Mathematics

18 pages, 329 KB

Open AccessFeature PaperArticle

by Edoardo Ballico and Elizabeth Gasparim

Mathematics 2025, 13(20), 3356; https://doi.org/10.3390/math13203356 - 21 Oct 2025

We discuss the complex-analytic subsets of the moduli spaces of rank 2 vector bundles on a classical Hopf surface formed by irregular bundles. We stratify the set of irregular bundles by weight (and irregular profiles). We provide the topological result (vanishing of higher [...] Read more.

We discuss the complex-analytic subsets of the moduli spaces of rank 2 vector bundles on a classical Hopf surface formed by irregular bundles. We stratify the set of irregular bundles by weight (and irregular profiles). We provide the topological result (vanishing of higher cohomology groups) on the part of the moduli spaces parameterizing regular bundles. Full article

20 pages, 1492 KB

Open AccessArticle

Interpretable Diagnostics with SHAP-Rule: Fuzzy Linguistic Explanations from SHAP Values

by Alexandra I. Khalyasmaa, Pavel V. Matrenin and Stanislav A. Eroshenko

Mathematics 2025, 13(20), 3355; https://doi.org/10.3390/math13203355 - 21 Oct 2025

Abstract

This study introduces SHAP-Rule, a novel explainable artificial intelligence method that integrates Shapley additive explanations with fuzzy logic to automatically generate interpretable linguistic IF-THEN rules for diagnostic tasks. Unlike purely numeric SHAP vectors, which are difficult for decision-makers to interpret, SHAP-Rule translates feature [...] Read more.

This study introduces SHAP-Rule, a novel explainable artificial intelligence method that integrates Shapley additive explanations with fuzzy logic to automatically generate interpretable linguistic IF-THEN rules for diagnostic tasks. Unlike purely numeric SHAP vectors, which are difficult for decision-makers to interpret, SHAP-Rule translates feature attributions into concise explanations that humans can understand. The method was rigorously evaluated and compared with baseline SHAP and AnchorTabular explanations across three distinct and representative datasets: the CWRU Bearing dataset for industrial predictive maintenance, a dataset for failure analysis in power transformers, and the medical Pima Indians Diabetes dataset. Experimental results demonstrated that SHAP-Rule consistently provided clearer and more easily comprehensible explanations, achieving high expert ratings for simplicity and understanding. Additionally, SHAP-Rule exhibited superior computational efficiency and robust consistency compared to alternative methods, making it particularly suitable for real-time diagnostic applications. Although SHAP-Rule showed minor trade-offs in coverage, it maintained high global fidelity, often approaching 100%. These findings highlight the significant practical advantages of linguistic fuzzy explanations generated by SHAP-Rule, emphasizing its strong potential for enhancing interpretability, efficiency, and reliability in diagnostic decision-support systems. Full article

(This article belongs to the Special Issue Mathematical Foundations and Advances in Machine Learning and Data Mining)

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18 pages, 329 KB

Open AccessArticle

Structures, Ranks and Minimal Distances of Cyclic Codes over Z_p2uZ_p2⁺

by Sami H. Saif

Mathematics 2025, 13(20), 3354; https://doi.org/10.3390/math13203354 - 21 Oct 2025

Abstract

Let p be a prime and

F_{p}

a finite field of order p. This paper investigates cyclic codes over the ring

R_{p^{2}, u} = Z_{p^{2}} + u Z_{p^{2}}

of order

p^{4}

, where [...] Read more.

Let p be a prime and

F_{p}

a finite field of order p. This paper investigates cyclic codes over the ring

R_{p^{2}, u} = Z_{p^{2}} + u Z_{p^{2}}

of order

p^{4}

, where the nilpotent element u satisfies

u^{2} = 0

and

p u \neq 0 .

