Journal Description
Mathematics
Mathematics
is a peer-reviewed, open access journal which provides an advanced forum for studies related to mathematics, and is published semimonthly online by MDPI. The European Society for Fuzzy Logic and Technology (EUSFLAT) and International Society for the Study of Information (IS4SI) are affiliated with Mathematics and their members receive a discount on article processing charges.
- Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
- High Visibility: indexed within Scopus, SCIE (Web of Science), RePEc, and other databases.
- Journal Rank: JCR - Q1 (Mathematics) / CiteScore - Q1 (General Mathematics)
- Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 16.9 days after submission; acceptance to publication is undertaken in 2.6 days (median values for papers published in this journal in the second half of 2023).
- Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
- Sections: published in 13 topical sections.
- Companion journals for Mathematics include: Foundations, AppliedMath, Analytics, International Journal of Topology, Geometry and Logics.
Impact Factor:
2.4 (2022);
5-Year Impact Factor:
2.3 (2022)
Latest Articles
Design, Modeling, and Experimental Validation of a Vision-Based Table Tennis Juggling Robot
Mathematics 2024, 12(11), 1634; https://doi.org/10.3390/math12111634 (registering DOI) - 23 May 2024
Abstract
This paper develops a new vision-based robot customized for table tennis juggling tasks. Specifically, the robot is equipped with two industrial cameras operating as a sensing system. An image-processing algorithm is proposed that allows the robot to balance a table tennis ball while
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This paper develops a new vision-based robot customized for table tennis juggling tasks. Specifically, the robot is equipped with two industrial cameras operating as a sensing system. An image-processing algorithm is proposed that allows the robot to balance a table tennis ball while controlling its bounce height. The robot adopts a parallel structure design, and the end effector employs three ball joints to increase the degree of freedom (DOF) of the parallel mechanism. In addition, we design a control scheme explicitly customized for this robotic system. Extensive real-time experiments are performed to show the effectiveness of the juggling robot at different jumping heights. Furthermore, the ability to consistently maintain a fixed preset bounce height is demonstrated. These experimental results confirm the efficacy of the developed robotic system.
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(This article belongs to the Special Issue Dynamics and Control of Complex Systems and Robots)
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On Curvature Pinching for Submanifolds with Parallel Normalized Mean Curvature Vector
by
Juanru Gu and Yao Lu
Mathematics 2024, 12(11), 1633; https://doi.org/10.3390/math12111633 (registering DOI) - 23 May 2024
Abstract
In this note, we investigate the pinching problem for oriented compact submanifolds of dimension n with parallel normalized mean curvature vector in a space form . We first prove a codimension reduction theorem for submanifolds under
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In this note, we investigate the pinching problem for oriented compact submanifolds of dimension n with parallel normalized mean curvature vector in a space form . We first prove a codimension reduction theorem for submanifolds under lower Ricci curvature bounds. Moreover, if the submanifolds have constant normalized scalar curvature , we obtain a classification theorem for submanifolds under lower Ricci curvature bounds. It should be emphasized that our Ricci pinching conditions are sharp for even n and .
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(This article belongs to the Special Issue Differential Geometry, Geometric Analysis and Their Related Applications)
Open AccessArticle
Painlevé Analysis of the Traveling Wave Reduction of the Third-Order Derivative Nonlinear Schrödinger Equation
by
Nikolay A. Kudryashov and Sofia F. Lavrova
Mathematics 2024, 12(11), 1632; https://doi.org/10.3390/math12111632 (registering DOI) - 23 May 2024
Abstract
The second partial differential equation from the Kaup–Newell hierarchy is considered. This equation can be employed to model pulse propagation in optical fiber, wave propagation in plasma, or high waves in the deep ocean. The integrability of the explored equation in traveling wave
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The second partial differential equation from the Kaup–Newell hierarchy is considered. This equation can be employed to model pulse propagation in optical fiber, wave propagation in plasma, or high waves in the deep ocean. The integrability of the explored equation in traveling wave variables is investigated using the Painlevé test. Periodic and solitary wave solutions of the studied equation are presented. The investigated equation belongs to the class of generalized nonlinear Schrödinger equations and may be used for the description of optical solitons in a nonlinear medium.
