Mathematics

18 pages, 5311 KiB

Open AccessArticle

Upper Extremity Motion-Based Telemanipulation with Component-Wise Rescaling of Spatial Twist and Parameter-Invariant Skeletal Kinematics

by Donghyeon Noh, Haegyeom Choi, Haneul Jeon, Taeho Kim and Donghun Lee

Mathematics 2024, 12(2), 358; https://doi.org/10.3390/math12020358 - 22 Jan 2024

Cited by 1 | Viewed by 1232

Abstract

This study introduces a framework to improve upper extremity motion-based telemanipulation by component-wise rescaling (CWR) of spatial twist. This method allows for separate adjustments of linear and angular scaling parameters, significantly improving precision and dexterity even when the operator’s heading direction changes. By [...] Read more.

This study introduces a framework to improve upper extremity motion-based telemanipulation by component-wise rescaling (CWR) of spatial twist. This method allows for separate adjustments of linear and angular scaling parameters, significantly improving precision and dexterity even when the operator’s heading direction changes. By finely controlling both the linear and angular velocities independently, the CWR method enables more accurate telemanipulation in tasks requiring diverse speed and accuracy based on personal preferences or task-specific demands. The study conducted experiments confirming that operators could precisely control the robot gripper with a steady, controlled motion even in confined spaces, irrespective of changes in the subject’s body-heading direction. The performance evaluation of the proposed motion-scaling-based telemanipulation leveraged Optitrack’s motion-capture system, comparing the trajectories of the operator’s hand and the manipulator’s end effector (EEF). This verification process solidified the efficacy of the developed framework in enhancing telemanipulation performance. Full article

(This article belongs to the Special Issue Mathematical Methods in Artificial Intelligence and Robotics)

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21 pages, 1614 KiB

Open AccessFeature PaperArticle

General Solutions for MHD Motions of Ordinary and Fractional Maxwell Fluids through Porous Medium When Differential Expressions of Shear Stress Are Prescribed on Boundary

by Dumitru Vieru and Constantin Fetecau

Mathematics 2024, 12(2), 357; https://doi.org/10.3390/math12020357 - 22 Jan 2024

Cited by 3 | Viewed by 1041

Abstract

Some MHD unidirectional motions of the electrically conducting incompressible Maxwell fluids between infinite horizontal parallel plates incorporated in a porous medium are analytically and graphically investigated when differential expressions of the non-trivial shear stress are prescribed on the boundary. Such boundary conditions are [...] Read more.

Some MHD unidirectional motions of the electrically conducting incompressible Maxwell fluids between infinite horizontal parallel plates incorporated in a porous medium are analytically and graphically investigated when differential expressions of the non-trivial shear stress are prescribed on the boundary. Such boundary conditions are usually necessary in order to formulate well-posed boundary value problems for motions of rate-type fluids. General closed-form expressions are established for the dimensionless fluid velocity, the corresponding shear stress, and Darcy’s resistance. For completion, as well as for comparison, all results are extended to a fractional model of Maxwell fluids in which the time fractional Caputo derivative is used. It is proven for the first time that a large class of unsteady motions of the fractional incompressible Maxwell fluids becomes steady in time. For illustration, three particular motions are considered, and the correctness of the results is graphically proven. They correspond to constant or oscillatory values of the differential expression of shear stress on the boundary. In the first case, the required time to reach the steady state is graphically determined. This time declines for increasing values of the fractional parameter. Consequently, the steady state is reached earlier for motions of the ordinary fluids in comparison with the fractional ones. Finally, the fluid velocity, shear stress, and Darcy’s resistance are graphically represented and discussed for the fractional model. Full article

(This article belongs to the Special Issue Applications of Mathematics to Fluid Dynamics)

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21 pages, 1183 KiB

Open AccessArticle

Mastery of “Monthly Effects”: Big Data Insights into Contrarian Strategies for DJI 30 and NDX 100 Stocks over a Two-Decade Period

by Chien-Liang Chiu, Paoyu Huang, Min-Yuh Day, Yensen Ni and Yuhsin Chen

Mathematics 2024, 12(2), 356; https://doi.org/10.3390/math12020356 - 22 Jan 2024

Cited by 1 | Viewed by 1162

Abstract

In contrast to finding better monthly performance shown in a specific month, such as the January effect (i.e., better stock price performance in January as opposed to other months), which has been extensively studied, the goal of this study is to determine whether [...] Read more.

In contrast to finding better monthly performance shown in a specific month, such as the January effect (i.e., better stock price performance in January as opposed to other months), which has been extensively studied, the goal of this study is to determine whether investors would obtain better subsequent performance as technical trading signals emitted in a specific month because, from the investment perspective, investors purchasing stocks now would not know their performance until later. We contend that our analysis emphasizes its critical role in steering investment decisions and enhancing profitability; nonetheless, this issue appears to be overlooked in the relevant literature. As such, utilizing big data to analyze the constituent stocks of the DJI 30 and NDX 100 indices from 2003 to 2022 (i.e., two-decade data), this study investigates whether trading these stocks as trading signals emitted via contrarian regulation of stochastic oscillator indicators (SOIs) and the relative strength index (RSI) in specific months would result in superior subsequent performance (hereafter referred to as “monthly effects”). This study discovers that the oversold signals generated by these two contrarian regulations in March were associated with higher subsequent performance for holding 100 to 250 trading days (roughly one year) than other months. These findings highlight the importance of the trading time and the superiority of the RSI over SOIs in generating profits. This study sheds light on the significance of oversold trading signals and suggests that the “monthly effect” is crucial for achieving higher returns. Full article

(This article belongs to the Special Issue Machine Learning, Statistics and Big Data)

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22 pages, 4668 KiB

Open AccessArticle

Cooperative Game-Based Digital Twin Drives Decision Making: Overall Framework, Basic Formalization and Application Case

by Fuwen Hu, Song Bi and Yuanzhi Zhu

Mathematics 2024, 12(2), 355; https://doi.org/10.3390/math12020355 - 22 Jan 2024

Viewed by 2108

Abstract

The emerging progress brought about by Industry 4.0 generates great opportunities for better decision making to cope with increasingly uncertain and complex industrial production. From the perspective of game theory, methods based on computational simulations and methods based on physical entities have their [...] Read more.

