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Math. Comput. Appl., Volume 29, Issue 3 (June 2024) – 17 articles

Cover Story (view full-size image): Nuclear abnormalities in avian erythrocytes have been used as biomarkers of genotoxicity in several species. Deep learning can be used in this context to improve standardization in identifying biological configurations of medical and veterinary importance. In this study, we present a deep learning model for identifying and classifying abnormal shapes in erythrocyte nuclei of the Kelp Gull. We trained convolutional neural networks (ResNet34 and ResNet50) to obtain models capable of detecting and classifying these abnormalities. The analysis was performed at three discrimination levels of classification, with broad categories subdivided into increasingly specific subcategories. The results evidenced a fast, efficient and standardized approach that could be replicated in similar contexts. View this paper
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15 pages, 499 KiB  
Article
Dynamic Mechanism Design for Repeated Markov Games with Hidden Actions: Computational Approach
by Julio B. Clempner
Math. Comput. Appl. 2024, 29(3), 46; https://doi.org/10.3390/mca29030046 - 10 Jun 2024
Viewed by 510
Abstract
This paper introduces a dynamic mechanism design tailored for uncertain environments where incentive schemes are challenged by the inability to observe players’ actions, known as moral hazard. In these scenarios, the system operates as a Markov game where outcomes depend on both the [...] Read more.
This paper introduces a dynamic mechanism design tailored for uncertain environments where incentive schemes are challenged by the inability to observe players’ actions, known as moral hazard. In these scenarios, the system operates as a Markov game where outcomes depend on both the state of payouts and players’ actions. Moral hazard and adverse selection further complicate decision-making. The proposed mechanism aims to incentivize players to truthfully reveal their states while maximizing their expected payoffs. This is achieved through players’ best-reply strategies, ensuring truthful state revelation despite moral hazard. The revelation principle, a core concept in mechanism design, is applied to models with both moral hazard and adverse selection, facilitating optimal reward structure identification. The research holds significant practical implications, addressing the challenge of designing reward structures for multiplayer Markov games with hidden actions. By utilizing dynamic mechanism design, researchers and practitioners can optimize incentive schemes in complex, uncertain environments affected by moral hazard. To demonstrate the approach, the paper includes a numerical example of solving an oligopoly problem. Oligopolies, with a few dominant market players, exhibit complex dynamics where individual actions impact market outcomes significantly. Using the dynamic mechanism design framework, the paper shows how to construct optimal reward structures that align players’ incentives with desirable market outcomes, mitigating moral hazard and adverse selection effects. This framework is crucial for optimizing incentive schemes in multiplayer Markov games, providing a robust approach to handling the intricacies of moral hazard and adverse selection. By leveraging this design, the research contributes to the literature by offering a method to construct effective reward structures even in complex and uncertain environments. The numerical example of oligopolies illustrates the practical application and effectiveness of this dynamic mechanism design. Full article
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17 pages, 3654 KiB  
Article
A Comprehensive Assessment and Classification of Acute Lymphocytic Leukemia
by Payal Bose and Samir Bandyopadhyay
Math. Comput. Appl. 2024, 29(3), 45; https://doi.org/10.3390/mca29030045 - 9 Jun 2024
Viewed by 677
Abstract
Leukemia is a form of blood cancer that results in an increase in the number of white blood cells in the body. The correct identification of leukemia at any stage is essential. The current traditional approaches rely mainly on field experts’ knowledge, which [...] Read more.
