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Mathematics
Volume 13
Issue 8
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Mathematics, Volume 13, Issue 8 (April-2 2025) – 147 articles

Cover Story (view full-size image): Processes in biotechnology are considered reliable if they produce samples that meet regulatory benchmarks. For example, laboratories may be required to demonstrate that the levels of an undesirable analyte rarely (e.g., in less than 5% of samples) exceed a tolerance threshold. This can be challenging when measurement systems feature a lower limit of detection, meaning that for some samples, the only available information is that the true measurement value lies below the minimum value detectable by the system. In this paper, we first investigate the implications of detection limits on commonly used location-scale model-based inference in reliability studies. To address the need for robust methods, we then introduce a flexible weakly parametric model in which the right tail of the response distribution is approximated using a piecewise-constant hazard model. View this paper
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16 pages, 2221 KiB  
Article
Efficient Training of Deep Spiking Neural Networks Using a Modified Learning Rate Scheduler
by Sung-Hyun Cha and Dong-Sun Kim
Mathematics 2025, 13(8), 1361; https://doi.org/10.3390/math13081361 - 21 Apr 2025
Abstract
Deep neural networks (DNNs) have achieved high accuracy in various applications, but with the rapid growth of AI and the increasing scale and complexity of datasets, their computational cost and power consumption have become even more significant challenges. Spiking neural networks (SNNs), inspired [...] Read more.
Deep neural networks (DNNs) have achieved high accuracy in various applications, but with the rapid growth of AI and the increasing scale and complexity of datasets, their computational cost and power consumption have become even more significant challenges. Spiking neural networks (SNNs), inspired by biological neurons, offer an energy-efficient alternative by using spike-based information processing. However, training SNNs is difficult due to the non-differentiability of their activation function and the challenges in constructing deep architectures. This study addresses these issues by integrating DNN-like backpropagation into SNNs using a supervised learning approach. A surrogate gradient descent based on the arctangent function is applied to approximate the non-differentiable activation function, enabling stable gradient-based learning. The study also explores the interplay between the spatial domain (layer-wise propagation) and the temporal domain (time step), ensuring proper gradient propagation using the chain rule. Additionally, mini-batch training, Adam optimization, and layer normalization are incorporated to improve training efficiency and mitigate gradient vanishing. A softmax-based probability representation and cross-entropy loss function are used to optimize classification performance. Along with these techniques, a deep SNN was designed to converge to the optimal point faster than other models in the early stages of training by utilizing a modified learning rate scheduler. The proposed learning method allows deep SNNs to achieve competitive accuracy while maintaining their inherent low-power characteristics. These findings contribute to making SNNs more practical for machine learning applications by combining the advantages of deep learning and biologically inspired computing. In summary, this study contributes to the field by analyzing and adapting deep learning techniques—such as dropout, layer normalization, mini-batch training, and Adam optimization—to the spiking domain, and by proposing a novel learning rate scheduler that enables faster convergence during early training phases with fewer epochs. Full article
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18 pages, 573 KiB  
Article
Finite Element Method for Solving the Screened Poisson Equation with a Delta Function
by Liang Tang and Yuhao Tang
Mathematics 2025, 13(8), 1360; https://doi.org/10.3390/math13081360 - 21 Apr 2025
Abstract
This paper presents a Finite Element Method (FEM) framework for solving the screened Poisson equation with a Dirac delta function as the forcing term. The singularity introduced by the delta function poses challenges for standard numerical methods, particularly in higher dimensions. To address [...] Read more.
This paper presents a Finite Element Method (FEM) framework for solving the screened Poisson equation with a Dirac delta function as the forcing term. The singularity introduced by the delta function poses challenges for standard numerical methods, particularly in higher dimensions. To address this, we employ integrated Legendre basis functions, which yield sparse and structured system matrices characterized by a Banded-Block-Banded-Arrowhead (B3-Arrowhead) form. In one dimension, the resulting linear system can be solved directly. In two and three dimensions, the equation can be efficiently solved using a generalized Alternating Direction Implicit (ADI) method combined with reverse Cholesky factorization. Numerical results in 1D, 2D, and 3D confirm that the method accurately captures the localized impulse response and reproduces the expected Green’s function behavior. The proposed approach offers a robust and scalable solution framework for partial differential equations with singular source terms and has potential applications in physics, engineering, and computational science. Full article
(This article belongs to the Special Issue Advances in Partial Differential Equations: Methods and Applications)
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17 pages, 3782 KiB  
Article
DRHT: A Hybrid Mathematical Model for Accurate Ultrasound Probe Calibration and Efficient 3D Reconstruction
by Xuquan Ji, Yonghong Zhang, Huaqing Shang, Lei Hu, Xiaozhi Qi and Wenyong Liu
Mathematics 2025, 13(8), 1359; https://doi.org/10.3390/math13081359 - 21 Apr 2025
Abstract
The calibration of ultrasound probes is essential for three-dimensional ultrasound reconstruction and navigation. However, the existing calibration methods are often cumbersome and inadequate in accuracy. In this paper, a hybrid mathematical model, Dimensionality Reduction and Homography Transformation (DRHT), is proposed. The model characterizes [...] Read more.
The calibration of ultrasound probes is essential for three-dimensional ultrasound reconstruction and navigation. However, the existing calibration methods are often cumbersome and inadequate in accuracy. In this paper, a hybrid mathematical model, Dimensionality Reduction and Homography Transformation (DRHT), is proposed. The model characterizes the relationship between the image plane of ultrasound and projected calibration lines and homography transformation. The homography transformation, which can be estimated using the singular value decomposition method, reduces the dimensionality of the calibration data and could significantly accelerate the computation of image points in ultrasonic three-dimensional reconstruction. Experiments comparing the DRHT method with the PLUS library demonstrated that DRHT outperformed the PLUS algorithm in terms of accuracy (0.89 mm vs. 0.92 mm) and efficiency (268 ms vs. 761 ms). Furthermore, high-precision calibration can be achieved with only four images, which greatly simplifies the calibration process and enhances the feasibility of the clinical application of this model. Full article
(This article belongs to the Special Issue Robust Perception and Control in Prognostic Systems)
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12 pages, 279 KiB  
Article
Construction of ε-Nets for the Space of Planar Convex Bodies Endowed with the Banach–Mazur Metric
by Yanmei Chen, Yunfang Lyu, Shenghua Gao and Senlin Wu
Mathematics 2025, 13(8), 1358; https://doi.org/10.3390/math13081358 - 21 Apr 2025
Abstract
In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry, the construction of ε-nets for the space of convex bodies endowed with the Banach–Mazur metric plays a crucial role. Recently, Gao et [...] Read more.
