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Editor’s Choice Articles

Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.

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15 pages, 4185 KiB  
Article
Outer Synchronization of Two Muti-Layer Dynamical Complex Networks with Intermittent Pinning Control
by Yi Liang, Yunyun Deng and Chuan Zhang
Mathematics 2023, 11(16), 3543; https://doi.org/10.3390/math11163543 - 16 Aug 2023
Cited by 4 | Viewed by 1045
Abstract
This paper regards the outer synchronization of multi-layer dynamical networks with additive couplings via aperiodically intermittent pinning control, in which different layers of each multi-layer network have different topological structures. First, a state-feedback intermittent pinning controller is designed in the drive and response [...] Read more.
This paper regards the outer synchronization of multi-layer dynamical networks with additive couplings via aperiodically intermittent pinning control, in which different layers of each multi-layer network have different topological structures. First, a state-feedback intermittent pinning controller is designed in the drive and response configuration, and sufficient conditions to achieve the outer synchronization are derived based on the Lyapunov stability theory and matrix inequalities. Second, outer synchronization problem of multi-layer networks is discussed by setting an adaptive intermittent pinning controller; an appropriate Lyapunov function is selected to prove the criteria of synchronization between the drive multi-layer network and the response multi-layer network. Finally, three simulation examples are given to show the effectiveness of our control schemes. Full article
(This article belongs to the Special Issue Complex Network Modeling: Theory and Applications, 2nd Edition)
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21 pages, 386 KiB  
Article
Optimization Models for the Vehicle Routing Problem under Disruptions
by Kai Huang and Michael Xu
Mathematics 2023, 11(16), 3521; https://doi.org/10.3390/math11163521 - 15 Aug 2023
Cited by 2 | Viewed by 1741
Abstract
In this paper, we study the role of disruptions in the multi-period vehicle routing problem (VRP), which naturally arises in humanitarian logistics and military applications. We assume that at any time during the delivery phase, each vehicle could have chance to be disrupted. [...] Read more.
In this paper, we study the role of disruptions in the multi-period vehicle routing problem (VRP), which naturally arises in humanitarian logistics and military applications. We assume that at any time during the delivery phase, each vehicle could have chance to be disrupted. When a disruption happens, vehicles will be unable to continue their journeys and supplies will be unable to be delivered. We model the occurrence of disruption as a given probability and consider the multi-period expected delivery. Our objective is to either minimize the total travel cost or maximize the demand fulfillment, depending on the supply quantity. This problem is denoted as the multi-period vehicle routing problem with disruption (VRPMD). VRPMD does not deal with disruptions in real-time and is more focused on the long-term performance of a single routing plan. We first prove that the proposed VRPMD problems are NP-hard. We then present some analytical properties related to the optimal solutions to these problems. We show that Dror and Trudeau’s property does not apply in our problem setting. Nevertheless, a generalization of Dror and Trudeau’s property holds. Finally, we present efficient heuristic algorithms to solve these problems and show the effectiveness of the proposed models and algorithms through numerical studies. Full article
(This article belongs to the Special Issue Machine Learning and Statistical Methods to Prediction and Optimal Decision-Making)
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13 pages, 295 KiB  
Article
Fuzzy Metrics in Terms of Fuzzy Relations
by Olga Grigorenko and Alexander Šostak
Mathematics 2023, 11(16), 3528; https://doi.org/10.3390/math11163528 - 15 Aug 2023
Cited by 1 | Viewed by 1267
Abstract
In this paper, we study the concept of fuzzy metrics from the perspective of fuzzy relations. Specifically, we analyze the commonly used definitions of fuzzy metrics. We begin by noting that crisp metrics can be uniquely characterized by linear order relations. Further, we [...] Read more.
In this paper, we study the concept of fuzzy metrics from the perspective of fuzzy relations. Specifically, we analyze the commonly used definitions of fuzzy metrics. We begin by noting that crisp metrics can be uniquely characterized by linear order relations. Further, we explore the criteria that crisp relations must satisfy in order to determine a crisp metric. Subsequently, we extend these conditions to obtain a fuzzy metric and investigate the additional axioms involved. Additionally, we introduce the definition of an extensional fuzzy metric or E-d-metric, which is a fuzzification of the expression d(x,y)=t. Thus, we examine fuzzy metrics from both the linear order and from the equivalence relation perspectives, where one argument is a value d(x,y) and the other is a number within the range [0,+∞). Full article
(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition)
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23 pages, 544 KiB  
Article
A Binary Black Widow Optimization Algorithm for Addressing the Cell Formation Problem Involving Alternative Routes and Machine Reliability
by Paulo Figueroa-Torrez, Orlando Durán, Broderick Crawford and Felipe Cisternas-Caneo
Mathematics 2023, 11(16), 3475; https://doi.org/10.3390/math11163475 - 11 Aug 2023
Cited by 5 | Viewed by 1373
Abstract
The Cell Formation Problem (CFP) involves the clustering of machines to enhance productivity and capitalize on various benefits. This study addresses a variant of the problem where alternative routes and machine reliability are included, which we call a Generalized Cell Formation Problem with [...] Read more.
The Cell Formation Problem (CFP) involves the clustering of machines to enhance productivity and capitalize on various benefits. This study addresses a variant of the problem where alternative routes and machine reliability are included, which we call a Generalized Cell Formation Problem with Machine Reliability (GCFP-MR). This problem is known to be NP-Hard, and finding efficient solutions is of utmost importance. Metaheuristics have been recognized as effective optimization techniques due to their adaptability and ability to generate high-quality solutions in a short time. Since BWO was originally designed for continuous optimization problems, its adaptation involves binarization. Accordingly, our proposal focuses on adapting the Black Widow Optimization (BWO) metaheuristic to tackle GCFP-MR, leading to a new approach named Binary Black Widow Optimization (B-BWO). We compare our proposal in two ways. Firstly, it is benchmarked against a previous Clonal Selection Algorithm approach. Secondly, we evaluate B-BWO with various parameter configurations. The experimental results indicate that the best configuration of parameters includes a population size (Pop) set to 100, and the number of iterations (Maxiter) defined as 75. Procreating Rate (PR) is set at 0.8, Cannibalism Rate (CR) is set at 0.4, and the Mutation Rate (PM) is also set at 0.4. Significantly, the proposed B-BWO outperforms the state-of-the-art literature’s best result, achieving a noteworthy improvement of 1.40%. This finding reveals the efficacy of B-BWO in solving GCFP-MR and its potential to produce superior solutions compared to alternative methods. Full article
(This article belongs to the Special Issue Recent Advances in Computational Intelligence Methodologies for Industries)
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29 pages, 618 KiB  
Article
A Symbolic Approach to Discrete Structural Optimization Using Quantum Annealing
by Kevin Wils and Boyang Chen
Mathematics 2023, 11(16), 3451; https://doi.org/10.3390/math11163451 - 9 Aug 2023
Cited by 5 | Viewed by 1358
Abstract
With the advent of novel quantum computing technologies and the new possibilities thereby offered, a prime opportunity has presented itself to investigate the practical application of quantum computing. This work investigates the feasibility of using quantum annealing for structural optimization. The target problem [...] Read more.
With the advent of novel quantum computing technologies and the new possibilities thereby offered, a prime opportunity has presented itself to investigate the practical application of quantum computing. This work investigates the feasibility of using quantum annealing for structural optimization. The target problem is the discrete truss sizing problem—the goal is to select the best size for each truss member so as to minimize a stress-based objective function. To make the problem compatible with quantum annealing devices, the objective function must be translated into a quadratic unconstrained binary optimization (QUBO) form. This work focuses on exploring the feasibility of making this translation. The practicality of using a quantum annealer for such optimization problems is also assessed. A method is eventually established to translate the objective function into a QUBO form and have it solved by a quantum annealer. However, scaling the method to larger problems faces some challenges that would require further research to address. Full article
(This article belongs to the Special Issue Advances in Quantum Computing and Applications)
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27 pages, 872 KiB  
Article
Machine Learning Alternatives to Response Surface Models
by Badih Ghattas and Diane Manzon
Mathematics 2023, 11(15), 3406; https://doi.org/10.3390/math11153406 - 4 Aug 2023
Cited by 7 | Viewed by 2595
Abstract
In the Design of Experiments, we seek to relate response variables to explanatory factors. Response Surface methodology (RSM) approximates the relation between output variables and a polynomial transform of the explanatory variables using a linear model. Some researchers have tried to adjust other [...] Read more.
In the Design of Experiments, we seek to relate response variables to explanatory factors. Response Surface methodology (RSM) approximates the relation between output variables and a polynomial transform of the explanatory variables using a linear model. Some researchers have tried to adjust other types of models, mainly nonlinear and nonparametric. We present a large panel of Machine Learning approaches that may be good alternatives to the classical RSM approximation. The state of the art of such approaches is given, including classification and regression trees, ensemble methods, support vector machines, neural networks and also direct multi-output approaches. We survey the subject and illustrate the use of ten such approaches using simulations and a real use case. In our simulations, the underlying model is linear in the explanatory factors for one response and nonlinear for the others. We focus on the advantages and disadvantages of the different approaches and show how their hyperparameters may be tuned. Our simulations show that even when the underlying relation between the response and the explanatory variables is linear, the RSM approach is outperformed by the direct neural network multivariate model, for any sample size (<50) and much more for very small samples (15 or 20). When the underlying relation is nonlinear, the RSM approach is outperformed by most of the machine learning approaches for small samples (n ≤ 30). Full article
(This article belongs to the Section Probability and Statistics)
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26 pages, 424 KiB  
Article
A New Instrumental-Type Estimator for Quantile Regression Models
by Li Tao, Lingnan Tai, Manling Qian and Maozai Tian
Mathematics 2023, 11(15), 3412; https://doi.org/10.3390/math11153412 - 4 Aug 2023
Cited by 1 | Viewed by 1212
Abstract
This paper proposes a new instrumental-type estimator of quantile regression models for panel data with fixed effects. The estimator is built upon the minimum distance, which is defined as the weighted average of the conventional individual instrumental variable quantile regression slope estimators. The [...] Read more.
