Mathematics

21 pages, 352 KB

Open AccessArticle

On α-ψ-Contractive Condition for Single-Valued and Multi-Valued Operators in Strong b-Metric Spaces

by Saud M. Alsulami and Thanaa A. Alarfaj

Mathematics 2025, 13(20), 3357; https://doi.org/10.3390/math13203357 - 21 Oct 2025

Viewed by 273

This paper aims to establish fixed point theorems in a complete strong b-metric space under the

α

-

ψ

-contractive condition imposed on single-valued mappings. Subsequently, we prove certain fixed point theorems, both locally and globally, under the

α_{*}

-

ψ

[...] Read more.

This paper aims to establish fixed point theorems in a complete strong b-metric space under the

α

-

ψ

-contractive condition imposed on single-valued mappings. Subsequently, we prove certain fixed point theorems, both locally and globally, under the

α_{*}

-

ψ

-contractive condition and the

α

-

ψ

-contractive condition on multi-valued mappings in a complete strong b-metric space. The theorems presented in this paper extend, generalize, and improve various existing results in the literature. To demonstrate the superiority of the results, we present multiple examples throughout this article and two applications: one in dynamic programming and another in ordinary differential equations. Moreover, the proposed results provide stronger and more general conclusions compared to several well-known fixed point theorems in the literature. In particular, our findings highlight the novelty and superiority of the

α

-

ψ

-contractive framework in the setting of strong b-metric spaces, offering broader applicability and deeper insight into both theoretical and applied contexts. Full article

(This article belongs to the Special Issue Fixed Point, Optimization, and Applications: 3rd Edition)

18 pages, 329 KB

Open AccessFeature PaperArticle

Irregular Bundles on Hopf Surfaces

by Edoardo Ballico and Elizabeth Gasparim

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

Viewed by 160

Abstract

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

Viewed by 348

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)

► Show Figures

Figure 1

18 pages, 330 KB

Open AccessArticle

Structures, Ranks and Minimal Distances of Cyclic Codes over

Z_{p^{2}} + u Z_{p^{2}}

by Sami H. Saif

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

Viewed by 150

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

Viewed by 186

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, 1497 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

Viewed by 376

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)

► Show Figures

Figure 1

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

Viewed by 532

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)

► Show Figures

Figure 1

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

Viewed by 226

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)

► Show Figures

Figure 1

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

Viewed by 379

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)

► Show Figures

Figure 1

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

Viewed by 257

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)

► Show Figures

Figure 1

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

Viewed by 220

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)

► Show Figures

Figure 1

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

Viewed by 353

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)

► Show Figures

Figure 1

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

Viewed by 330

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)

► Show Figures

Figure 1

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

Viewed by 447

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)

► Show Figures

Figure 1

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

Viewed by 273

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)

► Show Figures

Figure 1

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

Viewed by 308

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

Viewed by 270

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)

► Show Figures

Figure 1

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

Viewed by 253

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)

► Show Figures

Figure 1

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

Viewed by 152

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

► Show Figures

Figure 1

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

Viewed by 311

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

► Show Figures

Figure 1

20 pages, 334 KB

Open AccessArticle

Linear Quadratic Pursuit–Evasion Games on Time Scales

by Davis Funk, Richard Williams and Nick Wintz

Mathematics 2025, 13(20), 3337; https://doi.org/10.3390/math13203337 - 20 Oct 2025

Viewed by 263

Abstract

In this paper, we unify and extend the linear quadratic pursuit–evasion games to dynamic equations on time scales. A mixed strategy for a single pursuer and evader is studied in two settings. In the open-loop setting, the corresponding controls are expressed in terms [...] Read more.

In this paper, we unify and extend the linear quadratic pursuit–evasion games to dynamic equations on time scales. A mixed strategy for a single pursuer and evader is studied in two settings. In the open-loop setting, the corresponding controls are expressed in terms of a zero-input difference. In the closed-loop setting, the corresponding controls require a mixing feedback term when rewriting the system in extended state form. Finally, we offer a numerical simulation. Full article

(This article belongs to the Special Issue Recent Advances in Dynamic Equations on Time Scales)

► Show Figures

Figure 1

30 pages, 488 KB

Open AccessFeature PaperArticle

An Evolutionary Procedure for a Bi-Objective Assembly Line Balancing Problem

by Jordi Pereira and Mariona Vilà

Mathematics 2025, 13(20), 3336; https://doi.org/10.3390/math13203336 - 20 Oct 2025

Viewed by 321

Abstract

An assembly line is a manufacturing process commonly used in the production of commodity goods. The assembly process is divided into elementary tasks that are sequentially performed at serially arranged workstations. Among the various challenges that must be addressed during the design and [...] Read more.

