Mathematics

24 pages, 407 KB

Open AccessArticle

New Insights into the Multifractal Formalism of Branching Random Walks on Galton–Watson Tree

by Najmeddine Attia

Mathematics 2025, 13(17), 2904; https://doi.org/10.3390/math13172904 - 8 Sep 2025

In the present work, we consider three branching random walk

S_{n} Z (t),

Z \in {X, Y, Φ}

on a supercritical random Galton–Watson tree

\partial T

. We compute the Hausdorff and packing dimensions of [...] Read more.

In the present work, we consider three branching random walk

S_{n} Z (t),

Z \in {X, Y, Φ}

on a supercritical random Galton–Watson tree

\partial T

. We compute the Hausdorff and packing dimensions of the level set

E_{χ} (α, β) = \{t \in \partial T : \lim_{n \to \infty} \frac{S_{n} X (t)}{S_{n} Y (t)} = α and \lim_{n \to \infty} \frac{S_{n} Y (t)}{n} = β\},

where

\partial T

is endowed with random metric using

S_{n} Φ (t)

. This is achieved by constructing a suitable Mandelbrot measure supported on

E (α, β)

. In the case where

Φ = 1

, we develop a formalism that parallels Olsen’s framework (for measures) and Peyrière’s framework (for the vectorial case) within our setting. Full article

20 pages, 4920 KB

Open AccessArticle

A Complete Neural Network-Based Representation of High-Dimension Convolutional Neural Networks

by Ray-Ming Chen

Mathematics 2025, 13(17), 2903; https://doi.org/10.3390/math13172903 - 8 Sep 2025

Abstract

Convolutional Neural Networks (CNNs) are a highly used machine learning architecture in various fields. Typical descriptions of CNNs are based on low-dimension and tensor representations in the feature extraction part. In this article, we extend the setting of CNNs to any arbitrary dimension [...] Read more.

Convolutional Neural Networks (CNNs) are a highly used machine learning architecture in various fields. Typical descriptions of CNNs are based on low-dimension and tensor representations in the feature extraction part. In this article, we extend the setting of CNNs to any arbitrary dimension and linearize the whole setting via the typical layers of neurons. In essence, a partial and a full network construct the entire process of a standard CNN, with the partial network being used to linearize the feature extraction. By doing so, we link the tensor-style representation of CNNs with the pure network representation. The outcomes serve two main purposes: to relate CNNs with other machine learning frameworks and to facilitate intuitive representations. Full article

(This article belongs to the Special Issue Advanced Research in Neural Networks, Machine Learning, and Image Processing)

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27 pages, 2240 KB

Open AccessFeature PaperArticle

Hybrid Entropy-Based Metrics for k-Hop Environment Analysis in Complex Networks

by Csaba Biró

Mathematics 2025, 13(17), 2902; https://doi.org/10.3390/math13172902 - 8 Sep 2025

Abstract

Two hybrid, entropy-guided node metrics are proposed for the k-hop environment: Entropy-Weighted Redundancy (EWR) and Normalized Entropy Density (NED). The central idea is to couple local Shannon entropy with neighborhood density/redundancy so that structural heterogeneity around a vertex is captured even when [...] Read more.

Two hybrid, entropy-guided node metrics are proposed for the k-hop environment: Entropy-Weighted Redundancy (EWR) and Normalized Entropy Density (NED). The central idea is to couple local Shannon entropy with neighborhood density/redundancy so that structural heterogeneity around a vertex is captured even when classical indices (e.g., degree or clustering) are similar. The metrics are formally defined and shown to be bounded, isomorphism-invariant, and stable under small edge edits. Their behavior is assessed on representative topologies (Erdős–Rényi, Barabási–Albert, Watts–Strogatz, random geometric graphs, and the Zephyr quantum architecture). Across these settings, EWR and NED display predominantly negative correlation with degree and provide information largely orthogonal to standard centralities; vertices with identical degree can differ by factors of two to three in the proposed scores, revealing bridges and heterogeneous regions. These properties indicate utility for vulnerability assessment, topology-aware optimization, and layout heuristics in engineered and quantum networks. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

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13 pages, 1855 KB

Open AccessArticle

Synchronous Stability in Multiplex Network Subject to Higher-Order Intralayer Interactions

by Junqing Feng and Lixin Yang

Mathematics 2025, 13(17), 2901; https://doi.org/10.3390/math13172901 - 8 Sep 2025

Abstract

Recent research and instances have demonstrated that most real-world systems can be effectively schematized by multiplex networks. Moreover, the interactions within systems often emerge among triadic or tetradic interactions, or even interactions with more element combinations, in addition to pairwise interactions. Hypergraph coupling [...] Read more.