The condition

u^{2} = 0

with

p u \neq 0

is crucial, as it creates a nontrivial interaction between the components of the ring, allowing the construction of new codes with enhanced structural and distance properties. We provide explicit generating sets for cyclic codes over

R_{p^{2}, u}

and study fundamental parameters such as their rank and Hamming distance. In the case

gcd (n, p) = 1

, we show that cyclic codes can be generated by just two polynomials, which allows a complete determination of their rank and minimal Hamming distance distributions. Furthermore, using the Gray map from

R_{p^{2}, u}

to

F_{p}^{4}

, we construct all but one of the ternary optimal codes of length 12 as images of cyclic codes over

R_{3^{2}, u}

, with computations verified using the Magma system. Full article

15 pages, 275 KB

Open AccessArticle

On the Study of Nonlinear Capillarity Models Through Weighted Sobolev Spaces with Dirichlet Boundary Conditions

by Manal Badgaish, Lhoucine Hmidouch and Nacir Hmidouch

Mathematics 2025, 13(20), 3353; https://doi.org/10.3390/math13203353 - 21 Oct 2025

Abstract

We prove in this paper some existence and uniqueness results of weak solutions for some nonlinear degenerate parabolic problems arising from capillarity phenomena with the Dirichlet type boundary condition. Full article

26 pages, 1518 KB

Open AccessArticle

Lightweight End-to-End Diacritical Arabic Speech Recognition Using CTC-Transformer with Relative Positional Encoding

by Haifa Alaqel and Khalil El Hindi

Mathematics 2025, 13(20), 3352; https://doi.org/10.3390/math13203352 - 21 Oct 2025

Abstract

Arabic automatic speech recognition (ASR) faces distinct challenges due to its complex morphology, dialectal variations, and the presence of diacritical marks that strongly influence pronunciation and meaning. This study introduces a lightweight approach for diacritical Arabic ASR that employs a Transformer encoder architecture [...] Read more.

Arabic automatic speech recognition (ASR) faces distinct challenges due to its complex morphology, dialectal variations, and the presence of diacritical marks that strongly influence pronunciation and meaning. This study introduces a lightweight approach for diacritical Arabic ASR that employs a Transformer encoder architecture enhanced with Relative Positional Encoding (RPE) and Connectionist Temporal Classification (CTC) loss, eliminating the need for a conventional decoder. A two-stage training process was applied: initial pretraining on Modern Standard Arabic (MSA), followed by progressive three-phase fine-tuning on diacritical Arabic datasets. The proposed model achieves a WER of 22.01% on the SASSC dataset, improving over traditional systems (best 28.4% WER) while using only ≈14 M parameters. In comparison, XLSR-Large (300 M parameters) achieves a WER of 12.17% but requires over 20× more parameters and substantially higher training and inference costs. Although XLSR attains lower error rates, the proposed model is far more practical for resource-constrained environments, offering reduced complexity, faster training, and lower memory usage while maintaining competitive accuracy. These results show that encoder-only Transformers with RPE, combined with CTC training and systematic architectural optimization, can effectively model Arabic phonetic structure while maintaining computational efficiency. This work establishes a new benchmark for resource-efficient diacritical Arabic ASR, making the technology more accessible for real-world deployment. Full article

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

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14 pages, 957 KB

Open AccessArticle

TECP: Token-Entropy Conformal Prediction for LLMs

by Beining Xu and Yongming Lu

Mathematics 2025, 13(20), 3351; https://doi.org/10.3390/math13203351 - 21 Oct 2025

Abstract

Uncertainty quantification (UQ) for open-ended language generation remains a critical yet underexplored challenge, particularly in settings where token-level log-probabilities are available during decoding. We present Token-Entropy Conformal Prediction (TECP), which treats a log-probability-based token-entropy statistic as a nonconformity score and integrates it [...] Read more.

Uncertainty quantification (UQ) for open-ended language generation remains a critical yet underexplored challenge, particularly in settings where token-level log-probabilities are available during decoding. We present Token-Entropy Conformal Prediction (TECP), which treats a log-probability-based token-entropy statistic as a nonconformity score and integrates it with split conformal prediction to construct prediction sets with finite-sample coverage guarantees. We work in a white-box regime in which per-token log-probabilities are accessible during decoding. TECP estimates episodic uncertainty from the token-entropy structure of sampled generations and calibrates thresholds via conformal quantiles to ensure provable error control. Empirical evaluations across six large language models and two QA benchmarks (CoQA and TriviaQA) show that TECP consistently achieves reliable coverage and compact prediction sets, outperforming prior self-UQ methods. These results provide a principled and efficient solution for trustworthy generation in white-box, log-probability-accessible LLM settings. Full article

(This article belongs to the Topic Challenges and Solutions in Large Language Models)

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20 pages, 3817 KB

Open AccessArticle

Multiscale Contextual Fusion for Robust Airport Surveillance

by Fei Yan, Jiuxia Guo and Huawei Wang

Mathematics 2025, 13(20), 3350; https://doi.org/10.3390/math13203350 - 21 Oct 2025

Abstract

Object detection in airport surface surveillance presents significant challenges, primarily due to the extreme variation in object scales and the critical need for contextual information. To address these issues, we propose a novel deep learning architecture that integrates two specialized modules: the Poly [...] Read more.