An assembly line is a manufacturing process commonly used in the production of commodity goods. The assembly process is divided into elementary tasks that are sequentially performed at serially arranged workstations. Among the various challenges that must be addressed during the design and operation of an assembly line, the assembly line balancing problem involves the assignment of tasks to different workstations. In its simplest form, this problem aims to distribute assembly operations among the workstations efficiently. An efficient line is one that optimizes a specific objective function, usually associated with maximizing throughput or minimizing resource requirements. In this study, we adopt a bi-objective approach to find a Pareto set of efficient solutions balancing throughput and resource requirements. To address this problem, we propose a multi-objective evolutionary method, complemented by single- and multi-objective local search procedures that leverage a polynomially solvable case of the problem. We then compare the results of these methods, including their hybridizations, through a computational experiment demonstrating the ability to achieve high-quality solutions. Full article

(This article belongs to the Special Issue Application of Mathematical Methods in Multi-Objective Optimization and Evolutionary Algorithms)

► Show Figures

Figure 1

23 pages, 23803 KB

Open AccessArticle

An Improved Stiffness Model for Spur Gear with Surface Roughness Under Thermal Elastohydrodynamic Lubrication

by Shihua Zhou, Xuan Li, Chao An, Tengyuan Xu, Dongsheng Zhang, Ye Zhang and Zhaohui Ren

Mathematics 2025, 13(20), 3335; https://doi.org/10.3390/math13203335 - 20 Oct 2025

Viewed by 266

Abstract

To investigate the contact performances and meshing characteristics of gears systematically, an improved comprehensive meshing stiffness model of spur gears with consideration of the tooth surface morphology, lubrication, friction, and thermal effects is presented based on the thermal elastohydrodynamic lubrication (TEHL) theory. The [...] Read more.

To investigate the contact performances and meshing characteristics of gears systematically, an improved comprehensive meshing stiffness model of spur gears with consideration of the tooth surface morphology, lubrication, friction, and thermal effects is presented based on the thermal elastohydrodynamic lubrication (TEHL) theory. The fractal feature of the tooth surface morphology is verified experimentally and characterized by the Weierstrass–Mandelbrot fractal function. Based on this, the rough contact stiffness, oil film stiffness, and thermal stiffness are introduced into the stiffness model. Comparisons between smooth and rough models are carried out, and the maximum temperature rise is increased by 24.7%. Subsequently, the influences of the torque, rotational speed, and fractal parameters on the oil film pressure and thickness, friction and temperature rise, and contact stiffness and comprehensive meshing stiffness are investigated. The results show that the oil film pressure and the maximum temperature rise increase by 125.18% and 69.08%, respectively, with an increasing torque from 20 N·m to 300 N·m. As the rotational speed is increased, the oil film thickness sharply increases, the rough peak contact area and friction reduce, and the stiffness fluctuation weakens. For fractal parameters, the oil film pressure and film thickness, friction, and temperature rise are nonlinear with changes in the fractal dimension D and fractal scale characteristic G. The results reveal that this work provides a more reasonable analysis for understanding the meshing characteristics and the design and processing of gears. Full article

► Show Figures

Figure 1

26 pages, 2625 KB

Open AccessArticle

De Novo Single-Cell Biological Analysis of Drug Resistance in Human Melanoma Through a Novel Deep Learning-Powered Approach

by Sumaya Alghamdi, Turki Turki and Y-h. Taguchi

Mathematics 2025, 13(20), 3334; https://doi.org/10.3390/math13203334 - 20 Oct 2025

Viewed by 337

Abstract

Elucidating drug response mechanisms in human melanoma is crucial for improving treatment outcomes. Moreover, the existing tools intended to provide deeper insight into melanoma drug resistance have notable limitations. Therefore, we propose a deep learning (DL)-based approach that works as follows. First, we [...] Read more.