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(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)
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A Composite Half-Normal-Pareto Distribution with Applications to Income and Expenditure Data
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Neveka M. Olmos, Emilio Gómez-Déniz, Osvaldo Venegas and Héctor W. Gómez
Mathematics 2024, 12(11), 1631; https://doi.org/10.3390/math12111631 (registering DOI) - 23 May 2024
Abstract
The half-normal distribution is composited with the Pareto model to obtain a uni-parametric distribution with a heavy right tail, called the composite half-normal-Pareto distribution. This new distribution is useful for modeling positive data with atypical observations. We study the properties and the behavior
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The half-normal distribution is composited with the Pareto model to obtain a uni-parametric distribution with a heavy right tail, called the composite half-normal-Pareto distribution. This new distribution is useful for modeling positive data with atypical observations. We study the properties and the behavior of the right tail of this new distribution. We estimate the parameter using a method based on percentiles and the maximum likelihood method and assess the performance of the maximum likelihood estimator using Monte Carlo. We report three applications, one with simulated data and the others with income and expenditure data, in which the new distribution presents better performance than the Pareto distribution.
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(This article belongs to the Special Issue Statistics and Mathematics in Economics and Finance: Theory, Methods and Applications)
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Open AccessCorrection
Correction: Alkabaa et al. An Investigation on Spiking Neural Networks Based on the Izhikevich Neuronal Model: Spiking Processing and Hardware Approach. Mathematics 2022, 10, 612
by
Abdulaziz S. Alkabaa, Osman Taylan, Mustafa Tahsin Yilmaz, Ehsan Nazemi and El Mostafa Kalmoun
Mathematics 2024, 12(11), 1630; https://doi.org/10.3390/math12111630 - 23 May 2024
Abstract
In the original paper [...]
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Open AccessArticle
Some Estimation Methods for a Random Coefficient in the Gegenbauer Autoregressive Moving-Average Model
by
Oumaima Essefiani, Rachid El Halimi and Said Hamdoune
Mathematics 2024, 12(11), 1629; https://doi.org/10.3390/math12111629 - 22 May 2024
Abstract
The Gegenbauer autoregressive moving-average (GARMA) model is pivotal for addressing non-additivity, non-normality, and heteroscedasticity in real-world time-series data. While primarily recognized for its efficacy in various domains, including the health sector for forecasting COVID-19 cases, this study aims to assess its performance using
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The Gegenbauer autoregressive moving-average (GARMA) model is pivotal for addressing non-additivity, non-normality, and heteroscedasticity in real-world time-series data. While primarily recognized for its efficacy in various domains, including the health sector for forecasting COVID-19 cases, this study aims to assess its performance using yearly sunspot data. We evaluate the GARMA model’s goodness of fit and parameter estimation specifically within the domain of sunspots. To achieve this, we introduce the random coefficient generalized autoregressive moving-average (RCGARMA) model and develop methodologies utilizing conditional least squares (CLS) and conditional weighted least squares (CWLS) estimators. Employing the ratio of mean squared errors (RMSE) criterion, we compare the efficiency of these methods using simulation data. Notably, our findings highlight the superiority of the conditional weighted least squares method over the conditional least squares method. Finally, we provide an illustrative application using two real data examples, emphasizing the significance of the GARMA model in sunspot research.
Full article
(This article belongs to the Section Probability and Statistics)
Open AccessArticle
Well-Posedness of the Fixed Point Problem of Multifunctions of Metric Spaces
by
Nozara Sundus, Basit Ali and Maggie Aphane
Mathematics 2024, 12(11), 1628; https://doi.org/10.3390/math12111628 - 22 May 2024
Abstract
We consider a class of metrics which are equivalent to the Hausdorff metric in some sense to establish the well-posedness of fixed point problems associated with multifunctions of metric spaces, satisfying various generalized contraction conditions. Examples are provided to justify the applicability of
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We consider a class of metrics which are equivalent to the Hausdorff metric in some sense to establish the well-posedness of fixed point problems associated with multifunctions of metric spaces, satisfying various generalized contraction conditions. Examples are provided to justify the applicability of new results.