This paper proposes a new instrumental-type estimator of quantile regression models for panel data with fixed effects. The estimator is built upon the minimum distance, which is defined as the weighted average of the conventional individual instrumental variable quantile regression slope estimators. The weights assigned to each estimator are determined by the inverses of their corresponding individual variance–covariance matrices. The implementation of the estimation has many advantages in terms of computational efforts and simplifies the asymptotic distribution. Furthermore, the paper shows consistency and asymptotic normality for sequential and simultaneous asymptotics. Additionally, it presents an empirical application that investigates the income elasticity of health expenditures. Full article
(This article belongs to the Special Issue Statistical Analysis: Theory, Methods and Applications)
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21 pages, 3355 KiB  
Review
Advancements in Phase-Field Modeling for Fracture in Nonlinear Elastic Solids under Finite Deformations
by Gang Zhang, Cheng Tang, Peng Chen, Gongbo Long, Jiyin Cao and Shan Tang
Mathematics 2023, 11(15), 3366; https://doi.org/10.3390/math11153366 - 1 Aug 2023
Cited by 4 | Viewed by 2308
Abstract
The prediction of failure mechanisms in nonlinear elastic materials holds significant importance in engineering applications. In recent years, the phase-field model has emerged as an effective approach for addressing fracture problems. Compared with other discontinuous fracture methods, the phase-field method allows for the [...] Read more.
The prediction of failure mechanisms in nonlinear elastic materials holds significant importance in engineering applications. In recent years, the phase-field model has emerged as an effective approach for addressing fracture problems. Compared with other discontinuous fracture methods, the phase-field method allows for the easy simulation of complex fracture paths, including crack initiation, propagation, coalescence, and branching phenomena, through a scalar field known as the phase field. This method offers distinct advantages in tackling complex fracture problems in nonlinear elastic materials and exhibits substantial potential in material design and manufacturing. The current research has indicated that the energy distribution method employed in phase-field approaches significantly influences the simulated results of material fracture, such as crack initiation load, crack propagation path, crack branching, and so forth. This impact is particularly pronounced when simulating the fracture of nonlinear materials under finite deformation. Therefore, this review outlines various strain energy decomposition methods proposed by researchers for phase-field models of fracture in tension–compression symmetric nonlinear elastic materials. Additionally, the energy decomposition model for tension–compression asymmetric nonlinear elastic materials is also presented. Moreover, the fracture behavior of hydrogels is investigated through the application of the phase-field model with energy decomposition. In addition to summarizing the research on these types of nonlinear elastic body fractures, this review presents numerical benchmark examples from relevant studies to assess and validate the accuracy and effectiveness of the methods presented. Full article
(This article belongs to the Special Issue Recent Development and Application of Methods in Computational Mechanics)
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21 pages, 2650 KiB  
Article
A Methodology for Planning City Logistics Concepts Based on City-Dry Port Micro-Consolidation Centres
by Milovan Kovač, Snežana Tadić, Mladen Krstić and Miloš Veljović
Mathematics 2023, 11(15), 3347; https://doi.org/10.3390/math11153347 - 31 Jul 2023
Cited by 8 | Viewed by 1515
Abstract
The purpose of this study is to conceptualize a novel idea of potentially sustainable city logistics concepts—the development of urban consolidation centers (UCCs) on riverbanks and the establishment of city-dry port (DP) micro-consolidation centers (MCCs) as their displaced subsystems within the delivery zone. [...] Read more.
The purpose of this study is to conceptualize a novel idea of potentially sustainable city logistics concepts—the development of urban consolidation centers (UCCs) on riverbanks and the establishment of city-dry port (DP) micro-consolidation centers (MCCs) as their displaced subsystems within the delivery zone. The concept enables the application of river transportation in delivering goods to the UCC, where the modal shift to electric delivery vehicles takes place for delivering goods to city-DP MCCs. In the final delivery phase (from city-DP MCCs to flow generators), smaller eco-vehicles are utilized. An innovative methodology for the planning and selection of the most sustainable concept variant is developed. The methodology combines mathematical programming and the axial-distance-based aggregated measurement (ADAM) multi-criteria decision-making (MCDM) method. The application of the defined approach is demonstrated in a case study inspired by Belgrade, Serbia. The theoretical contribution of this study is in demonstrating how a wide set of potentially viable city logistics concepts can be defined, starting from an initial idea (city-DP MCC). The practical contribution lies in developing a robust methodology that considers all relevant tactical and operational-level planning questions and takes into account qualitative and quantitative criteria in evaluating different concept variants. Full article
(This article belongs to the Special Issue Mathematical Optimization and Decision Making)
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25 pages, 1124 KiB  
Article
An Inhomogeneous Model for Laser Welding of Industrial Interest
by Carmelo Filippo Munafò, Annunziata Palumbo and Mario Versaci
Mathematics 2023, 11(15), 3357; https://doi.org/10.3390/math11153357 - 31 Jul 2023
Cited by 9 | Viewed by 1750
Abstract
An innovative non-homogeneous dynamic model is presented for the recovery of temperature during the industrial laser welding process of Al-Si 5% alloy plates. It considers that, metallurgically, during welding, the alloy melts with the presence of solid/liquid phases until total melt is [...] Read more.
An innovative non-homogeneous dynamic model is presented for the recovery of temperature during the industrial laser welding process of Al-Si 5% alloy plates. It considers that, metallurgically, during welding, the alloy melts with the presence of solid/liquid phases until total melt is achieved, and afterwards it resolidifies with the reverse process. Further, a polynomial substitute thermal capacity of the alloy is chosen based on experimental evidence so that the volumetric solid-state fraction is identifiable. Moreover, to the usual radiative/convective boundary conditions, the contribution due to the positioning of the plates on the workbench is considered (endowing the model with Cauchy–Stefan–Boltzmann boundary conditions). Having verified the well-posedness of the problem, a Galerkin-FEM approach is implemented to recover the temperature maps, obtained by modeling the laser heat sources with formulations depending on the laser sliding speed. The results achieved show good adherence to the experimental evidence, opening up interesting future scenarios for technology transfer. Full article
(This article belongs to the Special Issue Mathematical Modeling in Industrial Engineering and Electrical Engineering, 2nd Edition)
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25 pages, 6506 KiB  
Article
An Improved Strength Pareto Evolutionary Algorithm 2 with Adaptive Crossover Operator for Bi-Objective Distributed Unmanned Aerial Vehicle Delivery
by Yu Song and Xi Fang
Mathematics 2023, 11(15), 3327; https://doi.org/10.3390/math11153327 - 28 Jul 2023
Cited by 3 | Viewed by 1399
Abstract
With the development of the e-commerce industry, using UAVs (unmanned aerial vehicles) to deliver goods has become more popular in transportation systems. This delivery method can reduce labor costs and improve the distribution efficiency, and UAVs can reach places that are difficult for [...] Read more.
With the development of the e-commerce industry, using UAVs (unmanned aerial vehicles) to deliver goods has become more popular in transportation systems. This delivery method can reduce labor costs and improve the distribution efficiency, and UAVs can reach places that are difficult for humans to reach. Because some goods are perishable, the quality of the delivery will have an impact on the customer satisfaction. At the same time, the delivery time should also meet the needs of customers as much as possible. Therefore, this paper takes the distribution distance and customer satisfaction as the objective functions, establishes a bi-objective dynamic programming model, and proposes an improved SPEA2 (strength Pareto evolutionary algorithm 2). The improved algorithm introduces the local search strategy, on the basis of the original algorithm. It conducts a local search for the better non-dominated solutions obtained in each iteration. The new dominated solutions and non-dominated solutions are determined, and the crossover operator is improved, so that the local search ability is improved, on the basis of ensuring its global search ability. The numerical experiment results show that the improved algorithm achieves an excellent performance in three aspects: the Pareto front, generation distance, and spacing, and would have a high application value in UAV cargo delivery and other MOPs (multi-objective optimization problems). The average spacing value of the improved algorithm is more than 20% smaller than SPEA2 + SDE (strength Pareto evolution algorithm 2–shift-based density estimation), which is the second-best algorithm. In the comparison of the average generation distance value, this number reaches 30%. Full article
(This article belongs to the Section Mathematics and Computer Science)
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24 pages, 368 KiB  
Article
Two-Round Multi-Signatures from Okamoto Signatures
by Kwangsu Lee and Hyoseung Kim
Mathematics 2023, 11(14), 3223; https://doi.org/10.3390/math11143223 - 22 Jul 2023
Cited by 4 | Viewed by 1377
Abstract
Multi-signatures (MS) are a special type of public-key signature (PKS) in which multiple signers participate cooperatively to generate a signature for a single message. Recently, applications that use an MS scheme to strengthen the security of blockchain wallets or to strengthen the security [...] Read more.