In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry, the construction of ε-nets for the space of convex bodies endowed with the Banach–Mazur metric plays a crucial role. Recently, Gao et al. provided a possible way of constructing ε-nets for Kn,dBM based on finite subsets of Zn theoretically. In this work, we present an algorithm to construct ε-nets for K2,dBM and a (1/4)-net for C2,dBM is constructed. To the best of our knowledge, this is the first concrete ε-net for C2,dBM for such a small ε. Full article
(This article belongs to the Section B: Geometry and Topology)
15 pages, 283 KiB  
Article
A Class of Non-Hopf Bi-Frobenius Algebras Generated by n Elements
by Zhan Fa and Yanhua Wang
Mathematics 2025, 13(8), 1357; https://doi.org/10.3390/math13081357 - 21 Apr 2025
Abstract
Bi-Frobenius algebras are a class of Frobenius algebras and Frobenius coalgebras with some compatible conditions. In this paper, we construct a class of bi-Frobenius algebras generated by n elements on graded algebra A. The comultiplication and counit are defined via a permutation [...] Read more.
Bi-Frobenius algebras are a class of Frobenius algebras and Frobenius coalgebras with some compatible conditions. In this paper, we construct a class of bi-Frobenius algebras generated by n elements on graded algebra A. The comultiplication and counit are defined via a permutation π on A, such that A becomes a bi-Frobenius algebra. For any n, these bi-Frobenius algebras are neither Hopf algebras nor S-type bi-Frobenius algebras. Full article
10 pages, 467 KiB  
Article
An Analysis of Nonlinear Axisymmetric Structural Vibrations of Circular Plates with the Extended Rayleigh–Ritz Method
by Jie Han, Xianglin Gong, Chencheng Lian, Huimin Jing, Bin Huang, Yangyang Zhang and Ji Wang
Mathematics 2025, 13(8), 1356; https://doi.org/10.3390/math13081356 - 21 Apr 2025
Abstract
The nonlinear deformation and vibrations of elastic plates represent a fundamental problem in structural vibration analysis, frequently encountered in engineering applications and classical mathematical studies. In the field of studying the nonlinear phenomena of elastic plates, numerous methods and techniques have emerged to [...] Read more.
The nonlinear deformation and vibrations of elastic plates represent a fundamental problem in structural vibration analysis, frequently encountered in engineering applications and classical mathematical studies. In the field of studying the nonlinear phenomena of elastic plates, numerous methods and techniques have emerged to obtain approximate and exact solutions for nonlinear differential equations. A particularly powerful and flexible method, known as the extended Rayleigh–Ritz method (ERRM), has been proposed. In this approach, the temporal variable is introduced as an additional dimension in the formulation. Through expanded integration across both the physical domain and a vibration period, the temporal variable is eliminated. The ERRM builds on the traditional RRM that offers a straightforward, sophisticated, and highly effective way to approximate solutions for nonlinear vibration and deformation issues in the realm of structural dynamics and vibration. In the case of circular plates, the method incorporates the linear displacement function along with high-frequency terms. As a result, it can accurately determine the nonlinear axisymmetric vibration frequencies of circular plates. For scenarios involving smaller deformations, its accuracy is on par with other approximate solution methods. This approach provides a valuable and novel procedure for the nonlinear analysis of circular structural vibrations. Full article
(This article belongs to the Special Issue Artificial Intelligence for Fault Detection in Manufacturing)
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20 pages, 2361 KiB  
Article
Mathematical and Computational Modeling of Catalytic Converter Using Navier–Stokes Equations in Curvilinear Coordinates
by Nurlan Temirbekov and Kerimakyn Ainur
Mathematics 2025, 13(8), 1355; https://doi.org/10.3390/math13081355 - 21 Apr 2025
Abstract
This article discusses the problem of numerically solving the Navier–Stokes equations, the heat conduction equation, and the transport equation in the orthogonal coordinates of a free curve. Since the numerical solution domain is complex, the curvilinear mesh method was used. To do so, [...] Read more.
This article discusses the problem of numerically solving the Navier–Stokes equations, the heat conduction equation, and the transport equation in the orthogonal coordinates of a free curve. Since the numerical solution domain is complex, the curvilinear mesh method was used. To do so, first, a boundary value problem was posed for the elliptic equation to automate the creation of orthogonal curved meshes. By numerically solving this problem, the program code for the curvilinear mesh generator was created. The motion of a liquid or gas through a porous medium was described by numerically solving the Navier–Stokes equations in freely curvilinear orthogonal coordinates. The transformation of the Navier–Stokes equation system, written in the stream function, vorticity variables, and cylindrical coordinates, into arbitrary curvilinear coordinates, was considered in detail by introducing metric coefficients. To solve these equations, the coefficients of which vary rapidly, a three-layer differential scheme was developed. The approximation, stability, and compactness of the differential scheme were previously studied. The considered problem was considered to be the mathematical model of a car catalytic converter, and computational experiments were conducted. Calculations were performed with the developed program code in different geometries of the computational domain and different values of grid size. The Reynolds number was changed from 100 to 10,000, and its effect on the size of the backflow in front of the porous medium was discussed. The software code, which is based on the differential equation of the Navier–Stokes equations written in the orthogonal coordinates of a curved line, and its calculation algorithm can be used for the mathematical and computer modeling of automobile catalytic converters and chemical reactors. Full article
(This article belongs to the Section E4: Mathematical Physics)
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17 pages, 2604 KiB  
Article
A Modified Nonlinear Lorentz Model for Third-Order Optical Nonlinearity
by Yao Xia and Jinjie Liu
Mathematics 2025, 13(8), 1354; https://doi.org/10.3390/math13081354 - 21 Apr 2025
Abstract
In this study, we propose a new nonlinear polarization model that modifies the polarization equation to account for the material’s nonlinear response. Specifically, the nonlinear restoring force in our model is reformulated as an electric field-dependent function, derived from the nonlinear Lorentz model. [...] Read more.