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(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition)
Open AccessArticle
Optimizing Variational Quantum Neural Networks Based on Collective Intelligence
by
Zitong Li, Tailong Xiao, Xiaoyang Deng, Guihua Zeng and Weimin Li
Mathematics 2024, 12(11), 1627; https://doi.org/10.3390/math12111627 - 22 May 2024
Abstract
Quantum machine learning stands out as one of the most promising applications of quantum computing, widely believed to possess potential quantum advantages. In the era of noisy intermediate-scale quantum, the scale and quality of quantum computers are limited, and quantum algorithms based on
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Quantum machine learning stands out as one of the most promising applications of quantum computing, widely believed to possess potential quantum advantages. In the era of noisy intermediate-scale quantum, the scale and quality of quantum computers are limited, and quantum algorithms based on fault-tolerant quantum computing paradigms cannot be experimentally verified in the short term. The variational quantum algorithm design paradigm can better adapt to the practical characteristics of noisy quantum hardware and is currently one of the most promising solutions. However, variational quantum algorithms, due to their highly entangled nature, encounter the phenomenon known as the “barren plateau” during the optimization and training processes, making effective optimization challenging. This paper addresses this challenging issue by researching a variational quantum neural network optimization method based on collective intelligence algorithms. The aim is to overcome optimization difficulties encountered by traditional methods such as gradient descent. We study two typical applications of using quantum neural networks: random 2D Hamiltonian ground state solving and quantum phase recognition. We find that the collective intelligence algorithm shows a better optimization compared to gradient descent. The solution accuracy of ground energy and phase classification is enhanced, and the optimization iterations are also reduced. We highlight that the collective intelligence algorithm has great potential in tackling the optimization of variational quantum algorithms.
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(This article belongs to the Special Issue Advances in Quantum Key Distribution and Quantum Information)
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Open AccessFeature PaperArticle
Ulam Stability for Boundary Value Problems of Differential Equations—Main Misunderstandings and How to Avoid Them
by
Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan
Mathematics 2024, 12(11), 1626; https://doi.org/10.3390/math12111626 - 22 May 2024
Abstract
Ulam type stability is an important property studied for different types of differential equations. When this type of stability is applied to boundary value problems, there are some misunderstandings in the literature. In connection with this, initially, we give a brief overview of
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Ulam type stability is an important property studied for different types of differential equations. When this type of stability is applied to boundary value problems, there are some misunderstandings in the literature. In connection with this, initially, we give a brief overview of the basic ideas of the application of Ulam type stability to initial value problems. We provide several examples with simulations to illustrate the main points in the application. Then, we focus on some misunderstandings in the application of Ulam stability to boundary value problems. We suggest a new way to avoid these misunderstandings and how to keep the main idea of Ulam type stability when it is applied to boundary value problems of differential equations. We present one possible way to connect both the solutions of the given problem and the solutions of the corresponding inequality. In addition, we provide several examples with simulations to illustrate the ideas for boundary value problems and we also show the necessity of the new way of applying the Ulam type stability. To illustrate the theoretical application of the suggested idea to Ulam type stability, we consider a linear boundary value problem for nonlinear impulsive fractional differential equations with the Caputo fractional derivative with respect to another function and piecewise-constant variable order. We define the Ulam–Hyers stability and obtain sufficient conditions on a finite interval. As partial cases, integral presentations of the solutions of boundary value problems for various types of fractional differential equations are obtained and their Ulam type stability is studied.
Full article
Open AccessArticle
A Comparative Analysis of Metaheuristic Algorithms for Enhanced Parameter Estimation on Inverted Pendulum System Dynamics
by
Daniel Sanin-Villa, Miguel Angel Rodriguez-Cabal, Luis Fernando Grisales-Noreña, Mario Ramirez-Neria and Juan C. Tejada
Mathematics 2024, 12(11), 1625; https://doi.org/10.3390/math12111625 - 22 May 2024
Abstract
This research explores the application of metaheuristic algorithms to refine parameter estimation in dynamic systems, with a focus on the inverted pendulum model. Three optimization techniques, Particle Swarm Optimization (PSO), Continuous Genetic Algorithm (CGA), and Salp Swarm Algorithm (SSA), are introduced to solve
[...] Read more.