Recent research and instances have demonstrated that most real-world systems can be effectively schematized by multiplex networks. Moreover, the interactions within systems often emerge among triadic or tetradic interactions, or even interactions with more element combinations, in addition to pairwise interactions. Hypergraph coupling structures are particularly well-suited for capturing such arbitrary higher-order interactions among nodes, thereby playing a key role in accurately depicting system dynamics. Meanwhile, the functionality of numerous complex systems depends on synchronization mechanisms. Therefore, this paper focuses on investigating the synchronous stability of a multiplex hypergraph. Specifically, we examine a three-layer network where intralayer interactions are represented by hyperedges, while the interlayer interactions are modeled through pairwise couplings. By generalizing the master stability function approach to the hypergraph structure, the synchronization phenomenon of such multiplex hypergraphs is analyzed. To verify our theoretical conclusions, we apply the proposed framework to networks of FitzHugh–Nagumo neurons and Rikitake two-disk dynamos. Simulation results unveil that the presence of higher-order interactions enhances the synchronous ability within the multiplex framework. Full article

(This article belongs to the Special Issue Dynamic Complex Networks: Models, Algorithms, and Applications)

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65 pages, 12354 KB

Open AccessReview

A Comprehensive Review of MPPT Strategies for Hybrid PV–TEG Systems: Advances, Challenges, and Future Directions

by AL-Wesabi Ibrahim, Hassan M. Hussein Farh and Abdullrahman A. Al-Shamma’a

Mathematics 2025, 13(17), 2900; https://doi.org/10.3390/math13172900 - 8 Sep 2025

Abstract

The pressing global transition to sustainable energy has intensified interest in overcoming the efficiency bottlenecks of conventional solar technologies. Hybrid photovoltaic–thermoelectric generator (PV–TEG) systems have recently emerged as a compelling solution, synergistically harvesting both electrical and thermal energy from solar radiation. By converting [...] Read more.

The pressing global transition to sustainable energy has intensified interest in overcoming the efficiency bottlenecks of conventional solar technologies. Hybrid photovoltaic–thermoelectric generator (PV–TEG) systems have recently emerged as a compelling solution, synergistically harvesting both electrical and thermal energy from solar radiation. By converting both sunlight and otherwise wasted heat, these integrated systems can substantially enhance total energy yield and overall conversion efficiency—mitigating the performance limitations of standalone PV panels. This review delivers a comprehensive, systematic assessment of maximum-power-point tracking (MPPT) methodologies specifically tailored for hybrid PV–TEG architectures. MPPT techniques are meticulously categorized and critically analyzed within the following six distinct groups: conventional algorithms, metaheuristic approaches, artificial intelligence (AI)-driven methods, mathematical models, hybrid strategies, and novel emerging solutions. For each category, we examine operational principles, implementation complexity, and adaptability to real-world phenomena such as partial shading and non-uniform temperature distribution. Through thorough comparative evaluation, the review uncovers existing research gaps, highlights ongoing challenges, and identifies promising directions for technological advancement. This work equips researchers and practitioners with an integrated knowledge base, fostering informed development and deployment of next-generation MPPT solutions for high-performance hybrid solar–thermal energy systems. Full article

(This article belongs to the Special Issue Artificial Intelligence and Optimization in Engineering Applications)

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15 pages, 4930 KB

Open AccessArticle

A Lightweight Hybrid CNN-ViT Network for Weed Recognition in Paddy Fields

by Tonglai Liu, Yixuan Wang, Chengcheng Yang, Youliu Zhang and Wanzhen Zhang

Mathematics 2025, 13(17), 2899; https://doi.org/10.3390/math13172899 - 8 Sep 2025

Abstract

Accurate identification of weed species is a fundamental task for promoting efficient farmland management. Existing recognition approaches are typically based on either conventional Convolutional Neural Networks (CNNs) or the more recent Vision Transformers (ViTs). CNNs demonstrate strong capability in capturing local spatial patterns, [...] Read more.