Multi-signatures (MS) are a special type of public-key signature (PKS) in which multiple signers participate cooperatively to generate a signature for a single message. Recently, applications that use an MS scheme to strengthen the security of blockchain wallets or to strengthen the security of blockchain consensus protocols are attracting a lot of attention. In this paper, we propose an efficient two-round MS scheme based on Okamoto signatures rather than Schnorr signatures. To this end, we first propose a new PKS scheme by modifying the Okamoto signature scheme and prove the unforgeability of our PKS scheme under the discrete logarithm assumption in the algebraic group model (AGM) and the non-programmable random oracle model (ROM). Next, we propose a two-round MS scheme based on the new PKS scheme and prove the unforgeability of our MS scheme under the discrete logarithm assumption in the AGM and the non-programmable ROM. Our MS scheme is the first one to prove security among two-round MS based on Okamoto signatures. Full article
(This article belongs to the Special Issue Advances in Cryptography and Blockchain: Mathematics for Healthcare Security)
11 pages, 1262 KiB  
Article
Dynamics and Embedded Solitons of Stochastic Quadratic and Cubic Nonlinear Susceptibilities with Multiplicative White Noise in the Itô Sense
by Zhao Li and Chen Peng
Mathematics 2023, 11(14), 3185; https://doi.org/10.3390/math11143185 - 20 Jul 2023
Cited by 11 | Viewed by 874
Abstract
The main purpose of this paper is to study the dynamics and embedded solitons of stochastic quadratic and cubic nonlinear susceptibilities in the Itô sense, which can further help researchers understand the propagation of soliton nonlinear systems. Firstly, a two-dimensional dynamics system and [...] Read more.
The main purpose of this paper is to study the dynamics and embedded solitons of stochastic quadratic and cubic nonlinear susceptibilities in the Itô sense, which can further help researchers understand the propagation of soliton nonlinear systems. Firstly, a two-dimensional dynamics system and its perturbation system are obtained by using a traveling wave transformation. Secondly, the phase portraits of the two-dimensional dynamics system are plotted. Furthermore, the chaotic behavior, two-dimensional phase portraits, three-dimensional phase portraits and sensitivity of the perturbation system are analyzed via Maple software. Finally, the embedded solitons of stochastic quadratic and cubic nonlinear susceptibilities are obtained. Moreover, three-dimensional and two-dimensional solitons of stochastic quadratic and cubic nonlinear susceptibilities are plotted. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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21 pages, 9043 KiB  
Article
An Efficient Numerical Approach for Solving Systems of Fractional Problems and Their Applications in Science
by Sondos M. Syam, Z. Siri, Sami H. Altoum and R. Md. Kasmani
Mathematics 2023, 11(14), 3132; https://doi.org/10.3390/math11143132 - 16 Jul 2023
Cited by 10 | Viewed by 1426
Abstract
In this article, we present a new numerical approach for solving a class of systems of fractional initial value problems based on the operational matrix method. We derive the method and provide a convergence analysis. To reduce computational cost, we transform the algebraic [...] Read more.
In this article, we present a new numerical approach for solving a class of systems of fractional initial value problems based on the operational matrix method. We derive the method and provide a convergence analysis. To reduce computational cost, we transform the algebraic problem produced by this approach into a set of 2×2 nonlinear equations, instead of solving a system of 2 m × 2 m equations. We apply our approach to three main applications in science: optimal control problems, Riccati equations, and clock reactions. We compare our results with those of other researchers, considering computational time, cost, and absolute errors. Additionally, we validate our numerical method by comparing our results with the integer model when the fractional order approaches one. We present numerous figures and tables to illustrate our findings. The results demonstrate the effectiveness of the proposed approach. Full article
(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)
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29 pages, 13241 KiB  
Article
Predicting Popularity of Viral Content in Social Media through a Temporal-Spatial Cascade Convolutional Learning Framework
by Zhixuan Xu and Minghui Qian
Mathematics 2023, 11(14), 3059; https://doi.org/10.3390/math11143059 - 11 Jul 2023
Cited by 7 | Viewed by 3730
Abstract
The viral spread of online content can lead to unexpected consequences such as extreme opinions about a brand or consumers’ enthusiasm for a product. This makes the prediction of viral content’s future popularity an important problem, especially for digital marketers, as well as [...] Read more.
The viral spread of online content can lead to unexpected consequences such as extreme opinions about a brand or consumers’ enthusiasm for a product. This makes the prediction of viral content’s future popularity an important problem, especially for digital marketers, as well as for managers of social platforms. It is not surprising that conventional methods, which heavily rely on either hand-crafted features or unrealistic assumptions, are insufficient in dealing with this challenging problem. Even state-of-art graph-based approaches are either inefficient to work with large-scale cascades or unable to explain what spread mechanisms are learned by the model. This paper presents a temporal-spatial cascade convolutional learning framework called ViralGCN, not only to address the challenges of existing approaches but also to try to provide some insights into actual mechanisms of viral spread from the perspective of artificial intelligence. We conduct experiments on the real-world dataset (i.e., to predict the retweet popularity of micro-blogs on Weibo). Compared to the existing approaches, ViralGCN possesses the following advantages: the flexible size of the input cascade graph, a coherent method for processing both structural and temporal information, and an intuitive and interpretable deep learning architecture. Moreover, the exploration of the learned features also provides valuable clues for managers to understand the elusive mechanisms of viral spread as well as to devise appropriate strategies at early stages. By using the visualization method, our approach finds that both broadcast and structural virality contribute to online content going viral; the cascade with a gradual descent or ascent-then-descent evolving pattern at the early stage is more likely to gain significant eventual popularity, and even the timing of users participating in the cascade has an effect on future popularity growth. Full article
(This article belongs to the Special Issue Artificial Intelligence and Data Science)
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20 pages, 331 KiB  
Article
A Stochastic Control Approach for Constrained Stochastic Differential Games with Jumps and Regimes
by Emel Savku
Mathematics 2023, 11(14), 3043; https://doi.org/10.3390/math11143043 - 9 Jul 2023
Cited by 8 | Viewed by 1673
Abstract
We develop an approach for two-player constraint zero-sum and nonzero-sum stochastic differential games, which are modeled by Markov regime-switching jump-diffusion processes. We provide the relations between a usual stochastic optimal control setting and a Lagrangian method. In this context, we prove corresponding theorems [...] Read more.
We develop an approach for two-player constraint zero-sum and nonzero-sum stochastic differential games, which are modeled by Markov regime-switching jump-diffusion processes. We provide the relations between a usual stochastic optimal control setting and a Lagrangian method. In this context, we prove corresponding theorems for two different types of constraints, which lead us to find real-valued and stochastic Lagrange multipliers, respectively. Then, we illustrate our results for a nonzero-sum game problem with the stochastic maximum principle technique. Our application is an example of cooperation between a bank and an insurance company, which is a popular, well-known business agreement type called Bancassurance. Full article
(This article belongs to the Special Issue Stochastic Analysis and Applications in Financial Mathematics)
19 pages, 4540 KiB  
Article
Numerical Solution of Thermal Phenomena in Welding Problems
by Mario Freire-Torres, Manuel Colera and Jaime Carpio
Mathematics 2023, 11(13), 3009; https://doi.org/10.3390/math11133009 - 6 Jul 2023
Viewed by 1463
Abstract
We present a novel finite element method to solve the thermal variables in welding problems. The mathematical model is based on the enthalpy formulation of the energy conservation law, which is simultaneously valid for the solid, liquid, and mushy regions. Both isothermal and [...] Read more.
We present a novel finite element method to solve the thermal variables in welding problems. The mathematical model is based on the enthalpy formulation of the energy conservation law, which is simultaneously valid for the solid, liquid, and mushy regions. Both isothermal and non-isothermal melting models are considered to relate the enthalpy with the temperature. Quadratic triangular elements with local anisotropic mesh adaptation are employed for the space discretization of the governing equation, and a second-order backward differentiation formula is employed for the time discretization. The resulting non-linear discretized system is solved with a simple Newton algorithm with two versions: the θ-Newton algorithm, which considers the temperature as the main unknown variable, as in most works in the literature, and the h-Newton algorithm, which considers the enthalpy, which is the main novelty of the present work. Then, we show via numerical experiments that the h-Newton method is robust and converges well to the solution, both for isothermal and non-isothermal melting. However, the θ-method can only be applied to the case of non-isothermal melting and converges only for a sufficiently large melting temperature range or sufficiently small time step. Numerical experiments also confirm that the method is able to adequately capture the discontinuities or sharp variations in the solution without the need for any kind of numerical dissipation. Full article
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36 pages, 537 KiB  
Review
Set-Based Particle Swarm Optimisation: A Review
by Jean-Pierre van Zyl and Andries Petrus Engelbrecht
Mathematics 2023, 11(13), 2980; https://doi.org/10.3390/math11132980 - 4 Jul 2023
Cited by 16 | Viewed by 2384
Abstract
The set-based particle swarm optimisation algorithm is a swarm-based meta-heuristic that has gained popularity in recent years. In contrast to the original particle swarm optimisation algorithm, the set-based particle swarm optimisation algorithm is used to solve discrete and combinatorial optimisation problems. The main [...] Read more.