This research explores the application of metaheuristic algorithms to refine parameter estimation in dynamic systems, with a focus on the inverted pendulum model. Three optimization techniques, Particle Swarm Optimization (PSO), Continuous Genetic Algorithm (CGA), and Salp Swarm Algorithm (SSA), are introduced to solve this problem. Through a thorough statistical evaluation, the optimal performance of each technique within the dynamic methodology is determined. Furthermore, the efficacy of these algorithms is demonstrated through experimental validation on a real prototype, providing practical insights into their performance. The outcomes of this study contribute to the advancement of control strategies by integrating precisely estimated physical parameters into various control algorithms, including PID controllers, fuzzy logic controllers, and model predictive controllers. Each algorithm ran 1000 times, and the SSA algorithm achieved the best performance, with the most accurate parameter estimation with a minimum error of 0.015 01 N m and a mean solution error of 0.015 06 N m. This precision was further underscored by its lowest standard deviation in RMSE (1.443 N m), indicating remarkable consistency across evaluations. The 95% confidence interval for error corroborated the algorithm’s reliability in deriving optimal solutions.
Full article
(This article belongs to the Special Issue Advances and Trends in Mathematical Modelling, Design, Control and Identification of Modern Vibrating Energy Conversion Systems)
Open AccessArticle
LAMBERT: Leveraging Attention Mechanisms to Improve the BERT Fine-Tuning Model for Encrypted Traffic Classification
by
Tao Liu, Xiting Ma, Ling Liu, Xin Liu, Yue Zhao, Ning Hu and Kayhan Zrar Ghafoor
Mathematics 2024, 12(11), 1624; https://doi.org/10.3390/math12111624 - 22 May 2024
Abstract
Encrypted traffic classification is a crucial part of privacy-preserving research. With the great success of artificial intelligence technology in fields such as image recognition and natural language processing, how to classify encrypted traffic based on AI technology has become an attractive topic in
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Encrypted traffic classification is a crucial part of privacy-preserving research. With the great success of artificial intelligence technology in fields such as image recognition and natural language processing, how to classify encrypted traffic based on AI technology has become an attractive topic in information security. With good generalization ability and high training accuracy, pre-training-based encrypted traffic classification methods have become the first option. The accuracy of this type of method depends highly on the fine-tuning model. However, it is a challenge for existing fine-tuned models to effectively integrate the representation of packet and byte features extracted via pre-training. A novel fine-tuning model, LAMBERT, is proposed in this article. By introducing an attention mechanism to capture the relationship between BiGRU and byte sequences, LAMBERT not only effectively improves the sequence loss phenomenon of BiGRU but also improves the processing performance of encrypted stream classification. LAMBERT can quickly and accurately classify multiple types of encrypted traffic. The experimental results show that our model performs well on datasets with uneven sample distribution, no pre-training, and large sample classification. LAMBERT was tested on four datasets, namely, ISCX-VPN-Service, ISCX-VPN-APP, USTC-TFC and CSTNET-TLS 1.3, and the F1 scores reached 99.15%, 99.52%, 99.30%, and 97.41%, respectively.
Full article
(This article belongs to the Special Issue Advanced Research on Information System Security and Privacy)
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Testing Informativeness of Covariate-Induced Group Sizes in Clustered Data
by
Hasika K. Wickrama Senevirathne and Sandipan Dutta
Mathematics 2024, 12(11), 1623; https://doi.org/10.3390/math12111623 - 22 May 2024
Abstract
Clustered data are a special type of correlated data where units within a cluster are correlated while units between different clusters are independent. The number of units in a cluster can be associated with that cluster’s outcome. This is called the informative cluster
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Clustered data are a special type of correlated data where units within a cluster are correlated while units between different clusters are independent. The number of units in a cluster can be associated with that cluster’s outcome. This is called the informative cluster size (ICS), which is known to impact clustered data inference. However, when comparing the outcomes from multiple groups of units in clustered data, investigating ICS may not be enough. This is because the number of units belonging to a particular group in a cluster can be associated with the outcome from that group in that cluster, leading to an informative intra-cluster group size or IICGS. This phenomenon of IICGS can exist even in the absence of ICS. Ignoring the existence of IICGS can result in a biased inference for group-based outcome comparisons in clustered data. In this article, we mathematically formulate the concept of IICGS while distinguishing it from ICS and propose a nonparametric bootstrap-based statistical hypothesis-testing mechanism for testing any claim of IICGS in a clustered data setting. Through simulations and real data applications, we demonstrate that our proposed statistical testing method can accurately identify IICGS, with substantial power, in clustered data.