Accurate identification of weed species is a fundamental task for promoting efficient farmland management. Existing recognition approaches are typically based on either conventional Convolutional Neural Networks (CNNs) or the more recent Vision Transformers (ViTs). CNNs demonstrate strong capability in capturing local spatial patterns, yet they are often limited in modeling long-range dependencies. In contrast, ViTs can effectively capture global contextual information through self-attention, but they may neglect fine-grained local features. These inherent shortcomings restrict the recognition performance of current models. To overcome these limitations, we propose a lightweight hybrid architecture, termed RepEfficientViT,which integrates convolutional operations with Transformer-based self-attention. This design enables the simultaneous aggregation of both local details and global dependencies. Furthermore, we employ a structural re-parameterization strategy to enhance the representational capacity of convolutional layers without introducing additional parameters or computational overhead. Experimental evaluations reveal that RepEfficientViT consistently surpasses state-of-the-art CNN and Transformer baselines. Specifically, the model achieves an accuracy of 94.77%, a precision of 94.75%, a recall of 94.93%, and an F1-score of 94.84%. In terms of efficiency, RepEfficientViT requires only 223.54 M FLOPs and 1.34 M parameters, while attaining an inference latency of merely 25.13 ms on CPU devices. These results demonstrate that the proposed model is well-suited for deployment in edge-computing scenarios subject to stringent computational and storage constraints. Full article

(This article belongs to the Special Issue Computational Intelligence, Computer Vision and Pattern Recognition)

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49 pages, 934 KB

Open AccessArticle

Analysis and Mean-Field Limit of a Hybrid PDE-ABM Modeling Angiogenesis-Regulated Resistance Evolution

by Louis Shuo Wang, Jiguang Yu, Shijia Li and Zonghao Liu

Mathematics 2025, 13(17), 2898; https://doi.org/10.3390/math13172898 - 8 Sep 2025

Abstract

Mathematical modeling is indispensable in oncology for unraveling the interplay between tumor growth, vascular remodeling, and therapeutic resistance. We present a hybrid modeling framework (continuum-discrete) and present its hybrid mathematical formulation as a coupled partial differential equation–agent-based (PDE-ABM) system. It couples reaction–diffusion fields [...] Read more.

Mathematical modeling is indispensable in oncology for unraveling the interplay between tumor growth, vascular remodeling, and therapeutic resistance. We present a hybrid modeling framework (continuum-discrete) and present its hybrid mathematical formulation as a coupled partial differential equation–agent-based (PDE-ABM) system. It couples reaction–diffusion fields for oxygen, drug, and tumor angiogenic factor (TAF) with discrete vessel agents and stochastic phenotype transitions in tumor cells. Stochastic phenotype switching is handled with an exact Gillespie algorithm (a Monte Carlo method that simulates random phenotype flips and their timing), while moment-closure methods (techniques that approximate higher-order statistical moments to obtain a closed, tractable PDE description) are used to derive mean-field PDE limits that connect microscale randomness to macroscopic dynamics. We provide existence/uniqueness results for the coupled PDE-ABM system, perform numerical analysis of discretization schemes, and derive analytically tractable continuum limits. By linking stochastic microdynamics and deterministic macrodynamics, this hybrid mathematical formulation—i.e., the coupled PDE-ABM system—captures bidirectional feedback between hypoxia-driven angiogenesis and resistance evolution and provides a rigorous foundation for predictive, multiscale oncology models. Full article

(This article belongs to the Special Issue Applied Mathematical Modeling in Oncology)

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15 pages, 240 KB

Open AccessFeature PaperArticle

The First Zagreb Index, the Laplacian Spectral Radius, and Some Hamiltonian Properties of Graphs

by Rao Li

Mathematics 2025, 13(17), 2897; https://doi.org/10.3390/math13172897 - 8 Sep 2025

Abstract

The first Zagreb index of a graph G is defined as the sum of the squares of the degrees of all the vertices in G. The Laplacian spectral radius of a graph G is defined as the largest eigenvalue of the Laplacian [...] Read more.