The set-based particle swarm optimisation algorithm is a swarm-based meta-heuristic that has gained popularity in recent years. In contrast to the original particle swarm optimisation algorithm, the set-based particle swarm optimisation algorithm is used to solve discrete and combinatorial optimisation problems. The main objective of this paper is to review the set-based particle swarm optimisation algorithm and to provide an overview of the problems to which the algorithm has been applied. This paper starts with an examination of previous attempts to create a set-based particle swarm optimisation algorithm and discusses the shortcomings of the existing attempts. The set-based particle swarm optimisation algorithm is established as the only suitable particle swarm variant that is both based on true set theory and does not require problem-specific modifications. In-depth explanations are given regarding the general position and velocity update equations, the mechanisms used to control the exploration–exploitation trade-off, and the quantifiers of swarm diversity. After the various existing applications of set-based particle swarm optimisation are presented, this paper concludes with a discussion on potential future research. Full article
(This article belongs to the Special Issue Combinatorial Optimization: Trends and Applications)
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10 pages, 264 KiB  
Article
Some Double q-Series by Telescoping
by Kwang-Wu Chen
Mathematics 2023, 11(13), 2949; https://doi.org/10.3390/math11132949 - 1 Jul 2023
Cited by 1 | Viewed by 861
Abstract
By means of the telescoping method, we derived two general double series formulas that encapsulate the Riemann zeta values ζ(s), the Catalan constant G, log(2), π and several other significant mathematical constants. Full article
24 pages, 517 KiB  
Article
Stability and Bifurcations in a Nutrient–Phytoplankton–Zooplankton Model with Delayed Nutrient Recycling with Gamma Distribution
by Mihaela Sterpu, Carmen Rocşoreanu, Raluca Efrem and Sue Ann Campbell
Mathematics 2023, 11(13), 2911; https://doi.org/10.3390/math11132911 - 28 Jun 2023
Viewed by 1873
Abstract
Two nutrient–phytoplankton–zooplankton (NZP) models for a closed ecosystem that incorporates a delay in nutrient recycling, obtained using the gamma distribution function with one or two degrees of freedom, are analysed. The models are described by systems of ordinary differential equations of four and [...] Read more.
Two nutrient–phytoplankton–zooplankton (NZP) models for a closed ecosystem that incorporates a delay in nutrient recycling, obtained using the gamma distribution function with one or two degrees of freedom, are analysed. The models are described by systems of ordinary differential equations of four and five dimensions. The purpose of this study is to investigate how the mean delay of the distribution and the total nutrients affect the stability of the equilibrium solutions. Local stability theory and bifurcation theory are used to determine the long-time dynamics of the models. It is found that both models exhibit comparable qualitative dynamics. There are a maximum of three equilibrium points in each of the two models, and at most one of them is locally asymptotically stable. The change of stability from one equilibrium to another takes place through a transcritical bifurcation. In some hypotheses on the functional response, the nutrient–phytoplankton–zooplankton equilibrium loses stability via a supercritical Hopf bifurcation, causing the apparition of a stable limit cycle. The way in which the results are consistent with prior research and how they extend them is discussed. Finally, various application-related consequences of the results of the theoretical study are deduced. Full article
(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 2nd Edition)
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10 pages, 297 KiB  
Article
Coefficients and Fekete–Szegö Functional Estimations of Bi-Univalent Subclasses Based on Gegenbauer Polynomials
by Abdulmtalb Hussen and Abdelbaset Zeyani
Mathematics 2023, 11(13), 2852; https://doi.org/10.3390/math11132852 - 25 Jun 2023
Cited by 10 | Viewed by 1066
Abstract
Subclasses of analytic and bi-univalent functions have been extensively improved and utilized for estimating the Taylor–Maclaurin coefficients and the Fekete–Szegö functional. In this paper, we consider a certain subclass of normalized analytic and bi-univalent functions. These functions have inverses that possess a bi-univalent [...] Read more.
Subclasses of analytic and bi-univalent functions have been extensively improved and utilized for estimating the Taylor–Maclaurin coefficients and the Fekete–Szegö functional. In this paper, we consider a certain subclass of normalized analytic and bi-univalent functions. These functions have inverses that possess a bi-univalent analytic continuation to an open unit disk and are associated with orthogonal polynomials; namely, Gegenbauer polynomials that satisfy subordination conditions on the open unit disk. We use this subclass to derive new approximations for the second and third Taylor–Maclaurin coefficients and the Fekete–Szegö functional. Furthermore, we discuss several new results that arise when we specialize the parameters used in our fundamental findings. Full article
8 pages, 255 KiB  
Article
Deriving Euler’s Equation for Rigid-Body Rotation via Lagrangian Dynamics with Generalized Coordinates
by Dennis S. Bernstein, Ankit Goel and Omran Kouba
Mathematics 2023, 11(12), 2727; https://doi.org/10.3390/math11122727 - 16 Jun 2023
Cited by 1 | Viewed by 3181
Abstract
Euler’s equation relates the change in angular momentum of a rigid body to the applied torque. This paper uses Lagrangian dynamics to derive Euler’s equation in terms of generalized coordinates. This is done by parameterizing the angular velocity vector in terms of 3-2-1 [...] Read more.
Euler’s equation relates the change in angular momentum of a rigid body to the applied torque. This paper uses Lagrangian dynamics to derive Euler’s equation in terms of generalized coordinates. This is done by parameterizing the angular velocity vector in terms of 3-2-1 and 3-1-3 Euler angles as well as Euler parameters, that is, quaternions. This paper fills a gap in the literature by using generalized coordinates to parameterize the angular velocity vector and thereby transform the dynamics obtained from Lagrangian dynamics into Euler’s equation for rigid-body rotation. Full article
(This article belongs to the Special Issue Mathematical Methods for Nonlinear Dynamics)
50 pages, 1073 KiB  
Review
Matrix Factorization Techniques in Machine Learning, Signal Processing, and Statistics
by Ke-Lin Du, M. N. S. Swamy, Zhang-Quan Wang and Wai Ho Mow
Mathematics 2023, 11(12), 2674; https://doi.org/10.3390/math11122674 - 12 Jun 2023
Cited by 11 | Viewed by 8951
Abstract
Compressed sensing is an alternative to Shannon/Nyquist sampling for acquiring sparse or compressible signals. Sparse coding represents a signal as a sparse linear combination of atoms, which are elementary signals derived from a predefined dictionary. Compressed sensing, sparse approximation, and dictionary learning are [...] Read more.
Compressed sensing is an alternative to Shannon/Nyquist sampling for acquiring sparse or compressible signals. Sparse coding represents a signal as a sparse linear combination of atoms, which are elementary signals derived from a predefined dictionary. Compressed sensing, sparse approximation, and dictionary learning are topics similar to sparse coding. Matrix completion is the process of recovering a data matrix from a subset of its entries, and it extends the principles of compressed sensing and sparse approximation. The nonnegative matrix factorization is a low-rank matrix factorization technique for nonnegative data. All of these low-rank matrix factorization techniques are unsupervised learning techniques, and can be used for data analysis tasks, such as dimension reduction, feature extraction, blind source separation, data compression, and knowledge discovery. In this paper, we survey a few emerging matrix factorization techniques that are receiving wide attention in machine learning, signal processing, and statistics. The treated topics are compressed sensing, dictionary learning, sparse representation, matrix completion and matrix recovery, nonnegative matrix factorization, the Nyström method, and CUR matrix decomposition in the machine learning framework. Some related topics, such as matrix factorization using metaheuristics or neurodynamics, are also introduced. A few topics are suggested for future investigation in this article. Full article
(This article belongs to the Special Issue Novel Mathematical Methods in Signal Processing and Its Applications)
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16 pages, 763 KiB  
Article
Digital Triplet: A Sequential Methodology for Digital Twin Learning
by Xueru Zhang, Dennis K. J. Lin and Lin Wang
Mathematics 2023, 11(12), 2661; https://doi.org/10.3390/math11122661 - 11 Jun 2023
Cited by 4 | Viewed by 2090
Abstract
A digital twin is a simulator of a physical system, which is built upon a series of models and computer programs with real-time data (from sensors or devices). Digital twins are used in various industries, such as manufacturing, healthcare, and transportation, to understand [...] Read more.
A digital twin is a simulator of a physical system, which is built upon a series of models and computer programs with real-time data (from sensors or devices). Digital twins are used in various industries, such as manufacturing, healthcare, and transportation, to understand complex physical systems and make informed decisions. However, predictions and optimizations with digital twins can be time-consuming due to the high computational requirements and complexity of the underlying computer programs. This poses significant challenges in making well-informed and timely decisions using digital twins. This paper proposes a novel methodology, called the “digital triplet”, to facilitate real-time prediction and decision-making. A digital triplet is an efficient representation of a digital twin, constructed using statistical models and effective experimental designs. It offers two noteworthy advantages. Firstly, by leveraging modern statistical models, a digital triplet can effectively capture and represent the complexities of a digital twin, resulting in accurate predictions and reliable decision-making. Secondly, a digital triplet adopts a sequential design and modeling approach, allowing real-time updates in conjunction with its corresponding digital twin. We conduct comprehensive simulation studies to explore the application of various statistical models and designs in constructing a digital triplet. It is shown that Gaussian process regression coupled with sequential MaxPro designs exhibits superior performance compared to other modeling and design techniques in accurately constructing the digital triplet. Full article
(This article belongs to the Special Issue Advances in Multivariate Statistics and Related Topics for the Fourth Industrial Revolution)
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14 pages, 417 KiB  
Article
Exponential Stability of a Class of Neutral Inertial Neural Networks with Multi-Proportional Delays and Leakage Delays
by Chao Wang, Yinfang Song, Fengjiao Zhang and Yuxiao Zhao
Mathematics 2023, 11(12), 2596; https://doi.org/10.3390/math11122596 - 6 Jun 2023
Cited by 8 | Viewed by 1031
Abstract
This paper investigates the exponential stability of a class of neutral inertial neural networks with multi-proportional delays and leakage delays. By utilizing the Lyapunov stability theory, the approach of parametric variation, and the differential inequality technique, some criteria are acquired that can guarantee [...] Read more.