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(This article belongs to the Special Issue Statistics and Data Science)
Open AccessArticle
Finite Element Analysis of Electromagnetic Forming Process and Optimization of Process Parameters Using RSM
by
Nilesh Satonkar, Gopalan Venkatachalam and Shenbaga Velu Pitchumani
Mathematics 2024, 12(11), 1622; https://doi.org/10.3390/math12111622 - 22 May 2024
Abstract
Aluminium can benefit from the high-speed forming technique known as electromagnetic forming (EMF). EMF is increasingly used in automotive applications as a result of this capability. This technology depends on Lorentz force (Magnetic force) in the practical forming application which relies on different
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Aluminium can benefit from the high-speed forming technique known as electromagnetic forming (EMF). EMF is increasingly used in automotive applications as a result of this capability. This technology depends on Lorentz force (Magnetic force) in the practical forming application which relies on different process parameters like forming a coil. A finite element model for the EMF process is built and studied in this work using the finite element analysis software ANSYS 2022 R1. The affecting process parameters are investigated using the Design of Experiments (DOE) approach. Response Surface Methodology (RSM) of the DOE approach is used by taking process parameters such as coil size, gap, and current density into account. The number of experiments is reduced by using Central Composite Design (CCD), an RSM model. To determine the optimal level of parameters, a magnetic force optimization study is carried out. The parameters of the EMF process (e.g., magnetic force) are investigated through a developed 2D finite element model and validated with available literature.
Full article
(This article belongs to the Special Issue Optimization and Simulation in Mechanical Engineering and Computer Aided Design)
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R-DDQN: Optimizing Algorithmic Trading Strategies Using a Reward Network in a Double DQN
by
Chujin Zhou, Yuling Huang, Kai Cui and Xiaoping Lu
Mathematics 2024, 12(11), 1621; https://doi.org/10.3390/math12111621 - 22 May 2024
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Algorithmic trading is playing an increasingly important role in the financial market, achieving more efficient trading strategies by replacing human decision-making. Among numerous trading algorithms, deep reinforcement learning is gradually replacing traditional high-frequency trading strategies and has become a mainstream research direction in
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Algorithmic trading is playing an increasingly important role in the financial market, achieving more efficient trading strategies by replacing human decision-making. Among numerous trading algorithms, deep reinforcement learning is gradually replacing traditional high-frequency trading strategies and has become a mainstream research direction in the field of algorithmic trading. This paper introduces a novel approach that leverages reinforcement learning with human feedback (RLHF) within the double DQN algorithm. Traditional reward functions in algorithmic trading heavily rely on expert knowledge, posing challenges in their design and implementation. To tackle this, the reward-driven double DQN (R-DDQN) algorithm is proposed, integrating human feedback via a reward function network trained on expert demonstrations. Additionally, a classification-based training method is employed for optimizing the reward function network. The experiments, conducted on datasets including HSI, IXIC, SP500, GOOGL, MSFT, and INTC, show that the proposed method outperforms all baselines across six datasets and achieves a maximum cumulative return of 1502% within 24 months.
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An Improved Expeditious Meta-Heuristic Clustering Method for Classifying Student Psychological Issues with Homogeneous Characteristics
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Muhammad Suhail Shaikh, Xiaoqing Dong, Gengzhong Zheng, Chang Wang and Yifan Lin
Mathematics 2024, 12(11), 1620; https://doi.org/10.3390/math12111620 - 22 May 2024
Abstract
Nowadays, cluster analyses are widely used in mental health research to categorize student stress levels. However, conventional clustering methods experience challenges with large datasets and complex issues, such as converging to local optima and sensitivity to initial random states. To address these limitations,
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Nowadays, cluster analyses are widely used in mental health research to categorize student stress levels. However, conventional clustering methods experience challenges with large datasets and complex issues, such as converging to local optima and sensitivity to initial random states. To address these limitations, this research work introduces an Improved Grey Wolf Clustering Algorithm (iGWCA). This improved approach aims to adjust the convergence rate and mitigate the risk of being trapped in local optima. The iGWCA algorithm provides a balanced technique for exploration and exploitation phases, alongside a local search mechanism around the optimal solution. To assess its efficiency, the proposed algorithm is verified on two different datasets. The dataset-I comprises 1100 individuals obtained from the Kaggle database, while dataset-II is based on 824 individuals obtained from the Mendeley database. The results demonstrate the competence of iGWCA in classifying student stress levels. The algorithm outperforms other methods in terms of lower intra-cluster distances, obtaining a reduction rate of 1.48% compared to Grey Wolf Optimization (GWO), 8.69% compared to Mayfly Optimization (MOA), 8.45% compared to the Firefly Algorithm (FFO), 2.45% Particle Swarm Optimization (PSO), 3.65%, Hybrid Sine Cosine with Cuckoo search (HSCCS), 8.20%, Hybrid Firefly and Genetic Algorithm (FAGA) and 8.68% Gravitational Search Algorithm (GSA). This demonstrates the effectiveness of the proposed algorithm in minimizing intra-cluster distances, making it a better choice for student stress classification. This research contributes to the advancement of understanding and managing student well-being within academic communities by providing a robust tool for stress level classification.