The emerging progress brought about by Industry 4.0 generates great opportunities for better decision making to cope with increasingly uncertain and complex industrial production. From the perspective of game theory, methods based on computational simulations and methods based on physical entities have their intrinsic drawbacks, such as partially accessible information, uncontrollable uncertainty and limitations of sample data. However, an insight that inspired us was that the digital twin modeling method induced interactive environments to allow decision makers to cooperatively learn from the immediate feedback from both cyberspace and physical spaces. To this end, a new decision-making method was put forward using game theory to autonomously ally the digital twin models in cyberspace with their physical counterparts in the real world. Firstly, the overall framework and basic formalization of the cooperative game-based decision making are presented, which used the negotiation objectives, alliance rules and negotiation strategy to ally the planning agents from the physical entities with the planning agents from the virtual simulations. Secondly, taking the assembly planning of large-scale composite skins as a proof of concept, a cooperative game prototype system was developed to marry the physical assembly-commissioning system with the virtual assembly-commissioning system. Finally, the experimental work clearly indicated that the coalitional game-based twinning method could make the decision making of composite assembly not only predictable but reliable and help to avoid stress concentration and secondary damage and achieve high-precision assembly. Obviously, this decision-making methodology that integrates the physical players and their digital twins into the game space can help them take full advantage of each other and make up for their intrinsic drawbacks, and it preliminarily demonstrates great potential to revolutionize the traditional decision-making methodology. Full article

(This article belongs to the Topic Uncertainty Quantification in Design, Manufacturing and Maintenance of Complex Systems)

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20 pages, 637 KiB

Open AccessArticle

Multidimensional Diffusion-Wave-Type Solutions to the Second-Order Evolutionary Equation

by Alexander Kazakov and Anna Lempert

Mathematics 2024, 12(2), 354; https://doi.org/10.3390/math12020354 - 22 Jan 2024

Cited by 1 | Viewed by 912

Abstract

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular [...] Read more.

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular case of it is the well-known porous medium equation. Unlike previous studies, we consider the case of several spatial variables. We construct and study solutions that describe disturbances propagating over a zero background with a finite speed, usually called ‘diffusion-wave-type solutions’. Such effects are atypical for parabolic equations and appear since the equation degenerates on manifolds where the desired function vanishes. The paper pays special attention to exact solutions of the required type, which can be expressed as either explicit or implicit formulas, as well as a reduction of the partial differential equation to an ordinary differential equation that cannot be integrated in quadratures. In this connection, Cauchy problems for second-order ordinary differential equations arise, inheriting the singularities of the original formulation. We prove the existence of continuously differentiable solutions for them. A new example, an analog of the classic example by S.V. Kovalevskaya for the considered case, is constructed. We also proved a new existence and uniqueness theorem of heat-wave-type solutions in the class of piece-wise analytic functions, generalizing previous ones. During the proof, we transit to the hodograph plane, which allows us to overcome the analytical difficulties. Full article

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

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15 pages, 1173 KiB

Open AccessArticle

Antiangiogenic Therapy Efficacy Can Be Tumor-Size Dependent, as Mathematical Modeling Suggests

by Maxim Kuznetsov and Andrey Kolobov

Mathematics 2024, 12(2), 353; https://doi.org/10.3390/math12020353 - 22 Jan 2024

Viewed by 1182

Abstract

Antiangiogenic therapy (AAT) is an indirect oncological modality that is aimed at the disruption of cancer cell nutrient supply. Invasive tumors have been shown to possess inherent resistance to this treatment, while compactly growing benign tumors react to it by shrinking. It is [...] Read more.

Antiangiogenic therapy (AAT) is an indirect oncological modality that is aimed at the disruption of cancer cell nutrient supply. Invasive tumors have been shown to possess inherent resistance to this treatment, while compactly growing benign tumors react to it by shrinking. It is generally accepted that AAT by itself is not curative. This study presents a mathematical model of non-invasive tumor growth with a physiologically justified account of microvasculature alteration and the biomechanical aspects of importance during tumor growth and AAT. In the untreated setting, the model reproduces tumor growth with saturation, where the maximum tumor volume depends on the level of angiogenesis. The outcomes of the AAT simulations depend on the tumor size at the moment of treatment initiation. If it is close to the stable size of an avascular tumor grown in the absence of angiogenesis, then the tumor is rapidly stabilized by AAT. The treatment of large tumors is accompanied by the displacement of normal tissue due to tumor shrinkage. During this, microvasculature undergoes distortion, the degree of which depends on the displacement distance. As it affects tumor nutrient supply, the stable size of a tumor that undergoes AAT negatively correlates with its size at the beginning of treatment. For sufficiently large initial tumors, the long-term survival of tumor cells is compromised by competition with normal cells for the severely limited inflow of nutrients, which makes AAT effectively curative. Full article

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

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23 pages, 346 KiB

Open AccessArticle

The Augmented Weak Sharpness of Solution Sets in Equilibrium Problems

by Ruyu Wang, Wenling Zhao, Daojin Song and Yaozhong Hu

Mathematics 2024, 12(2), 352; https://doi.org/10.3390/math12020352 - 22 Jan 2024

Cited by 1 | Viewed by 961

Abstract

This study considers equilibrium problems, focusing on identifying finite solutions for feasible solution sequences. We introduce an innovative extension of the weak sharp minimum concept from convex programming to equilibrium problems, coining this as weak sharpness for solution sets. Recognizing situations where the [...] Read more.

This study considers equilibrium problems, focusing on identifying finite solutions for feasible solution sequences. We introduce an innovative extension of the weak sharp minimum concept from convex programming to equilibrium problems, coining this as weak sharpness for solution sets. Recognizing situations where the solution set may not exhibit weak sharpness, we propose an augmented mapping approach to mitigate this limitation. The core of our research is the formulation of augmented weak sharpness for the solution set. This comprehensive concept encapsulates both weak sharpness and strong non-degeneracy within feasible solution sequences. Crucially, we identify a necessary and sufficient condition for the finite termination of these sequences under the premise of augmented weak sharpness for the solution set in equilibrium problems. This condition significantly broadens the scope of the existing literature, which often assumes the solution set to be weakly sharp or strongly non-degenerate, especially in mathematical programming and variational inequality problems. Our findings not only shed light on the termination conditions in equilibrium problems but also introduce a less stringent sufficient condition for the finite termination of various optimization algorithms. This research, therefore, makes a substantial contribution to the field by enhancing our understanding of termination conditions in equilibrium problems and expanding the applicability of established theories to a wider range of optimization scenarios. Full article