This paper investigates the exponential stability of a class of neutral inertial neural networks with multi-proportional delays and leakage delays. By utilizing the Lyapunov stability theory, the approach of parametric variation, and the differential inequality technique, some criteria are acquired that can guarantee that all solutions of the addressed system converge exponentially to the equilibrium point. In particular, the neutral term, multi-proportional delays, and leakage delays are incorporated simultaneously, resulting in a more general model, and the findings are novel and refine the previous works. Finally, one example is provided to indicate that the dynamic behavior is consistent with the theoretical analysis. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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15 pages, 312 KiB  
Article
On the Rate of Convergence of Greedy Algorithms
by Vladimir Temlyakov
Mathematics 2023, 11(11), 2559; https://doi.org/10.3390/math11112559 - 2 Jun 2023
Cited by 2 | Viewed by 1532
Abstract
In this paper, a new criterion for the evaluation of the theoretical efficiency of a greedy algorithm is suggested. Using this criterion, we prove some results on the rate of convergence of greedy algorithms, which provide expansions. We consider both the case of [...] Read more.
In this paper, a new criterion for the evaluation of the theoretical efficiency of a greedy algorithm is suggested. Using this criterion, we prove some results on the rate of convergence of greedy algorithms, which provide expansions. We consider both the case of Hilbert spaces and the more general case of Banach spaces. The new component of this paper is that we bound the error of approximation by the product of two norms—the norm of f and the A1-norm of f. Typically, only the A1-norm of f is used. In particular, we establish that some greedy algorithms (Pure Greedy Algorithm (PGA) and its modifications) are as good as the Orthogonal Greedy Algorithm (OGA) in this new sense of the rate of convergence, while it is known that the PGA is much worse than the OGA in the standard sense. Our new results provide better bounds for the accuracy than known results in the case of small ∥f∥. Full article
(This article belongs to the Special Issue Fourier Analysis, Approximation Theory and Applications)
19 pages, 484 KiB  
Article
Graphical Local Genetic Algorithm for High-Dimensional Log-Linear Models
by Lyndsay Roach and Xin Gao
Mathematics 2023, 11(11), 2514; https://doi.org/10.3390/math11112514 - 30 May 2023
Cited by 2 | Viewed by 2274
Abstract
Graphical log-linear models are effective for representing complex structures that emerge from high-dimensional data. It is challenging to fit an appropriate model in the high-dimensional setting and many existing methods rely on a convenient class of models, called decomposable models, which lend well [...] Read more.
Graphical log-linear models are effective for representing complex structures that emerge from high-dimensional data. It is challenging to fit an appropriate model in the high-dimensional setting and many existing methods rely on a convenient class of models, called decomposable models, which lend well to a stepwise approach. However, these methods restrict the pool of candidate models from which they can search, and these methods are difficult to scale. It can be shown that a non-decomposable model can be approximated by the decomposable model which is its minimal triangulation, thus extending the convenient computational properties of decomposable models to any model. In this paper, we propose a local genetic algorithm with a crossover-hill-climbing operator, adapted for log-linear graphical models. We show that the graphical local genetic algorithm can be used successfully to fit non-decomposable models for both a low number of variables and a high number of variables. We use the posterior probability as a measure of fitness and parallel computing to decrease the computation time. Full article
(This article belongs to the Special Issue Machine Learning and Statistical Methods to Prediction and Optimal Decision-Making)
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28 pages, 7316 KiB  
Article
Supply Chain Demand Forecasting and Price Optimisation Models with Substitution Effect
by Keun Hee Lee, Mali Abdollahian, Sergei Schreider and Sona Taheri
Mathematics 2023, 11(11), 2502; https://doi.org/10.3390/math11112502 - 29 May 2023
Cited by 3 | Viewed by 8341
Abstract
Determining the optimal price of products is essential, as it plays a critical role in improving a company’s profitability and market competitiveness. This requires the ability to calculate customers’ demand in the Fast Moving Consumer Goods (FMCG) industry as various effects exist between [...] Read more.
Determining the optimal price of products is essential, as it plays a critical role in improving a company’s profitability and market competitiveness. This requires the ability to calculate customers’ demand in the Fast Moving Consumer Goods (FMCG) industry as various effects exist between multiple products within a product category. The substitution effect is one of the challenging effects at retail stores, as it requires investigating an exponential number of combinations of price changes and the availability of other products. This paper suggests a systematic price decision support tool for demand prediction and price optimise in online and stationary retailers considering the substitution effect. Two procedures reflecting the product price changes and the demand correlation structure are introduced for demand prediction and price optimisation models. First, the developed demand prediction procedure is carried out considering the combination of price changes of all products reflecting the effect of substitution. Time series and different well-known machine learning approaches with hyperparameter tuning and rolling forecasting methods are utilised to select each product’s best demand forecast. Demand forecast results are used as input in the price optimisation model. Second, the developed price optimisation procedure is a constraint programming problem based on a week time frame and a product category level aggregation and is capable of maximising profit out of the many price combinations. The results using real-world transaction data with 12 products and 4 discount rates demonstrate that including some business rules as constraints in the proposed price optimisation model reduces the number of price combinations from 11,274,924 to 19,440 and execution time from 129.59 to 25.831 min. The utilisation of the presented price optimisation support tool enables the supply chain managers to identify the optimal discount rate for individual products in a timely manner, resulting in a net profit increase. Full article
(This article belongs to the Special Issue Advances of Machine Learning and Data Mining Using Mathematical Optimization in Honor of the 65th Birthday of Prof. Adil M. Bagirov)
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10 pages, 267 KiB  
Article
On Ulam Stability of a Partial Differential Operator in Banach Spaces
by Adela Novac, Diana Otrocol and Dorian Popa
Mathematics 2023, 11(11), 2488; https://doi.org/10.3390/math11112488 - 28 May 2023
Viewed by 1135
Abstract
In this paper, we prove that, if infx∈A|f(x)|=m>0, then the partial differential operator D defined by [...] Read more.
In this paper, we prove that, if infx∈A|f(x)|=m>0, then the partial differential operator D defined by D(u)=∑k=1nfk∂u∂xk−fu, where f,fi∈C(A,R),u∈C1(A,X),i=1,…,n,I⊂R is an interval, A=I×Rn−1 and X is a Banach space, is Ulam stable with the Ulam constant K=1m. Moreover, if infx∈A|f(x)|=0, we prove that D is not generally Ulam stable. Full article
(This article belongs to the Section Difference and Differential Equations)
20 pages, 2248 KiB  
Review
A Review of High-Performance Computing Methods for Power Flow Analysis
by Shadi G. Alawneh, Lei Zeng and Seyed Ali Arefifar
Mathematics 2023, 11(11), 2461; https://doi.org/10.3390/math11112461 - 26 May 2023
Cited by 5 | Viewed by 2576
Abstract
Power flow analysis is critical for power systems due to the development of multiple energy supplies. For safety, stability, and real-time response in grid operation, grid planning, and analysis of power systems, it requires designing high-performance computing methods, accelerating power flow calculation, obtaining [...] Read more.
Power flow analysis is critical for power systems due to the development of multiple energy supplies. For safety, stability, and real-time response in grid operation, grid planning, and analysis of power systems, it requires designing high-performance computing methods, accelerating power flow calculation, obtaining the voltage magnitude and phase angle of buses inside the power system, and coping with the increasingly complex large-scale power system. This paper provides an overview of the available parallel methods to fix the issues. Specifically, these methods can be classified into three categories from a hardware perspective: multi-cores, hybrid CPU-GPU architecture, and FPGA. In addition, from the perspective of numerical computation, the power flow algorithm is generally classified into iterative and direct methods. This review paper introduces models of power flow and hardware computing architectures and then compares their performance in parallel power flow calculations depending on parallel numerical methods on different computing platforms. Furthermore, this paper analyzes the challenges and pros and cons of these methods and provides guidance on how to exploit the parallelism of future power flow applications. Full article
(This article belongs to the Section Computational and Applied Mathematics)
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19 pages, 386 KiB  
Article
A Mathematical Interpretation of Autoregressive Generative Pre-Trained Transformer and Self-Supervised Learning
by Minhyeok Lee
Mathematics 2023, 11(11), 2451; https://doi.org/10.3390/math11112451 - 25 May 2023
Cited by 14 | Viewed by 7632
Abstract
In this paper, we present a rigorous mathematical examination of generative pre-trained transformer (GPT) models and their autoregressive self-supervised learning mechanisms. We begin by defining natural language space and knowledge space, which are two key concepts for understanding the dimensionality reduction process in [...] Read more.
