Mathematics

23 pages, 376 KiB

Open AccessFeature PaperArticle

by Rédina Berkachy and Laurent Donzé

Mathematics 2024, 12(24), 4042; https://doi.org/10.3390/math12244042 - 23 Dec 2024

Viewed by 639

This paper presents a comprehensive study of the signed distance metric for fuzzy numbers. Due to the property of directionality, this measure has been widely used. However, it has a main drawback in handling asymmetry and irregular shapes in fuzzy numbers. To overcome [...] Read more.

This paper presents a comprehensive study of the signed distance metric for fuzzy numbers. Due to the property of directionality, this measure has been widely used. However, it has a main drawback in handling asymmetry and irregular shapes in fuzzy numbers. To overcome this rather bad feature, we introduce two new distances, the balanced signed distance (BSGD) and the generalised signed distance (GSGD), seen as generalisations of the classical signed distance. The developed distances successfully and effectively take into account the shape, the asymmetry and the overlap of fuzzy numbers. The GSGD is additionally directional, while the BSGD satisfies the requirements for being a metric of fuzzy quantities. Analytical simplifications of both distances in the case of often-used particular types of fuzzy numbers are provided to simplify the computation process, making them as simple as the classical signed distance but more realistic and precise. We empirically analyse the sensitivity of these distances. Considering several scenarios of fuzzy numbers, we also numerically compare these distances against established metrics, highlighting the advantages of the BSGD and the GSGD in capturing the shape properties of fuzzy numbers. One main finding of this research is that the defended distances capture with great precision the distance between fuzzy numbers; additionally, they are theoretically appealing and are computationally easy for traditional fuzzy numbers such as triangular, trapezoidal, Gaussian, etc., making these metrics promising. Full article

(This article belongs to the Special Issue Research and Application of Fuzzy Statistics)

18 pages, 4029 KiB

Open AccessArticle

An Integrated Algorithm with Feature Selection, Data Augmentation, and XGBoost for Ovarian Cancer

by Jingxun Cai, Zne-Jung Lee, Zhihxian Lin, Chih-Hung Hsu and Yun Lin

Mathematics 2024, 12(24), 4041; https://doi.org/10.3390/math12244041 - 23 Dec 2024

Cited by 1 | Viewed by 829

Abstract

Ovarian cancer is one of the most aggressive gynecological cancers due to its high invasion and chemoresistance. It not only has a high incidence rate but also tops the list of mortality rates. Its subtle early symptoms make subsequent diagnosis difficult, significantly delaying [...] Read more.

Ovarian cancer is one of the most aggressive gynecological cancers due to its high invasion and chemoresistance. It not only has a high incidence rate but also tops the list of mortality rates. Its subtle early symptoms make subsequent diagnosis difficult, significantly delaying timely treatment for patients. Once ovarian cancer reaches an advanced stage, the complexity and difficulty of treatment increase substantially, affecting patient survival rates. Therefore, it is crucial for both medical professionals and patients to remain highly vigilant about the early signs of ovarian cancer to ensure timely intervention. In recent years, ovarian cancer prediction research has advanced, allowing for the analysis of the likelihood and type of cancer based on patients’ genetic data. With the rapid development of machine learning, numerous efficient classification prediction models have emerged. These new technologies offer significant opportunities and potential for developing ovarian cancer diagnostic prediction methods. However, traditional approaches often struggle to achieve satisfactory classification accuracy in high-dimensional genetic datasets with small sample sizes. This research offers a prediction model utilizing genomic data to enhance the early diagnosis rate of ovarian cancer, incorporating feature selection, data augmentation through adversarial conditional generative adversarial networks (AC-GAN), and an extreme gradient boosting (XGBoost) classifier. First, we can simplify the original genetic dataset through feature selection methods, removing irrelevant variables and noise, thereby improving the model’s predictive accuracy. Following dimensionality reduction, AC-GAN enriches the data, producing more realistic genetic samples to enhance the model’s generalization capacity. Finally, the XGBoost classifier is applied to classify the augmented data, achieving efficient predictions for ovarian cancer. These research findings strongly demonstrate that the diagnostic method proposed in this paper has a significant advantage in the predictive diagnosis of ovarian cancer, with an accuracy of 99.01% that surpasses the current technologies in use. Additionally, the algorithm identifies twelve genes highly relevant to ovarian cancer, providing valuable insights for physicians during diagnosis. Full article

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22 pages, 11825 KiB

Open AccessArticle

Analytical Solutions and Computer Modeling of a Boundary Value Problem for a Nonstationary System of Nernst–Planck–Poisson Equations in a Diffusion Layer

by Savva Kovalenko, Evgenia Kirillova, Vladimir Chekanov, Aminat Uzdenova and Mahamet Urtenov

Mathematics 2024, 12(24), 4040; https://doi.org/10.3390/math12244040 - 23 Dec 2024

Viewed by 545

Abstract

This article proposes various new approximate analytical solutions of the boundary value problem for the non-stationary system of Nernst–Planck–Poisson (NPP) equations in the diffusion layer of an ideally selective ion-exchange membrane at overlimiting current densities. As is known, the diffusion layer in the [...] Read more.

This article proposes various new approximate analytical solutions of the boundary value problem for the non-stationary system of Nernst–Planck–Poisson (NPP) equations in the diffusion layer of an ideally selective ion-exchange membrane at overlimiting current densities. As is known, the diffusion layer in the general case consists of a space charge region and a region of local electroneutrality. The proposed analytical solutions of the boundary value problems for the non-stationary system of Nernst–Planck–Poisson equations are based on the derivation of a new singularly perturbed nonlinear partial differential equation for the potential in the space charge region (SCR). This equation can be reduced to a singularly perturbed inhomogeneous Burgers equation, which, by the Hopf–Cole transformation, is reduced to an inhomogeneous singularly perturbed linear equation of parabolic type. Inside the extended SCR, there is a sufficiently accurate analytical approximation to the solution of the original boundary value problem. The electroneutrality region has a curvilinear boundary with the SCR, and with an unknown boundary condition on it. The article proposes a solution to this problem. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transfer in membrane systems. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transport in membrane systems. Full article

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

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16 pages, 545 KiB

Open AccessArticle

Fuzzy Rough Set Models Based on Fuzzy Similarity Relation and Information Granularity in Multi-Source Mixed Information Systems

by Pengfei Zhang, Yuxin Zhao, Dexian Wang, Yujie Zhang and Zheng Yu

Mathematics 2024, 12(24), 4039; https://doi.org/10.3390/math12244039 - 23 Dec 2024

Viewed by 783

Abstract

As a pivotal research method in the field of granular computing (GrC), fuzzy rough sets (FRSs) have garnered significant attention due to their successful overcoming of the limitations of traditional rough sets in handling continuous data. This paper is dedicated to exploring the [...] Read more.