In this study, we propose a new nonlinear polarization model that modifies the polarization equation to account for the material’s nonlinear response. Specifically, the nonlinear restoring force in our model is reformulated as an electric field-dependent function, derived from the nonlinear Lorentz model. Additionally, we perform a comparative analysis of the Kerr model, the Duffing model, the nonlinear Lorentz model, and our modified nonlinear Lorentz model (MNL) by solving Maxwell’s equations using the finite-difference time-domain (FDTD) method. This research focuses on the third-order nonlinearity of these models under varying light intensities and different ratios of resonant frequency to carrier frequency. First, in the example we studied, our results show that the MNL model produces results closer to the Kerr model when the light intensity is significantly high. Second, the comparison under different resonant frequencies reveals that all models converge to the Kerr model when the carrier frequency is much lower than the resonant frequency. However, when the carrier frequency significantly exceeds the resonant frequency, the differences between the Kerr model and the other models become more noticeable. The third-order nonlinearity of our MNL model aligns more closely with the Kerr model than the nonlinear Lorentz and Duffing models do when the ratio of resonant frequency to carrier frequency is between 1 and 2. Full article
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20 pages, 1581 KiB  
Article
Heterogeneous Spillover Networks and Spatial–Temporal Dynamics of Systemic Risk Transmission: Evidence from G20 Financial Risk Stress Index
by Xing Wang, Jiahui Zhang, Xiaolong Chen, Hongfeng Zhang, Cora Un In Wong and Thomas Chan
Mathematics 2025, 13(8), 1353; https://doi.org/10.3390/math13081353 - 21 Apr 2025
Abstract
With the continuous integration of globalization and financial markets, the linkage of global financial risks has increased significantly. This study examines the risk spillover effects and transmission dynamics among the financial markets in G20 countries, which together represent over 80% of global GDP. [...] Read more.
With the continuous integration of globalization and financial markets, the linkage of global financial risks has increased significantly. This study examines the risk spillover effects and transmission dynamics among the financial markets in G20 countries, which together represent over 80% of global GDP. With increasing globalization and the interconnectedness of financial markets, understanding risk transmission mechanisms has become critical for effective risk management. Previous research has primarily focused on price volatility to measure financial risks, often overlooking other critical dimensions such as liquidity, credit, and operational risks. This paper addresses this gap by utilizing the vector autoregressive (VAR) model to explore the spillover effects and the temporal and spatial characteristics of risk transmission. Specifically, we employ global and local Moran indices to analyze spatial dependencies across markets. Our findings reveal that the risk linkages among the G20 financial markets exhibit significant time-varying characteristics, with spatial risk distribution showing weaker dispersion. By constructing a comprehensive financial risk index system and applying a network-based spillover analysis, this study enhances the measurement of financial market risk and uncovers the complex transmission pathways between sub-markets and countries. These results not only deepen our understanding of global financial market dynamics but also provide valuable insights for the design of effective cross-border financial regulatory policies. The study’s contributions lie in enriching the empirical literature on multi-dimensional financial risks, advancing policy formulation by identifying key risk transmission channels, and supporting international risk management strategies through the detection and mitigation of potential contagion effects. Full article
(This article belongs to the Special Issue Machine Learning Methods and Mathematical Modeling with Applications)
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17 pages, 6456 KiB  
Article
A Mathematical Model for RNA 3D Structures
by Sixiang Zhang and Liming Cai
Mathematics 2025, 13(8), 1352; https://doi.org/10.3390/math13081352 - 21 Apr 2025
Abstract
The computational prediction of RNA three-dimensional (3D) structures remains a significant challenge, largely due to the limited understanding of RNA folding pathways. Although the scarcity of resolved native RNA structures has hindered the effectiveness of machine learning-based prediction methods, small, local structural motifs [...] Read more.
The computational prediction of RNA three-dimensional (3D) structures remains a significant challenge, largely due to the limited understanding of RNA folding pathways. Although the scarcity of resolved native RNA structures has hindered the effectiveness of machine learning-based prediction methods, small, local structural motifs are both recurring and abundant in the available data. Precisely modeling these geometric motifs presents a promising approach to improving 3D structure prediction. In this paper, we introduce a novel mathematical model that represents RNA 3D structures as collections of interacting helices with concise geometric descriptions. By using a small set of parameters for each modeled helix, our method maps RNA strand segments onto helices within a 3D space, facilitating the effective assembly of large RNA structures. Preliminary tests on RNA sequences from the Protein Data Bank demonstrated the model’s potential in predicting key structural elements, including double helices, hairpin loops, and bulges. Full article
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35 pages, 5812 KiB  
Article
A Chemistry-Based Optimization Algorithm for Quality of Service-Aware Multi-Cloud Service Compositions
by Mona Aldakheel and Heba Kurdi
Mathematics 2025, 13(8), 1351; https://doi.org/10.3390/math13081351 - 21 Apr 2025
Abstract
The increasing complexity of cloud service composition demands innovative approaches that can efficiently optimize both functional requirements and quality of service (QoS) parameters. While several methods exist, they struggle to simultaneously minimize the number of combined clouds, examined services, and execution time while [...] Read more.