Leukemia is a form of blood cancer that results in an increase in the number of white blood cells in the body. The correct identification of leukemia at any stage is essential. The current traditional approaches rely mainly on field experts’ knowledge, which is time consuming. A lengthy testing interval combined with inadequate comprehension could harm a person’s health. In this situation, an automated leukemia identification delivers more reliable and accurate diagnostic information. To effectively diagnose acute lymphoblastic leukemia from blood smear pictures, a new strategy based on traditional image analysis techniques with machine learning techniques and a composite learning approach were constructed in this experiment. The diagnostic process is separated into two parts: detection and identification. The traditional image analysis approach was utilized to identify leukemia cells from smear images. Finally, four widely recognized machine learning algorithms were used to identify the specific type of acute leukemia. It was discovered that Support Vector Machine (SVM) provides the highest accuracy in this scenario. To boost the performance, a deep learning model Resnet50 was hybridized with this model. Finally, it was revealed that this composite approach achieved 99.9% accuracy. Full article
(This article belongs to the Section Engineering)
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22 pages, 1264 KiB  
Article
Bitcoin versus S&P 500 Index: Return and Risk Analysis
by Aubain Nzokem and Daniel Maposa
Math. Comput. Appl. 2024, 29(3), 44; https://doi.org/10.3390/mca29030044 - 9 Jun 2024
Viewed by 516
Abstract
The S&P 500 Index is considered the most popular trading instrument in financial markets. With the rise of cryptocurrencies over the past few years, Bitcoin has grown in popularity and adoption. This study analyzes the daily return distribution of Bitcoin and the S&P [...] Read more.
The S&P 500 Index is considered the most popular trading instrument in financial markets. With the rise of cryptocurrencies over the past few years, Bitcoin has grown in popularity and adoption. This study analyzes the daily return distribution of Bitcoin and the S&P 500 Index and assesses their tail probabilities using two financial risk measures. As a methodology, we use Bitcoin and S&P 500 Index daily return data to fit the seven-parameter General Tempered Stable (GTS) distribution using the advanced fast fractional Fourier transform (FRFT) scheme developed by combining the fast fractional Fourier transform algorithm and the 12-point composite Newton–Cotes rule. The findings show that peakedness is the main characteristic of the S&P 500 Index return distribution, whereas heavy-tailedness is the main characteristic of Bitcoin return distribution. The GTS distribution shows that 80.05% of S&P 500 returns are within 1.06% and 1.23% against only 40.32% of Bitcoin returns. At a risk level (α), the severity of the loss (AVaRα(X)) on the left side of the distribution is larger than the severity of the profit (AVaR1α(X)) on the right side of the distribution. Compared to the S&P 500 Index, Bitcoin has 39.73% more prevalence to produce high daily returns (more than 1.23% or less than 1.06%). The severity analysis shows that, at α risk level, the average value-at-risk (AVaR(X)) of Bitcoin returns at one significant figure is four times larger than that of the S&P 500 Index returns at the same risk. Full article
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22 pages, 673 KiB  
Article
New Lie Symmetries and Exact Solutions of a Mathematical Model Describing Solute Transport in Poroelastic Materials
by Roman Cherniha, Vasyl’ Davydovych and Alla Vorobyova
Math. Comput. Appl. 2024, 29(3), 43; https://doi.org/10.3390/mca29030043 - 3 Jun 2024
Viewed by 418
Abstract
A one-dimensional model for fluid and solute transport in poroelastic materials (PEMs) is studied. Although the model was recently derived and some exact solutions, in particular steady-state solutions and their applications, were studied, special cases occurring when some parameters vanish were not analysed [...] Read more.
A one-dimensional model for fluid and solute transport in poroelastic materials (PEMs) is studied. Although the model was recently derived and some exact solutions, in particular steady-state solutions and their applications, were studied, special cases occurring when some parameters vanish were not analysed earlier. Since the governing equations are nonintegrable in nonstationary cases, the Lie symmetry method and modern tools for solving ODE systems are applied in order to construct time-dependent exact solutions. Depending on parameters arising in the governing equations, several special cases with new Lie symmetries are identified. Some of them have a highly nontrivial structure that cannot be predicted from a physical point of view or using Lie symmetries of other real-world models. Applying the symmetries obtained, multiparameter families of exact solutions are constructed, including those in terms of elementary and special functions (hypergeometric, Whittaker, Bessel and modified Bessel functions). A possible application of the solutions obtained is demonstrated, and it is shown that some exact solutions can describe (at least qualitatively) the solute transport in PEM. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems. Full article
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21 pages, 1461 KiB  
Article
DSTree: A Spatio-Temporal Indexing Data Structure for Distributed Networks
by Majid Hojati, Steven Roberts and Colin Robertson
Math. Comput. Appl. 2024, 29(3), 42; https://doi.org/10.3390/mca29030042 - 31 May 2024
Viewed by 539
Abstract
The widespread availability of tools to collect and share spatial data enables us to produce a large amount of geographic information on a daily basis. This enormous production of spatial data requires scalable data management systems. Geospatial architectures have changed from clusters to [...] Read more.