The first Zagreb index of a graph G is defined as the sum of the squares of the degrees of all the vertices in G. The Laplacian spectral radius of a graph G is defined as the largest eigenvalue of the Laplacian matrix of the graph G. In this paper, we first establish inequalities on the first Zagreb index and the Laplacian spectral radius of a graph. Using the ideas of proving the inequalities, we present sufficient conditions involving the first Zagreb index and the Laplacian spectral radius for some Hamiltonian properties of graphs. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

24 pages, 643 KB

Open AccessArticle

Development of Viscosity Iterative Techniques for Split Variational-like Inequalities and Fixed Points Related to Pseudo-Contractions

by Ghada AlNemer, Mohammad Farid and Rehan Ali

Mathematics 2025, 13(17), 2896; https://doi.org/10.3390/math13172896 - 8 Sep 2025

Abstract

This work presents an extragradient-type iterative process combined with the viscosity method to find a common solution to a split generalized variational-like inequality, a variational inequality, and a fixed point problem associated with a family of

ε

-strict pseudo-contractive mappings and a nonexpansive [...] Read more.

This work presents an extragradient-type iterative process combined with the viscosity method to find a common solution to a split generalized variational-like inequality, a variational inequality, and a fixed point problem associated with a family of

ε

-strict pseudo-contractive mappings and a nonexpansive operator in Hilbert spaces. Strong convergence of the proposed algorithm is established, with some remarks derived from the main theorem. Numerical experiments are carried out to verify the applicability of the method and provide comparative observations. The results broaden and unify a range of existing contributions in this field. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications: 2nd Edition)

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19 pages, 1407 KB

Open AccessArticle

Eigenvector Distance-Modulated Graph Neural Network: Spectral Weighting for Enhanced Node Classification

by Ahmed Begga, Francisco Escolano and Miguel Ángel Lozano

Mathematics 2025, 13(17), 2895; https://doi.org/10.3390/math13172895 - 8 Sep 2025

Abstract

Graph Neural Networks (GNNs) face significant challenges in node classification across diverse graph structures. Traditional message passing mechanisms often fail to adaptively weight node relationships, thereby limiting performance in both homophilic and heterophilic graph settings. We propose the Eigenvector Distance-Modulated Graph Neural Network [...] Read more.

Graph Neural Networks (GNNs) face significant challenges in node classification across diverse graph structures. Traditional message passing mechanisms often fail to adaptively weight node relationships, thereby limiting performance in both homophilic and heterophilic graph settings. We propose the Eigenvector Distance-Modulated Graph Neural Network (EDM-GNN), which enhances message passing by incorporating spectral information from the graph’s eigenvectors. Our method introduces a novel weighting scheme that modulates information flow based on a combined similarity measure. This measure balances feature-based similarity with structural similarity derived from eigenvector distances. This approach creates a more discriminative aggregation process that adapts to the underlying graph topology. It does not require prior knowledge of homophily characteristics. We implement a hierarchical neighborhood aggregation framework that utilizes these spectral weights across multiple powers of the adjacency matrix. Experimental results on benchmark datasets demonstrate that EDM-GNN achieves competitive performance with state-of-the-art methods across both homophilic and heterophilic settings. Our approach provides a unified solution for node classification problems with strong theoretical foundations in spectral graph theory and significant empirical improvements in classification accuracy. Full article

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13 pages, 473 KB

Open AccessArticle

Multi-Objective Batch Energy-Entropy Acquisition Function for Bayesian Optimization

by Hangyu Zhu and Xilu Wang

Mathematics 2025, 13(17), 2894; https://doi.org/10.3390/math13172894 - 8 Sep 2025

Viewed by 66

Abstract

Bayesian Optimization (BO) provides an efficient framework for optimizing expensive black-box functions by employing a surrogate model (typically a Gaussian Process) to approximate the objective function and an acquisition function to guide the search for optimal points. Batch BO extends this paradigm by [...] Read more.

Bayesian Optimization (BO) provides an efficient framework for optimizing expensive black-box functions by employing a surrogate model (typically a Gaussian Process) to approximate the objective function and an acquisition function to guide the search for optimal points. Batch BO extends this paradigm by selecting and evaluating multiple candidate points simultaneously, which improves computational efficiency but introduces challenges in optimizing the resulting high-dimensional acquisition functions. Among existing acquisition functions for batch Bayesian Optimization, entropy-based methods are considered to be state-of-the-art methods due to their ability to enable more globally efficient while avoiding redundant evaluations. However, they often fail to fully capture the dependencies and interactions among the selected batch points. In this work, we propose a Multi-Objective Batch Energy–Entropy acquisition function for Bayesian Optimization (MOBEEBO), which adaptively exploits the correlations among batch points. In addition, MOBEEBO incorporates multiple types of acquisition functions as objectives in a unified framework to achieve more effective batch diversity and quality. Empirical results demonstrate that the proposed algorithm is applicable to a wide range of optimization problems and achieves competitive performance. Full article