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

36 pages, 4271 KiB

Open AccessArticle

Automated Classification of Agricultural Species through Parallel Artificial Multiple Intelligence System–Ensemble Deep Learning

by Keartisak Sriprateep, Surajet Khonjun, Paulina Golinska-Dawson, Rapeepan Pitakaso, Peerawat Luesak, Thanatkij Srichok, Somphop Chiaranai, Sarayut Gonwirat and Budsaba Buakum

Mathematics 2024, 12(2), 351; https://doi.org/10.3390/math12020351 - 22 Jan 2024

Viewed by 1978

Abstract

The classification of certain agricultural species poses a formidable challenge due to their inherent resemblance and the absence of dependable visual discriminators. The accurate identification of these plants holds substantial importance in industries such as cosmetics, pharmaceuticals, and herbal medicine, where the optimization [...] Read more.

The classification of certain agricultural species poses a formidable challenge due to their inherent resemblance and the absence of dependable visual discriminators. The accurate identification of these plants holds substantial importance in industries such as cosmetics, pharmaceuticals, and herbal medicine, where the optimization of essential compound yields and product quality is paramount. In response to this challenge, we have devised an automated classification system based on deep learning principles, designed to achieve precision and efficiency in species classification. Our approach leverages a diverse dataset encompassing various cultivars and employs the Parallel Artificial Multiple Intelligence System–Ensemble Deep Learning model (P-AMIS-E). This model integrates ensemble image segmentation techniques, including U-Net and Mask-R-CNN, alongside image augmentation and convolutional neural network (CNN) architectures such as SqueezeNet, ShuffleNetv2 1.0x, MobileNetV3, and InceptionV1. The culmination of these elements results in the P-AMIS-E model, enhanced by an Artificial Multiple Intelligence System (AMIS) for decision fusion, ultimately achieving an impressive accuracy rate of 98.41%. This accuracy notably surpasses the performance of existing methods, such as ResNet-101 and Xception, which attain 93.74% accuracy on the testing dataset. Moreover, when applied to an unseen dataset, the P-AMIS-E model demonstrates a substantial advantage, yielding accuracy rates ranging from 4.45% to 31.16% higher than those of the compared methods. It is worth highlighting that our heterogeneous ensemble approach consistently outperforms both single large models and homogeneous ensemble methods, achieving an average improvement of 13.45%. This paper provides a case study focused on the Centella Asiatica Urban (CAU) cultivar to exemplify the practical application of our approach. By integrating image segmentation, augmentation, and decision fusion, we have significantly enhanced accuracy and efficiency. This research holds theoretical implications for the advancement of deep learning techniques in image classification tasks while also offering practical benefits for industries reliant on precise species identification. Full article

(This article belongs to the Special Issue Artificial Intelligence and Machine Learning Based Methods and Applications)

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17 pages, 4105 KiB

Open AccessArticle

A Copula-Based Bivariate Composite Model for Modelling Claim Costs

by Girish Aradhye, George Tzougas and Deepesh Bhati

Mathematics 2024, 12(2), 350; https://doi.org/10.3390/math12020350 - 22 Jan 2024

Cited by 1 | Viewed by 1133

Abstract

This paper aims to develop a new family of bivariate distributions for modelling different types of claims and their associated costs jointly in a flexible manner. The proposed bivariate distributions can be viewed as a continuous copula distribution paired with two marginals based [...] Read more.

This paper aims to develop a new family of bivariate distributions for modelling different types of claims and their associated costs jointly in a flexible manner. The proposed bivariate distributions can be viewed as a continuous copula distribution paired with two marginals based on composite distributions. For expository purposes, the details of one of the proposed bivarite composite distributions is provided. The dependence measures for the resulting bivariate copula-based composite distribution are studied, and its fitting is compared with other bivariate composite distributions and existing bivariate distributions. The parameters of the proposed bivariate composite model are estimated via the inference functions for margins (IFM) method. The suitability of the proposed bivariate distribution is examined using two real-world insurance datasets, namely the motor third-party liability (MTPL) insurance dataset and Danish fire insurance dataset. Full article

(This article belongs to the Special Issue Recent Advances in Mathematical Methods for Economics)

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14 pages, 4556 KiB

Open AccessArticle

The Impact of Thermal Radiation on Mixed Convective Unsteady Nanofluid Flow in a Revolving Vertical Cone

by Shweta Mishra, Hiranmoy Mondal, Ramandeep Behl and Mehdi Salimi

Mathematics 2024, 12(2), 349; https://doi.org/10.3390/math12020349 - 22 Jan 2024

Cited by 1 | Viewed by 983

Abstract

This study investigates the effects of an unsteady mixed convection nanofluid flow in a rotating vertical cone submerged in spinning nanofluid. Our analysis considered the impacts of heat flux, chemical reactions, and thermal radiation, with the thermal and concentration Biot numbers serving as [...] Read more.

This study investigates the effects of an unsteady mixed convection nanofluid flow in a rotating vertical cone submerged in spinning nanofluid. Our analysis considered the impacts of heat flux, chemical reactions, and thermal radiation, with the thermal and concentration Biot numbers serving as constraints at the boundary. The governing unsteady and coupled partial differential equations were solved through appropriate similarity transformations, addressing the nonlinear terms inherent in these equations. The spectral quasi-linearisation method (SQLM) was employed to solve the higher-order nonlinear differential equations. This study elucidates and assesses the impact of diverse physical constraints and parameters through the use of graphical representations. Notably, the temperature distribution of the liquefied substance was intensified as the thermal and solutal Biot numbers increased. Full article

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19 pages, 878 KiB

Open AccessReview

Sustainable Rail/Road Unimodal Transportation of Bulk Cargo in Zambia: A Review of Algorithm-Based Optimization Techniques

by Fines Miyoba, Egbert Mujuni, Musa Ndiaye, Hastings M. Libati and Adnan M. Abu-Mahfouz

Mathematics 2024, 12(2), 348; https://doi.org/10.3390/math12020348 - 21 Jan 2024

Cited by 1 | Viewed by 1866

Abstract

Modern rail/road transportation systems are critical to global travel and commercial transportation. The improvement of transport systems that are needed for efficient cargo movements possesses further challenges. For instance, diesel-powered trucks and goods trains are widely used in long-haul unimodal transportation of heavy [...] Read more.