As a pivotal research method in the field of granular computing (GrC), fuzzy rough sets (FRSs) have garnered significant attention due to their successful overcoming of the limitations of traditional rough sets in handling continuous data. This paper is dedicated to exploring the application potential of FRS models within the framework of multi-source complex information systems, which undoubtedly holds profound research significance. Firstly, a novel multi-source mixed information system (MsMIS), encompassing five distinct data types, is introduced, thereby enriching the dimensions of data processing. Subsequently, a similarity function, designed based on the unique attributes of the data, is utilized to accurately quantify the similarity relations among objects. Building on this foundation, fuzzy T-norm operators are employed to integrate the similarity matrices derived from different data types into a cohesive whole. This integration not only lays a solid foundation for subsequent model construction but also highlights the value of multi-source information fusion in the analysis of the MsMIS. The integrated results are subsequently utilized to develop FRS models. Through rigorous examination from the perspective of information granularity, the rationality of the FRS model is proven, and its mathematical properties are explored. This paper contributes to the theoretical advancement of FRS models in GrC and offers promising prospects for their practical implementation. Full article

(This article belongs to the Special Issue Intelligent Data Mining and Decision Making for Prognostics and Health Management)

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26 pages, 327 KiB

Open AccessArticle

SBE-Algebras via Intuitionistic Fuzzy Structures

by Tahsin Oner, Hashem Bordbar, Neelamegarajan Rajesh and Akbar Rezaei

Mathematics 2024, 12(24), 4038; https://doi.org/10.3390/math12244038 - 23 Dec 2024

Viewed by 541

Abstract

The study introduces the concept of intuitionistic fuzzy SBE-subalgebras, ideals, and filters, along with level sets of intuitionistic fuzzy sets within the framework of Sheffer stroke BE-algebras. These concepts are shown to be crucial for understanding the behavior of intuitionistic fuzzy logic in [...] Read more.

The study introduces the concept of intuitionistic fuzzy SBE-subalgebras, ideals, and filters, along with level sets of intuitionistic fuzzy sets within the framework of Sheffer stroke BE-algebras. These concepts are shown to be crucial for understanding the behavior of intuitionistic fuzzy logic in this algebraic structure. The study further establishes a bidirectional relationship between subalgebras, ideals, filters, and their respective level sets on Sheffer stroke BE-algebras, demonstrating that the level set of an intuitionistic fuzzy SBE-subalgebra, ideal, or filter is itself a subalgebra, ideal, or filter on this algebra, and vice versa. Full article

28 pages, 9780 KiB

Open AccessArticle

Dynamic Multi-Energy Optimization for Unit Commitment Integrating PEVs and Renewable Energy: A DO3LSO Algorithm

by Linxin Zhang, Zuobin Ying, Zhile Yang and Yuanjun Guo

Mathematics 2024, 12(24), 4037; https://doi.org/10.3390/math12244037 - 23 Dec 2024

Viewed by 563

Abstract

The global energy crisis and the pursuit of carbon neutrality have introduced significant challenges to the optimal dispatch of power systems. Despite advancements in optimization techniques, existing methods often struggle to efficiently handle the uncertainties introduced by renewable energy sources and the dynamic [...] Read more.

The global energy crisis and the pursuit of carbon neutrality have introduced significant challenges to the optimal dispatch of power systems. Despite advancements in optimization techniques, existing methods often struggle to efficiently handle the uncertainties introduced by renewable energy sources and the dynamic behavior of plug-in electric vehicles (PEVs). This study presents a multi-energy collaborative optimization approach based on a dynamic opposite level-based learning optimization swarm algorithm (DO3LSO). The methodology explores the impact of integrating PEVs and renewable energy sources, including photovoltaic and wind power, on unit commitment (UC) problems. By incorporating the bidirectional charging and discharging capabilities of PEVs and addressing the volatility of renewable energy, the proposed method demonstrates the ability to reduce reliance on traditional fossil fuel power generation, decrease carbon emissions, stabilize power output, and achieve a 7.01% reduction in costs. Comparative analysis with other optimization algorithms highlights the effectiveness of DO3LSO in achieving rapid convergence and precise optimization through hierarchical learning and dynamic opposite strategies, showcasing superior adaptability in complex load scenarios. The findings underscore the importance of multi-energy collaborative optimization as a pivotal solution for addressing the energy crisis, facilitating low-carbon transitions, and providing essential support for the development of intelligent and sustainable power systems. Full article

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19 pages, 2634 KiB

Open AccessArticle

GDSMOTE: A Novel Synthetic Oversampling Method for High-Dimensional Imbalanced Financial Data

by Libin Hu and Yunfeng Zhang

Mathematics 2024, 12(24), 4036; https://doi.org/10.3390/math12244036 - 23 Dec 2024

Viewed by 614

Abstract

Synthetic oversampling methods for dealing with imbalanced classification problems have been widely studied. However, the current synthetic oversampling methods still cannot perform well when facing high-dimensional imbalanced financial data. The failure of distance measurement in high-dimensional space, error accumulation caused by noise samples, [...] Read more.

Synthetic oversampling methods for dealing with imbalanced classification problems have been widely studied. However, the current synthetic oversampling methods still cannot perform well when facing high-dimensional imbalanced financial data. The failure of distance measurement in high-dimensional space, error accumulation caused by noise samples, and the reduction of recognition accuracy of majority samples caused by the distribution of synthetic samples are the main reasons that limit the performance of current methods. Taking these factors into consideration, a novel synthetic oversampling method is proposed, namely the gradient distribution-based synthetic minority oversampling technique (GDSMOTE). Firstly, the concept of gradient contribution was used to assign the minority-class samples to different gradient intervals instead of relying on the spatial distance. Secondly, the root sample selection strategy of GDSMOTE avoids the error accumulation caused by noise samples and a new concept of nearest neighbor was proposed to determine the auxiliary samples. Finally, a safety gradient distribution approximation strategy based on cosine similarity was designed to determine the number of samples to be synthesized in each safety gradient interval. Experiments on high-dimensional imbalanced financial datasets show that GDSMOTE can achieve a higher F1-Score and MCC metrics than baseline methods while achieving a higher recall score. This means that our method has the characteristics of improving the recognition accuracy of minority-class samples without sacrificing the recognition accuracy of majority-class samples and has good adaptability to data decision-making tasks in the financial field. Full article

(This article belongs to the Special Issue Advancement of Mathematical Methods in Feature Representation Learning for Artificial Intelligence, Data Mining and Robotics, 2nd Edition)

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32 pages, 382 KiB

Open AccessFeature PaperArticle

Classical Gasses with Singular Densities

by Luca Di Persio, Yuri Kondratiev and Viktorya Vardanyan

Mathematics 2024, 12(24), 4035; https://doi.org/10.3390/math12244035 - 23 Dec 2024

Viewed by 559

Abstract

We investigate classical continuous systems characterized by singular velocity distributions, where the corresponding Radon measures are defined over the entire space with infinite mass. These singular distributions are used to model particle velocities in systems where traditional velocity distributions do not apply. As [...] Read more.