The increasing complexity of cloud service composition demands innovative approaches that can efficiently optimize both functional requirements and quality of service (QoS) parameters. While several methods exist, they struggle to simultaneously minimize the number of combined clouds, examined services, and execution time while maintaining a high QoS. This novelty of this paper is the chemistry-based approach (CA) that draws inspiration from the periodic table’s organizational principles and electron shell theory to systematically reduce the complexity associated with service composition. As chemical elements are organized in the periodic table and electrons organize themselves in atomic shells based on energy levels, the proposed approach organizes cloud services in hierarchical structures based on their cloud number, composition frequencies, cloud quality, and QoS levels. By mapping chemical principles to cloud service attributes—where service quality levels correspond to electron shells and service combinations mirror molecular bonds—an efficient framework for service composition is created that simultaneously addresses multiple objectives in QoS, NC, NEC, NES, and execution time. The experimental results demonstrated significant improvements over existing methods, such as Genetic Algorithms (GAs), Simulated Annealing (SA), and Tabu Search (TS), across multiple performance metrics, i.e., reductions of 14–33% are observed in combined clouds, while reductions of 20–85% are observed in examined clouds, and reductions of 74–98% are observed in examined services. Also, a reduction of 10–99% is observed in execution time, while fitness levels are enhanced by 1–14% compared to benchmarks. These results validate the proposed approach’s effectiveness in optimizing service composition while minimizing computational overhead in multi-cloud environments. Full article
(This article belongs to the Special Issue Computational Intelligence: Theory and Applications, 2nd Edition)
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20 pages, 2756 KiB  
Article
Data-Driven Robust Attitude Tracking Control of Unmanned Underwater Vehicles with Performance Constraints
by He-Ning Zhang, Run-Ze Chen, Zi-Yi Liu, Zhi-Fu Zhang and Yi-Zhe Huang
Mathematics 2025, 13(8), 1350; https://doi.org/10.3390/math13081350 - 21 Apr 2025
Abstract
This paper studies the data-driven attitude tracking control issue for an unmanned underwater vehicle (UUV) with disturbances. First, a new polynomial finite-time prescribed performance function (FTPF) is introduced to avoid the problem of the computation number increasing as the exponential term increases in [...] Read more.
This paper studies the data-driven attitude tracking control issue for an unmanned underwater vehicle (UUV) with disturbances. First, a new polynomial finite-time prescribed performance function (FTPF) is introduced to avoid the problem of the computation number increasing as the exponential term increases in the conventional exponential FTPF. By using the new polynomial FTPF, the tracking error is converted into a constrained form. Then, an estimator is designed for estimating the unknown pseudo-partitioned Jacobian matrix (PJM) in the linearization model, and a discrete-time nonlinear disturbance observer (DNDO) is adopted for observing unknown disturbances. It is worth noting that the DNDO can avoid the large overshoot by introducing a saturated function. With the help of the estimator for the PJM, the DNDO, and the constrained error, a data-driven robust control strategy with performance constraints is designed to fulfill accurate attitude tracking control of the UUV, which ensures that the tracking error draws into a prescribed region in a predetermined time. Eventually, the control strategy is verified by numerical simulations. Full article
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18 pages, 685 KiB  
Article
Multiscale Fuzzy Temporal Pattern Mining: A Block-Decomposition Algorithm for Partial Periodic Associations in Event Data
by Aihua Zhu, Haote Zhang, Xingqian Chen and Dingkun Zhu
Mathematics 2025, 13(8), 1349; https://doi.org/10.3390/math13081349 - 20 Apr 2025
Abstract
This paper introduces a dual-strategy model based on temporal transformation and fuzzy theory, and designs a partitioned mining algorithm for periodic frequent patterns in large-scale event data (3P-TFT). The model reconstructs original event data through temporal reorganization and attribute fuzzification, preserving data continuity [...] Read more.
This paper introduces a dual-strategy model based on temporal transformation and fuzzy theory, and designs a partitioned mining algorithm for periodic frequent patterns in large-scale event data (3P-TFT). The model reconstructs original event data through temporal reorganization and attribute fuzzification, preserving data continuity distribution characteristics while enabling efficient processing of multidimensional attributes within a multi-temporal granularity calendar framework. The 3P-TFT algorithm employs temporal interval and object attribute partitioning strategies to achieve distributed mining of large-scale data. Experimental results demonstrate that this method effectively reveals hidden periodic patterns in stock trading events at specific temporal granularities, with volume–price association rules providing significant predictive and decision-making value. Furthermore, comparative algorithm experiments confirm that the 3P-TFT algorithm exhibits exceptional stability and adaptability across event databases with various cycle lengths, offering a novel theoretical tool for complex event data mining. Full article
(This article belongs to the Special Issue Artificial Intelligence Applications with Advanced Mathematical Methods)
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17 pages, 2517 KiB  
Article
Multi-Condition Rolling Bearing Fault Denoising Method and Application Based on RIME-VMD
by Xin Zhao, Xuebin Liu and Hanshan Li
Mathematics 2025, 13(8), 1348; https://doi.org/10.3390/math13081348 - 20 Apr 2025
Abstract
To improve the stability of rolling bearing fault signal denoising under different working conditions, this study proposes a multi-condition rolling bearing fault denoising method based on RIME-VMD. According to the characteristics of rolling bearing fault signal, RIME (frost and ice algorithm) is utilized [...] Read more.
To improve the stability of rolling bearing fault signal denoising under different working conditions, this study proposes a multi-condition rolling bearing fault denoising method based on RIME-VMD. According to the characteristics of rolling bearing fault signal, RIME (frost and ice algorithm) is utilized to obtain adaptive optimization of the modal number and penalty factors in VMD (variational mode decomposition) algorithm, and then the optimized core parameters are input into VMD to decompose the rolling bearing fault signal. The fitness function is established by introducing the fusion of power spectrum entropy and kurtosis value; these intrinsic modal functions (IMFs) with high correlation based on original rolling bearing fault signal are selected to reconstruct the rolling bearing fault denoising signal. The experimental results show that the RIME-VMD method can effectively remove most of the noise in the rolling bearing fault signal, and the performance evaluation indexes of this method are better than other existing optimization algorithms. The research achievement of this study can provide effective data support for the fault diagnosis of bearing equipment. Full article
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21 pages, 9224 KiB  
Article
A Multi-Scale Fusion Convolutional Network for Time-Series Silicon Prediction in Blast Furnaces
by Qiancheng Hao, Wenjing Liu, Wenze Gao and Xianpeng Wang
Mathematics 2025, 13(8), 1347; https://doi.org/10.3390/math13081347 - 20 Apr 2025
Abstract
In steel production, the blast furnace is a critical element. In this process, precisely controlling the temperature of the molten iron is indispensable for attaining efficient operations and high-grade products. This temperature is often indirectly reflected by the silicon content in the hot [...] Read more.
