in which Markov Chains can intervene in equations describing the stochastic behavior of any many-particle system, but also, as a preliminary methodological step, to point out the way in which the notion of a Markov Chain can be suitably generalized to this aim. As a final result of the discussion, we find how to develop new very plausible and likely ways to take into account possible effects of the external world on a non-isolated many-particle system, with particular attention paid to socio-economic problems. Full article

(This article belongs to the Special Issue Kinetic Models of Collective Phenomena and Data Science)

22 pages, 860 KB

Open AccessEditor’s ChoiceArticle

A Dynamic Programming Approach to the Collision Avoidance of Autonomous Ships

by Raphael Zaccone

Mathematics 2024, 12(10), 1546; https://doi.org/10.3390/math12101546 - 15 May 2024

Cited by 28 | Viewed by 4179

Abstract

The advancement of autonomous capabilities in maritime navigation has gained significant attention, with a trajectory moving from decision support systems to full autonomy. This push towards autonomy has led to extensive research focusing on collision avoidance, a critical aspect of safe navigation. Among [...] Read more.

The advancement of autonomous capabilities in maritime navigation has gained significant attention, with a trajectory moving from decision support systems to full autonomy. This push towards autonomy has led to extensive research focusing on collision avoidance, a critical aspect of safe navigation. Among the various possible approaches, dynamic programming is a promising tool for optimizing collision avoidance maneuvers. This paper presents a DP formulation for the collision avoidance of autonomous vessels. We set up the problem framework, formulate it as a multi-stage decision process, define cost functions and constraints focusing on the actual requirements a marine maneuver must comply with, and propose a solution algorithm leveraging parallel computing. Additionally, we present a greedy approximation to reduce algorithm complexity. We put the proposed algorithms to the test in realistic navigation scenarios and also develop an extensive test on a large set of randomly generated scenarios, comparing them with the RRT* algorithm using performance metrics proposed in the literature. The results show the potential benefits of an autonomous navigation or decision support framework. Full article

(This article belongs to the Special Issue Dynamic Programming)

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16 pages, 309 KB

Open AccessEditor’s ChoiceArticle

Almost Sure Central Limit Theorem for Error Variance Estimator in Pth-Order Nonlinear Autoregressive Processes

by Kaiyu Liang and Yong Zhang

Mathematics 2024, 12(10), 1482; https://doi.org/10.3390/math12101482 - 10 May 2024

Cited by 2 | Viewed by 1258

Abstract

In this paper, under some suitable assumptions, using the Taylor expansion, Borel–Cantelli lemma and the almost sure central limit theorem for independent random variables, the almost sure central limit theorem for error variance estimator in the pth-order nonlinear autoregressive processes with independent and [...] Read more.

In this paper, under some suitable assumptions, using the Taylor expansion, Borel–Cantelli lemma and the almost sure central limit theorem for independent random variables, the almost sure central limit theorem for error variance estimator in the pth-order nonlinear autoregressive processes with independent and identical distributed errors was established. Four examples, first-order autoregressive processes, self-exciting threshold autoregressive processes, threshold-exponential AR progresses and multilayer perceptrons progress, are given to verify the results. Full article

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

18 pages, 393 KB

Open AccessFeature PaperEditor’s ChoiceArticle

An Ad Hoc Procedure for Testing Serial Correlation in Spatial Fixed-Effects Panels

by Giovanni Millo

Mathematics 2024, 12(10), 1475; https://doi.org/10.3390/math12101475 - 9 May 2024

Viewed by 2372

Abstract

We consider testing for error persistence in spatial panels with (potentially correlated) individual heterogeneity. We propose two variants of an ad hoc testing procedure based on first transforming out the individual effects, either by time-demeaning or by taking forward orthogonal deviations, then estimating [...] Read more.