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(This article belongs to the Special Issue Deep Learning and Adaptive Control, 3rd Edition)
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Enhanced Unmanned Aerial Vehicle Localization in Dynamic Environments Using Monocular Simultaneous Localization and Mapping and Object Tracking
by
Youssef El Gaouti, Fouad Khenfri, Mehdi Mcharek and Cherif Larouci
Mathematics 2024, 12(11), 1619; https://doi.org/10.3390/math12111619 - 22 May 2024
Abstract
This work proposes an innovative approach to enhance the localization of unmanned aerial vehicles (UAVs) in dynamic environments. The methodology integrates a sophisticated object-tracking algorithm to augment the established simultaneous localization and mapping (ORB-SLAM) framework, utilizing only a monocular camera setup. Moving objects
[...] Read more.
This work proposes an innovative approach to enhance the localization of unmanned aerial vehicles (UAVs) in dynamic environments. The methodology integrates a sophisticated object-tracking algorithm to augment the established simultaneous localization and mapping (ORB-SLAM) framework, utilizing only a monocular camera setup. Moving objects are detected by harnessing the power of YOLOv4, and a specialized Kalman filter is employed for tracking. The algorithm is integrated into the ORB-SLAM framework to improve UAV pose estimation by correcting the impact of moving elements and effectively removing features connected to dynamic elements from the ORB-SLAM process. Finally, the results obtained are recorded using the TUM RGB-D dataset. The results demonstrate that the proposed algorithm can effectively enhance the accuracy of pose estimation and exhibits high accuracy and robustness in real dynamic scenes.
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(This article belongs to the Special Issue Advanced Machine Vision with Mathematics)
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Open AccessArticle
Newtonian Property of Subgradient Method with Optimization of Metric Matrix Parameter Correction
by
Elena Tovbis, Vladimir Krutikov and Lev Kazakovtsev
Mathematics 2024, 12(11), 1618; https://doi.org/10.3390/math12111618 - 22 May 2024
Abstract
The work proves that under conditions of instability of the second derivatives of the function in the minimization region, the estimate of the convergence rate of Newton’s method is determined by the parameters of the irreducible part of the conditionality degree of the
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The work proves that under conditions of instability of the second derivatives of the function in the minimization region, the estimate of the convergence rate of Newton’s method is determined by the parameters of the irreducible part of the conditionality degree of the problem. These parameters represent the degree of difference between eigenvalues of the matrices of the second derivatives in the coordinate system, where this difference is minimal, and the resulting estimate of the convergence rate subsequently acts as a standard. The paper studies the convergence rate of the relaxation subgradient method (RSM) with optimization of the parameters of two-rank correction of metric matrices on smooth strongly convex functions with a Lipschitz gradient without assumptions about the existence of second derivatives of the function. The considered RSM is similar in structure to quasi-Newton minimization methods. Unlike the latter, its metric matrix is not an approximation of the inverse matrix of second derivatives but is adjusted in such a way that it enables one to find the descent direction that takes the method beyond a certain neighborhood of the current minimum as a result of one-dimensional minimization along it. This means that the metric matrix enables one to turn the current gradient into a direction that is gradient-consistent with the set of gradients of some neighborhood of the current minimum. Under broad assumptions on the parameters of transformations of metric matrices, an estimate of the convergence rate of the studied RSM and an estimate of its ability to exclude removable linear background are obtained. The obtained estimates turn out to be qualitatively similar to estimates for Newton’s method. In this case, the assumption of the existence of second derivatives of the function is not required. A computational experiment was carried out in which the quasi-Newton BFGS method and the subgradient method under study were compared on various types of smooth functions. The testing results indicate the effectiveness of the subgradient method in minimizing smooth functions with a high degree of conditionality of the problem and its ability to eliminate the linear background that worsens the convergence.