In steel production, the blast furnace is a critical element. In this process, precisely controlling the temperature of the molten iron is indispensable for attaining efficient operations and high-grade products. This temperature is often indirectly reflected by the silicon content in the hot metal. However, due to the dynamic nature and inherent delays of the ironmaking process, real-time prediction of silicon content remains a significant challenge, and traditional methods often suffer from insufficient prediction accuracy. This study presents a novel Multi-Scale Fusion Convolutional Neural Network (MSF-CNN) to accurately predict the silicon content during the blast furnace smelting process, addressing the limitations of existing data-driven approaches. The proposed MSF-CNN model extracts temporal features at two distinct scales. The first scale utilizes a Convolutional Block Attention Module, which captures local temporal dependencies by focusing on the most relevant features across adjacent time steps. The second scale employs a Multi-Head Self-Attention mechanism to model long-term temporal dependencies, overcoming the inherent delay issues in the blast furnace process. By combining these two scales, the model effectively captures both short-term and long-term temporal dependencies, thereby enhancing prediction accuracy and real-time applicability. Validation using real blast furnace data demonstrates that MSF-CNN outperforms recurrent neural network models such as Long Short-Term Memory (LSTM) and the Gated Recurrent Unit (GRU). Compared with LSTM and the GRU, MSF-CNN reduces the Root Mean Square Error (RMSE) by approximately 22% and 21%, respectively, and improves the Hit Rate (HR) by over 3.5% and 4%, highlighting its superiority in capturing complex temporal dependencies. These results indicate that the MSF-CNN adapts better to the blast furnace’s dynamic variations and inherent delays, achieving significant improvements in prediction precision and robustness compared to state-of-the-art recurrent models. Full article
(This article belongs to the Topic AI and Data-Driven Advancements in Industry 4.0, 2nd Edition)
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19 pages, 1414 KiB  
Article
Wavelet and Deep Learning Framework for Predicting Commodity Prices Under Economic and Financial Uncertainty
by Lyubov Doroshenko, Loretta Mastroeni and Alessandro Mazzoccoli
Mathematics 2025, 13(8), 1346; https://doi.org/10.3390/math13081346 - 20 Apr 2025
Abstract
The analysis of commodity markets—particularly in the energy and metals sectors—is essential for understanding economic dynamics and guiding decision-making. Financial and economic uncertainty indices provide valuable insights that help reduce price uncertainty. This study employs wavelet analyses and wavelet energy-based measures to investigate [...] Read more.
The analysis of commodity markets—particularly in the energy and metals sectors—is essential for understanding economic dynamics and guiding decision-making. Financial and economic uncertainty indices provide valuable insights that help reduce price uncertainty. This study employs wavelet analyses and wavelet energy-based measures to investigate the relationship between these indices and commodity prices across multiple time scales. The wavelet approach captures complex, time-varying dependencies, offering a more nuanced understanding of how uncertainty indices influence commodity price fluctuations. By integrating this analysis with predictability measures, we assess how uncertainty indices enhance forecasting accuracy. We further incorporate deep learning models capable of capturing sequential patterns in financial time series into our analysis to better evaluate their predictive potential. Our findings highlight the varying impact of financial and economic uncertainty on the predictability of commodity prices, showing that while some indices offer valuable forecasting information, others display strong correlations without significant predictive power. These results underscore the need for tailored predictive models, as different commodities react differently to the same financial conditions. By combining wavelet-based measures with machine learning techniques, this study presents a comprehensive framework for evaluating the role of uncertainty in commodity markets. The insights gained can support investors, policymakers, and market analysts in making more informed decisions. Full article
(This article belongs to the Special Issue Advancements in Applied Mathematics for Economic Data Analytics: Models, Methods, and Applications)
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13 pages, 264 KiB  
Article
Parameter Estimation of Geographically and Temporally Weighted Elastic Net Ordinal Logistic Regression
by Margaretha Ohyver, Purhadi and Achmad Choiruddin
Mathematics 2025, 13(8), 1345; https://doi.org/10.3390/math13081345 - 19 Apr 2025
Viewed by 84
Abstract
Geographically and Temporally Weighted Elastic Net Ordinal Logistic Regression is a parsimonious ordinal logistic regression with consideration of the existence of spatial and temporal effects. This model has been developed with the following three considerations: the spatial effect, the temporal effect, and predictor [...] Read more.
Geographically and Temporally Weighted Elastic Net Ordinal Logistic Regression is a parsimonious ordinal logistic regression with consideration of the existence of spatial and temporal effects. This model has been developed with the following three considerations: the spatial effect, the temporal effect, and predictor selection. The last point prompted the use of Elastic Net regularization in choosing predictors while handling multicollinearity, which often arises when there are many predictors involved. The Elastic Net penalty combines ridge and LASSO penalties, leading to the determination of the appropriate λEN and αEN. Therefore, the objective of this study is to determine the parameter estimator using Maximum Likelihood Estimation. The estimation process comprises defining the likelihood function, determining the natural logarithm of the likelihood function, and maximizing the function using Berndt–Hall–Hall–Hausman. These steps continue until the estimator converges on the values that maximize the likelihood function. This study contributes by developing an estimation framework that integrates spatial and temporal effects with Elastic Net regularization, allowing for improved model interpretation and stability. The findings provide an advanced methodological approach for ordinal logistic regression models that incorporate spatial and temporal dependencies. This framework is particularly useful for applications in fields such as economic forecasting, epidemiology, and environmental studies, where ordinal responses exhibit spatial and temporal patterns. Full article
(This article belongs to the Special Issue Spatial Statistics Methods and Modeling)
16 pages, 3434 KiB  
Article
Adaptive Terminal Sliding Mode Control for a Quadrotor System with Barrier Function Switching Law
by Jiangting Zhu, Xionghui Long and Quan Yuan
Mathematics 2025, 13(8), 1344; https://doi.org/10.3390/math13081344 - 19 Apr 2025
Viewed by 131
Abstract
This study presents a novel finite-time robust control framework for quadrotor systems subjected to model uncertainties and unknown external disturbances. A fast terminal sliding mode (FTSM) manifold is first constructed to achieve finite-time convergence of tracking errors. To address the challenges posed by [...] Read more.