We consider testing for error persistence in spatial panels with (potentially correlated) individual heterogeneity. We propose two variants of an ad hoc testing procedure based on first transforming out the individual effects, either by time-demeaning or by taking forward orthogonal deviations, then estimating an encompassing spatio-temporal model. The procedure can also be employed under the random effects assumption, in which case, although suboptimal, it can be computationally cheaper and safer than existing tests. We report Monte Carlo simulations demonstrating satisfactory empirical properties. Full article

(This article belongs to the Special Issue Mathematical Economics and Spatial Econometrics)

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15 pages, 269 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Secure Key Exchange in Tropical Cryptography: Leveraging Efficiency with Advanced Block Matrix Protocols

by Mariana Durcheva and Kiril Danilchenko

Mathematics 2024, 12(10), 1429; https://doi.org/10.3390/math12101429 - 7 May 2024

Cited by 6 | Viewed by 3080

Abstract

In the quest for robust and efficient digital communication, this paper introduces cutting-edge key exchange protocols leveraging the computational prowess of tropical semirings and the structural resilience of block matrices. Moving away from the conventional use of finite fields, these protocols deliver markedly [...] Read more.

In the quest for robust and efficient digital communication, this paper introduces cutting-edge key exchange protocols leveraging the computational prowess of tropical semirings and the structural resilience of block matrices. Moving away from the conventional use of finite fields, these protocols deliver markedly faster processing speeds and heightened security. We present two implementations of our concept, each utilizing a different platform for the set of commuting matrices: one employing tropical polynomials of matrices and the other employing Linde–de la Puente matrices. The inherent simplicity of tropical semirings leads to a decrease in operational complexity, while using block matrices enhances our protocols’ security profile. The security of these protocols relies on the Matrix Decomposition Problem. In addition, we provide a comparative analysis of our protocols against existing matrix block-based protocols in finite fields. This research marks a significant shift in cryptographic protocol design, is specifically tailored for demanding engineering applications, and sets a new standard in secure and efficient digital communication. Full article

(This article belongs to the Special Issue Chaos-Based Secure Communication and Cryptography, 2nd Edition)

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15 pages, 1155 KB

Open AccessEditor’s ChoiceArticle

Enhancing Bitcoin Price Volatility Estimator Predictions: A Four-Step Methodological Approach Utilizing Elastic Net Regression

by Georgia Zournatzidou, Ioannis Mallidis, Dimitrios Farazakis and Christos Floros

Mathematics 2024, 12(9), 1392; https://doi.org/10.3390/math12091392 - 2 May 2024

Cited by 22 | Viewed by 10228

Abstract

This paper provides a computationally efficient and novel four-step methodological approach for predicting volatility estimators derived from bitcoin prices. In the first step, open, high, low, and close bitcoin prices are transformed into volatility estimators using Brownian motion assumptions and logarithmic transformations. The [...] Read more.

This paper provides a computationally efficient and novel four-step methodological approach for predicting volatility estimators derived from bitcoin prices. In the first step, open, high, low, and close bitcoin prices are transformed into volatility estimators using Brownian motion assumptions and logarithmic transformations. The second step determines the optimal number of time-series lags required for converting the series into an autoregressive model. This selection process utilizes random forest regression, evaluating the importance of each lag using the Mean Decrease in Impurity (MDI) criterion and optimizing the number of lags considering an 85% cumulative importance threshold. The third step of the developed methodological approach fits the Elastic Net Regression (ENR) to the volatility estimator’s dataset, while the final fourth step assesses the predictive accuracy of ENR, compared to decision tree (DTR), random forest (RFR), and support vector regression (SVR). The results reveal that the ENR prevails in its predictive accuracy for open and close prices, as these prices may be linear and less susceptible to sudden, non-linear shifts typically seen during trading hours. On the other hand, SVR prevails for high and low prices as these prices often experience spikes and drops driven by transient news and intra-day market sentiments, forming complex patterns that do not align well with linear modelling. Full article

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13 pages, 287 KB

Open AccessEditor’s ChoiceArticle

Metric Dimension of Circulant Graphs with 5 Consecutive Generators

by Martin Knor, Riste Škrekovski and Tomáš Vetrík

Mathematics 2024, 12(9), 1384; https://doi.org/10.3390/math12091384 - 1 May 2024

Cited by 2 | Viewed by 2629

Abstract

The problem of finding the metric dimension of circulant graphs with t generators

1, 2, \dots, t

(and their inverses) has been extensively studied. The problem is solved for

t = 2, 3, 4

, and some exact [...] Read more.