Object detection in airport surface surveillance presents significant challenges, primarily due to the extreme variation in object scales and the critical need for contextual information. To address these issues, we propose a novel deep learning architecture that integrates two specialized modules: the Poly Kernel Inception (PKI) module and the Context Anchor Attention (CAA) module. The PKI module is designed to effectively capture multi-scale features, enabling the accurate detection of objects ranging from large aircraft to small staff members. Concurrently, the CAA module leverages long-range contextual information, which significantly enhances the model’s ability to precisely localize and identify targets within complex scenes. The synergistic integration of these two modules demonstrates a substantial improvement in feature extraction performance, leading to enhanced detection accuracy on our publicly available ASS dataset. This work provides a robust and effective solution for the challenging task of airport surface object detection, establishing a strong foundation for future research in this domain. Full article

(This article belongs to the Special Issue Optimization and Machine Learning-Based Methods in Air Traffic Management and Aeronautical Domains)

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18 pages, 1825 KB

Open AccessFeature PaperArticle

Fast Deep Belief Propagation: An Efficient Learning-Based Algorithm for Solving Constraint Optimization Problems

by Shufeng Kong, Feifan Chen, Zijie Wang and Caihua Liu

Mathematics 2025, 13(20), 3349; https://doi.org/10.3390/math13203349 - 21 Oct 2025

Abstract

Belief Propagation (BP) is a fundamental heuristic for solving Constraint Optimization Problems (COPs), yet its practical applicability is constrained by slow convergence and instability in loopy factor graphs. While Damped BP (DBP) improves convergence by using manually tuned damping factors, its reliance on [...] Read more.

Belief Propagation (BP) is a fundamental heuristic for solving Constraint Optimization Problems (COPs), yet its practical applicability is constrained by slow convergence and instability in loopy factor graphs. While Damped BP (DBP) improves convergence by using manually tuned damping factors, its reliance on labor-intensive hyperparameter optimization limits scalability. Deep Attentive BP (DABP) addresses this by automating damping through recurrent neural networks (RNNs), but introduces significant memory overhead and sequential computation bottlenecks. To reduce memory usage and accelerate deep belief propagation, this paper introduces Fast Deep Belief Propagation (FDBP), a deep learning framework that improves COP solving through online self-supervised learning and graphics processing unit (GPU) acceleration. FDBP decouples the learning of damping factors from BP message passing, inferring all parameters for an entire BP iteration in a single step, and leverages mixed precision to further optimize GPU memory usage. This approach substantially improves both the efficiency and scalability of BP optimization. Extensive evaluations on synthetic and real-world benchmarks highlight the superiority of FDBP, especially for large-scale instances where DABP fails due to memory constraints. Moreover, FDBP achieves an average speedup of 2.87× over DABP with the same restart counts. Because BP for COPs is a mathematically grounded GPU-parallel message-passing framework that bridges applied mathematics, computing, and machine learning, and is widely applicable across science and engineering, our work offers a promising step toward more efficient solutions to these problems. Full article

(This article belongs to the Special Issue Applied Mathematics, Computing, and Machine Learning)

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18 pages, 2757 KB

Open AccessFeature PaperArticle

Robust Bias Compensation LMS Algorithms Under Colored Gaussian Input Noise and Impulse Observation Noise Environments

by Ying-Ren Chien, Han-En Hsieh and Guobing Qian

Mathematics 2025, 13(20), 3348; https://doi.org/10.3390/math13203348 - 21 Oct 2025

Abstract

Adaptive filtering algorithms often suffer from biased parameter estimation and performance degradation in the presence of colored input noise and impulsive observation noise, both of which are common in practical sensor and communication systems. Existing bias-compensated least mean square (LMS) algorithms generally assume [...] Read more.