Elucidating drug response mechanisms in human melanoma is crucial for improving treatment outcomes. Moreover, the existing tools intended to provide deeper insight into melanoma drug resistance have notable limitations. Therefore, we propose a deep learning (DL)-based approach that works as follows. First, we processed two single-cell datasets related to human melanoma from the GEO (GSE108383_A375 and GSE108383_451Lu) database and trained a fully connected neural network with five adapted methods (L1-Regularization, DeepLIFT, SHAP, IG, and LRP). We then identified 100 genes by ranking all genes from the highest to the lowest based on the sum of absolute values for corresponding weights across all neurons in the first hidden layer. From a biological perspective, compared to existing bioinformatics tools, the presented DL-based methods identified a higher number of expressed genes in four well-established melanoma cell lines: MALME-3M, MDA-MB435, SK-MEL-28, and SK-MEL-5. Furthermore, we identified FDA-approved melanoma drugs (e.g., Vemurafenib and Dabrafenib), critical genes such as ARAF, SOX10, DCT, and AXL, and key TFs including MITF and TFAP2A. From a classification perspective, we utilized five-fold cross-validation and provided gene sets using all the abovementioned methods to three randomly selected machine learning algorithms, namely, support vector machines, random forests, and neural networks with different hyperparameters. The results demonstrate that the integrated gradients (IG) method adapted in our DL approach contributed to 2.2% and 0.5% overall performance improvements over the best-performing baselines when using A375 and 451Lu cell line datasets. Additional comparison against no gene selection demonstrated that IG is the only method to generate statistically significant results, with 14.4% and 11.7% overall performance improvements. Full article

(This article belongs to the Special Issue Computational Intelligence for Bioinformatics)

► Show Figures

Figure 1

27 pages, 1452 KB

Open AccessArticle

The Alternative Prioritization and Assessment System (ALPAS) Method for Environmental Performance Evaluation

by Alptekin Ulutaş, Ayşe Topal and Fatih Ecer

Mathematics 2025, 13(20), 3333; https://doi.org/10.3390/math13203333 - 20 Oct 2025

Viewed by 470

Abstract

This study aims to evaluate the environmental performance of G7 countries using the Environmental Performance Index. To do this, we introduce a novel ranking multi-criteria method, Alternative Prioritization and Assessment System, for the first time in the literature. It offers a useful contribution [...] Read more.

This study aims to evaluate the environmental performance of G7 countries using the Environmental Performance Index. To do this, we introduce a novel ranking multi-criteria method, Alternative Prioritization and Assessment System, for the first time in the literature. It offers a useful contribution to the multi-criteria decision-making field by tackling several ranking problems, such as low interpretability, a lack of dual evaluation metrics, and limited flexibility in data-driven scenarios. Moreover, three advanced multi-criteria decision-making weighting methods are used to assign weights to the environmental performance criteria. Therefore, the proposed Alternative Prioritization and Assessment System-based methodology evaluates the environmental performance of G7 countries in reaching sustainable development goals. The results show that the waste recovery rate is the paramount indicator, while unsafe drinking water has the least significance. Germany is ranked as the top-performing country, while Japan is ranked lowest. The key contribution of this research lies in the development and implementation of the Alternative Prioritization and Assessment System method, offering enhanced ranking stability, transparency, and dual-perspective evaluation. The use of the Environmental Performance Index further supports replicability and policy relevance. The proposed model can guide environmental policy formulation and benchmarking efforts among industrialized nations. It also provides a robust framework for cross-national sustainability comparisons in future research. Full article

(This article belongs to the Special Issue Advances in Multiple-Criteria Decision Making: New Trends and Applications)

► Show Figures

Figure 1

21 pages, 1461 KB

Open AccessArticle

Modeling SSE 50 ETF Returns and Option Pricing: Evidence from a Score-Driven GARCH-Jump Approach

by Mingfu Shi, Chuanhai Zhang, Qingqing Chen and Wolfgang Karl Härdle

Mathematics 2025, 13(20), 3332; https://doi.org/10.3390/math13203332 - 19 Oct 2025

Viewed by 423

Abstract

Modeling stock returns and option pricing in the presence of jumps remains a central challenge in financial economics. This paper employs a novel score-driven GARCH-jump model to analyze SSE (Shanghai Stock Exchange) 50 ETF returns and option pricing. The main findings are as [...] Read more.