We investigate classical continuous systems characterized by singular velocity distributions, where the corresponding Radon measures are defined over the entire space with infinite mass. These singular distributions are used to model particle velocities in systems where traditional velocity distributions do not apply. As a result, the particle positions in such systems no longer conform to conventional configurations in physical space. This necessitates the development of novel analytical tools to understand the underlying models. To address this, we introduce a new conceptual framework that redefines particle configurations in phase space, where each particle is represented by its spatial position and a velocity vector. The key idea is the construction of the Plato space, which is designed to represent idealized particle configurations where the total velocity remains bounded within any compact subset of phase space. This space serves as a crucial bridge to the space of vector-valued discrete Radon measures, where each measure captures the velocity distribution over the entire system. Given the inherent complexity of analyzing infinite-dimensional spaces, we tackle the problem by reformulating it onto a finite-dimensional configuration space. This is achieved by decomposing the infinite space into smaller, more manageable components. A central tool in this reformulation is the K-transform, which is pivotal in enabling harmonic analysis of the space. The K-transform allows us to represent the system in terms of components that are more amenable to analysis, thus simplifying the study of the system’s dynamics. Furthermore, we extend previous results in the study of correlation functions by developing correlation measures tailored for these vector-valued Radon measures. These generalized functions provide deeper insights into the correlations between particle positions and velocities, expanding the range of analysis to systems with singular velocity distributions. Through this approach, we develop a robust mathematical framework that sheds light on the structure and dynamics of complex particle systems, especially those characterized by singular velocity distributions. Our results offer a new perspective on systems with non-traditional velocity distributions, advancing the theory and methodology of particle systems in both classical and modern contexts. Full article

16 pages, 1604 KiB

Open AccessArticle

Crude Oil Futures Price Forecasting Based on Variational and Empirical Mode Decompositions and Transformer Model

by Linya Huang, Xite Yang, Yongzeng Lai, Ankang Zou and Jilin Zhang

Mathematics 2024, 12(24), 4034; https://doi.org/10.3390/math12244034 - 23 Dec 2024

Viewed by 1179

Abstract

Crude oil is a raw and natural, but nonrenewable, resource. It is one of the world’s most important commodities, and its price can have ripple effects throughout the broader economy. Accurately predicting crude oil prices is vital for investment decisions but it remains [...] Read more.

Crude oil is a raw and natural, but nonrenewable, resource. It is one of the world’s most important commodities, and its price can have ripple effects throughout the broader economy. Accurately predicting crude oil prices is vital for investment decisions but it remains challenging. Due to the deficiencies neglecting residual factors when forecasting using conventional combination models, such as the autoregressive moving average and the long short-term memory for prediction, the variational mode decomposition (VMD)-empirical mode decomposition (EMD)-Transformer model is proposed to predict crude oil prices in this study. This model integrates a second decomposition and Transformer model-based machine learning method. More specifically, we employ the VMD technique to decompose the original sequence into variational mode filtering (VMF) and a residual sequence, followed by using EMD to decompose the residual sequence. Ultimately, we apply the Transformer model to predict the decomposed modal components and superimpose the results to produce the final forecasted prices. Further empirical test results demonstrate that the proposed quadratic decomposition composite model can comprehensively identify the characteristics of WTI and Brent crude oil futures daily price series. The test results illustrate that the proposed VMD–EMD–Transformer model outperforms the other three models—long short-term memory (LSTM), Transformer, and VMD–Transformer in forecasting crude oil prices. Details are presented in the empirical study part. Full article

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20 pages, 594 KiB

Open AccessArticle

Solution of a Nonlinear Integral Equation Arising in the Moment Approximation of Spatial Logistic Dynamics

by Mikhail Nikolaev, Alexey Nikitin and Ulf Dieckmann

Mathematics 2024, 12(24), 4033; https://doi.org/10.3390/math12244033 - 23 Dec 2024

Viewed by 627

Abstract

We investigate a nonlinear integral equation derived through moment approximation from the individual-based representation of spatial logistic dynamics. The equation describes how the densities of pairs of individuals represented by points in continuous space are expected to equilibrate under spatially explicit birth–death processes [...] Read more.

We investigate a nonlinear integral equation derived through moment approximation from the individual-based representation of spatial logistic dynamics. The equation describes how the densities of pairs of individuals represented by points in continuous space are expected to equilibrate under spatially explicit birth–death processes characterized by constant fecundity with local natal dispersal and variable mortality determined by local competition. The equation is derived from a moment hierarchy truncated by a moment closure expressing the densities of triplets as a function of the densities of pairs. Focusing on results for individuals inhabiting two-dimensional habitats, we explore the solvability of the equation by introducing a dedicated space of functions that are integrable up to a constant. Using this function space, we establish sufficient conditions for the existence of solutions of the equation within a zero-centered ball. For illustration and further insights, we complement our analytical findings with numerical results. Full article

(This article belongs to the Collection Theoretical and Mathematical Ecology)

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23 pages, 3832 KiB

Open AccessFeature PaperReview

Higher-Order Spectral Analysis and Artificial Intelligence for Diagnosing Faults in Electrical Machines: An Overview

by Miguel Enrique Iglesias Martínez, Jose A. Antonino-Daviu, Larisa Dunai, J. Alberto Conejero and Pedro Fernández de Córdoba

Mathematics 2024, 12(24), 4032; https://doi.org/10.3390/math12244032 - 23 Dec 2024

Cited by 1 | Viewed by 1074

Abstract

Fault diagnosis in electrical machines is a cornerstone of operational reliability and cost-effective maintenance strategies. This review provides a comprehensive exploration of the integration of higher-order spectral analysis (HOSA) techniques—such as a bispectrum, spectral kurtosis, and multifractal wavelet analysis—with advanced artificial intelligence (AI) [...] Read more.