Modern rail/road transportation systems are critical to global travel and commercial transportation. The improvement of transport systems that are needed for efficient cargo movements possesses further challenges. For instance, diesel-powered trucks and goods trains are widely used in long-haul unimodal transportation of heavy cargo in most landlocked and developing countries, a situation that leads to concerns of greenhouse gases (GHGs) such as carbon dioxide coming from diesel fuel combustion. In this context, it is critical to understand aspects such as the use of some parameters, variables and constraints in the formulation of mathematical models, optimization techniques and algorithms that directly contribute to sustainable transportation solutions. In seeking sustainable solutions to the bulk cargo long-haul transportation problems in Zambia, we conduct a systematic review of various transportation modes and related mathematical models, and optimization approaches. In this paper, we provide an updated survey of various transport models for bulk cargo and their associated optimized combinations. We identify key research challenges and notable issues to be considered for further studies in transport system optimization, especially when dealing with long-haul unimodal or single-mode heavy cargo movement in countries that are yet to implement intermodal and multimodal systems. Full article

(This article belongs to the Special Issue Application of Machine Learning and Data Mining)

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16 pages, 4072 KiB

Open AccessArticle

OCT Retinopathy Classification via a Semi-Supervised Pseudo-Label Sub-Domain Adaptation and Fine-Tuning Method

by Zhicong Tan, Qinqin Zhang, Gongpu Lan, Jingjiang Xu, Chubin Ou, Lin An, Jia Qin and Yanping Huang

Mathematics 2024, 12(2), 347; https://doi.org/10.3390/math12020347 - 21 Jan 2024

Cited by 1 | Viewed by 1530

Abstract

Conventional OCT retinal disease classification methods primarily rely on fully supervised learning, which requires a large number of labeled images. However, sometimes the number of labeled images in a private domain is small but there exists a large annotated open dataset in the [...] Read more.

Conventional OCT retinal disease classification methods primarily rely on fully supervised learning, which requires a large number of labeled images. However, sometimes the number of labeled images in a private domain is small but there exists a large annotated open dataset in the public domain. In response to this scenario, a new transfer learning method based on sub-domain adaptation (TLSDA), which involves a first sub-domain adaptation and then fine-tuning, was proposed in this study. Firstly, a modified deep sub-domain adaptation network with pseudo-label (DSAN-PL) was proposed to align the feature spaces of a public domain (labeled) and a private domain (unlabeled). The DSAN-PL model was then fine-tuned using a small amount of labeled OCT data from the private domain. We tested our method on three open OCT datasets, using one as the public domain and the other two as the private domains. Remarkably, with only 10% labeled OCT images (~100 images per category), TLSDA achieved classification accuracies of 93.63% and 96.59% on the two private datasets, significantly outperforming conventional transfer learning approaches. With the Gradient-weighted Class Activation Map (Grad-CAM) technique, it was observed that the proposed method could more precisely localize the subtle lesion regions for OCT image classification. TLSDA could be a potential technique for applications where only a small number of images is labeled in a private domain and there exists a public database having a large number of labeled images with domain difference. Full article

(This article belongs to the Special Issue Applications of Artificial Intelligence in Biomedical Image Processing)

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31 pages, 6204 KiB

Open AccessArticle

A Formalization of Multilabel Classification in Terms of Lattice Theory and Information Theory: Concerning Datasets

by Francisco J. Valverde-Albacete and Carmen Peláez-Moreno

Mathematics 2024, 12(2), 346; https://doi.org/10.3390/math12020346 - 21 Jan 2024

Cited by 2 | Viewed by 1032

Abstract

Multilabel classification is a recently conceptualized task in machine learning. Contrary to most of the research that has so far focused on classification machinery, we take a data-centric approach and provide an integrative framework that blends qualitative and quantitative descriptions of multilabel data [...] Read more.

Multilabel classification is a recently conceptualized task in machine learning. Contrary to most of the research that has so far focused on classification machinery, we take a data-centric approach and provide an integrative framework that blends qualitative and quantitative descriptions of multilabel data sources. By combining lattice theory, in the form of formal concept analysis, and entropy triangles, obtained from information theory, we explain from first principles the fundamental issues of multilabel datasets such as the dependencies of the labels, their imbalances, or the effects of the presence of hapaxes. This allows us to provide guidelines for resampling and new data collection and their relationship with broad modelling approaches. We have empirically validated our framework using 56 open datasets, challenging previous characterizations that prove that our formalization brings useful insights into the task of multilabel classification. Further work will consider the extension of this formalization to understand the relationship between the data sources, the classification methods, and ways to assess their performance. Full article

(This article belongs to the Special Issue Applications of Mathematics in Neural Networks and Machine Learning)

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26 pages, 3248 KiB

Open AccessArticle

Application of the Improved Cuckoo Algorithm in Differential Equations

by Yan Sun

Mathematics 2024, 12(2), 345; https://doi.org/10.3390/math12020345 - 21 Jan 2024

Cited by 1 | Viewed by 1256

Abstract

To address the drawbacks of the slow convergence speed and lack of individual information exchange in the cuckoo search (CS) algorithm, this study proposes an improved cuckoo search algorithm based on a sharing mechanism (ICSABOSM). The enhanced algorithm reinforces information sharing among individuals [...] Read more.

To address the drawbacks of the slow convergence speed and lack of individual information exchange in the cuckoo search (CS) algorithm, this study proposes an improved cuckoo search algorithm based on a sharing mechanism (ICSABOSM). The enhanced algorithm reinforces information sharing among individuals through the utilization of a sharing mechanism. Additionally, new search strategies are introduced in both the global and local searches of the CS. The results from numerical experiments on four standard test functions indicate that the improved algorithm outperforms the original CS in terms of search capability and performance. Building upon the improved algorithm, this paper introduces a numerical solution approach for differential equations involving the coupling of function approximation and intelligent algorithms. By constructing an approximate function using Fourier series to satisfy the conditions of the given differential equation and boundary conditions with minimal error, the proposed method minimizes errors while satisfying the differential equation and boundary conditions. The problem of solving the differential equation is then transformed into an optimization problem with the coefficients of the approximate function as variables. Furthermore, the improved cuckoo search algorithm is employed to solve this optimization problem. The specific steps of applying the improved algorithm to solve differential equations are illustrated through examples. The research outcomes broaden the application scope of the cuckoo optimization algorithm and provide a new perspective for solving differential equations. Full article

(This article belongs to the Special Issue Smart Computing, Optimization and Operations Research)

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18 pages, 274 KiB

Open AccessArticle

Solutions of Umbral Dirac-Type Equations

by Hongfen Yuan and Valery Karachik

Mathematics 2024, 12(2), 344; https://doi.org/10.3390/math12020344 - 20 Jan 2024

Viewed by 1128

Abstract

The aim of this work is to study the method of the normalized systems of functions. The normalized systems of functions with respect to the Dirac operator in the umbral Clifford analysis are constructed. Furthermore, the solutions of umbral Dirac-type equations are investigated [...] Read more.