(This article belongs to the Special Issue Multi-Objective Optimizations and Their Applications)

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14 pages, 1313 KB

Open AccessArticle

A Fast and Privacy-Preserving Outsourced Approach for K-Means Clustering Based on Symmetric Homomorphic Encryption

by Wanqi Tang and Shiwei Xu

Mathematics 2025, 13(17), 2893; https://doi.org/10.3390/math13172893 - 8 Sep 2025

Abstract

Training a machine learning (ML) model always needs many computing resources, and cloud-based outsourced training is a good solution to address the issue of a computing resources shortage. However, the cloud may be untrustworthy, and it may pose a privacy threat to the [...] Read more.

Training a machine learning (ML) model always needs many computing resources, and cloud-based outsourced training is a good solution to address the issue of a computing resources shortage. However, the cloud may be untrustworthy, and it may pose a privacy threat to the training process. Currently, most work makes use of multi-party computation protocols and lattice-based homomorphic encryption algorithms to solve the privacy problem, but these tools are inefficient in communication or computation. Therefore, in this paper, we focus on the k-means and propose a fast and privacy-preserving method for outsourced clustering of k-means models based on symmetric homomorphic encryption (SHE), which is used to encrypt the clustering dataset and model parameters in our scheme. We design an interactive protocol and use various tools to optimize the protocol time overheads. We perform security analysis and detailed evaluation on the performance of our scheme, and the experimental results show that our scheme has better prediction accuracy, as well as lower computation and total overheads. Full article

(This article belongs to the Special Issue Advanced Neural Network and Machine Learning Algorithms, Models and Architectures in Data Mining)

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1 pages, 162 KB

Open AccessCorrection

Correction: Fanchi, J.R. Probabilistic Basis of Parametrized Relativistic Quantum Theory in Curved Spacetime. Mathematics 2025, 13, 1657

by John R. Fanchi

Mathematics 2025, 13(17), 2892; https://doi.org/10.3390/math13172892 - 8 Sep 2025

Viewed by 68

Abstract

The term with

Γ_{ν μ}^{μ}

was omitted in the version of Ref [...] Full article

(This article belongs to the Section E4: Mathematical Physics)

14 pages, 303 KB

Open AccessArticle

Luxemburg Norm Characterizations of BLO Spaces in General Metric Measure Frameworks

by Liping Yang and Xin Jiang

Mathematics 2025, 13(17), 2891; https://doi.org/10.3390/math13172891 - 7 Sep 2025

Viewed by 310

Abstract

This study provides new equivalent descriptions of the Bounded Lower Oscillation (

BLO

) space through Luxemburg-type

L^{φ}

integrability conditions, where

φ

is a nonnegative function with either convexity or concavity. The framework accommodates various representative forms of

φ

, such as [...] Read more.

This study provides new equivalent descriptions of the Bounded Lower Oscillation (

BLO

) space through Luxemburg-type

L^{φ}

integrability conditions, where

φ

is a nonnegative function with either convexity or concavity. The framework accommodates various representative forms of

φ

, such as the power function

φ (t) = t^{p}

, exponential-type functions

φ (t) = e^{p t} - 1

, and logarithmic functions

φ (t) = {log}_{+}^{k} t

, with parameters

p \in (0, \infty)

and

k \in N

. These results unify and extend existing characterizations of

BLO

by encompassing a broad class of generating functions. Full article

20 pages, 302 KB

Open AccessFeature PaperArticle

A Unified Approach to Implicit Fractional Differential Equations with Anti-Periodic Boundary Conditions

by Ricardo Almeida

Mathematics 2025, 13(17), 2890; https://doi.org/10.3390/math13172890 - 7 Sep 2025

Viewed by 204

Abstract

This paper develops a unified analytical framework for implicit fractional differential equations subject to anti-periodic boundary conditions. The study considers two main cases: fractional derivatives of order

α \in (0, 1)

and

α \in (1, 2)

, [...] Read more.