The widespread availability of tools to collect and share spatial data enables us to produce a large amount of geographic information on a daily basis. This enormous production of spatial data requires scalable data management systems. Geospatial architectures have changed from clusters to cloud architectures and more parallel and distributed processing platforms to be able to tackle these challenges. Peer-to-peer (P2P) systems as a backbone of distributed systems have been established in several application areas such as web3, blockchains, and crypto-currencies. Unlike centralized systems, data storage in P2P networks is distributed across network nodes, providing scalability and no single point of failure. However, managing and processing queries on these networks has always been challenging. In this work, we propose a spatio-temporal indexing data structure, DSTree. DSTree does not require additional Distributed Hash Trees (DHTs) to perform multi-dimensional range queries. Inserting a piece of new geographic information updates only a portion of the tree structure and does not impact the entire graph of the data. For example, for time-series data, such as storing sensor data, the DSTree performs around 40% faster in spatio-temporal queries for small and medium datasets. Despite the advantages of our proposed framework, challenges such as 20% slower insertion speed or semantic query capabilities remain. We conclude that more significant research effort from GIScience and related fields in developing decentralized applications is needed. The need for the standardization of different geographic information when sharing data on the IPFS network is one of the requirements. Full article
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13 pages, 3700 KiB  
Article
Integrating Deep Learning into Genotoxicity Biomarker Detection for Avian Erythrocytes: A Case Study in a Hemispheric Seabird
by Martín G. Frixione, Facundo Roffet, Miguel A. Adami, Marcelo Bertellotti, Verónica L. D’Amico, Claudio Delrieux and Débora Pollicelli
Math. Comput. Appl. 2024, 29(3), 41; https://doi.org/10.3390/mca29030041 - 28 May 2024
Viewed by 805
Abstract
Recently, nuclear abnormalities in avian erythrocytes have been used as biomarkers of genotoxicity in several species. Anomalous shapes are usually detected in the nuclei by means of microscopy inspection. However, due to inter- and intra-observer variability, the classification of these blood cell abnormalities [...] Read more.
Recently, nuclear abnormalities in avian erythrocytes have been used as biomarkers of genotoxicity in several species. Anomalous shapes are usually detected in the nuclei by means of microscopy inspection. However, due to inter- and intra-observer variability, the classification of these blood cell abnormalities could be problematic for replicating research. Deep learning, as a powerful image analysis technique, can be used in this context to improve standardization in identifying the biological configurations of medical and veterinary importance. In this study, we present a standardized deep learning model for identifying and classifying abnormal shapes in erythrocyte nuclei in blood smears of the hemispheric and synanthropic Kelp Gull (Larus dominicanus). We trained three convolutional backbones (ResNet34 and ResNet50 architectures) to obtain models capable of detecting and classifying these abnormalities in blood cells. The analysis was performed at three discrimination levels of classification, with broad categories subdivided into increasingly specific subcategories (level 1: “normal”, “abnormal”, “other”; level 2: “normal”, “ENAs”, “micronucleus”, “other”; level 3: “normal”, “irregular”, “displaced”, “enucleated”, “micronucleus”, “other”). The results were more than adequate and very similar in levels 1 and 2 (F1-score 84.6% and 83.6%, and accuracy 83.9% and 82.6%). In level 3, performance was lower (F1-score 65.9% and accuracy 80.8%). It can be concluded that the level 2 analysis should be considered the most appropriate as it is more specific than level 1, with similar quality of performance. This method has proven to be a fast, efficient, and standardized approach that reduces the dependence on human supervision in the classification of nuclear abnormalities in avian erythrocytes, and can be adapted to be used in similar contexts with reduced effort. Full article
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33 pages, 594 KiB  
Review
A Review on Large-Scale Data Processing with Parallel and Distributed Randomized Extreme Learning Machine Neural Networks
by Elkin Gelvez-Almeida, Marco Mora, Ricardo J. Barrientos, Ruber Hernández-García, Karina Vilches-Ponce and Miguel Vera
Math. Comput. Appl. 2024, 29(3), 40; https://doi.org/10.3390/mca29030040 - 27 May 2024
Viewed by 667
Abstract
The randomization-based feedforward neural network has raised great interest in the scientific community due to its simplicity, training speed, and accuracy comparable to traditional learning algorithms. The basic algorithm consists of randomly determining the weights and biases of the hidden layer and analytically [...] Read more.