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(This article belongs to the Section Mathematics and Computer Science)
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Hermite Finite Element Method for One-Dimensional Fourth-Order Boundary Value Problems
by
Bangmin Wu and Jiali Qiu
Mathematics 2024, 12(11), 1613; https://doi.org/10.3390/math12111613 - 22 May 2024
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One-dimensional fourth-order boundary value problems (BVPs) play a critical role in engineering applications, particularly in the analysis of beams. Current numerical investigations primarily concentrate on homogeneous boundary conditions. In addition to its high precision advantages, the Hermite finite element method (HFEM) is capable
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One-dimensional fourth-order boundary value problems (BVPs) play a critical role in engineering applications, particularly in the analysis of beams. Current numerical investigations primarily concentrate on homogeneous boundary conditions. In addition to its high precision advantages, the Hermite finite element method (HFEM) is capable of directly computing both the function value and its derivatives. In this paper, both the cubic and quintic HFEM are employed to address two prevalent non-homogeneous fourth-order BVPs. Furthermore, a priori error estimations are established for both BVPs, demonstrating the optimal error convergence order in semi-norm and norm. Finally, a numerical simulation is presented to validate the theoretical results.
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Open AccessArticle
RBF-Assisted Hybrid Neural Network for Solving Partial Differential Equations
by
Ying Li, Wei Gao and Shihui Ying
Mathematics 2024, 12(11), 1617; https://doi.org/10.3390/math12111617 - 21 May 2024
Abstract
In scientific computing, neural networks have been widely used to solve partial differential equations (PDEs). In this paper, we propose a novel RBF-assisted hybrid neural network for approximating solutions to PDEs. Inspired by the tendency of physics-informed neural networks (PINNs) to become local
[...] Read more.
In scientific computing, neural networks have been widely used to solve partial differential equations (PDEs). In this paper, we propose a novel RBF-assisted hybrid neural network for approximating solutions to PDEs. Inspired by the tendency of physics-informed neural networks (PINNs) to become local approximations after training, the proposed method utilizes a radial basis function (RBF) to provide the normalization and localization properties to the input data. The objective of this strategy is to assist the network in solving PDEs more effectively. During the RBF-assisted processing part, the method selects the center points and collocation points separately to effectively manage data size and computational complexity. Subsequently, the RBF processed data are put into the network for predicting the solutions to PDEs. Finally, a series of experiments are conducted to evaluate the novel method. The numerical results confirm that the proposed method can accelerate the convergence speed of the loss function and improve predictive accuracy.
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(This article belongs to the Special Issue Mathematical Modeling and Numerical Simulation in Engineering, 2nd Edition)
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Open AccessArticle
Existence and Nonexistence of Positive Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents
by
Lin-Lin Wang and Yong-Hong Fan
Mathematics 2024, 12(11), 1616; https://doi.org/10.3390/math12111616 - 21 May 2024
Abstract
The following semi-linear elliptic equations involving Hardy–Sobolev critical exponents
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The following semi-linear elliptic equations involving Hardy–Sobolev critical exponents have been investigated, where is an open-bounded domain in , with a smooth boundary , , and is the Hardy–Sobolev critical exponent. This problem comes from the study of standing waves in the anisotropic Schrödinger equation; it is very important in the fields of hydrodynamics, glaciology, quantum field theory, and statistical mechanics. Under some deterministic conditions on , by a detailed estimation of the extremum function and using mountain pass lemma with conditions, we obtained that: (a) If , and then the above problem has at least a positive solution in ; (b) If , then when , the above problem has at least a positive solution in ; (c) if and , then the above problem has no positive solution for These results are extensions of E. Jannelli’s research ( ).
Full article
(This article belongs to the Section Difference and Differential Equations)
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Fractal and Design of Multipoint Iterative Methods for Nonlinear Problems
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Algorithms, Computation, Information, Mathematics
Complex Networks and Social Networks
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Research on Data Mining of Electronic Health Records Using Deep Learning Methods
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New Trends on Boundary Value Problems
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Applications of Fuzzy Modeling in Risk Management
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Computational Statistical Methods and Extreme Value Theory
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Dynamical System and Stochastic Analysis
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