This study presents a novel finite-time robust control framework for quadrotor systems subjected to model uncertainties and unknown external disturbances. A fast terminal sliding mode (FTSM) manifold is first constructed to achieve finite-time convergence of tracking errors. To address the challenges posed by uncertain system dynamics, a radial basis function neural network (RBFNN) is integrated for real-time approximation of unknown nonlinearities. In addition, an adaptive gain regulation mechanism based on a barrier Lyapunov function (BLF) is developed to ensure boundedness of system trajectories while enhancing robustness without requiring prior knowledge of disturbance bounds. The proposed control scheme guarantees finite-time stability, strong robustness, and precise trajectory tracking. Numerical simulations substantiate the efficacy and superiority of the proposed method in comparison with existing control approaches. Full article
(This article belongs to the Special Issue Deep Learning and Adaptive Control, 3rd Edition)
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25 pages, 380 KiB  
Article
Limit Theorems for the Non-Convex Multispecies Curie–Weiss Model
by Francesco Camilli, Emanuele Mingione and Godwin Osabutey
Mathematics 2025, 13(8), 1343; https://doi.org/10.3390/math13081343 - 19 Apr 2025
Viewed by 68
Abstract
We study the thermodynamic properties of the generalized non-convex multispecies Curie–Weiss model, where interactions among different types of particles (forming the species) are encoded in a generic matrix. For spins with a generic prior distribution, we compute the thermodynamic limit of the generating [...] Read more.
We study the thermodynamic properties of the generalized non-convex multispecies Curie–Weiss model, where interactions among different types of particles (forming the species) are encoded in a generic matrix. For spins with a generic prior distribution, we compute the thermodynamic limit of the generating functional for the moments of the Boltzmann–Gibbs measure using simple interpolation techniques. For Ising spins, we further analyze the fluctuations of the magnetization in the thermodynamic limit under the Boltzmann–Gibbs measure. It is shown that a central limit theorem (CLT) holds for a rescaled and centered vector of species magnetizations, which converges to either a centered or non-centered multivariate normal distribution, depending on the rate of convergence of the relative sizes of the species. Full article
(This article belongs to the Section E4: Mathematical Physics)
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16 pages, 12134 KiB  
Article
Intelligent Dynamic Multi-Dimensional Heterogeneous Resource Scheduling Optimization Strategy Based on Kubernetes
by Jialin Cai, Hui Zeng, Feifei Liu and Junming Chen
Mathematics 2025, 13(8), 1342; https://doi.org/10.3390/math13081342 - 19 Apr 2025
Viewed by 42
Abstract
In this paper, we tackle the challenge of optimizing resource utilization and demand-driven allocation in dynamic, multi-dimensional heterogeneous environments. Traditional containerized task scheduling systems, like Kubernetes, typically rely on default schedulers that primarily focus on CPU and memory, overlooking the multi-dimensional nature of [...] Read more.
In this paper, we tackle the challenge of optimizing resource utilization and demand-driven allocation in dynamic, multi-dimensional heterogeneous environments. Traditional containerized task scheduling systems, like Kubernetes, typically rely on default schedulers that primarily focus on CPU and memory, overlooking the multi-dimensional nature of heterogeneous resources such as GPUs, network I/O, and disk I/O. This results in suboptimal scheduling and underutilization of resources. To address this, we propose a dynamic scheduling method for heterogeneous resources using an enhanced Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) algorithm that adjusts weights in real time and applies nonlinear normalization. Leveraging parallel computing, approximation, incremental computation, local updates, and hardware acceleration, the method minimizes overhead and ensures efficiency. Experimental results showed that, under low-load conditions, our method reduced task response times by 31–36%, increased throughput by 20–50%, and boosted resource utilization by over 20% compared to both the default Kubernetes scheduler and the Kubernetes Container Scheduling Strategy (KCSS) algorithm. These improvements were tested across diverse workloads, utilizing CPU, memory, GPU, and I/O resources, in a large-scale cluster environment, demonstrating the method’s robustness. These enhancements optimize cluster performance and resource efficiency, offering valuable insights for task scheduling in containerized cloud platforms. Full article
(This article belongs to the Special Issue Advances in Artificial Intelligence, Machine Learning and Optimization)
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20 pages, 6392 KiB  
Article
A Rotational Speed Sensor Based on Flux-Switching Principle
by Duy-Tinh Hoang, Joon-Ku Lee, Kwang-Il Jeong, Kyung-Hun Shin and Jang-Young Choi
Mathematics 2025, 13(8), 1341; https://doi.org/10.3390/math13081341 - 19 Apr 2025
Viewed by 29
Abstract
This study proposes a rotational speed measurement machine based on the flux-switching principle with a 6-stator-slot/19-rotor-pole (6s/19p) topology. With a rotor shape similar to a variable reluctance sensor (VRS), the proposed machine features a simple and robust structure while ensuring the same output [...] Read more.
This study proposes a rotational speed measurement machine based on the flux-switching principle with a 6-stator-slot/19-rotor-pole (6s/19p) topology. With a rotor shape similar to a variable reluctance sensor (VRS), the proposed machine features a simple and robust structure while ensuring the same output frequency as VRS. Additionally, compared to the conventional 12s/10p topology, the 6s/19p configuration reduces permanent magnet (PM) consumption by half while maintaining high induced voltage characteristics. A nonlinear analytical model (NAM), which incorporates the harmonic modeling (HM) technique and an iterative process, is presented. This model more accurately captures the rectangular shape of the PM and stator teeth while accounting for core saturation effects. Based on this model, the optimal dimensions of the proposed machine are investigated to achieve the best performance for speed measurement applications. A coupling FEA simulation between Ansys Maxwell and Twin Builder further analyzes the machine’s performance. Compared to a commercial product of the same size, the proposed machine achieves 31.5% higher output voltage while ensuring lower linearity errors. Moreover, superior load characteristics are observed, with a voltage drop of only 1.58% at 1500 rpm and 30 mA. The proposed machine and NAM provide an improved solution and analytical tool for speed measurement applications. Full article
(This article belongs to the Special Issue Mathematical Modelling and Numerical Analysis in Electrical Engineering, 2nd Edition)
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23 pages, 2486 KiB  
Article
Learning High-Order Features for Fine-Grained Visual Categorization with Causal Inference
by Yuhang Zhang, Yuan Wan, Jiahui Hao, Zaili Yang and Huanhuan Li
Mathematics 2025, 13(8), 1340; https://doi.org/10.3390/math13081340 - 19 Apr 2025
Viewed by 36
Abstract
Recently, causal models have gained significant attention in natural language processing (NLP) and computer vision (CV) due to their capability of capturing features with causal relationships. This study addresses Fine-Grained Visual Categorization (FGVC) by incorporating high-order feature fusions to improve the representation of [...] Read more.