The randomization-based feedforward neural network has raised great interest in the scientific community due to its simplicity, training speed, and accuracy comparable to traditional learning algorithms. The basic algorithm consists of randomly determining the weights and biases of the hidden layer and analytically calculating the weights of the output layer by solving a linear overdetermined system using the Moore–Penrose generalized inverse. When processing large volumes of data, randomization-based feedforward neural network models consume large amounts of memory and drastically increase training time. To efficiently solve the above problems, parallel and distributed models have recently been proposed. Previous reviews of randomization-based feedforward neural network models have mainly focused on categorizing and describing the evolution of the algorithms presented in the literature. The main contribution of this paper is to approach the topic from the perspective of the handling of large volumes of data. In this sense, we present a current and extensive review of the parallel and distributed models of randomized feedforward neural networks, focusing on extreme learning machine. In particular, we review the mathematical foundations (Moore–Penrose generalized inverse and solution of linear systems using parallel and distributed methods) and hardware and software technologies considered in current implementations. Full article
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13 pages, 7450 KiB  
Article
Numerical Solution of Natural Convection Problems Using Radial Point Interpolation Meshless (RPIM) Method Combined with Artificial-Compressibility Model
by Pranowo, Albertus Joko Santoso and Agung Tri Wijayanta
Math. Comput. Appl. 2024, 29(3), 39; https://doi.org/10.3390/mca29030039 - 20 May 2024
Viewed by 610
Abstract
A numerical method is used to solve the thermal analysis of natural convection in enclosures. This paper proposes the use of an implicit artificial-compressibility model in conjunction with the Radial Point Interpolation Meshless (RPIM) method to mimic laminar natural convective heat transport. The [...] Read more.
A numerical method is used to solve the thermal analysis of natural convection in enclosures. This paper proposes the use of an implicit artificial-compressibility model in conjunction with the Radial Point Interpolation Meshless (RPIM) method to mimic laminar natural convective heat transport. The technique couples the pressure with the velocity components using an artificial compressibility model. The RPIM is used to discretize the spatial terms of the governing equation. We solve the semi-algebraic system implicitly in backward Euler pseudo-time. The proposed method solves two test problems—natural convection in the annulus of concentric circular cylinders and trapezoidal cavity. Additionally, the results are validated using experimental and numerical data available in the literature. Excellent agreement was seen between the numerical results acquired with the suggested method and those obtained through the standard techniques found in the literature. Full article
(This article belongs to the Section Engineering)
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15 pages, 3611 KiB  
Article
Detailed Investigation of the Eddy Current and Core Losses in Coaxial Magnetic Gears through a Two-Dimensional Analytical Model
by Nikolina Nikolarea, Panteleimon Tzouganakis, Vasilios Gakos, Christos Papalexis, Antonios Tsolakis and Vasilios Spitas
Math. Comput. Appl. 2024, 29(3), 38; https://doi.org/10.3390/mca29030038 - 18 May 2024
Viewed by 644
Abstract
This work introduces a 2D model that calculates power losses in coaxial magnetic gears (CMGs). The eddy current losses of the magnets are computed analytically, whereas the core losses of the ferromagnetic segments are computed using an analytical–finite element hybrid model. The results [...] Read more.