Adaptive filtering algorithms often suffer from biased parameter estimation and performance degradation in the presence of colored input noise and impulsive observation noise, both of which are common in practical sensor and communication systems. Existing bias-compensated least mean square (LMS) algorithms generally assume white Gaussian input noise, thereby limiting their applicability in real-world scenarios. This paper introduces a robust convex combination bias-compensated LMS (CC-BC-LMS) algorithm designed to address both colored Gaussian input noise and impulsive observation noise. The proposed algorithm achieves bias compensation through robust estimation of the input noise autocorrelation matrix and employs a modified Huber function to mitigate the influence of impulsive noise. A convex combination of fast and slow adaptive filters enables variable step-size adaptation, effectively balancing rapid convergence and low steady-state error. Extensive simulation results demonstrate that the proposed CC-BC-LMS algorithm provides substantial improvements in normalized mean square deviation (NMSD), surpassing state-of-the-art bias-compensated and robust adaptive filtering techniques by

4.48

dB to

11.4

dB under various noise conditions. These results confirm the effectiveness of the proposed approach for reliable adaptive filtering in challenging noisy environments. Full article

(This article belongs to the Special Issue Mathematical Foundations and Distributed Applications of Adaptive Signal Processing)

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18 pages, 912 KB

Open AccessArticle

Coupled Dynamical Systems for Solving Linear Inverse Problems

by Ryosuke Kasai, Omar M. Abou Al-Ola and Tetsuya Yoshinaga

Mathematics 2025, 13(20), 3347; https://doi.org/10.3390/math13203347 - 21 Oct 2025

Abstract

We propose a class of coupled dynamical systems for solving linear inverse problems, treating both the unknown variable and an auxiliary variable representing measurement dynamics as state variables. This framework does not rely on probabilistic modeling or explicit regularization; instead, it achieves noise [...] Read more.

We propose a class of coupled dynamical systems for solving linear inverse problems, treating both the unknown variable and an auxiliary variable representing measurement dynamics as state variables. This framework does not rely on probabilistic modeling or explicit regularization; instead, it achieves noise suppression through deterministic interactions between system variables. We analyze the theoretical properties of the systems, including stability, equilibrium behavior, and convergence for the linear system, and equilibrium stability for the two nonlinear variants. The nonlinear extensions incorporate state-dependent mechanisms that preserve equilibrium stability while enhancing convergence and robustness in practice. Numerical experiments illustrate the effectiveness of the proposed approach in estimating the unknown variable and mitigating measurement noise. Full article

(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications, 3rd Edition)

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28 pages, 547 KB

Open AccessArticle

State-DynAttn: A Hybrid State-Space and Dynamic Graph Attention Architecture for Robust Air Traffic Flow Prediction Under Weather Disruptions

by Fei Yan and Huawei Wang

Mathematics 2025, 13(20), 3346; https://doi.org/10.3390/math13203346 - 21 Oct 2025

Abstract

We propose State-DynAttn, a hybrid architecture for robust air traffic flow prediction under weather disruptions, which integrates state-space models (SSMs) with dynamic graph attention to address the challenges of long-range dependency modeling and adaptive spatial–temporal relationship learning. The increasing complexity of air traffic [...] Read more.

We propose State-DynAttn, a hybrid architecture for robust air traffic flow prediction under weather disruptions, which integrates state-space models (SSMs) with dynamic graph attention to address the challenges of long-range dependency modeling and adaptive spatial–temporal relationship learning. The increasing complexity of air traffic systems, exacerbated by unpredictable weather events, demands methods that can simultaneously capture global temporal patterns and localized disruptions; existing approaches often struggle to balance these requirements efficiently. The proposed method employs two parallel branches: an SSM branch for continuous-time recurrent modeling of long-range dependencies with linear complexity, and a dynamic graph attention branch that adaptively computes node-pair weights while incorporating weather severity features through sparsification strategies for scalability. These branches are fused via a data-dependent gating mechanism, enabling the model to dynamically prioritize either global temporal dynamics or localized spatial interactions based on input conditions. Moreover, the architecture leverages memory-efficient attention computation and HiPPO initialization to ensure stable training and inference. Experiments on real-world air traffic datasets demonstrate that State-DynAttn outperforms existing baselines in prediction accuracy and robustness, particularly under severe weather scenarios. The framework’s ability to handle both gradual traffic evolution and abrupt disruption-induced changes makes it suitable for real-world deployment in air traffic management systems. Furthermore, the design principles of State-DynAttn can be extended to other spatiotemporal prediction tasks where long-range dependencies and dynamic relational structures coexist. This work contributes a principled approach to hybridizing state-space models with graph-based attention, offering insights into the trade-offs between computational efficiency and modeling flexibility in complex dynamical systems. Full article