In this paper, we present a rigorous mathematical examination of generative pre-trained transformer (GPT) models and their autoregressive self-supervised learning mechanisms. We begin by defining natural language space and knowledge space, which are two key concepts for understanding the dimensionality reduction process in GPT-based large language models (LLMs). By exploring projection functions and their inverses, we establish a framework for analyzing the language generation capabilities of these models. We then investigate the GPT representation space, examining its implications for the models’ approximation properties. Finally, we discuss the limitations and challenges of GPT models and their learning mechanisms, considering trade-offs between complexity and generalization, as well as the implications of incomplete inverse projection functions. Our findings demonstrate that GPT models possess the capability to encode knowledge into low-dimensional vectors through their autoregressive self-supervised learning mechanism. This comprehensive analysis provides a solid mathematical foundation for future advancements in GPT-based LLMs, promising advancements in natural language processing tasks such as language translation, text summarization, and question answering due to improved understanding and optimization of model training and performance. Full article
(This article belongs to the Special Issue Current Trends in Natural Language Processing (NLP) and Human Language Technology (HLT))
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13 pages, 532 KiB  
Article
Spectral Analysis of the Infinite-Dimensional Sonic Drillstring Dynamics
by Kaïs Ammari and Lotfi Beji
Mathematics 2023, 11(11), 2426; https://doi.org/10.3390/math11112426 - 24 May 2023
Viewed by 1015
Abstract
By deploying sonic drilling for soil structure fracturing in the presence of consolidated/ unconsolidated formations, this technique greatly reduces the friction on the drillstring and bit by using energetic resonance, a bit-bouncing high-frequency axial vibration. While resonance must be avoided, to our knowledge, [...] Read more.
By deploying sonic drilling for soil structure fracturing in the presence of consolidated/ unconsolidated formations, this technique greatly reduces the friction on the drillstring and bit by using energetic resonance, a bit-bouncing high-frequency axial vibration. While resonance must be avoided, to our knowledge, drilling is the only application area where resonance is necessary to break up the rocks. The problem is that the machine’s tool can encounter several different geological layers with many varieties of density. Hence, keeping the resonance of the tool plays an important role in drill processes, especially in tunnel or infrastructure shoring. In this paper, we analyze the sonic drillstring dynamics as an infinite-dimensional system from another viewpoint using the frequency domain approach. From the operator theory in defining the adequate function spaces, we show the system well-posedness. The hydraulic produced axial force that should preserve the resonant drillstring mode is defined from the spectrum study of the constructed linear operator guided by the ratio control from the top to tip boundary magnitudes. Full article
(This article belongs to the Special Issue Advances in Complex Systems and Their Control Principles)
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23 pages, 797 KiB  
Article
HAP-Assisted RSMA-Enabled Vehicular Edge Computing: A DRL-Based Optimization Framework
by Tri-Hai Nguyen and Laihyuk Park
Mathematics 2023, 11(10), 2376; https://doi.org/10.3390/math11102376 - 19 May 2023
Cited by 8 | Viewed by 1717
Abstract
In recent years, the demand for vehicular edge computing (VEC) has grown rapidly due to the increasing need for low-latency and high-throughput applications such as autonomous driving and smart transportation systems. Nevertheless, offering VEC services in rural locations remains a difficulty owing to [...] Read more.
In recent years, the demand for vehicular edge computing (VEC) has grown rapidly due to the increasing need for low-latency and high-throughput applications such as autonomous driving and smart transportation systems. Nevertheless, offering VEC services in rural locations remains a difficulty owing to a lack of network facilities. We tackle this issue by taking advantage of high-altitude platforms (HAPs) and rate-splitting multiple access (RSMA) techniques to propose an HAP-assisted RSMA-enabled VEC system, which can enhance connectivity and provide computational capacity in rural locations. We also introduce a deep deterministic policy gradient (DDPG)-based framework that optimizes the allocation of resources and task offloading by jointly considering the offloading rate, splitting rate, transmission power, and decoding order parameters. Via results from extensive simulations, the proposed framework shows superior performance in comparison with conventional schemes regarding task success rate and energy consumption. Full article
(This article belongs to the Special Issue Intelligent Systems, Communications, and Networks for the Next-Generation Internet of Things)
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13 pages, 289 KiB  
Article
Some Conformal Transformations on Finsler Warped Product Manifolds
by Yuze Ren, Xiaoling Zhang and Lili Zhao
Mathematics 2023, 11(10), 2361; https://doi.org/10.3390/math11102361 - 18 May 2023
Viewed by 1184
Abstract
The conformal transformation, which preserves Einstein metrics on Finsler warped product manifolds, is studied in this paper. We obtain sufficient and necessary conditions of a conformal transformation preserving Einstein metrics. In addition, we provide nontrivial examples of conformal transformations. Furthermore, we completely classify [...] Read more.
The conformal transformation, which preserves Einstein metrics on Finsler warped product manifolds, is studied in this paper. We obtain sufficient and necessary conditions of a conformal transformation preserving Einstein metrics. In addition, we provide nontrivial examples of conformal transformations. Furthermore, we completely classify Einstein Riemannian warped product metrics and obtain the existence of a nontrivial conformal transformation that preserves Einstein metrics. Full article
(This article belongs to the Section Algebra, Geometry and Topology)
17 pages, 436 KiB  
Article
A Mathematical Investigation of Hallucination and Creativity in GPT Models
by Minhyeok Lee
Mathematics 2023, 11(10), 2320; https://doi.org/10.3390/math11102320 - 16 May 2023
Cited by 38 | Viewed by 9431
Abstract
In this paper, we present a comprehensive mathematical analysis of the hallucination phenomenon in generative pretrained transformer (GPT) models. We rigorously define and measure hallucination and creativity using concepts from probability theory and information theory. By introducing a parametric family of GPT models, [...] Read more.
In this paper, we present a comprehensive mathematical analysis of the hallucination phenomenon in generative pretrained transformer (GPT) models. We rigorously define and measure hallucination and creativity using concepts from probability theory and information theory. By introducing a parametric family of GPT models, we characterize the trade-off between hallucination and creativity and identify an optimal balance that maximizes model performance across various tasks. Our work offers a novel mathematical framework for understanding the origins and implications of hallucination in GPT models and paves the way for future research and development in the field of large language models (LLMs). Full article
(This article belongs to the Special Issue Deep Learning for Natural Language Processing: Advances and Challenges)
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15 pages, 316 KiB  
Article
Formulation of Impulsive Ecological Systems Using the Conformable Calculus Approach: Qualitative Analysis
by Anatoliy Martynyuk, Gani Stamov, Ivanka Stamova and Ekaterina Gospodinova
Mathematics 2023, 11(10), 2221; https://doi.org/10.3390/math11102221 - 9 May 2023
Cited by 2 | Viewed by 1279
Abstract
In this paper, an impulsive conformable fractional Lotka–Volterra model with dispersion is introduced. Since the concept of conformable derivatives avoids some limitations of the classical fractional-order derivatives, it is more suitable for applied problems. The impulsive control approach which is common for population [...] Read more.
In this paper, an impulsive conformable fractional Lotka–Volterra model with dispersion is introduced. Since the concept of conformable derivatives avoids some limitations of the classical fractional-order derivatives, it is more suitable for applied problems. The impulsive control approach which is common for population dynamics’ models is applied and fixed moments impulsive perturbations are considered. The combined concept of practical stability with respect to manifolds is adapted to the introduced model. Sufficient conditions for boundedness and generalized practical stability of the solutions are obtained by using an analogue of the Lyapunov function method. The uncertain case is also studied. Examples are given to demonstrate the effectiveness of the established results. Full article
(This article belongs to the Special Issue Stability Analysis of Fractional Systems-II)
25 pages, 954 KiB  
Article
From Cell–Cell Interaction to Stochastic and Deterministic Descriptions of a Cancer–Immune System Competition Model
by Gabriel Morgado, Annie Lemarchand and Carlo Bianca
Mathematics 2023, 11(9), 2188; https://doi.org/10.3390/math11092188 - 6 May 2023
Cited by 2 | Viewed by 1650
Abstract
We consider a cell–cell interaction model of competition between cancer cells and immune system cells, first introduced in the framework of the thermostatted kinetic theory, and derive a master equation for the probability of the number of cancer cells and immune system cells [...] Read more.
We consider a cell–cell interaction model of competition between cancer cells and immune system cells, first introduced in the framework of the thermostatted kinetic theory, and derive a master equation for the probability of the number of cancer cells and immune system cells for a given activity. Macroscopic deterministic equations for the concentrations and mean activities of cancer cells and immune system cells are deduced from the kinetic equations. The conditions for which the 3Es of immunotherapy (elimination, equilibrium, and escape) are reproduced are discussed. Apparent elimination of cancer followed by a long pseudo-equilibrium phase and the eventual escape of cancer from the control of the immune system are observed in the three descriptions. The macroscopic equations provide an analytical approach to the transition observed in the simulations of both the kinetic equations and the master equation. For efficient control of activity fluctuations, the steady states associated with the elimination of either cancer or immune system disappear and are replaced by a steady state in which cancer is controlled by the immune system. Full article
(This article belongs to the Section Mathematical Biology)
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14 pages, 739 KiB  
Article
A Depth-Progressive Initialization Strategy for Quantum Approximate Optimization Algorithm
by Xinwei Lee, Ningyi Xie, Dongsheng Cai, Yoshiyuki Saito and Nobuyoshi Asai
Mathematics 2023, 11(9), 2176; https://doi.org/10.3390/math11092176 - 5 May 2023
Cited by 7 | Viewed by 1866
Abstract
The quantum approximate optimization algorithm (QAOA) is known for its capability and universality in solving combinatorial optimization problems on near-term quantum devices. The results yielded by QAOA depend strongly on its initial variational parameters. Hence, parameter selection for QAOA becomes an active area [...] Read more.