The aim of this work is to study the method of the normalized systems of functions. The normalized systems of functions with respect to the Dirac operator in the umbral Clifford analysis are constructed. Furthermore, the solutions of umbral Dirac-type equations are investigated by the normalized systems. Full article

(This article belongs to the Special Issue Differential Equations with Boundary Value Problems: Theory and Applications)

28 pages, 2329 KiB

Open AccessArticle

Application of Structural Equation Modelling to Cybersecurity Risk Analysis in the Era of Industry 4.0

by Miroslav Gombár, Alena Vagaská, Antonín Korauš and Pavlína Račková

Mathematics 2024, 12(2), 343; https://doi.org/10.3390/math12020343 - 20 Jan 2024

Cited by 3 | Viewed by 1861

Abstract

In the current digital transformation to Industry 4.0, the demands on the ability of countries to react responsibly and effectively to threats in the field of cyber security (CS) are increasing. Cyber safety is one of the pillars and concepts of Industry 4.0, [...] Read more.

In the current digital transformation to Industry 4.0, the demands on the ability of countries to react responsibly and effectively to threats in the field of cyber security (CS) are increasing. Cyber safety is one of the pillars and concepts of Industry 4.0, as digitization brings convergence and integration of information technologies (IT) and operational technologies (OT), IT/OT systems, and data. Collecting and connecting a large amount of data in smart factories and cities poses risks, in a broader context for the entire state. The authors focus attention on the issue of CS, where, despite all digitization, the human factor plays a key role—an actor of risk as well as strengthening the sustainability and resilience of CS. It is obvious that in accordance with how the individuals (decision-makers) perceive the risk, thus they subsequently evaluate the situation and countermeasures. Perceiving cyber threats/risks in their complexity as a part of hybrid threats (HT) helps decision-makers prevent and manage them. Due to the growing trend of HT, the need for research focused on the perception of threats by individuals and companies is increasing. Moreover, the literature review points out a lack of methodology and evaluation strategy. This study presents the results of the research aimed at the mathematical modelling of risk perception of threats to the state and industry through the disruption of CS. The authors provide the developed factor model of cyber security (FMCS), i.e., the model of CS threat risk perception. When creating the FMCS, the researchers applied SEM (structural equation modelling) and confirmatory factor analysis to the data obtained by the implementation of the research tool (a questionnaire designed by the authors). The pillars and sub-pillars of CS defined within the questionnaire enable quantification in the perception of the level of risk of CS as well as differentiation and comparison between the analyzed groups of respondents (students of considered universities in SK and CZ). The convergent and discriminant validity of the research instrument is verified, and its reliability is confirmed (Cronbach’s alpha = 0.95047). The influence of the individual pillars is demonstrated as significant at the significance level of α = 5%. For the entire research set N = 964, the highest share of risk perception of CS threats is achieved by the DISRIT pillar (disruption or reduction of the resistance of IT infrastructure). Full article

(This article belongs to the Special Issue Recent Advances of Mathematics in Industrial Engineering)

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20 pages, 695 KiB

Open AccessArticle

Linear Disassembly Line Balancing Problem with Tool Deterioration and Solution by Discrete Migratory Bird Optimizer

by Shujin Qin, Jiaxin Wang, Jiacun Wang, Xiwang Guo, Liang Qi and Yaping Fu

Mathematics 2024, 12(2), 342; https://doi.org/10.3390/math12020342 - 20 Jan 2024

Cited by 2 | Viewed by 1143

Abstract

In recent years, the global resource shortage has become a serious issue. Recycling end-of-life (EOL) products is conducive to resource reuse and circular economy and can mitigate the resource shortage issue. The disassembly of EOL products is the first step for resource reuse. [...] Read more.

In recent years, the global resource shortage has become a serious issue. Recycling end-of-life (EOL) products is conducive to resource reuse and circular economy and can mitigate the resource shortage issue. The disassembly of EOL products is the first step for resource reuse. Disassembly activities need tools, and tool deterioration occurs inevitably during the disassembly process. This work studies the influence of tool deterioration on disassembly efficiency. A disassembly line balancing model with the goal of maximizing disassembly profits is established, in which tool selection and assignment is a critical part. A modified discrete migratory bird optimizer is proposed to solve optimization problems. The well-known IBM CPLEX optimizer is used to verify the correctness of the model. Six real-world products are used for disassembly experiments. The popular fruit fly optimization algorithm, whale optimization algorithm and salp swarm algorithm are used for search performance comparison. The results show that the discrete migratory bird optimizer outperforms all three other algorithms in all disassembly instances. Full article

(This article belongs to the Special Issue Ensemble Evolutionary Algorithms and Machine Learning for Solving Complex Optimization and Scheduling Problems)

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3 pages, 161 KiB

Open AccessEditorial

Mathematical Modeling and Simulation in Mechanics and Dynamic Systems, 2nd Edition

by Maria Luminita Scutaru and Catalin-Iulian Pruncu

Mathematics 2024, 12(2), 341; https://doi.org/10.3390/math12020341 - 19 Jan 2024

Viewed by 1618

Abstract

Although it has been considered difficult to make further contributions in the field of mechanics, the spectacular evolution of technology and numerical calculation techniques has made these opinions shift, and increasingly sophisticated models have been developed, which should predict, as accurately as possible, [...] Read more.