This paper develops a unified analytical framework for implicit fractional differential equations subject to anti-periodic boundary conditions. The study considers two main cases: fractional derivatives of order

α \in (0, 1)

and

α \in (1, 2)

, both defined with respect to a general kernel function. The existence and uniqueness of solutions are established using Banach’s and Schaefer’s fixed-point theorems under suitable Lipschitz conditions. Furthermore, Ulam–Hyers stability and generalized Ulam–Hyers stability are investigated for each problem. Examples are provided to illustrate the applicability of the main results. Full article

(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)

17 pages, 1078 KB

Open AccessArticle

Prototype-Based Two-Stage Few-Shot Instance Segmentation with Flexible Novel Class Adaptation

by Qinying Zhu, Yilin Zhang, Peng Xiao, Mengxi Ying, Lei Zhu and Chengyuan Zhang

Mathematics 2025, 13(17), 2889; https://doi.org/10.3390/math13172889 - 7 Sep 2025

Viewed by 459

Abstract

Few-shot instance segmentation (FSIS) is devised to address the intricate challenge of instance segmentation when labeled data for novel classes is scant. Nevertheless, existing methodologies encounter notable constraints in the agile expansion of novel classes and the management of memory overhead. The integration [...] Read more.

Few-shot instance segmentation (FSIS) is devised to address the intricate challenge of instance segmentation when labeled data for novel classes is scant. Nevertheless, existing methodologies encounter notable constraints in the agile expansion of novel classes and the management of memory overhead. The integration workflow for novel classes is inflexible, and given the necessity of retaining class exemplars during both training and inference stages, considerable memory consumption ensues. To surmount these challenges, this study introduces an innovative framework encompassing a two-stage “base training-novel class fine-tuning” paradigm. It acquires discriminative instance-level embedding representations. Concretely, instance embeddings are aggregated into class prototypes, and the storage of embedding vectors as opposed to images inherently mitigates the issue of memory overload. Via a Region of Interest (RoI)-level cosine similarity matching mechanism, the flexible augmentation of novel classes is realized, devoid of the requirement for supplementary training and independent of historical data. Experimental validations attest that this approach significantly outperforms state-of-the-art techniques in mainstream benchmark evaluations. More crucially, its memory-optimized attributes facilitate, for the first time, the conjoint assessment of FSIS performance across all classes within the COCO dataset. Visualized instances (incorporating colored masks and class annotations of objects across diverse scenarios) further substantiate the efficacy of the method in real-world complex contexts. Full article

(This article belongs to the Special Issue Structural Networks for Image Application)

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50 pages, 5419 KB

Open AccessArticle

MSAPO: A Multi-Strategy Fusion Artificial Protozoa Optimizer for Solving Real-World Problems

by Hanyu Bo, Jiajia Wu and Gang Hu

Mathematics 2025, 13(17), 2888; https://doi.org/10.3390/math13172888 - 6 Sep 2025

Viewed by 215

Abstract

Artificial protozoa optimizer (APO), as a newly proposed meta-heuristic algorithm, is inspired by the foraging, dormancy, and reproduction behaviors of protozoa in nature. Compared with traditional optimization algorithms, APO demonstrates strong competitive advantages; nevertheless, it is not without inherent limitations, such as slow [...] Read more.

Artificial protozoa optimizer (APO), as a newly proposed meta-heuristic algorithm, is inspired by the foraging, dormancy, and reproduction behaviors of protozoa in nature. Compared with traditional optimization algorithms, APO demonstrates strong competitive advantages; nevertheless, it is not without inherent limitations, such as slow convergence and a proclivity towards local optimization. In order to enhance the efficacy of the algorithm, this paper puts forth a multi-strategy fusion artificial protozoa optimizer, referred to as MSAPO. In the initialization stage, MSAPO employs the piecewise chaotic opposition-based learning strategy, which results in a uniform population distribution, circumvents initialization bias, and enhances the global exploration capability of the algorithm. Subsequently, cyclone foraging strategy is implemented during the heterotrophic foraging phase. enabling the algorithm to identify the optimal search direction with greater precision, guided by the globally optimal individuals. This reduces random wandering, significantly accelerating the optimization search and enhancing the ability to jump out of the local optimal solutions. Furthermore, the incorporation of hybrid mutation strategy in the reproduction stage enables the algorithm to adaptively transform the mutation patterns during the iteration process, facilitating a strategic balance between rapid escape from local optima in the initial stages and precise convergence in the subsequent stages. Ultimately, crisscross strategy is incorporated at the conclusion of the algorithm’s iteration. This not only enhances the algorithm’s global search capacity but also augments its capability to circumvent local optima through the integrated application of horizontal and vertical crossover techniques. This paper presents a comparative analysis of MSAPO with other prominent optimization algorithms on the three-dimensional CEC2017 and the highest-dimensional CEC2022 test sets, and the results of numerical experiments show that MSAPO outperforms the compared algorithms, and ranks first in the performance evaluation in a comprehensive way. In addition, in eight real-world engineering design problem experiments, MSAPO almost always achieves the theoretical optimal value, which fully confirms its high efficiency and applicability, thus verifying the great potential of MSAPO in solving complex optimization problems. Full article