(This article belongs to the Special Issue Optimization and Machine Learning-Based Methods in Air Traffic Management and Aeronautical Domains)

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15 pages, 3574 KB

Open AccessArticle

A Credit Risk Identification Model Based on the Minimax Probability Machine with Generative Adversarial Networks

by Yutong Zhang, Xiaodong Zhao and Hailong Huang

Mathematics 2025, 13(20), 3345; https://doi.org/10.3390/math13203345 - 20 Oct 2025

Abstract

In the context of industrial transitions and tariff frictions, financial markets are experiencing frequent defaults, emphasizing the urgency of upgrading credit scoring methodologies. A novel credit risk identification model integrating generative adversarial networks (GAN) and the minimax probability machine (MPM) is proposed. GAN [...] Read more.

In the context of industrial transitions and tariff frictions, financial markets are experiencing frequent defaults, emphasizing the urgency of upgrading credit scoring methodologies. A novel credit risk identification model integrating generative adversarial networks (GAN) and the minimax probability machine (MPM) is proposed. GAN generates realistic augmented samples to alleviate class imbalance in the credit score dataset, while the MPM optimizes the classification hyperplane by reformulating probability constraints into second-order cone problems via the multivariate Chebyshev inequality. Numerical experiments conducted on the South German Credit dataset, which represents individual (consumer) credit risk, demonstrate that the proposed generative adversarial network’s minimax probability machine (GAN-MPM) model achieves 76.13%, 60.93%, 71.78%, and 72.03% for accuracy, F1-score, sensitivity, and AUC, respectively, significantly outperforming support vector machines, random forests, and XGBoost. Furthermore, SHAP analysis reveals that the installment rate in percentage of disposable income, housing type, duration in month, and status of existing checking accounts are the most influential features. These findings demonstrate the effectiveness and interpretability of the GAN-MPM model, offering a more accurate and reliable tool for credit risk management. Full article

(This article belongs to the Section E2: Control Theory and Mechanics)

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24 pages, 22545 KB

Open AccessArticle

Eliminating Packing-Aware Masking via LoRA-Based Supervised Fine-Tuning of Large Language Models

by Jeong Woo Seo and Ho-Young Jung

Mathematics 2025, 13(20), 3344; https://doi.org/10.3390/math13203344 - 20 Oct 2025

Abstract

Packing approaches enhance training efficiency by filling the padding space in each batch with shorter sequences, thereby reducing the total number of batches per epoch. This approach has proven effective in both pre-training and supervised fine-tuning of large language models (LLMs). However, most [...] Read more.

Packing approaches enhance training efficiency by filling the padding space in each batch with shorter sequences, thereby reducing the total number of batches per epoch. This approach has proven effective in both pre-training and supervised fine-tuning of large language models (LLMs). However, most packing methods necessitate a packing-aware masking (PAM) mechanism to prevent cross-contamination between different text segments in the multi-head attention (MHA) layers. This masking ensures that the scaled dot-product attention operates only within segment boundaries. Despite its functional utility, PAM introduces significant implementation complexity and computational overhead during training. In this paper, we propose a novel method that eliminates the need for PAM during supervised fine-tuning with packing. Instead of masking, we introduce a learnable tensor derived from Low-Rank Adaptation (LoRA) with the query and value parameters of the attention mechanism. This tensor is trained to attenuate the subspace corresponding to cross-contamination, effectively replacing the function of PAM. Through component-wise decomposition of attention head outputs, we isolate the contamination component and demonstrate that it can be attenuated using the LoRA-derived tensor. Empirical evaluations on 7B-scale LLMs show that our method reduces training time and runtime overhead by completely removing the implementation associated with PAM. This enables more scalable and efficient supervised fine-tuning with packing, without compromising model integrity. Full article

(This article belongs to the Special Issue Advances in Intelligent Computing, Machine Learning and Pattern Recognition)

18 pages, 3244 KB

Open AccessFeature PaperArticle

Achieving Distributional Robustness with Group-Wise Flat Minima

by Seowon Ji, Seunghyun Moon, Jiyoon Shin and Sangwoo Hong

Mathematics 2025, 13(20), 3343; https://doi.org/10.3390/math13203343 - 20 Oct 2025

Abstract

Improving robustness under distributional shifts remains a central challenge in machine learning. Although Sharpness-Aware Minimization (SAM) has proven effective in finding flatter minima for better generalization, it overlooks the heterogeneity in sharpness across different subpopulations, which can exacerbate performance gaps for minority or [...] Read more.