The quantum approximate optimization algorithm (QAOA) is known for its capability and universality in solving combinatorial optimization problems on near-term quantum devices. The results yielded by QAOA depend strongly on its initial variational parameters. Hence, parameter selection for QAOA becomes an active area of research, as bad initialization might deteriorate the quality of the results, especially at great circuit depths. We first discuss the patterns of optimal parameters in QAOA in two directions: the angle index and the circuit depth. Then, we discuss the symmetries and periodicity of the expectation that is used to determine the bounds of the search space. Based on the patterns in optimal parameters and the bounds restriction, we propose a strategy that predicts the new initial parameters by taking the difference between the previous optimal parameters. Unlike most other strategies, the strategy we propose does not require multiple trials to ensure success. It only requires one prediction when progressing to the next depth. We compare this strategy with our previously proposed strategy and the layerwise strategy for solving the Max-cut problem in terms of the approximation ratio and the optimization cost. We also address the non-optimality in previous parameters, which is seldom discussed in other works despite its importance in explaining the behavior of variational quantum algorithms. Full article
(This article belongs to the Special Issue Advances in Quantum Computing and Applications)
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12 pages, 2790 KiB  
Article
Antithetic Power Transformation in Monte Carlo Simulation: Correcting Hidden Errors in the Response Variable
by Dennis Ridley and Pierre Ngnepieba
Mathematics 2023, 11(9), 2097; https://doi.org/10.3390/math11092097 - 28 Apr 2023
Cited by 1 | Viewed by 1238
Abstract
Monte Carlo simulation is performed with uniformly distributed U(0,1) pseudo-random numbers. Because the numbers are generated from a mathematical formula, they will contain some serial correlation, even if very small. This serial correlation becomes embedded in the correlation structure of the response variable. [...] Read more.
Monte Carlo simulation is performed with uniformly distributed U(0,1) pseudo-random numbers. Because the numbers are generated from a mathematical formula, they will contain some serial correlation, even if very small. This serial correlation becomes embedded in the correlation structure of the response variable. The response variable becomes an asynchronous time series. This leads to hidden errors in the response variable. The purpose of this paper is to illustrate how this happens and how it can be corrected. The method is demonstrated for the case of a simple queue for which the time in the system is known exactly from theory. The paper derives the correlation between an exponential random variable and its antithetic counterpart obtained by power transform with an infinitesimal negative exponent. Full article
(This article belongs to the Special Issue Modelling and Analysis in Time Series and Econometrics)
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18 pages, 10054 KiB  
Article
Stock Price Prediction Using CNN-BiLSTM-Attention Model
by Jilin Zhang, Lishi Ye and Yongzeng Lai
Mathematics 2023, 11(9), 1985; https://doi.org/10.3390/math11091985 - 23 Apr 2023
Cited by 21 | Viewed by 11680
Abstract
Accurate stock price prediction has an important role in stock investment. Because stock price data are characterized by high frequency, nonlinearity, and long memory, predicting stock prices precisely is challenging. Various forecasting methods have been proposed, from classical time series methods to machine-learning-based [...] Read more.
Accurate stock price prediction has an important role in stock investment. Because stock price data are characterized by high frequency, nonlinearity, and long memory, predicting stock prices precisely is challenging. Various forecasting methods have been proposed, from classical time series methods to machine-learning-based methods, such as random forest (RF), recurrent neural network (RNN), convolutional neural network (CNN), Long Short-Term Memory (LSTM) neural networks and their variants, etc. Each method can reach a certain level of accuracy but also has its limitations. In this paper, a CNN-BiLSTM-Attention-based model is proposed to boost the accuracy of predicting stock prices and indices. First, the temporal features of sequence data are extracted using a convolutional neural network (CNN) and bi-directional long and short-term memory (BiLSTM) network. Then, an attention mechanism is introduced to fit weight assignments to the information features automatically; and finally, the final prediction results are output through the dense layer. The proposed method was first used to predict the price of the Chinese stock index—the CSI300 index and was found to be more accurate than any of the other three methods—LSTM, CNN-LSTM, CNN-LSTM-Attention. In order to investigate whether the proposed model is robustly effective in predicting stock indices, three other stock indices in China and eight international stock indices were selected to test, and the robust effectiveness of the CNN-BiLSTM-Attention model in predicting stock prices was confirmed. Comparing this method with the LSTM, CNN-LSTM, and CNN-LSTM-Attention models, it is found that the accuracy of stock price prediction is highest using the CNN-BiLSTM-Attention model in almost all cases. Full article
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10 pages, 1766 KiB  
Article
Phi, Fei, Fo, Fum: Effect Sizes for Categorical Data That Use the Chi-Squared Statistic
by Mattan S. Ben-Shachar, Indrajeet Patil, Rémi Thériault, Brenton M. Wiernik and Daniel Lüdecke
Mathematics 2023, 11(9), 1982; https://doi.org/10.3390/math11091982 - 22 Apr 2023
Cited by 10 | Viewed by 6071
Abstract
In both theoretical and applied research, it is often of interest to assess the strength of an observed association. Existing guidelines also frequently recommend going beyond null-hypothesis significance testing and reporting effect sizes and their confidence intervals. As such, measures of effect sizes [...] Read more.
In both theoretical and applied research, it is often of interest to assess the strength of an observed association. Existing guidelines also frequently recommend going beyond null-hypothesis significance testing and reporting effect sizes and their confidence intervals. As such, measures of effect sizes are increasingly reported, valued, and understood. Beyond their value in shaping the interpretation of the results from a given study, reporting effect sizes is critical for meta-analyses, which rely on their aggregation. We review the most common effect sizes for analyses of categorical variables that use the χ2 (chi-square) statistic and introduce a new effect size—פ (Fei, pronounced “fay”). We demonstrate the implementation of these measures and their confidence intervals via the effectsize package in the R programming language. Full article
(This article belongs to the Special Issue Advances in Statistical Computing)
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18 pages, 3920 KiB  
Article
Multi-Scale Annulus Clustering for Multi-Label Classification
by Yan Liu, Changshun Liu, Jingjing Song, Xibei Yang, Taihua Xu and Pingxin Wang
Mathematics 2023, 11(8), 1969; https://doi.org/10.3390/math11081969 - 21 Apr 2023
Cited by 2 | Viewed by 1379
Abstract
Label-specific feature learning has become a hot topic as it induces classification models by accounting for the underlying features of each label. Compared with single-label annotations, multi-label annotations can describe samples from more comprehensive perspectives. It is generally believed that the compelling classification [...] Read more.
Label-specific feature learning has become a hot topic as it induces classification models by accounting for the underlying features of each label. Compared with single-label annotations, multi-label annotations can describe samples from more comprehensive perspectives. It is generally believed that the compelling classification features of a data set often exist in the aggregation of label distribution. In this in-depth study of a multi-label data set, we find that the distance between all samples and the sample center is a Gaussian distribution, which means that the label distribution has the tendency to cluster from the center and spread to the surroundings. Accordingly, the double annulus field based on this distribution trend, named DEPT for double annulusfield and label-specific features for multi-label classification, is proposed in this paper. The double annulus field emphasizes that samples of a specific size can reflect some unique features of the data set. Through intra-annulus clustering for each layer of annuluses, the distinctive feature space of these labels is captured and formed. Then, the final classification model is obtained by training the feature space. Contrastive experiments on 10 benchmark multi-label data sets verify the effectiveness of the proposed algorithm. Full article
(This article belongs to the Section Fuzzy Sets, Systems and Decision Making)
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11 pages, 303 KiB  
Article
Generalized Halanay Inequalities and Relative Application to Time-Delay Dynamical Systems
by Chunsheng Wang, Xiangdong Liu, Feng Jiao, Hong Mai, Han Chen and Runpeng Lin
Mathematics 2023, 11(8), 1940; https://doi.org/10.3390/math11081940 - 20 Apr 2023
Cited by 9 | Viewed by 1207
Abstract
A class of generalized Halanay inequalities is studied via the Banach fixed point method and comparison principle. The conditions to ensure the boundedness and stability of the zero solution are obtained in this study. This research provides a new approach to the study [...] Read more.
A class of generalized Halanay inequalities is studied via the Banach fixed point method and comparison principle. The conditions to ensure the boundedness and stability of the zero solution are obtained in this study. This research provides a new approach to the study of the boundedness and stability of Halanay inequality. Numerical examples and simulation results verify the validity and superiority of the conclusions obtained in this study. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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24 pages, 4680 KiB  
Article
AdaBoost Algorithm Could Lead to Weak Results for Data with Certain Characteristics
by Olivér Hornyák and László Barna Iantovics
Mathematics 2023, 11(8), 1801; https://doi.org/10.3390/math11081801 - 10 Apr 2023
Cited by 13 | Viewed by 2546
Abstract
There are many state-of-the-art algorithms presented in the literature that perform very well on some evaluation data but are not studied with the data properties on which they are applied; therefore, they could have low performance on data with other characteristics. In this [...] Read more.