Modeling stock returns and option pricing in the presence of jumps remains a central challenge in financial economics. This paper employs a novel score-driven GARCH-jump model to analyze SSE (Shanghai Stock Exchange) 50 ETF returns and option pricing. The main findings are as follows. First, we use 50 ETF spot returns to estimate conditional volatility and jump intensity, and find that the SDSDJ (score-driven separate dynamic jumps) model significantly outperforms conventional GARCH-jump models in model fitting. Second, we evaluate both in-sample and out-of-sample pricing performance using data from 50 ETF options, and find that the SDSDJ model achieves the lowest in-sample pricing error among all benchmarks, while its simplified variant—the SDJ (score-driven jumps) model—delivers the most accurate out-of-sample results. Third, the superior pricing performance of both models is robust across different levels of moneyness and DTM (days-to-maturity). Full article

(This article belongs to the Special Issue Recent Advances in Time Series Analysis: Methods, Theory, and Applications)

► Show Figures

Figure 1

18 pages, 1596 KB

Open AccessArticle

New Multiscale Approach of Complex Modelling Chordae Tendineae Considering Strain-Dependent Modulus of Elasticity

by Alicia Menéndez Hurtado, Sergejus Borodinas, Olga Chabarova, Jelena Selivonec and Eugeniuš Stupak

Mathematics 2025, 13(20), 3331; https://doi.org/10.3390/math13203331 - 19 Oct 2025

Viewed by 240

Abstract

Understanding the nonlinear mechanical behaviour of mitral valve chordae tendineae is critical for accurate biomechanical modelling in cardiac simulations. This study integrates high-resolution 3D finite element analysis with experimentally derived Cauchy stress–Green–Lagrange strain data to capture both material and geometric nonlinearities. A one-dimensional [...] Read more.

Understanding the nonlinear mechanical behaviour of mitral valve chordae tendineae is critical for accurate biomechanical modelling in cardiac simulations. This study integrates high-resolution 3D finite element analysis with experimentally derived Cauchy stress–Green–Lagrange strain data to capture both material and geometric nonlinearities. A one-dimensional formulation incorporating strain-dependent elasticity and large deformation kinematics was developed and validated against 3D simulations in COMSOL Multiphysics. Calibrated using experimental stress–strain data and validated against high-fidelity 3D finite element simulations in COMSOL, it reveals that neglecting transverse deformation overestimates axial force by 7%. Cross-sectional area reduction during stretch remained consistently around 12%, underscoring the importance of Poisson effects. A polynomial fit to the strain-dependent modulus of elasticity enables efficient force prediction with excellent agreement to experimental data. These results advance the mathematical modelling of biological tissues with nonlinear hyperelastic behaviour, providing a foundation for patient-specific simulations and real-time predictive tools in cardiovascular engineering. Full article

(This article belongs to the Special Issue Advanced Mathematical Modeling and Numerical Solutions in Applied Mechanics and Engineering, 2nd Edition)

► Show Figures

Figure 1

28 pages, 10678 KB

Open AccessArticle

Deep-DSO: Improving Mapping of Direct Sparse Odometry Using CNN-Based Single-Image Depth Estimation

by Erick P. Herrera-Granda, Juan C. Torres-Cantero, Israel D. Herrera-Granda, José F. Lucio-Naranjo, Andrés Rosales, Javier Revelo-Fuelagán and Diego H. Peluffo-Ordóñez

Mathematics 2025, 13(20), 3330; https://doi.org/10.3390/math13203330 - 19 Oct 2025

Viewed by 486

Abstract

In recent years, SLAM, visual odometry, and structure-from-motion approaches have widely addressed the problems of 3D reconstruction and ego-motion estimation. Of the many input modalities that can be used to solve these ill-posed problems, the pure visual alternative using a single monocular RGB [...] Read more.

In recent years, SLAM, visual odometry, and structure-from-motion approaches have widely addressed the problems of 3D reconstruction and ego-motion estimation. Of the many input modalities that can be used to solve these ill-posed problems, the pure visual alternative using a single monocular RGB camera has attracted the attention of multiple researchers due to its low cost and widespread availability in handheld devices. One of the best proposals currently available is the Direct Sparse Odometry (DSO) system, which has demonstrated the ability to accurately recover trajectories and depth maps using monocular sequences as the only source of information. Given the impressive advances in single-image depth estimation using neural networks, this work proposes an extension of the DSO system, named DeepDSO. DeepDSO effectively integrates the state-of-the-art NeW CRF neural network as a depth estimation module, providing depth prior information for each candidate point. This reduces the point search interval over the epipolar line. This integration improves the DSO algorithm’s depth point initialization and allows each proposed point to converge faster to its true depth. Experimentation carried out in the TUM-Mono dataset demonstrated that adding the neural network depth estimation module to the DSO pipeline significantly reduced rotation, translation, scale, start-segment alignment, end-segment alignment, and RMSE errors. Full article