Although it has been considered difficult to make further contributions in the field of mechanics, the spectacular evolution of technology and numerical calculation techniques has made these opinions shift, and increasingly sophisticated models have been developed, which should predict, as accurately as possible, the phenomena that take place in dynamic systems [...] Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems, 2nd Edition)

20 pages, 522 KiB

Open AccessArticle

Predicting Typhoon Flood in Macau Using Dynamic Gaussian Bayesian Network and Surface Confluence Analysis

by Shujie Zou, Chiawei Chu, Weijun Dai, Ning Shen, Jia Ren and Weiping Ding

Mathematics 2024, 12(2), 340; https://doi.org/10.3390/math12020340 - 19 Jan 2024

Viewed by 1722

Abstract

A typhoon passing through or making landfall in a coastal city may result in seawater intrusion and continuous rainfall, which may cause urban flooding. The urban flood disaster caused by a typhoon is a dynamic process that changes over time, and a dynamic [...] Read more.

A typhoon passing through or making landfall in a coastal city may result in seawater intrusion and continuous rainfall, which may cause urban flooding. The urban flood disaster caused by a typhoon is a dynamic process that changes over time, and a dynamic Gaussian Bayesian network (DGBN) is used to model the time series events in this paper. The scene data generated by each typhoon are different, which means that each typhoon has different characteristics. This paper establishes multiple DGBNs based on the historical data of Macau flooding caused by multiple typhoons, and similar analysis is made between the scene data related to the current flooding to be predicted and the scene data of historical flooding. The DGBN most similar to the scene characteristics of the current flooding is selected as the predicting network of the current flooding. According to the topography, the influence of the surface confluence is considered, and the Manning formula analysis method is proposed. The Manning formula is combined with the DGBN to obtain the final prediction model, DGBN-m, which takes into account the effects of time series and non-time-series factors. The flooding data provided by the Macau Meteorological Bureau are used to carry out experiments, and it is proved that the proposed model can predict the flooding depth well in a specific area of Macau under the condition of a small amount of data and that the best predicting accuracy can reach 84%. Finally, generalization analysis is performed to further confirm the validity of the proposed model. Full article

(This article belongs to the Special Issue Hybrid Data Processing by Combining Machine Learning, Expert, Safety and Security)

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23 pages, 7387 KiB

Open AccessArticle

Solving Heterogeneous USV Scheduling Problems by Problem-Specific Knowledge Based Meta-Heuristics with Q-Learning

by Zhenfang Ma, Kaizhou Gao, Hui Yu and Naiqi Wu

Mathematics 2024, 12(2), 339; https://doi.org/10.3390/math12020339 - 19 Jan 2024

Cited by 5 | Viewed by 1549

Abstract

This study focuses on the scheduling problem of heterogeneous unmanned surface vehicles (USVs) with obstacle avoidance pretreatment. The goal is to minimize the overall maximum completion time of USVs. First, we develop a mathematical model for the problem. Second, with obstacles, an A* [...] Read more.

This study focuses on the scheduling problem of heterogeneous unmanned surface vehicles (USVs) with obstacle avoidance pretreatment. The goal is to minimize the overall maximum completion time of USVs. First, we develop a mathematical model for the problem. Second, with obstacles, an A* algorithm is employed to generate a path between two points where tasks need to be performed. Third, three meta-heuristics, i.e., simulated annealing (SA), genetic algorithm (GA), and harmony search (HS), are employed and improved to solve the problems. Based on problem-specific knowledge, nine local search operators are designed to improve the performance of the proposed algorithms. In each iteration, three Q-learning strategies are used to select high-quality local search operators. We aim to improve the performance of meta-heuristics by using Q-learning-based local search operators. Finally, 13 instances with different scales are adopted to validate the effectiveness of the proposed strategies. We compare with the classical meta-heuristics and the existing meta-heuristics. The proposed meta-heuristics with Q-learning are overall better than the compared ones. The results and comparisons show that HS with the second Q-learning, HS + QL2, exhibits the strongest competitiveness (the smallest mean rank value 1.00) among 15 algorithms. Full article

(This article belongs to the Special Issue Ensemble Evolutionary Algorithms and Machine Learning for Solving Complex Optimization and Scheduling Problems)

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25 pages, 848 KiB

Open AccessArticle

Proposal of a Framework for Evaluating the Importance of Production and Maintenance Integration Supported by the Use of Ordinal Linguistic Fuzzy Modeling

by Ronald Díaz Cazañas, Daynier Rolando Delgado Sobrino, Estrella María De La Paz Martínez, Jana Petru and Carlos Daniel Díaz Tejeda

Mathematics 2024, 12(2), 338; https://doi.org/10.3390/math12020338 - 19 Jan 2024

Viewed by 1139

Abstract

Over the years, the integration of Production Management and Maintenance Management has gained significant attention from the scientific community due to its benefits for the company. When searching through the states of the art and practice, it is possible to understand that one [...] Read more.

Over the years, the integration of Production Management and Maintenance Management has gained significant attention from the scientific community due to its benefits for the company. When searching through the states of the art and practice, it is possible to understand that one of the main challenges for the integration is the lack of systematic, methodological, and scientific approaches and evaluation systems that lead companies into a successful implementation and a clear understanding of the benefits and drawbacks of the process. This paper introduces an original framework that conducts the processes of evaluation, weighting, and aggregation of set of novel indicators proposed by the authors. The main output of the proposal is an integral index that allows us to qualify, in a linguistic domain, the importance of the Production and Maintenance Management integration. At the same time, the proposed framework includes a methodology to evaluate the consensus of the experts, based on the use of linguistic terms with a membership function of the triangular type, which attempts to overcome some deficiencies of previous models identified by the authors in a detailed and complex analysis of the scientific literature. The proposed framework is applied in a plant of the Cuban mechanical industry. The results of this application are clearly presented and discussed, allowing us to verify and validate the proposal while also contributing to its ease of understanding and ultimately to the successful integration of the production and maintenance tasks in the given company. Full article

(This article belongs to the Special Issue Fuzzy Decision Making and Soft Computing Applications: Future Perspectives)

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25 pages, 2911 KiB

Open AccessArticle

A Double-Layer Coupled Network Model of Network Density Effects on Multi-Stage Innovation Efficiency Dynamics: Agent-Based Modeling Methods

by Jing Han, Wenjing Zhang, Jiutian Wang and Songmei Li

Mathematics 2024, 12(2), 337; https://doi.org/10.3390/math12020337 - 19 Jan 2024

Viewed by 1482

Abstract

This paper proposes a double-layer coupled network model to analyze the multi-stage innovation activities of online, and the model consists of two layers: the online layer, which represents the virtual interactions among innovators, and the offline layer, which represents the physical interactions among [...] Read more.