This work introduces a 2D model that calculates power losses in coaxial magnetic gears (CMGs). The eddy current losses of the magnets are computed analytically, whereas the core losses of the ferromagnetic segments are computed using an analytical–finite element hybrid model. The results were within 1.51% and 3.18% of those obtained from an FEA for the eddy current and core losses in the CMG for an indicative inner rotor speed of 2500 rpm. In addition, the significance of the circumferential magnet segmentation is demonstrated in the CMGs. Furthermore, a parametric investigation of the efficiency of the system for different applied external loads is carried out. Finally, a mesh sensitivity analysis is performed, along with the computation of the average power losses throughout one full period, resulting in an at least 80% reduction in computational costs with a negligible effect on accuracy. The developed model could be a valuable tool for the minimization of power losses in CMGs since it combines high accuracy with a low computational cost. Full article
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27 pages, 1287 KiB  
Article
Exploring Trust Dynamics in Online Social Networks: A Social Network Analysis Perspective
by Stavroula Kridera and Andreas Kanavos
Math. Comput. Appl. 2024, 29(3), 37; https://doi.org/10.3390/mca29030037 - 15 May 2024
Viewed by 1084
Abstract
This study explores trust dynamics within online social networks, blending social science theories with advanced machine-learning (ML) techniques. We examine trust’s multifaceted nature—definitions, types, and mechanisms for its establishment and maintenance—and analyze social network structures through graph theory. Employing a diverse array of [...] Read more.
This study explores trust dynamics within online social networks, blending social science theories with advanced machine-learning (ML) techniques. We examine trust’s multifaceted nature—definitions, types, and mechanisms for its establishment and maintenance—and analyze social network structures through graph theory. Employing a diverse array of ML models (e.g., KNN, SVM, Naive Bayes, Gradient Boosting, and Neural Networks), we predict connection strengths on Facebook, focusing on model performance metrics such as accuracy, precision, recall, and F1-score. Our methodology, executed in Python using the Anaconda distribution, unveils insights into trust formation and sustainability on social media, highlighting the potent application of ML in understanding these dynamics. Challenges, including the complexity of modeling social behaviors and ethical data use concerns, are discussed, emphasizing the need for continued innovation. Our findings contribute to the discourse on trust in social networks and suggest future research directions, including the application of our methodologies to other platforms and the study of online trust over time. This work not only advances the academic understanding of digital social interactions but also offers practical implications for developers, policymakers, and online communities. Full article
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13 pages, 660 KiB  
Article
Periodic Solutions in a Simple Delay Differential Equation
by Anatoli Ivanov and Sergiy Shelyag
Math. Comput. Appl. 2024, 29(3), 36; https://doi.org/10.3390/mca29030036 - 12 May 2024
Viewed by 769
Abstract
A simple-form scalar differential equation with delay and nonlinear negative periodic feedback is considered. The existence of several types of slowly oscillating periodic solutions is shown with the same and double periods of the feedback coefficient. The periodic solutions are built explicitly in [...] Read more.
A simple-form scalar differential equation with delay and nonlinear negative periodic feedback is considered. The existence of several types of slowly oscillating periodic solutions is shown with the same and double periods of the feedback coefficient. The periodic solutions are built explicitly in the case with piecewise constant nonlinearities involved. The periodic dynamics are shown to persist under small perturbations of the equation, which make it smooth. The theoretical results are verified through extensive numerical simulations. Full article
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16 pages, 3437 KiB  
Article
Clustering of Wind Speed Time Series as a Tool for Wind Farm Diagnosis
by Ana Alexandra Martins, Daniel C. Vaz, Tiago A. N. Silva, Margarida Cardoso and Alda Carvalho
Math. Comput. Appl. 2024, 29(3), 35; https://doi.org/10.3390/mca29030035 - 9 May 2024
Viewed by 861
Abstract
In several industrial fields, environmental and operational data are acquired with numerous purposes, potentially generating a huge quantity of data containing valuable information for management actions. This work proposes a methodology for clustering time series based on the K-medoids algorithm using a convex [...] Read more.