Recently, causal models have gained significant attention in natural language processing (NLP) and computer vision (CV) due to their capability of capturing features with causal relationships. This study addresses Fine-Grained Visual Categorization (FGVC) by incorporating high-order feature fusions to improve the representation of feature interactions while mitigating the influence of confounding factors through causal inference. A novel high-order feature learning framework with causal inference is developed to enhance FGVC. A causal graph tailored to FGVC is constructed, and the causal assumptions of baseline models are analyzed to identify confounding factors. A reconstructed causal structure establishes meaningful interactions between individual images and image pairs. Causal interventions are applied by severing specific causal links, effectively reducing confounding effects and enhancing model robustness. The framework combines high-order feature fusion with interventional fine-grained learning by performing causal interventions on both classifiers and categories. The experimental results demonstrate that the proposed method achieves accuracies of 90.7% on CUB-200, 92.0% on FGVC-Aircraft, and 94.8% on Stanford Cars, highlighting its effectiveness and robustness across these widely used fine-grained recognition datasets. Comprehensive evaluations of these three widely used fine-grained recognition datasets demonstrate the proposed framework’s effectiveness and robustness. Full article
(This article belongs to the Special Issue Advances, Challenges, and Applications of Deep Learning Models in Computer Vision and Image Processing and Analysis)
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19 pages, 1789 KiB  
Article
Optimization of Temporal Feature Attribution and Sequential Dependency Modeling for High-Precision Multi-Step Resource Forecasting: A Methodological Framework and Empirical Evaluation
by Jiaqi Shen, Peiwen Qin, Rui Zhong and Peiyao Han
Mathematics 2025, 13(8), 1339; https://doi.org/10.3390/math13081339 - 19 Apr 2025
Viewed by 35
Abstract
This paper presents a comprehensive time-series analysis framework leveraging the Temporal Fusion Transformer (TFT) architecture to address the challenge of multi-horizon forecasting in complex ecological systems, specifically focusing on global fishery resources. Using global fishery data spanning 70 years (1950–2020), enhanced with key [...] Read more.
This paper presents a comprehensive time-series analysis framework leveraging the Temporal Fusion Transformer (TFT) architecture to address the challenge of multi-horizon forecasting in complex ecological systems, specifically focusing on global fishery resources. Using global fishery data spanning 70 years (1950–2020), enhanced with key climate indicators, we develop a methodology for predicting time-dependent patterns across three-year, five-year, and extended seven-year horizons. Our approach integrates static metadata with temporal features, including historical catch and climate data, through a specialized architecture incorporating variable selection networks, multi-head attention mechanisms, and bidirectional encoding layers. A comparative analysis demonstrates the TFT model’s robust performance against traditional methods (ARIMA), standard deep learning models (MLP, LSTM), and contemporary architectures (TCN, XGBoost). While competitive across different horizons, TFT excels in the 7-year forecast, achieving a mean absolute percentage error (MAPE) of 13.7%, outperforming the next best model (LSTM, 15.1%). Through a sensitivity analysis, we identify the optimal temporal granularity and historical context length for maximizing prediction accuracy. The variable selection component reveals differential weighting, with recent market observations (past 1-year catch: 31%) and climate signals (ONI index: 15%, SST anomaly: 10%) playing significant roles. A species-specific analysis uncovers variations in predictability patterns. Ablation experiments quantify the contributions of the architectural components. The proposed methodology offers practical applications for resource management and theoretical insights into modeling temporal dependencies in complex ecological data. Full article
(This article belongs to the Special Issue Deep Neural Network: Theory, Algorithms and Applications)
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32 pages, 2387 KiB  
Article
Stability and Optimal Control Analysis for a Fractional-Order Industrial Virus-Propagation Model Based on SCADA System
by Luping Huang, Dapeng Gao, Shiqiang Feng and Jindong Li
Mathematics 2025, 13(8), 1338; https://doi.org/10.3390/math13081338 - 19 Apr 2025
Viewed by 136
Abstract
The increasing reliance on and remote accessibility of automated industrial systems have shifted SCADA networks from being strictly isolated to becoming highly interconnected systems. The growing interconnectivity among systems enhances operational efficiency and also increases network security threats, especially attacks from industrial viruses. [...] Read more.
The increasing reliance on and remote accessibility of automated industrial systems have shifted SCADA networks from being strictly isolated to becoming highly interconnected systems. The growing interconnectivity among systems enhances operational efficiency and also increases network security threats, especially attacks from industrial viruses. This paper focuses on the stability analysis and optimal control analysis for a fractional-order industrial virus-propagation model based on a SCADA system. Firstly, we prove the existence, uniqueness, non-negativity and boundedness of the solutions for the proposed model. Secondly, the basic reproduction number R0α is determined, which suggests the conditions for ensuring the persistence and elimination of the virus. Moreover, we investigate the local and global asymptotic stability of the derived virus-free and virus-present equilibrium points. As is known to all, there is no unified method to establish a Lyapunov function. In this paper, by constructing an appropriate Lyapunov function and applying the method of undetermined coefficients, we prove the global asymptotic stability for all possible equilibrium points. Thirdly, we formulate our system as an optimal control problem by introducing appropriate control variables and derive the corresponding optimality conditions. Lastly, a set of numerical simulations are conducted to validate the findings, followed by a summary of the overall study. Full article
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10 pages, 255 KiB  
Article
Choice Numbers of Chordal, Chordless, and Some Non-Chordal Graphs
by Julian Allagan, Jianning Su, Weizheng Gao and Shanzhen Gao
Mathematics 2025, 13(8), 1337; https://doi.org/10.3390/math13081337 - 19 Apr 2025
Viewed by 89
Abstract
We prove that chordal graphs are chromatic-choosable and present a decomposition theorem to help estimate the choice numbers for certain classes of chordless and non-chordal graphs. Full article
(This article belongs to the Section E1: Mathematics and Computer Science)
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15 pages, 248 KiB  
Article
Existence and Nonexistence of Positive Solutions for Fractional Boundary Value Problems with Lidstone-Inspired Fractional Conditions
by Jeffrey W. Lyons, Jeffrey T. Neugebauer and Aaron G. Wingo
Mathematics 2025, 13(8), 1336; https://doi.org/10.3390/math13081336 - 19 Apr 2025
Viewed by 73
Abstract
This paper investigates the existence and nonexistence of positive solutions for a class of nonlinear Riemann–Liouville fractional boundary value problems of order α+2n, where α∈(m−1,m] with m≥3 and [...] Read more.