Fault diagnosis in electrical machines is a cornerstone of operational reliability and cost-effective maintenance strategies. This review provides a comprehensive exploration of the integration of higher-order spectral analysis (HOSA) techniques—such as a bispectrum, spectral kurtosis, and multifractal wavelet analysis—with advanced artificial intelligence (AI) methodologies, including deep learning, clustering algorithms, Transformer models, and transfer learning. The synergy between HOSA’s robustness in noisy and transient environments and AI’s automation of complex classifications has significantly advanced fault diagnosis in synchronous and DC motors. The novelty of this work lies in its detailed examination of the latest AI advancements, and the hybrid framework combining HOSA-derived features with AI techniques. The proposed approaches address challenges such as computational efficiency and scalability for industrial-scale applications, while offering innovative solutions for predictive maintenance. By leveraging these hybrid methodologies, the work charts a transformative path for improving the reliability and adaptability of industrial-grade electrical machine systems. Full article

(This article belongs to the Special Issue Signal Processing and Machine Learning in Real-Life Processes)

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20 pages, 9510 KiB

Open AccessArticle

Generalized Type-2 Fuzzy Approach for Parameter Adaptation in the Whale Optimization Algorithm

by Leticia Amador-Angulo, Oscar Castillo, Patricia Melin and Zong Woo Geem

Mathematics 2024, 12(24), 4031; https://doi.org/10.3390/math12244031 - 22 Dec 2024

Cited by 1 | Viewed by 1102

Abstract

An enhanced whale optimization algorithm (WOA) through the implementation of a generalized type-2 fuzzy logic system (GT2FLS) is outlined. The initial idea is to improve the efficacy of the original WOA using a GT2FLS to find the optimal values of the [...] Read more.

An enhanced whale optimization algorithm (WOA) through the implementation of a generalized type-2 fuzzy logic system (GT2FLS) is outlined. The initial idea is to improve the efficacy of the original WOA using a GT2FLS to find the optimal values of the

{\vec{r}}_{1}

and

{\vec{r}}_{2}

parameters of the WOA, for the case of optimizing mathematical functions. In the WOA algorithm,

{\vec{r}}_{1}

is a variable that affects the new position of the whale in the search space, in this case, affecting the exploration, and

{\vec{r}}_{2}

is a variable that has an effect on finding the local optima, which is an important factor for the exploration. The efficiency of a fuzzy WOA with a GT2FLS (FWOA-GT2FLS) is highlighted by presenting the excellent results of the case study of the benchmark function optimization. A relevant analysis and comparison with a bio-inspired algorithm based on artificial bees is also presented. Statistical tests and comparisons with other bio-inspired algorithms and the initial WOA, with type-1 FLS (FWOA-T1FLS) and interval type-2 FLS (FWOA-IT2FLS), are presented. For each of the methodologies, the metric for evaluation is the average of the minimum squared errors. Full article

(This article belongs to the Special Issue Advances in Artificial Intelligence: Models, Optimization, and Machine Learning, 3rd Edition)

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26 pages, 23809 KiB

Open AccessArticle

A Navigation Algorithm Based on the Reinforcement Learning Reward System and Optimised with Genetic Algorithm

by Mireya Cabezas-Olivenza, Ekaitz Zulueta, Iker Azurmendi-Marquinez, Unai Fernandez-Gamiz and Danel Rico-Melgosa

Mathematics 2024, 12(24), 4030; https://doi.org/10.3390/math12244030 - 22 Dec 2024

Viewed by 951

Abstract

Regarding autonomous vehicle navigation, reinforcement learning is a technique that has demonstrated significant results. Nevertheless, it is a technique with a high number of parameters that need to be optimised without prior information, and correctly performing this is a complicated task. In this [...] Read more.

Regarding autonomous vehicle navigation, reinforcement learning is a technique that has demonstrated significant results. Nevertheless, it is a technique with a high number of parameters that need to be optimised without prior information, and correctly performing this is a complicated task. In this research study, a system based on the principles of reinforcement learning, specifically on the concept of rewards, is presented. A mathematical expression was proposed to control the vehicle’s direction based on its position, the obstacles in the environment and the destination. In this equation proposal, there was only one unknown parameter that regulated the degree of the action to be taken, and this was optimised through the genetic algorithm. In this way, a less computationally expensive navigation algorithm was presented, as it avoided the use of neural networks. The controller’s time to obtain the navigation instructions was around 6.201·10⁻⁴ s. This algorithm is an efficient and accurate system which manages not to collide with obstacles and to reach the destination from any position. Moreover, in most cases, it has been found that the proposed navigations are also optimal. Full article

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12 pages, 7751 KiB

Open AccessFeature PaperArticle

Considering a Classical Upper Bound on the Frobenius Number

by Aled Williams and Daiki Haijima

Mathematics 2024, 12(24), 4029; https://doi.org/10.3390/math12244029 - 22 Dec 2024

Viewed by 468

Abstract

In this paper, we study the (classical) Frobenius problem, namely the problem of finding the largest integer that cannot be represented as a nonnegative integer combination of given, relatively prime, (strictly) positive integers (known as the Frobenius number). The main contribution of this [...] Read more.

In this paper, we study the (classical) Frobenius problem, namely the problem of finding the largest integer that cannot be represented as a nonnegative integer combination of given, relatively prime, (strictly) positive integers (known as the Frobenius number). The main contribution of this paper are observations regarding a previously known upper bound on the Frobenius number where, in particular, we observe that a previously presented argument features a subtle error, which alters the value of the upper bound. Despite this, we demonstrate that the subtle error does not impact upon on the validity of the upper bound, although it does impact on the upper bounds tightness. Notably, we formally state the corrected result and additionally compare the relative tightness of the corrected upper bound with the original. In particular, we show that the updated bound is tighter in all but only a relatively “small” number of cases using both formal techniques and via Monte Carlo simulation techniques. Full article

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

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27 pages, 12355 KiB

Open AccessArticle

Low-Light Image Enhancement Using CycleGAN-Based Near-Infrared Image Generation and Fusion

by Min-Han Lee, Young-Ho Go, Seung-Hwan Lee and Sung-Hak Lee

Mathematics 2024, 12(24), 4028; https://doi.org/10.3390/math12244028 - 22 Dec 2024

Viewed by 1173

Abstract

Image visibility is often degraded under challenging conditions such as low light, backlighting, and inadequate contrast. To mitigate these issues, techniques like histogram equalization, high dynamic range (HDR) tone mapping and near-infrared (NIR)–visible image fusion are widely employed. However, these methods have inherent [...] Read more.