In several industrial fields, environmental and operational data are acquired with numerous purposes, potentially generating a huge quantity of data containing valuable information for management actions. This work proposes a methodology for clustering time series based on the K-medoids algorithm using a convex combination of different time series correlation metrics, the COMB distance. The multidimensional scaling procedure is used to enhance the visualization of the clustering results, and a matrix plot display is proposed as an efficient visualization tool to interpret the COMB distance components. This is a general-purpose methodology that is intended to ease time series interpretation; however, due to the relevance of the field, this study explores the clustering of time series judiciously collected from data of a wind farm located on a complex terrain. Using the COMB distance for wind speed time bands, clustering exposes operational similarities and dissimilarities among neighboring turbines which are influenced by the turbines’ relative positions and terrain features and regarding the direction of oncoming wind. In a significant number of cases, clustering does not coincide with the natural geographic grouping of the turbines. A novel representation of the contributing distances—the COMB distance matrix plot—provides a quick way to compare pairs of time bands (turbines) regarding various features. Full article
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13 pages, 270 KiB  
Article
New Model for Hill’s Problem in the Framework of Continuation Fractional Potential
by Elbaz I. Abouelmagd
Math. Comput. Appl. 2024, 29(3), 34; https://doi.org/10.3390/mca29030034 - 2 May 2024
Viewed by 1066
Abstract
In this work, we derived a new type model for spatial Hill’s system considering the created perturbation by the parameter effect of the continuation fractional potential. The new model is considered a reduced system from the restricted three-body problem under the same effect [...] Read more.
In this work, we derived a new type model for spatial Hill’s system considering the created perturbation by the parameter effect of the continuation fractional potential. The new model is considered a reduced system from the restricted three-body problem under the same effect for describing Hill’s problem. We identified the associated Lagrangian and Hamiltonian functions of the new system, and used them to verify the existence of the new equations of motion. We also proved that the new model has different six valid solutions under different six symmetries transformations as well as the original solution, where the new model is an invariant under these transformations. The several symmetries of Hill’s model can extremely simplify the calculation and analysis of preparatory studies for the dynamical behavior of the system. Finally, we confirm that these symmetries also authorize us to explore the similarities and differences among many classes of paths that otherwise differ from the obtained trajectories by restricted three-body problem. Full article
17 pages, 10189 KiB  
Article
Evaluation of Aortic Valve Pressure Gradients for Increasing Severities of Rheumatic and Calcific Stenosis Using Empirical and Numerical Approaches
by Lindi Grobler, Ryno Laubscher, Johan van der Merwe and Philip G. Herbst
Math. Comput. Appl. 2024, 29(3), 33; https://doi.org/10.3390/mca29030033 - 28 Apr 2024
Viewed by 1134
Abstract
The evaluation and accurate diagnosis of the type and severity of aortic stenosis relies on the precision of medical imaging technology and clinical correlations and the expertise of medical professionals. The application of the clinical correlation to different aortic stenosis morphologies and severities [...] Read more.
The evaluation and accurate diagnosis of the type and severity of aortic stenosis relies on the precision of medical imaging technology and clinical correlations and the expertise of medical professionals. The application of the clinical correlation to different aortic stenosis morphologies and severities is investigated. The manner in which numerical techniques can be used to simulate the blood flow through pathological aortic valves was analysed and compared to the ground-truth CFD model. Larger pressure gradients are estimated in all severities of rheumatic aortic valves compared to calcific aortic valves. The zero-dimensional morphology-insensitive model underpredicted the transvalvular pressure gradient with the greatest error. The 1D model underestimated the pressure gradient in rheumatic cases and overestimated the pressure gradient in calcific cases. The pressure gradients estimated by the clinical approach depends on the location of the flow vena contracta and is sensitive to the severity and type of valve lesion. Through the analysis of entropy generation within the flow domain, the dominant parameters and regions driving adverse pressure gradients were identified. It is concluded that sudden expansion is the dominant parameter leading to higher pressure gradients in rheumatic heart valves compared to calcific ones. Full article
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15 pages, 512 KiB  
Article
Recognizable Languages of k-Forcing Automata
by Marzieh Shamsizadeh, Mohammad Mehdi Zahedi, Khadijeh Abolpour and Manuel De la Sen
Math. Comput. Appl. 2024, 29(3), 32; https://doi.org/10.3390/mca29030032 - 25 Apr 2024
Viewed by 701
Abstract
In this study, we show that automata theory is also a suitable tool for analyzing a more complex type of the k-forcing process. First, the definition of k-forcing automata is presented according to the definition of k-forcing for graphs. Moreover, we study and [...] Read more.