This paper investigates the existence and nonexistence of positive solutions for a class of nonlinear Riemann–Liouville fractional boundary value problems of order α+2n, where α∈(m−1,m] with m≥3 and m,n∈N. The conjugate fractional boundary conditions are inspired by Lidstone conditions. The nonlinearity depends on a positive parameter on which we identify constraints that determine the existence or nonexistence of positive solutions. Our method involves constructing Green’s function by convolving the Green functions of a lower-order fractional boundary value problem and a conjugate boundary value problem and using properties of this Green function to apply the Guo–Krasnosel’skii fixed-point theorem. Illustrative examples are provided to demonstrate existence and nonexistence intervals. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)
12 pages, 336 KiB  
Article
Marking Algorithms in Permutation Tableaux and Transformations on Linked Partitions
by Carol Jian Wang and Meryl Nan Wang
Mathematics 2025, 13(8), 1335; https://doi.org/10.3390/math13081335 - 19 Apr 2025
Viewed by 83
Abstract
In this paper, we focus on the internal structural characteristics of permutation tableaux and their correspondence with linked partitions. We begin by introducing new statistics for permutation tableaux, designed to thoroughly describe various positional relationships among the topmost 1s and the rightmost restricted [...] Read more.
In this paper, we focus on the internal structural characteristics of permutation tableaux and their correspondence with linked partitions. We begin by introducing new statistics for permutation tableaux, designed to thoroughly describe various positional relationships among the topmost 1s and the rightmost restricted 0s. Subsequently, we develop two marking algorithms for permutation tableaux, each from the perspective of columns and rows. Additionally, we introduce tugging and rebound transformations, which elucidate the generative relationship from original partitions to linked partitions. As a result, we demonstrate that the construction of these two marking algorithms in permutation tableaux provides a straightforward method for enumerating the crossing number and nesting number of the corresponding linked partitions. Full article
(This article belongs to the Section A: Algebra and Logic)
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11 pages, 244 KiB  
Article
Invariant Geometric Objects of the Equitorsion Canonical Biholomorphically Projective Mappings of Generalized Riemannian Space in the Eisenhart Sense
by Vladislava M. Milenković, Mića S. Stanković and Nenad O. Vesić
Mathematics 2025, 13(8), 1334; https://doi.org/10.3390/math13081334 - 18 Apr 2025
Viewed by 76
Abstract
The study of the equitorsion biholomorphically projective mappings between two generalized Riemannian spaces in the sense of Eisenhart’s definition is continued. Some new invariant geometric objects of an equitorsion canonical biholomorphically projective mapping are found, as well as some relations between these objects. [...] Read more.
The study of the equitorsion biholomorphically projective mappings between two generalized Riemannian spaces in the sense of Eisenhart’s definition is continued. Some new invariant geometric objects of an equitorsion canonical biholomorphically projective mapping are found, as well as some relations between these objects. At the end, the linear independence of the obtained invariants is examined. Full article
(This article belongs to the Special Issue Differential Geometry, Geometric Analysis and Their Related Applications)
22 pages, 6364 KiB  
Article
Multi-Frame Joint Detection Approach for Foreign Object Detection in Large-Volume Parenterals
by Ziqi Li, Dongyao Jia, Zihao He and Nengkai Wu
Mathematics 2025, 13(8), 1333; https://doi.org/10.3390/math13081333 - 18 Apr 2025
Viewed by 125
Abstract
Large-volume parenterals (LVPs), as essential medical products, are widely used in healthcare settings, making their safety inspection crucial. Current methods for detecting foreign particles in LVP solutions through image analysis primarily rely on single-frame detection or simple temporal smoothing strategies, which fail to [...] Read more.
Large-volume parenterals (LVPs), as essential medical products, are widely used in healthcare settings, making their safety inspection crucial. Current methods for detecting foreign particles in LVP solutions through image analysis primarily rely on single-frame detection or simple temporal smoothing strategies, which fail to effectively utilize spatiotemporal correlations across multiple frames. Factors such as occlusion, motion blur, and refractive distortion can significantly impact detection accuracy. To address these challenges, this paper proposes a multi-frame object detection framework based on spatiotemporal collaborative learning, incorporating three key innovations: a YOLO network optimized with deformable convolution, a differentiable cross-frame association module, and an uncertainty-aware feature fusion and re-identification module. Experimental results demonstrate that our method achieves a 97% detection rate for contaminated LVP solutions on the LVPD dataset. Furthermore, the proposed method enables end-to-end training and processes five bottles per second, meeting the requirements for real-time pipeline applications. Full article
(This article belongs to the Topic New Applications of Big Data Technology: Integration of Data Mining and Artificial Intelligence)
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17 pages, 710 KiB  
Article
Optimal Feedback Policy for the Tracking Control of Markovian Jump Boolean Control Networks over a Finite Horizon
by Bingquan Chen, Yuyi Xue and Aiju Shi
Mathematics 2025, 13(8), 1332; https://doi.org/10.3390/math13081332 - 18 Apr 2025
Viewed by 54
Abstract
This paper aims to find optimal feedback policies for the tracking control of Markovian jump Boolean control networks (MJBCNs) over a finite horizon. The tracking objective is a predetermined time-varying trajectory with a finite length. To minimize the expected total tracking error between [...] Read more.
This paper aims to find optimal feedback policies for the tracking control of Markovian jump Boolean control networks (MJBCNs) over a finite horizon. The tracking objective is a predetermined time-varying trajectory with a finite length. To minimize the expected total tracking error between the output trajectory of MJBCN and the reference trajectory, an algorithm is proposed to determine the optimal policy for the system. Furthermore, considering the penalty for control input changes, a new objective function is obtained by weighted summing the total tracking error with the total variation of control input. Certain optimal policies sre designed using an algorithm to minimize the expectation of the new objective function. Finally, the methodology is applied to two simplified biological models to demonstrate its effectiveness. Full article
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