Image visibility is often degraded under challenging conditions such as low light, backlighting, and inadequate contrast. To mitigate these issues, techniques like histogram equalization, high dynamic range (HDR) tone mapping and near-infrared (NIR)–visible image fusion are widely employed. However, these methods have inherent drawbacks: histogram equalization frequently causes oversaturation and detail loss, while visible–NIR fusion requires complex and error-prone images. The proposed algorithm of a complementary cycle-consistent generative adversarial network (CycleGAN)-based training with visible and NIR images, leverages CycleGAN to generate fake NIR images by blending the characteristics of visible and NIR images. This approach presents tone compression and preserves fine details, effectively addressing the limitations of traditional methods. Experimental results demonstrate that the proposed method outperforms conventional algorithms, delivering superior quality and detail retention. This advancement holds substantial promise for applications where dependable image visibility is critical, such as autonomous driving and CCTV (Closed-Circuit Television) surveillance systems. Full article

(This article belongs to the Special Issue New Advances and Applications in Image Processing and Computer Vision)

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7 pages, 233 KiB

Open AccessArticle

On Diophantine Equations 2^x ± (2^kp)^y = z² and −2^x + (2^k3)^y = z²

by Yuan Li, Torre Lloyd and Angel Clinton

Mathematics 2024, 12(24), 4027; https://doi.org/10.3390/math12244027 - 22 Dec 2024

Viewed by 413

Abstract

In this paper, we solve three Diophantine equations:

2^{x} \pm {(2^{k} p)}^{y} = z^{2}

and

- 2^{x} + {(2^{k} 3)}^{y} = z^{2}

with

k \geq 0

and prime [...] Read more.

In this paper, we solve three Diophantine equations:

2^{x} \pm {(2^{k} p)}^{y} = z^{2}

and

- 2^{x} + {(2^{k} 3)}^{y} = z^{2}

with

k \geq 0

and prime

p \equiv \pm 3

(mod 8)

. We obtain all the non-negative integer solutions by using elementary methods and the database of elliptic curves in “The L-functions and modular forms database” (LMFDB). Full article

21 pages, 445 KiB

Open AccessArticle

Analysis of Block Adaptive Type-II Progressive Hybrid Censoring with Weibull Distribution

by Kundan Singh, Yogesh Mani Tripathi, Liang Wang and Shuo-Jye Wu

Mathematics 2024, 12(24), 4026; https://doi.org/10.3390/math12244026 - 22 Dec 2024

Viewed by 694

Abstract

The estimation of unknown model parameters and reliability characteristics is considered under a block adaptive progressive hybrid censoring scheme, where data are observed from a Weibull model. This censoring scheme enhances experimental efficiency by conducting experiments across different testing facilities. Point and interval [...] Read more.

The estimation of unknown model parameters and reliability characteristics is considered under a block adaptive progressive hybrid censoring scheme, where data are observed from a Weibull model. This censoring scheme enhances experimental efficiency by conducting experiments across different testing facilities. Point and interval estimates for parameters and reliability assessments are derived using both classical and Bayesian approaches. The existence and uniqueness of maximum likelihood estimates are established. Consequently, reliability performance and differences across different testing facilities are analyzed. In addition, a Metropolis–Hastings sampling algorithm is developed to approximate complex posterior computations. Approximate confidence intervals and highest posterior density credible intervals are obtained for the parametric functions. The performance of all estimators is evaluated through an extensive simulation study, and observations are discussed. A cancer dataset is analyzed to illustrate the findings under the block adaptive censoring scheme. Full article

(This article belongs to the Special Issue Statistical Simulation and Computation: 3rd Edition)

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17 pages, 16110 KiB

Open AccessArticle

Low-Light Image Enhancement Integrating Retinex-Inspired Extended Decomposition with a Plug-and-Play Framework

by Chenping Zhao, Wenlong Yue, Yingjun Wang, Jianping Wang, Shousheng Luo, Huazhu Chen and Yan Wang

Mathematics 2024, 12(24), 4025; https://doi.org/10.3390/math12244025 - 22 Dec 2024

Viewed by 655

Abstract

Images captured under low-light conditions often suffer from serious degradation due to insufficient light, which adversely impacts subsequent computer vision tasks. Retinex-based methods have demonstrated strong potential in low-light image enhancement. However, existing approaches often directly design prior regularization functions for either illumination [...] Read more.

Images captured under low-light conditions often suffer from serious degradation due to insufficient light, which adversely impacts subsequent computer vision tasks. Retinex-based methods have demonstrated strong potential in low-light image enhancement. However, existing approaches often directly design prior regularization functions for either illumination or reflectance components, which may unintentionally introduce noise. To address these limitations, this paper presents an enhancement method by integrating a Plug-and-Play strategy into an extended decomposition model. The proposed model consists of three main components: an extended decomposition term, an iterative reweighting regularization function for the illumination component, and a Plug-and-Play refinement term applied to the reflectance component. The extended decomposition enables a more precise representation of image components, while the iterative reweighting mechanism allows for gentle smoothing near edges and brighter areas while applying more pronounced smoothing in darker regions. Additionally, the Plug-and-Play framework incorporates off-the-shelf image denoising filters to effectively suppress noise and preserve useful image details. Extensive experiments on several datasets confirm that the proposed method consistently outperforms existing techniques. Full article

(This article belongs to the Special Issue Mathematical Methods for Machine Learning and Computer Vision)

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27 pages, 529 KiB

Open AccessArticle

The Impact of Digital Economy Development on Regional Income Gaps: A Perspective on Multidimensional Inequality Decomposition and Threshold Effects

by Jing Ruan, Lu Zou, Ruiqi Liu and Hongyun Pan

Mathematics 2024, 12(24), 4024; https://doi.org/10.3390/math12244024 - 22 Dec 2024

Viewed by 1367

Abstract

This paper investigates the impact of digital economy development on regional income gaps within provinces in China, focusing on the underlying mechanisms and multidimensional threshold effects. Using provincial panel data from 2011 to 2021, this study finds that the relationship between digital economy [...] Read more.