There are many state-of-the-art algorithms presented in the literature that perform very well on some evaluation data but are not studied with the data properties on which they are applied; therefore, they could have low performance on data with other characteristics. In this paper, the results of comprehensive research regarding the prediction with the frequently applied AdaBoost algorithm on real-world sensor data are presented. The chosen dataset has some specific characteristics, and it contains error and failure data of several machines and their components. The research aims to investigate whether the AdaBoost algorithm has the capability of predicting failures, thus providing the necessary information for monitoring and condition-based maintenance (CBM). The dataset is analyzed, and the principal characteristics are presented. Performance evaluations of the AdaBoost algorithm that we present show a prediction capability below expectations for this algorithm. The specificity of this study is that it indicates the limitation of the AdaBoost algorithm, which could perform very well on some data, but not so well on others. Based on this research and some others that we performed, and actual research from worldwide studies, we must outline that the mathematical analysis of the data is especially important to develop or adapt algorithms to be very efficient. Full article
(This article belongs to the Special Issue Industrial Big Data and Process Modelling for Smart Manufacturing)
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12 pages, 611 KiB  
Article
Non-Associative Structures and Their Applications in Differential Equations
by Yakov Krasnov
Mathematics 2023, 11(8), 1790; https://doi.org/10.3390/math11081790 - 9 Apr 2023
Cited by 3 | Viewed by 1939
Abstract
This article establishes a connection between nonlinear DEs and linear PDEs on the one hand, and non-associative algebra structures on the other. Such a connection simplifies the formulation of many results of DEs and the methods of their solution. The main link between [...] Read more.
This article establishes a connection between nonlinear DEs and linear PDEs on the one hand, and non-associative algebra structures on the other. Such a connection simplifies the formulation of many results of DEs and the methods of their solution. The main link between these theories is the nonlinear spectral theory developed for algebra and homogeneous differential equations. A nonlinear spectral method is used to prove the existence of an algebraic first integral, interpretations of various phase zones, and the separatrices construction for ODEs. In algebra, the same methods exploit subalgebra construction and explain fusion rules. In conclusion, perturbation methods may also be interpreted for near-Jordan algebra construction. Full article
(This article belongs to the Special Issue Non-associative Structures and Their Applications in Physics and Geometry)
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20 pages, 1646 KiB  
Article
Output Feedback Robust Tracking Control for a Variable-Speed Pump-Controlled Hydraulic System Subject to Mismatched Uncertainties
by Manh Hung Nguyen and Kyoung Kwan Ahn
Mathematics 2023, 11(8), 1783; https://doi.org/10.3390/math11081783 - 8 Apr 2023
Cited by 9 | Viewed by 1743
Abstract
In this paper, a novel simple, but effective output feedback robust control (OFRC) for achieving a highly accurate position tracking of a pump-controlled electro-hydraulic system is presented. To cope with the unavailability of all system state information, an extended state observer (ESO) was [...] Read more.
In this paper, a novel simple, but effective output feedback robust control (OFRC) for achieving a highly accurate position tracking of a pump-controlled electro-hydraulic system is presented. To cope with the unavailability of all system state information, an extended state observer (ESO) was adopted to estimate the angular velocity and load-pressure-related state variable of the actuator and total matched disturbance, which enters the system through the same channel as the control input in the system dynamics. In addition, for the first time, another ESO acting as a disturbance observer (DOB) was skillfully integrated to effectively compensate for the adverse effects of the lumped mismatched uncertainty caused by parameter perturbation and external loads in the velocity dynamics. Then, a dynamic surface-control-based backstepping controller (DSC-BC) based on the constructed ESOs for the tracking control of the studied electro-hydraulic system was synthesized to guarantee that the system output closely tracks the desired trajectory and avoid the inherent computational burden of the conventional backstepping method because of repetitive analytical derivative calculation at each backstepping iteration. Furthermore, the stability of the two observes and overall closed-loop system was verified by using the Lyapunov theory. Finally, several extensive comparative experiments were carried out to demonstrate the advantage of the recommended control approach in comparison with some reference control methods. Full article
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54 pages, 3508 KiB  
Review
Auto-Encoders in Deep Learning—A Review with New Perspectives
by Shuangshuang Chen and Wei Guo
Mathematics 2023, 11(8), 1777; https://doi.org/10.3390/math11081777 - 7 Apr 2023
Cited by 73 | Viewed by 22571
Abstract
Deep learning, which is a subfield of machine learning, has opened a new era for the development of neural networks. The auto-encoder is a key component of deep structure, which can be used to realize transfer learning and plays an important role in [...] Read more.
Deep learning, which is a subfield of machine learning, has opened a new era for the development of neural networks. The auto-encoder is a key component of deep structure, which can be used to realize transfer learning and plays an important role in both unsupervised learning and non-linear feature extraction. By highlighting the contributions and challenges of recent research papers, this work aims to review state-of-the-art auto-encoder algorithms. Firstly, we introduce the basic auto-encoder as well as its basic concept and structure. Secondly, we present a comprehensive summarization of different variants of the auto-encoder. Thirdly, we analyze and study auto-encoders from three different perspectives. We also discuss the relationships between auto-encoders, shallow models and other deep learning models. The auto-encoder and its variants have successfully been applied in a wide range of fields, such as pattern recognition, computer vision, data generation, recommender systems, etc. Then, we focus on the available toolkits for auto-encoders. Finally, this paper summarizes the future trends and challenges in designing and training auto-encoders. We hope that this survey will provide a good reference when using and designing AE models. Full article
(This article belongs to the Special Issue Mathematical Methods and Applications for Artificial Intelligence and Computer Vision)
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12 pages, 402 KiB  
Article
On Optimal Embeddings in 3-Ary n-Cubes
by S. Rajeshwari and M. Rajesh
Mathematics 2023, 11(7), 1711; https://doi.org/10.3390/math11071711 - 3 Apr 2023
Cited by 2 | Viewed by 1403
Abstract
The efficiency of a graph embedding problem when simulating one interconnection network in another interconnection network is characterized by the influential parameter of wirelength. Obtaining the minimum wirelength in an embedding problem determines the quality of that embedding. In this paper, we obtained [...] Read more.
The efficiency of a graph embedding problem when simulating one interconnection network in another interconnection network is characterized by the influential parameter of wirelength. Obtaining the minimum wirelength in an embedding problem determines the quality of that embedding. In this paper, we obtained the convex edge partition of 3-Ary n-Cubes and the minimized wirelength of the embeddings of both 3-Ary n-Cubes and circulant networks. Full article
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18 pages, 446 KiB  
Article
Hybrid Impulsive Pinning Control for Mean Square Synchronization of Uncertain Multi-Link Complex Networks with Stochastic Characteristics and Hybrid Delays
by Yong Tang, Lang Zhou, Jiahui Tang, Yue Rao, Hongguang Fan and Jihong Zhu
Mathematics 2023, 11(7), 1697; https://doi.org/10.3390/math11071697 - 2 Apr 2023
Cited by 25 | Viewed by 1821
Abstract
This study explores the synchronization issue for uncertain multi-link complex networks incorporating stochastic characteristics and hybrid delays. Unlike previous works, internal delays, coupling delays, and stochastic delays considered in our model change over time; meanwhile, the impulse strength and position change with time [...] Read more.
This study explores the synchronization issue for uncertain multi-link complex networks incorporating stochastic characteristics and hybrid delays. Unlike previous works, internal delays, coupling delays, and stochastic delays considered in our model change over time; meanwhile, the impulse strength and position change with time evolution. To actualize network synchronization, a strategy called hybrid impulsive pinning control is applied, which combines the virtue of impulsive control and pinning control as well as two categories of impulses (i.e., synchronization and desynchronization). By decomposing the complicated topological structures into diagonal items and off-diagonal items, multiple nonlinear coupling terms are linearly decomposed in the process of theoretical analysis. Combining inequality technology and matrix decomposition theory, several novel synchronization criteria have been gained to ensure synchronization for the concerning multi-link model. The criteria get in touch with the uncertain strengths, coupling strengths, hybrid impulse strengths, delay sizes, impulsive intervals, and network topologies. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)
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16 pages, 1170 KiB  
Article
A Novel Fractional-Order RothC Model
by Vsevolod Bohaienko, Fasma Diele, Carmela Marangi, Cristiano Tamborrino, Sebastian Aleksandrowicz and Edyta Woźniak
Mathematics 2023, 11(7), 1677; https://doi.org/10.3390/math11071677 - 31 Mar 2023
Viewed by 2245
Abstract
A new fractional q-order variation of the RothC model for the dynamics of soil organic carbon is introduced. A computational method based on the discretization of the analytic solution along with the finite-difference technique are suggested and the stability results for the [...] Read more.
A new fractional q-order variation of the RothC model for the dynamics of soil organic carbon is introduced. A computational method based on the discretization of the analytic solution along with the finite-difference technique are suggested and the stability results for the latter are given. The accuracy of the scheme, in terms of the temporal step size h, is confirmed through numerical testing of a constructed analytic solution. The effectiveness of the proposed discrete method is compared with that of the classical discrete RothC model. Results from real-world experiments show that, by adjusting the fractional order q and the multiplier term ζ(t,q), a better match between simulated and actual data can be achieved compared to the traditional integer-order model. Full article
(This article belongs to the Special Issue Differential Equation Models in Applied Mathematics: Theoretical and Numerical Challenges-II)
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