The problem of finding the metric dimension of circulant graphs with t generators

1, 2, \dots, t

(and their inverses) has been extensively studied. The problem is solved for

t = 2, 3, 4

, and some exact values and bounds are known also for

t = 5

. We solve all the open cases for

t = 5

. Full article

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

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23 pages, 2149 KB

Open AccessEditor’s ChoiceArticle

Inter-Departure Time Correlations in PH/G/1 Queues

by Ruth Sagron and Uri Yechiali

Mathematics 2024, 12(9), 1362; https://doi.org/10.3390/math12091362 - 30 Apr 2024

Cited by 2 | Viewed by 7808

Abstract

In non-Markovian tandem queueing networks the output process of one site, which is the input process to the next site, is not renewal. Consequently, the correlation analysis of that output processes is essential when studying such networks. A correlation analysis in the M/G/1 [...] Read more.

In non-Markovian tandem queueing networks the output process of one site, which is the input process to the next site, is not renewal. Consequently, the correlation analysis of that output processes is essential when studying such networks. A correlation analysis in the M/G/1 queue has been studied in the literature via derivation of the joint Laplace-Stieltjes transform (LST) of the sum of two consecutive inter-departure times. That LST is obtained by considering all possible cases at departure epochs. However, those epochs are expressed via dependent variables. In this paper, we first extend the analysis to the more general PH/G/1 queue, and investigate various queues, such as E₂/G/1 and C₂/C₂/1. Then, we consider the lag-n correlation, which requires derivation of the joint LST of sum of n + 1 consecutive inter-departure times. Yet, deriving this LST by the common approach becomes impractical for n + 1 ≥ 3, as the number of all possible cases at departure epochs increases significantly. To overcome this obstacle, we derive a corresponding single-parameter LST, which expresses the sum of n + 1 consecutive inter-departure times via the (n + 1)-st departure epoch only. Consequently, the latter LST is expressed via a much fewer number of possible cases, and not less important, as a function of independent variables only, eliminating the need to derive the corresponding joint density. Considering the M/G/1 and the E₂/G/1 queues, we demonstrate that the joint LST can be reconstructed directly via the corresponding single-parameter LST when n + 1 = 2. We further conjecture that the multi-parameter joint LST can be reconstructed from the corresponding single-parameter LST in more general queues and for values of n + 1 > 2. The conjecture is validated for various PH/G/1 queues and proved for n + 1 = 3 in the M/G/1 case. The new approach facilitates the calculation of lag-n correlation of the departure process from PH/G/1 queue for n + 1 ≥ 3. Our analysis illuminates the cases when using renewal approximation of the output process provides a proper approximation when studying non-Markovian stochastic networks. Full article

(This article belongs to the Special Issue Advances in Queueing Theory, 2nd Edition)

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20 pages, 529 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Extending Undirected Graph Techniques to Directed Graphs via Category Theory

by Sebastian Pardo-Guerra, Vivek Kurien George, Vikash Morar, Joshua Roldan and Gabriel Alex Silva

Mathematics 2024, 12(9), 1357; https://doi.org/10.3390/math12091357 - 29 Apr 2024

Cited by 7 | Viewed by 3245

Abstract

We use Category Theory to construct a ‘bridge’ relating directed graphs with undirected graphs, such that the notion of direction is preserved. Specifically, we provide an isomorphism between the category of simple directed graphs and a category we call ‘prime graphs category’; this [...] Read more.