This paper investigates the impact of digital economy development on regional income gaps within provinces in China, focusing on the underlying mechanisms and multidimensional threshold effects. Using provincial panel data from 2011 to 2021, this study finds that the relationship between digital economy development and income gaps follows a U-shape, initially reducing the gap and later widening it. This analysis identifies internet penetration and digital transaction growth as key contributors to multidimensional inequality in digital economy development. Moreover, moderate levels of multidimensional inequality in digital economy development enhance its positive effect on narrowing regional income gaps, while excessive multidimensional inequality may diminish or even reverse this effect. These findings highlight the importance of managing multidimensional inequality to maximize digital economies’ potential for fostering more balanced regional development. Full article

(This article belongs to the Special Issue Advances in Statistical Analysis for Health, Finance, Industry and Digital Economy)

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33 pages, 7390 KiB

Open AccessArticle

Optimizing Multi-Depot Mixed Fleet Vehicle–Drone Routing Under a Carbon Trading Mechanism

by Yong Peng, Yanlong Zhang, Dennis Z. Yu, Song Liu, Yali Zhang and Yangyan Shi

Mathematics 2024, 12(24), 4023; https://doi.org/10.3390/math12244023 - 22 Dec 2024

Viewed by 894

Abstract

The global pursuit of carbon neutrality requires the reduction of carbon emissions in logistics and distribution. The integration of electric vehicles (EVs) and drones in a collaborative delivery model revolutionizes last-mile delivery by significantly reducing operating costs and enhancing delivery efficiency while supporting [...] Read more.

The global pursuit of carbon neutrality requires the reduction of carbon emissions in logistics and distribution. The integration of electric vehicles (EVs) and drones in a collaborative delivery model revolutionizes last-mile delivery by significantly reducing operating costs and enhancing delivery efficiency while supporting environmental objectives. This paper presents a cost-minimization model that addresses transportation, energy, and carbon trade costs within a cap-and-trade framework. We develop a multi-depot mixed fleet, including electric and fuel vehicles, and a drone collaborative delivery routing optimization model. This model incorporates key factors such as nonlinear EV charging times, time-dependent travel conditions, and energy consumption. We propose an adaptive large neighborhood search algorithm integrating spatiotemporal distance (ALNS-STD) to solve this complex model. This algorithm introduces five domain-specific operators and an adaptive adjustment mechanism to improve solution quality and efficiency. Our computational experiments demonstrate the effectiveness of the ALNS-STD, showing its ability to optimize routes by accounting for both spatial and temporal factors. Furthermore, we analyze the influence of charging station distribution and carbon trading mechanisms on overall delivery costs and route planning, underscoring the global significance of our findings. Full article

(This article belongs to the Special Issue Data-Driven Algorithms for Optimal Decision Making in Logistics and Supply Chain Management)

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18 pages, 318 KiB

Open AccessArticle

Spinor Equations of Smarandache Curves in E³

by Zeynep İsabeyoǧlu, Tülay Erişir and Ayşe Zeynep Azak

Mathematics 2024, 12(24), 4022; https://doi.org/10.3390/math12244022 - 22 Dec 2024

Viewed by 628

Abstract

This study examines the spinor representations of

TN

(tangent and normal),

NB

(normal and binormal),

TB

(tangent and binormal) and

TNB

(tangent, normal and binormal)–Smarandache curves in three-dimensional Euclidean space

E^{3}

. Spinors are complex column vectors and move on Pauli spin [...] Read more.

This study examines the spinor representations of

TN

(tangent and normal),

NB

(normal and binormal),

TB

(tangent and binormal) and

TNB

(tangent, normal and binormal)–Smarandache curves in three-dimensional Euclidean space

E^{3}

. Spinors are complex column vectors and move on Pauli spin matrices. Isotropic vectors in the

C^{3}

complex vector space form a two-dimensional surface in the

C^{2}

complex space. Additionally, each isotropic vector in

C^{3}

space corresponds to two vectors in

C^{2}

space, called spinors. Based on this information, our goal is to establish a relationship between curve theory in differential geometry and spinor space by matching a spinor with an isotropic vector and a real vector generated from the vectors of the Frenet–Serret frame of a curve in three-dimensional Euclidean space. Accordingly, we initially assume two spinors corresponding to the Frenet–Serret frames of the main curve and its (

TN

,

NB

,

TB

and

TNB

)–Smarandache curves. Then, we utilize the relationships between the Frenet frames of these curves to examine the connections between the two spinors corresponding to these curves. Thus, we give the relationships between spinors corresponding to these Smarandache curves. For this reason, this study creates a bridge between mathematics and physics. This study can also serve as a reference for new studies in geometry and physics as a geometric interpretation of a physical expression. Full article

(This article belongs to the Special Issue Differential Geometric Structures and Their Applications)

33 pages, 4748 KiB

Open AccessArticle

A New Class of Bivariate Distributions: Properties, Estimation, and Modeling

by Jumanah Ahmed Darwish, Saman Hanif Shahbaz and Muhammad Qaiser Shahbaz

Mathematics 2024, 12(24), 4021; https://doi.org/10.3390/math12244021 - 22 Dec 2024

Viewed by 575

Abstract

The development of bivariate distributions is a challenging area of research, and a novel class of bivariate distributions is proposed in this paper. Certain useful mathematical characteristics of the proposed class of bivariate distributions are explored in general. These properties include the joint [...] Read more.

The development of bivariate distributions is a challenging area of research, and a novel class of bivariate distributions is proposed in this paper. Certain useful mathematical characteristics of the proposed class of bivariate distributions are explored in general. These properties include the joint and conditional and joint survival function moments and measures of dependence. The parameters of the proposed class of distributions are estimated using the maximum likelihood method of estimation. A multivariate version of the proposed class is also proposed, and some necessary properties are discussed. The multivariate dependence measures are obtained for the proposed multivariate class of distributions. A new bivariate power function distribution is proposed using the power function distribution as a baseline distribution in the proposed bivariate class. Some useful properties of the proposed bivariate distribution are studied. The parameters of the proposed bivariate distribution are estimated using the maximum likelihood method. An extensive simulation study has been conducted to see the performance of the estimation method. The new bivariate power function distribution is used to model some real data. Full article

(This article belongs to the Special Issue Advances in Statistical Methods with Applications)

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25 pages, 992 KiB

Open AccessArticle

A Self-Rewarding Mechanism in Deep Reinforcement Learning for Trading Strategy Optimization

by Yuling Huang, Chujin Zhou, Lin Zhang and Xiaoping Lu

Mathematics 2024, 12(24), 4020; https://doi.org/10.3390/math12244020 - 22 Dec 2024

Cited by 1 | Viewed by 4665

Abstract

Reinforcement Learning (RL) is increasingly being applied to complex decision-making tasks such as financial trading. However, designing effective reward functions remains a significant challenge. Traditional static reward functions often fail to adapt to dynamic environments, leading to inefficiencies in learning. This paper presents [...] Read more.