This paper proposes a double-layer coupled network model to analyze the multi-stage innovation activities of online, and the model consists of two layers: the online layer, which represents the virtual interactions among innovators, and the offline layer, which represents the physical interactions among innovators. The model assumes that the innovation activities are influenced by both the online and offline network structures, as well as the coupling effect between them. And it simulates the entire innovation process including knowledge diffusion and knowledge recombination. The model also incorporates the concept of network density, which measures the degree of network connectivity and cohesion (network structure). Observing the network density influence on innovation efficiency during the innovation process is realized through setting the selection mechanism and the knowledge recombination mechanism. The coupling relationship between the two layers of network density on the three stages of innovation is further discussed under the theoretical framework of the innovation value chain. Simulation and experimental results suggest that when the offline network density is constant, a higher online network density is not always better. When the online network density is low, the sparse structure of the online network reduces innovation efficiency. When the online network density is high, the structural redundancy caused by the tight network structure prevents innovation efficiency from improving. The results of the study help enterprises to adjust and optimize the internal cooperation network structure at different stages of innovation in order to maximize its effectiveness and improve the innovation efficiency of enterprises. Full article

(This article belongs to the Special Issue Computational and Mathematical Methods in Information Science and Engineering, 2nd Edition)

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25 pages, 7232 KiB

Open AccessArticle

An Icon-Based Methodology for the Design of a Prototype of a Multi-Process, Multi-Product, Aggregated Production Planning Software

by Erick Miranda-Meza, Iván Derpich and Juan M. Sepúlveda

Mathematics 2024, 12(2), 336; https://doi.org/10.3390/math12020336 - 19 Jan 2024

Viewed by 1241

Abstract

This paper proposes an icon-based methodology for the design of prototype aggregated production planning software that addresses the complexity of multi-process and multi-product production. Aggregate planning is a critical task in production management, which involves coordinating the production of multiple products in different [...] Read more.

This paper proposes an icon-based methodology for the design of prototype aggregated production planning software that addresses the complexity of multi-process and multi-product production. Aggregate planning is a critical task in production management, which involves coordinating the production of multiple products in different processes to meet demand efficiently. The approach focuses on the use of visual icons to represent key elements of the production process, such as products, processes, resources, and constraints. These icons allow an intuitive representation of information and facilitate communication between production team members. In addition, this paper presents a conceptual structure that defines the relationships between the icons and how they are used to model and simulate aggregate production planning. The prototype software based on a conceptual foundation allows planners to easily create and adjust production plans in a visual environment. This method improves the ability to make informed and rapid decisions in response to changes in demand or production capacity. The prototype is based on icons and programmed in Excel spreadsheets to facilitate the planner’s planning. At the end of the document, the application of a case study is shown. Full article

(This article belongs to the Special Issue Industrial Big Data and Process Modelling for Smart Manufacturing)

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23 pages, 1072 KiB

Open AccessArticle

Composite and Mixture Distributions for Heavy-Tailed Data—An Application to Insurance Claims

by Walena Anesu Marambakuyana and Sandile Charles Shongwe

Mathematics 2024, 12(2), 335; https://doi.org/10.3390/math12020335 - 19 Jan 2024

Cited by 3 | Viewed by 1504

Abstract

This research provides a comprehensive analysis of two-component non-Gaussian composite models and mixture models for insurance claims data. These models have gained attraction in actuarial literature because they provide flexible methods for curve-fitting. We consider 256 composite models and 256 mixture models derived [...] Read more.

This research provides a comprehensive analysis of two-component non-Gaussian composite models and mixture models for insurance claims data. These models have gained attraction in actuarial literature because they provide flexible methods for curve-fitting. We consider 256 composite models and 256 mixture models derived from 16 popular parametric distributions. The composite models are developed by piecing together two distributions at a threshold value, while the mixture models are developed as convex combinations of two distributions on the same domain. Two real insurance datasets from different industries are considered. Model selection criteria and risk metrics of the top 20 models in each category (composite/mixture) are provided by using the ‘single-best model’ approach. Finally, for each of the datasets, composite models seem to provide better risk estimates. Full article

(This article belongs to the Special Issue Mathematical Data Science with Applications in Business, Industry, and Medicine)

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14 pages, 5476 KiB

Open AccessArticle

Archimedean Copulas-Based Estimation under One-Parameter Distributions in Coherent Systems

by Ioannis S. Triantafyllou

Mathematics 2024, 12(2), 334; https://doi.org/10.3390/math12020334 - 19 Jan 2024

Viewed by 1001

Abstract

In the present work we provide a signature-based framework for delivering the estimated mean lifetime along with the variance of the continuous distribution of a coherent system consisting of exchangeable components. The dependency of the components is modelled by the aid of well-known [...] Read more.

In the present work we provide a signature-based framework for delivering the estimated mean lifetime along with the variance of the continuous distribution of a coherent system consisting of exchangeable components. The dependency of the components is modelled by the aid of well-known Archimedean multivariate copulas. The estimated results are calculated under two different copulas, namely the so-called Frank copula and the Joe copula. A numerical experimentation is carried out for illustrating the proposed procedure under all possible coherent systems with three components. Full article

(This article belongs to the Special Issue Mathematical Data Science with Applications in Business, Industry, and Medicine)

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25 pages, 2824 KiB

Open AccessArticle

Knowledge Granularity Attribute Reduction Algorithm for Incomplete Systems in a Clustering Context

by Baohua Liang, Erli Jin, Liangfen Wei and Rongyao Hu

Mathematics 2024, 12(2), 333; https://doi.org/10.3390/math12020333 - 19 Jan 2024

Cited by 2 | Viewed by 969

Abstract

The phenomenon of missing data can be seen everywhere in reality. Most typical attribute reduction models are only suitable for complete systems. But for incomplete systems, we cannot obtain the effective reduction rules. Even if there are a few reduction approaches, the classification [...] Read more.