We use Category Theory to construct a ‘bridge’ relating directed graphs with undirected graphs, such that the notion of direction is preserved. Specifically, we provide an isomorphism between the category of simple directed graphs and a category we call ‘prime graphs category’; this has as objects labeled undirected bipartite graphs (which we call prime graphs), and as morphisms undirected graph morphisms that preserve the labeling (which we call prime graph morphisms). This theoretical bridge allows us to extend undirected graph techniques to directed graphs by converting the directed graphs into prime graphs. To give a proof of concept, we show that our construction preserves topological features when applied to the problems of network alignment and spectral graph clustering. Full article

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17 pages, 1480 KB

Open AccessFeature PaperEditor’s ChoiceArticle

A Three-Dimensional Velocity Field Related to a Generalized Third-Grade Fluid Model

by Fernando Carapau, Paulo Correia and Gracino Rodrigues

Mathematics 2024, 12(9), 1326; https://doi.org/10.3390/math12091326 - 26 Apr 2024

Cited by 1 | Viewed by 2075

Abstract

In this work, we propose a new three-dimensional constitutive equation related to a third-grade fluid. This proposal is based on experimental work for which the viscosity term and the terms related to viscoelasticity may depend on the shear rate, in accordance with a [...] Read more.

In this work, we propose a new three-dimensional constitutive equation related to a third-grade fluid. This proposal is based on experimental work for which the viscosity term and the terms related to viscoelasticity may depend on the shear rate, in accordance with a power-law type model. The numerical implementation of this fluid model is rather demanding in terms of computational calculation and, in this sense, we use the Cosserat theory related to fluid dynamics, which makes the transition from the three-dimensional fluid model to a one-dimensional fluid model for a specific geometry under study which, in this case, is a straight tube with constant circular cross-section. Based on this approximation theory, the one-dimensional fluid model is solved by assuming an ordinary differential equation involving: an unsteady mean pressure gradient; an unsteady volume flow rate; the Womersley number; and the viscosity and viscoelasticity parameters. Consequently, for specific data, and using the Runge–Kutta method, we can obtain the solution for the unsteady volume flow rate and we can present simulations to the three-dimensional velocity field. Full article

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

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17 pages, 363 KB

Open AccessFeature PaperEditor’s ChoiceReview

Feynman Diagrams beyond Physics: From Biology to Economy

by Nicolò Cangiotti

Mathematics 2024, 12(9), 1295; https://doi.org/10.3390/math12091295 - 25 Apr 2024

Cited by 1 | Viewed by 5546

Abstract

Feynman diagrams represent one of the most powerful and fascinating tools developed in theoretical physics in the last century. Introduced within the framework of quantum electrodynamics as a suitable method for computing the amplitude of a physical process, they rapidly became a fundamental [...] Read more.

Feynman diagrams represent one of the most powerful and fascinating tools developed in theoretical physics in the last century. Introduced within the framework of quantum electrodynamics as a suitable method for computing the amplitude of a physical process, they rapidly became a fundamental mathematical object in quantum field theory. However, their abstract nature seems to suggest a wider usage, which actually exceeds the physical context. Indeed, as mathematical objects, they could simply be considered graphs that depict not only physical quantities but also biological or economic entities. We survey the analytical and algebraic properties of such diagrams to understand their utility in several areas of science, eventually providing some examples of recent applications. Full article

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

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20 pages, 382 KB

Open AccessEditor’s ChoiceArticle

Joins, Secant Varieties and Their Associated Grassmannians

by Edoardo Ballico

Mathematics 2024, 12(9), 1274; https://doi.org/10.3390/math12091274 - 23 Apr 2024

Cited by 3 | Viewed by 1648

Abstract

We prove a strong theorem on the partial non-defectivity of secant varieties of embedded homogeneous varieties developing a general set-up for families of subvarieties of Grassmannians. We study this type of problem in the more general set-up of joins of embedded varieties. Joins [...] Read more.

We prove a strong theorem on the partial non-defectivity of secant varieties of embedded homogeneous varieties developing a general set-up for families of subvarieties of Grassmannians. We study this type of problem in the more general set-up of joins of embedded varieties. Joins are defined by taking a closure. We study the set obtained before making the closure (often called the open part of the join) and the set added after making the closure (called the boundary of the join). For a point q of the open part, we give conditions for the uniqueness of the set proving that q is in the open part. Full article

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

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