Reinforcement Learning (RL) is increasingly being applied to complex decision-making tasks such as financial trading. However, designing effective reward functions remains a significant challenge. Traditional static reward functions often fail to adapt to dynamic environments, leading to inefficiencies in learning. This paper presents a novel approach, called Self-Rewarding Deep Reinforcement Learning (SRDRL), which integrates a self-rewarding network within the RL framework. The SRDRL mechanism operates in two primary phases: First, supervised learning techniques are used to learn from expert knowledge by employing advanced time-series feature extraction models, including TimesNet and WFTNet. This step refines the self-rewarding network parameters by comparing predicted rewards with expert-labeled rewards, which are based on metrics such as Min-Max, Sharpe Ratio, and Return. In the second phase, the model selects the higher value between the expert-labeled and predicted rewards as the RL reward, storing it in the replay buffer. This combination of expert knowledge and predicted rewards enhances the performance of trading strategies. The proposed implementation, called Self-Rewarding Double DQN (SRDDQN), demonstrates that the self-rewarding mechanism improves learning and optimizes trading decisions. Experiments conducted on datasets including DJI, IXIC, and SP500 show that SRDDQN achieves a cumulative return of 1124.23% on the IXIC dataset, significantly outperforming the next best method, Fire (DQN-HER), which achieved 51.87%. SRDDQN also enhances the stability and efficiency of trading strategies, providing notable improvements over traditional RL methods. The integration of a self-rewarding mechanism within RL addresses a critical limitation in reward function design and offers a scalable, adaptable solution for complex, dynamic trading environments. Full article

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

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27 pages, 3052 KiB

Open AccessArticle

Sensitivity Analysis of Factors Influencing Coal Prices in China

by Jingye Lyu, Chong Li, Wenwen Zhou and Jinsuo Zhang

Mathematics 2024, 12(24), 4019; https://doi.org/10.3390/math12244019 - 21 Dec 2024

Viewed by 937

Abstract

A scientific assessment of the sensitivity of the Chinese coal market has become an important research topic. This paper combines Gaussian Process Regression (GPR) and Sobol sensitivity analysis to construct a GPR–Sobol hybrid model innovatively applied to the Chinese coal market, thus addressing [...] Read more.

A scientific assessment of the sensitivity of the Chinese coal market has become an important research topic. This paper combines Gaussian Process Regression (GPR) and Sobol sensitivity analysis to construct a GPR–Sobol hybrid model innovatively applied to the Chinese coal market, thus addressing a gap in the economic applications of this method. The model is used to analyze the sensitivity of factors influencing coal prices in China. The GPR–Sobol model effectively handles nonlinear relationships, enabling an in-depth exploration of key factors affecting price volatility and quantifying their impacts, thus overcoming the limitations of traditional econometric models in nonlinear data processing. The results indicate that economic growth, energy prices, interest rates, exchange rates, and uncertainty factors exhibit high sensitivity and significantly impact coal price fluctuations in China. Coal prices in northwest China are more sensitive to interest rates and geopolitical risks, while prices in east and south China are more responsive to exchange rates but less so to economic policy uncertainty. Additionally, coal prices in north, south, and east China are highly sensitive to international energy prices, indicating that coal prices are dominated by the global energy market, yet their weak response to macroeconomic indicators suggests these regions is still insufficiently mature. Full article

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15 pages, 331 KiB

Open AccessFeature PaperArticle

Analyzing Sample Size in Information-Theoretic Models

by D. Bernal-Casas and J. M. Oller

Mathematics 2024, 12(24), 4018; https://doi.org/10.3390/math12244018 - 21 Dec 2024

Viewed by 635

Abstract

In this paper, we delve into the complexities of information-theoretic models, specifically focusing on how we can model sample size and how it affects our previous findings. This question is fundamental and intricate, posing a significant intellectual challenge to our research. While previous [...] Read more.

In this paper, we delve into the complexities of information-theoretic models, specifically focusing on how we can model sample size and how it affects our previous findings. This question is fundamental and intricate, posing a significant intellectual challenge to our research. While previous studies have considered a fixed sample size, this work explores other possible alternatives to assess its impact on the mathematical approach. To ensure that our framework aligns with the principles of quantum theory, specific conditions related to sample size must be met, as they are inherently linked to information quantities. The arbitrary nature of sample size presents a significant challenge in achieving this alignment, which we thoroughly investigate in this study. Full article

17 pages, 1494 KiB

Open AccessArticle

Improved Genetic Algorithm for Solving Robot Path Planning Based on Grid Maps

by Jie Zhu and Dazhi Pan

Mathematics 2024, 12(24), 4017; https://doi.org/10.3390/math12244017 - 21 Dec 2024

Cited by 1 | Viewed by 1184

Abstract

Aiming at some shortcomings of the genetic algorithm to solve the path planning in a global static environment, such as a low efficiency of population initialization, slow convergence speed, and easy-to-fall-into the local optimum, an improved genetic algorithm is proposed to solve the [...] Read more.

Aiming at some shortcomings of the genetic algorithm to solve the path planning in a global static environment, such as a low efficiency of population initialization, slow convergence speed, and easy-to-fall-into the local optimum, an improved genetic algorithm is proposed to solve the path planning problem. Firstly, the environment model is established by using the grid method; secondly, in order to overcome the difficulty of a low efficiency of population initialization, a population initialization method with directional guidance is proposed; finally, in order to balance the global and local optimization searching and to speed up the solution speed, the proposed non-common point crossover operator, range mutation operator, and simplification operator are used in combination with the one-point crossover operator and one-point mutation operator in the traditional genetic algorithm to obtain an improved genetic algorithm. In the simulation experiment, Experiment 1 verifies the effectiveness of the population initialization method proposed in this paper. The success rates in Map 1, Map 2, Map 3, and Map 4 were 56.3854%, 55.851%, 34.1%, and 24.1514%, respectively, which were higher than the two initialization methods compared. Experiment 2 verifies the effectiveness of the genetic algorithm (IGA) improved in this paper for path planning. In four maps, the path planning is compared with the five algorithms and the shortest distance is achieved in all of them. The two experiments show that the improved genetic algorithm in this paper has advantages in path planning. Full article

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20 pages, 322 KiB

Open AccessFeature PaperArticle

Summed Series Involving ₁F₂ Hypergeometric Functions

by Jack C. Straton

Mathematics 2024, 12(24), 4016; https://doi.org/10.3390/math12244016 - 21 Dec 2024

Cited by 1 | Viewed by 580

Abstract

Summation of infinite series has played a significant role in a broad range of problems in the physical sciences and is of interest in a purely mathematical context. In a prior paper, we found that the Fourier–Legendre series of a Bessel function of [...] Read more.