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

► Show Figures

Figure 1

17 pages, 10635 KB

Open AccessArticle

Hybrid Convolutional Transformer with Dynamic Prompting for Adaptive Image Restoration

by Jinmei Zhang, Guorong Chen, Junliang Yang, Qingru Zhang, Shaofeng Liu and Weijie Zhang

Mathematics 2025, 13(20), 3329; https://doi.org/10.3390/math13203329 - 19 Oct 2025

Viewed by 281

Abstract

High-quality image restoration (IR) is a fundamental task in computer vision, aiming to recover a clear image from its degraded version. Prevailing methods typically employ a static inference pipeline, neglecting the spatial variability of image content and degradation, which makes it difficult for [...] Read more.

High-quality image restoration (IR) is a fundamental task in computer vision, aiming to recover a clear image from its degraded version. Prevailing methods typically employ a static inference pipeline, neglecting the spatial variability of image content and degradation, which makes it difficult for them to adaptively handle complex and diverse restoration scenarios. To address this issue, we propose a novel adaptive image restoration framework named Hybrid Convolutional Transformer with Dynamic Prompting (HCTDP). Our approach introduces two key architectural innovations: a Spatially Aware Dynamic Prompt Head Attention (SADPHA) module, which performs fine-grained local restoration by generating spatially variant prompts through real-time analysis of image content and a Gated Skip-Connection (GSC) module that refines multi-scale feature flow using efficient channel attention. To guide the network in generating more visually plausible results, the framework is optimized with a hybrid objective function that combines a pixel-wise L1 loss and a feature-level perceptual loss. Extensive experiments on multiple public benchmarks, including image deraining, dehazing, and denoising, demonstrate that our proposed HCTDP exhibits superior performance in both quantitative and qualitative evaluations, validating the effectiveness of the adaptive restoration framework while utilizing fewer parameters than key competitors. Full article

(This article belongs to the Special Issue Intelligent Mathematics and Applications)

► Show Figures

Figure 1

17 pages, 340 KB

Open AccessArticle

O

-Regular Mappings on

C (C)

: A Structured Operator–Theoretic Framework

by Ji Eun Kim

Mathematics 2025, 13(20), 3328; https://doi.org/10.3390/math13203328 - 18 Oct 2025

Viewed by 245

Abstract

Motivation. Analytic function theory on commutative complex extensions calls for an operator–theoretic calculus that simultaneously sees the algebra-induced coupling among components and supports boundary-to-interior mechanisms. Gap. While Dirac-type frameworks are classical in several complex variables and Clifford analysis, a coherent calculus aligning structural [...] Read more.

Motivation. Analytic function theory on commutative complex extensions calls for an operator–theoretic calculus that simultaneously sees the algebra-induced coupling among components and supports boundary-to-interior mechanisms. Gap. While Dirac-type frameworks are classical in several complex variables and Clifford analysis, a coherent calculus aligning structural CR systems, a canonical first derivative, and a Cauchy-type boundary identity on the commutative model

C (C) ≅ C^{4}

has not been systematically developed. Purpose and Aims. This paper develops such a calculus for

O

-regular mappings on

C (C)

and establishes three pillars of the theory. Main Results. (i) A fully coupled Cauchy–Riemann system characterizing

O

-regularity; (ii) identification of a canonical first derivative

g^{'} (z) = \partial_{x_{0}} g (z)

; and (iii) a Stokes-driven boundary annihilation law

\int_{\partial Ω} τ g = 0

for a canonical 7-form

τ

. On (pseudo)convex domains,

\bar{\partial}

-methods yield solvability under natural compatibility and regularity assumptions. Stability (under algebra-preserving maps), Liouville-type, and removability results are also obtained, and function spaces suited to this algebra are outlined. Significance. The results show that a large portion of the classical holomorphic toolkit survives, in algebra-aware form, on

C (C)

. Full article

(This article belongs to the Special Issue Clifford Analysis: Theory, Methods, and Multidisciplinary Applications)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Mathematics, Volume 13, Issue 20 (October-2 2025) – 119 articles

Further Information

Guidelines

MDPI Initiatives

Follow MDPI