Improving robustness under distributional shifts remains a central challenge in machine learning. Although Sharpness-Aware Minimization (SAM) has proven effective in finding flatter minima for better generalization, it overlooks the heterogeneity in sharpness across different subpopulations, which can exacerbate performance gaps for minority or vulnerable groups. To address this challenge, we propose Group-gap Guided SAM (G²-SAM), a new optimization framework that promotes distributional robustness by steering flatness-seeking directions according to intergroup loss disparities. Our method estimates group-wise sharpness and adaptively refines perturbation strategies to minimize the worst-group loss while preserving model consistency. Through comprehensive experiments across various datasets, we show that G²-SAM achieves superior Worst-Group Accuracy and robustness, outperforming previous baselines. These findings highlight the importance of addressing group-specific geometry in the loss landscape to build more reliable and equitable neural networks. Full article

(This article belongs to the Special Issue Advancement of Mathematical Methods in Feature Representation Learning for Artificial Intelligence, Data Mining and Robotics, 3rd Edition)

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20 pages, 317 KB

Open AccessFeature PaperArticle

Majorization Inequalities for n-Convex Functions with Applications to 3-Convex Functions

by László Horváth

Mathematics 2025, 13(20), 3342; https://doi.org/10.3390/math13203342 - 20 Oct 2025

Abstract

In this paper, we study majorization-type inequalities for n-convex (specifically 3-convex) functions. Numerous papers deal with such integral inequalities, in which n-convex functions are defined on compact intervals and nonnegative measures are used in the integrals. The main goal of this [...] Read more.

In this paper, we study majorization-type inequalities for n-convex (specifically 3-convex) functions. Numerous papers deal with such integral inequalities, in which n-convex functions are defined on compact intervals and nonnegative measures are used in the integrals. The main goal of this paper is to formulate similar results for noncompact intervals and signed measures. We follow a well-known method often used for compact intervals: approximation of n-convex functions with simple n-convex functions. After some preliminary results, we present new approximation theorems, some of which extend classical results, while others are completely unique approximations. Then we obtain some novel majorization-type inequalities, which can be applied under more general conditions than those currently known. Finally, we illustrate the applicability of our results by answering problems from different areas: discrete majorization-type inequalities, specifically one-dimensional inequality of Sherman for n-convex functions; characterization of Steffensen–Popoviciu measures for nonnegative, continuous, and increasing 3-convex functions; Hermite–Hadamard-type inequalities for 3-convex functions. Full article

14 pages, 395 KB

Open AccessArticle

Bayesian Approach to Simultaneous Variable Selection and Estimation in a Linear Regression Model with Applications in Driver Telematics

by Himchan Jeong and Minwoo Kim

Mathematics 2025, 13(20), 3341; https://doi.org/10.3390/math13203341 - 20 Oct 2025

Abstract

This article proposes a novel application of the Bayesian variable selection framework for driver telematics data. Unlike the traditional LASSO, the Bayesian variable selection framework allows us to incorporate the importance of certain features in advance in the variable selection procedure so that [...] Read more.

This article proposes a novel application of the Bayesian variable selection framework for driver telematics data. Unlike the traditional LASSO, the Bayesian variable selection framework allows us to incorporate the importance of certain features in advance in the variable selection procedure so that the traditional features more likely remain in the ratemaking models. The applicability of the proposed framework in the ratemaking practices is also validated via synthetic telematics data. Full article

(This article belongs to the Special Issue Actuarial Statistical Modeling and Applications)

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14 pages, 449 KB

Open AccessArticle

Local Quantum Uncertainty and Entanglement in the Hyperfine Structure of the Hydrogen Atom: A Lindblad Approach

by Kamal Berrada and Smail Bougouffa

Mathematics 2025, 13(20), 3340; https://doi.org/10.3390/math13203340 - 20 Oct 2025

Abstract

In this work, we investigate quantum correlations, including entanglement and quantum discord, within the hyperfine structure of the hydrogen atom using the Lindblad master equation to model its dynamics as an open quantum system interacting with an environment. By incorporating realistic environmental influences, [...] Read more.