Summation of infinite series has played a significant role in a broad range of problems in the physical sciences and is of interest in a purely mathematical context. In a prior paper, we found that the Fourier–Legendre series of a Bessel function of the first kind

J_{N} (k x)

and modified Bessel functions of the first kind

I_{N} (k x)

lead to an infinite set of series involving

{}_{1}F_{2}

hypergeometric functions (extracted therefrom) that could be summed, having values that are inverse powers of the eight primes

1 / (2^{i} 3^{j} 5^{k} 7^{l} 11^{m} 13^{n} 17^{o} 19^{p})

multiplying powers of the coefficient k, for the first 22 terms in each series. The present paper shows how to generate additional, doubly infinite summed series involving

{}_{1}F_{2}

hypergeometric functions from Chebyshev polynomial expansions of Bessel functions, and trebly infinite sets of summed series involving

{}_{1}F_{2}

hypergeometric functions from Gegenbauer polynomial expansions of Bessel functions. That the parameters in these new cases can be varied at will significantly expands the landscape of applications for which they could provide a solution. Full article

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15 pages, 755 KiB

Open AccessArticle

High-Order Control Lyapunov–Barrier Functions for Real-Time Optimal Control of Constrained Non-Affine Systems

by Alaa Eddine Chriat and Chuangchuang Sun

Mathematics 2024, 12(24), 4015; https://doi.org/10.3390/math12244015 - 21 Dec 2024

Viewed by 847

Abstract

This paper presents a synthesis of higher-order control Lyapunov functions (HOCLFs) and higher-order control barrier functions (HOCBFs) capable of controlling nonlinear dynamic systems while maintaining safety. Building on previous Lyapunov and barrier formulations, we first investigate the feasibility of the Lyapunov and barrier [...] Read more.

This paper presents a synthesis of higher-order control Lyapunov functions (HOCLFs) and higher-order control barrier functions (HOCBFs) capable of controlling nonlinear dynamic systems while maintaining safety. Building on previous Lyapunov and barrier formulations, we first investigate the feasibility of the Lyapunov and barrier function approach in controlling a non-affine dynamic system under certain convexity conditions. Then we propose an HOCLF form that ensures convergence of non-convex dynamics with convex control inputs to target states. We combine the HOCLF with the HOCBF to ensure forward invariance of admissible sets and guarantee safety. This online non-convex optimal control problem is then formulated as a convex Quadratic Program (QP) that can be efficiently solved on board for real-time applications. Lastly, we determine the HOCLBF coefficients using a heuristic approach where the parameters are tuned and automatically decided to ensure the feasibility of the QPs, an inherent major limitation of high-order CBFs. The efficacy of the suggested algorithm is demonstrated on the real-time six-degree-of-freedom powered descent optimal control problem, where simulation results were run efficiently on a standard laptop. Full article

(This article belongs to the Special Issue Advances in Decision Making, Control, and Optimization)

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27 pages, 3581 KiB

Open AccessArticle

Sustainable Design Factors and Solutions Analysis and Assessment for the Graphic Design Industry: A Hybrid Fuzzy AHP–Fuzzy MARCOS Approach

by Chia-Liang Lin

Mathematics 2024, 12(24), 4014; https://doi.org/10.3390/math12244014 - 21 Dec 2024

Viewed by 843

Abstract

Within the realm of graphic design sustainability, selecting appropriate solutions has become a crucial strategic decision for organizations aiming to optimize their operations. This paper presents a novel hybrid multi-criteria decision-making (MCDM) approach, integrating a fuzzy analytical hierarchy process (FAHP) and fuzzy measurement [...] Read more.

Within the realm of graphic design sustainability, selecting appropriate solutions has become a crucial strategic decision for organizations aiming to optimize their operations. This paper presents a novel hybrid multi-criteria decision-making (MCDM) approach, integrating a fuzzy analytical hierarchy process (FAHP) and fuzzy measurement alternatives and ranking according to compromise solution (FMARCOS). Evaluation criteria for graphic design sustainability are determined through consultation with experts, with their judgments expressed using linguistic terms based on fuzzy numbers. Criteria weights are calculated using FAHP, and the ranking and selection of the optimal potential solution are determined using FMARCOS. Subsequently, sensitivity analysis of the criteria weights is conducted to validate the results. Findings indicate that the integrated FAHP and FMARCOS model provides a robust and adaptable assessment framework for graphic design sustainability, enabling companies to navigate complexities strategically and effectively. The key contribution of this research is its emphasis on a systematic and objective model, offering practical insights relevant to the industry. It also serves as a valuable benchmark for future research in similar fields. Full article

(This article belongs to the Special Issue Fuzzy Decision Making and Applications)

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33 pages, 13737 KiB

Open AccessArticle

Nonlinear Adaptive Optimal Control Design and Implementation for Trajectory Tracking of Four-Wheeled Mecanum Mobile Robots

by Yung-Hsiang Chen

Mathematics 2024, 12(24), 4013; https://doi.org/10.3390/math12244013 - 21 Dec 2024

Viewed by 691

Abstract

This study proposes a nonlinear adaptive optimal control method, the adaptive H₂ control method, applied to the trajectory tracking problem of the wheeled mobile robot (WMR) with four-wheel mecanum wheels. From the perspective of solving mathematical problems, finding an analytical adaptive control [...] Read more.

This study proposes a nonlinear adaptive optimal control method, the adaptive H₂ control method, applied to the trajectory tracking problem of the wheeled mobile robot (WMR) with four-wheel mecanum wheels. From the perspective of solving mathematical problems, finding an analytical adaptive control solution that satisfies the adaptive H₂ performance criterion for the trajectory tracking problem of the WMR with four-wheel mecanum wheels is an extremely challenging task due to the high complexity of the dynamic system. To analytically derive the control law and adaptive control law for this trajectory tracking problem, a proportional-derivative (PD) type transformation is employed to formalize the trajectory tracking error dynamics between the WMR and the desired trajectory (DT). Based on an in-depth analysis of the trajectory tracking error dynamics, a closed-form adaptive control law is analytically derived from the highly complex nonlinear dynamic system equations. This control law provides a solution to the trajectory tracking problem of the WMR while satisfying the adaptive H₂ performance criterion. The proposed adaptive nonlinear control method offers a simple control structure and advantages such as improved energy efficiency. Finally, simulations and experimental implementations were conducted to verify the performance of the proposed adaptive H₂ control method and the H₂ control method in tracking the DT. The results demonstrate that, compared to the H₂ control method, the adaptive H₂ control method exhibits superior trajectory tracking performance, particularly in the presence of significant model uncertainties. Full article

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

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