In this study, we show that automata theory is also a suitable tool for analyzing a more complex type of the k-forcing process. First, the definition of k-forcing automata is presented according to the definition of k-forcing for graphs. Moreover, we study and discuss the language of k-forcing automata for particular graphs. Also, for some graphs with different k-forcing sets, we study the languages of their k-forcing automata. In addition, for some given recognizable languages, we study the structure of graphs. After that, we show that k-forcing automata arising from isomorph graphs are also isomorph. Also, we present the style of words that can be recognized with k-forcing automata. Moreover, we introduce the structure of graphs the k-forcing automata arising from which recognize some particular languages. To clarify the notions and the results obtained in this study, some examples are submitted as well. Full article
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15 pages, 313 KiB  
Article
A New Generalized Definition of Fractal–Fractional Derivative with Some Applications
by Francisco Martínez and Mohammed K. A. Kaabar
Math. Comput. Appl. 2024, 29(3), 31; https://doi.org/10.3390/mca29030031 - 25 Apr 2024
Viewed by 799
Abstract
In this study, a new generalized fractal–fractional (FF) derivative is proposed. By applying this definition to some elementary functions, we show its compatibility with the results of the FF derivative in the Caputo sense with the power law. The main elements of classical [...] Read more.
In this study, a new generalized fractal–fractional (FF) derivative is proposed. By applying this definition to some elementary functions, we show its compatibility with the results of the FF derivative in the Caputo sense with the power law. The main elements of classical differential calculus are introduced in terms of this new derivative. Thus, we establish and demonstrate the basic operations with derivatives, chain rule, mean value theorems with their immediate applications and inverse function’s derivative. We complete the theory of generalized FF calculus by proposing a notion of integration and presenting two important results of integral calculus: the fundamental theorem and Barrow’s rule. Finally, we analytically solve interesting FF ordinary differential equations by applying our proposed definition. Full article
16 pages, 4439 KiB  
Article
Simulation of Temperature Field in Steel Billets during Reheating in Pusher-Type Furnace by Meshless Method
by Qingguo Liu, Umut Hanoglu, Zlatko Rek and Božidar Šarler
Math. Comput. Appl. 2024, 29(3), 30; https://doi.org/10.3390/mca29030030 - 24 Apr 2024
Viewed by 976
Abstract
Using a meshless method, a simulation of steel billets in a pusher-type reheating furnace is carried out for the first time. The simulation represents an affordable way to replace the measurements. The heat transfer from the billets with convection and radiation is considered. [...] Read more.
Using a meshless method, a simulation of steel billets in a pusher-type reheating furnace is carried out for the first time. The simulation represents an affordable way to replace the measurements. The heat transfer from the billets with convection and radiation is considered. Inside each of the billets, the heat diffusion equation is solved on a two-dimensional central slice of the billet. The diffusion equation is solved in a strong form by the Local Radial Basis Function Collocation Method (LRBFCM) with explicit time-stepping. The ray tracing procedure solves the radiation, where the view factors are computed with the Monte Carlo method. The changing number of billets in the furnace at the start and the end of the loading and unloading of the furnace is considered. A sensitivity study on billets’ temperature evolution is performed as a function of a different number of rays used in the Monte Carlo method, different stopping times of the billets in the furnace, and different spacing between the billets. The temperature field simulation is also essential for automatically optimizing the furnace’s productivity, energy consumption, and the billet’s quality. For the first time, the LRBFCM is successfully demonstrated for solving such a complex industrial problem. Full article
(This article belongs to the Special Issue Radial Basis Functions)
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