The phenomenon of missing data can be seen everywhere in reality. Most typical attribute reduction models are only suitable for complete systems. But for incomplete systems, we cannot obtain the effective reduction rules. Even if there are a few reduction approaches, the classification accuracy of their reduction sets still needs to be improved. In order to overcome these shortcomings, this paper first defines the similarities of intra-cluster objects and inter-cluster objects based on the tolerance principle and the mechanism of knowledge granularity. Secondly, attributes are selected on the principle that the similarity of inter-cluster objects is small and the similarity of intra-cluster objects is large, and then the knowledge granularity attribute model is proposed under the background of clustering; then, the IKAR algorithm program is designed. Finally, a series of comparative experiments about reduction size, running time, and classification accuracy are conducted with twelve UCI datasets to evaluate the performance of IKAR algorithms; then, the stability of the Friedman test and Bonferroni–Dunn tests are conducted. The experimental results indicate that the proposed algorithms are efficient and feasible. Full article

(This article belongs to the Special Issue Advances in Data Mining, Machine Learning and Causal Inference and Their Applications)

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27 pages, 568 KiB

Open AccessArticle

A Routing Model for the Distribution of Perishable Food in a Green Cold Chain

by Gilberto Pérez-Lechuga, José Francisco Martínez-Sánchez, Francisco Venegas-Martínez and Karla Nataly Madrid-Fernández

Mathematics 2024, 12(2), 332; https://doi.org/10.3390/math12020332 - 19 Jan 2024

Cited by 2 | Viewed by 1750

Abstract

In this research, we develop an extension of the stochastic routing model with a fixed capacity for the distribution of perishable products with a time window. We use theoretical probability distributions to model the life of transported products and travel times in the [...] Read more.

In this research, we develop an extension of the stochastic routing model with a fixed capacity for the distribution of perishable products with a time window. We use theoretical probability distributions to model the life of transported products and travel times in the network. Our main objective is to maximize the probability of delivering products within the established deadline with a certain level of customer service. Our project is justified from the perspective of reducing the pollution caused by greenhouse gases generated in the process. To optimize the proposed model, we use a Generic Random Search Algorithm. Finally, we apply the idea to a real problem of designing strategies for the optimal management of perishable food distribution routes that involve a time window, the objective being to maximize the probability of meeting the time limit assigned to the route problem by reducing, in this way, the pollution generated by refrigerated transport. Full article

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20 pages, 1215 KiB

Open AccessArticle

Sentiment Analysis Based on Heterogeneous Multi-Relation Signed Network

by Qin Zhao, Chenglei Yu, Jingyi Huang, Jie Lian and Dongdong An

Mathematics 2024, 12(2), 331; https://doi.org/10.3390/math12020331 - 19 Jan 2024

Cited by 2 | Viewed by 1370

Abstract

Existing sentiment prediction methods often only classify users’ emotions into a few categories and cannot predict the variation of emotions under different topics. Meanwhile, network embedding methods that consider structural information often assume that links represent positive relationships, ignoring the possibility of negative [...] Read more.

Existing sentiment prediction methods often only classify users’ emotions into a few categories and cannot predict the variation of emotions under different topics. Meanwhile, network embedding methods that consider structural information often assume that links represent positive relationships, ignoring the possibility of negative relationships. To address these challenges, we present an innovative approach in sentiment analysis, focusing on the construction of a denser heterogeneous signed information network from sparse heterogeneous data. We explore the extraction of latent relationships between similar node types, integrating emotional reversal and meta-path similarity for relationship prediction. Our approach uniquely handles user-entity and topic-entity relationships, offering a tailored methodology for diverse entity types within heterogeneous networks. We contribute to a deeper understanding of emotional expressions and interactions in social networks, enhancing sentiment analysis techniques. Experimental results on four publicly available datasets demonstrate the superiority of our proposed model over state-of-the-art approaches. Full article

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

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19 pages, 2440 KiB

Open AccessArticle

A Comprehensive Evaluation Method of Machining Center Components’ Importance Based on Combined Variable Weight

by Lan Luan, Guixiang Shen and Yingzhi Zhang

Mathematics 2024, 12(2), 330; https://doi.org/10.3390/math12020330 - 19 Jan 2024

Viewed by 844

Abstract

The fault transitivity of machining center components causes their fault propagation indexes to demonstrate dynamic time variability, which affects their importance. The method proposed in this study overcomes the biases of existing methods of evaluating the importance of system components, as they are [...] Read more.

The fault transitivity of machining center components causes their fault propagation indexes to demonstrate dynamic time variability, which affects their importance. The method proposed in this study overcomes the biases of existing methods of evaluating the importance of system components, as they are mostly based on single indexes; the fault propagation probability and fault propagation risk are selected to perform a comprehensive evaluation. This study first establishes a network hierarchical structure model for machining center components, and the degree of influence of fault propagation among the components is calculated. On this basis, the improved adjacent spreading paths (ASP) algorithm is used to calculate the fault propagation index of each component; from the two perspectives of fault propagation probability and fault propagation risk, an evaluation mechanism involving the combined variable weight is used to comprehensively evaluate components’ importance. Taking a certain type of machining center as an example, through a comparison with ranking results from other node importance methods, it is verified that the proposed method can more effectively distinguish the differences in the importance of each component, thus illustrating the effectiveness and practical value of this method. Full article

(This article belongs to the Special Issue Applied Mathematics to Mechanisms and Machines II)

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16 pages, 902 KiB

Open AccessArticle

Control the Coefficient of a Differential Equation as an Inverse Problem in Time

by Vladimir Ternovski and Victor Ilyutko

Mathematics 2024, 12(2), 329; https://doi.org/10.3390/math12020329 - 19 Jan 2024

Viewed by 925

Abstract

There are many problems based on solving nonautonomous differential equations of the form

\ddot{x} (t) + ω^{2} (t) x (t) = 0,

where

x (t)

represents the coordinate of a material point [...] Read more.

There are many problems based on solving nonautonomous differential equations of the form

\ddot{x} (t) + ω^{2} (t) x (t) = 0,

where

x (t)

represents the coordinate of a material point and

ω

is the angular frequency. The inverse problem involves finding the bounded coefficient

ω

. Continuity of the function

ω (t)

is not required. The trajectory

x (t)

is also unknown, but the initial and final values of the phase variables are given. The variation principle of the minimum time for the entire dynamic process allows for the determination of the optimal solution. Thus, the inverse problem is an optimal control problem. No simplifying assumptions were made. Full article

(This article belongs to the Section Mathematical Physics)

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