In this work, we investigate quantum correlations, including entanglement and quantum discord, within the hyperfine structure of the hydrogen atom using the Lindblad master equation to model its dynamics as an open quantum system interacting with an environment. By incorporating realistic environmental influences, we examine the time evolution of two key measures of quantum correlations: concurrence, which quantifies entanglement, and local quantum uncertainty (LQU), a broader indicator of quantumness. Our analysis spans various initial states, including coherent superpositions of hyperfine states, to capture a wide range of possible configurations and demonstrate how these measures capture distinct aspects of quantum behavior. The results reveal the robustness of LQU in regimes where entanglement may vanish. This resilience of LQU underscores its utility as a robust measure of quantum correlations beyond entanglement alone in the hydrogen atom. By elucidating the dynamics of quantum correlations in the hydrogen atom under realistic conditions, this work not only deepens our fundamental understanding of atomic systems but also highlights their potential relevance to quantum information science and the development of quantum technologies. Full article

(This article belongs to the Special Issue Advances in Mathematics for Quantum Mechanics)

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17 pages, 1129 KB

Open AccessArticle

Stability and Bifurcation in a Delayed Malaria Model with Threshold Control

by Ying Qiao, Yuelin Gao, Jimin Li, Zhixin Han and Bo Zhang

Mathematics 2025, 13(20), 3339; https://doi.org/10.3390/math13203339 - 20 Oct 2025

Abstract

In this paper, we develop a delayed malaria model that integrates a discrete time delay and a non-smooth threshold-based control strategy. Using the time delay τ as a bifurcation parameter, we investigate the local stability of the endemic equilibrium through analysis of the [...] Read more.

In this paper, we develop a delayed malaria model that integrates a discrete time delay and a non-smooth threshold-based control strategy. Using the time delay τ as a bifurcation parameter, we investigate the local stability of the endemic equilibrium through analysis of the characteristic equation. We establish sufficient conditions for the occurrence of Hopf bifurcation, demonstrating how stability switches emerge as τ varies. Furthermore, when the infected population exceeds a critical threshold I_m, a sliding mode domain arises. We analyze the dynamics within this sliding region using the Utkin equivalent control method. Numerical simulations are provided to support the theoretical findings, illustrating the complex dynamical behaviors induced by both delay and threshold control. Full article

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24 pages, 10663 KB

Open AccessArticle

Feature Decomposition-Based Framework for Source-Free Universal Domain Adaptation in Mechanical Equipment Fault Diagnosis

by Peiyi Zhou, Weige Liang, Shiyan Sun and Qizheng Zhou

Mathematics 2025, 13(20), 3338; https://doi.org/10.3390/math13203338 - 20 Oct 2025

Abstract

Aiming at the problems of high complexity in source domain data, inaccessibility of target domain data, and unknown fault patterns in real-world industrial scenarios for mechanical fault diagnosis, this paper proposes a Feature Decomposition-based Source-Free Universal Domain Adaptation (FD-SFUniDA) framework for mechanical equipment [...] Read more.

Aiming at the problems of high complexity in source domain data, inaccessibility of target domain data, and unknown fault patterns in real-world industrial scenarios for mechanical fault diagnosis, this paper proposes a Feature Decomposition-based Source-Free Universal Domain Adaptation (FD-SFUniDA) framework for mechanical equipment fault diagnosis. First, the CBAM attention module is incorporated to enhance the ResNet-50 convolutional network for extracting feature information from source domain data. During the target domain adaptation phase, singular value decomposition is applied to the weights of the pre-trained model’s classification layer, orthogonally decoupling the feature space into a source-known subspace and a target-private subspace. Then, based on the magnitude of feature projections, a dynamic decision boundary is constructed and combined with an entropy threshold mechanism to accurately distinguish between known and unknown class samples. Furthermore, intra-class feature consistency is strengthened through neighborhood-expanded contrastive learning, and semantic weight calibration is employed to reconstruct the feature space, thereby suppressing the negative transfer effect. Finally, extensive experiments under multiple operating conditions on rolling bearing and reciprocating mechanism datasets demonstrate that the proposed method excels in addressing source-free fault diagnosis problems for mechanical equipment and shows promising potential for practical engineering applications in fault classification tasks. Full article

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