(This article belongs to the Special Issue Advances in Metaheuristic Optimization Algorithms)

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15 pages, 851 KB

Open AccessArticle

Third-Order Hankel Determinant for a Class of Bi-Univalent Functions Associated with Sine Function

by Mohammad El-Ityan, Mustafa A. Sabri, Suha Hammad, Basem Frasin, Tariq Al-Hawary and Feras Yousef

Mathematics 2025, 13(17), 2887; https://doi.org/10.3390/math13172887 - 6 Sep 2025

Viewed by 192

Abstract

This paper investigates a new subclass of bi-univalent analytic functions defined on the open unit disk in the complex plane, associated with the subordination to

1 + s i n z

. Coefficient bounds are obtained for the initial Taylor–Maclaurin coefficients, with a [...] Read more.

This paper investigates a new subclass of bi-univalent analytic functions defined on the open unit disk in the complex plane, associated with the subordination to

1 + s i n z

. Coefficient bounds are obtained for the initial Taylor–Maclaurin coefficients, with a particular focus on the second- and third-order Hankel determinants. To illustrate the non-emptiness of the proposed class, we consider the function

\sqrt{1 + tanh z}

, which maps the unit disk onto a bean-shaped domain. This function satisfies the required subordination condition and hence serves as an explicit member of the class. A graphical depiction of the image domain is provided to highlight its geometric characteristics. The results obtained in this work confirm that the class under study is non-trivial and possesses rich geometric structure, making it suitable for further development in the theory of geometric function classes and coefficient estimation problems. Full article

(This article belongs to the Special Issue New Trends in Polynomials and Mathematical Analysis)

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20 pages, 910 KB

Open AccessArticle

The Instability in the Dimensions of Polynomial Splines of Mixed Smoothness over T-Meshes

by Pengxiao Wang

Mathematics 2025, 13(17), 2886; https://doi.org/10.3390/math13172886 - 6 Sep 2025

Viewed by 193

Abstract

Mixed-smoothness splines facilitate localized control over smoothness; however, the issue of dimensional instability in mixed-smoothness spline spaces remains unstudied in the existing literature. This paper studies such instabilities over T-meshes, where different orders of smoothness are required across interior mesh segments. Using the [...] Read more.

Mixed-smoothness splines facilitate localized control over smoothness; however, the issue of dimensional instability in mixed-smoothness spline spaces remains unstudied in the existing literature. This paper studies such instabilities over T-meshes, where different orders of smoothness are required across interior mesh segments. Using the smoothing cofactor-conformality method, we introduce a constraint on T-meshes to derive a stable dimension formula for mixed-smoothness spline spaces. Furthermore, we show dimensional instability in cases involving T-cycles and nested T-cycles. By defining a singularity factor for each T-cycle, we demonstrate that both dimensional instabilities and structural degenerations are associated with these singularity factors. The work contributes to a deeper understanding of spline spaces defined over non-tensor-product structures. Full article

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26 pages, 4813 KB

Open AccessArticle

Nonlinear Dynamics Analysis of the Wheel-Side Planetary Reducer with Tooth Wear for the In-Wheel Motored Electric Vehicle

by Dehua Shi, Le Sun, Qirui Zhang, Shaohua Wang, Kaimei Zhang, Chunfang Yin and Chun Li

Mathematics 2025, 13(17), 2885; https://doi.org/10.3390/math13172885 - 6 Sep 2025

Viewed by 226

Abstract

This paper investigates the nonlinear dynamics of the wheel-side planetary reducer, considering the tooth wear effect. The tooth wear model based on the Archard adhesion wear theory is established, and the impact of tooth wear on meshing stiffness and piecewise-linear backlash of the [...] Read more.

This paper investigates the nonlinear dynamics of the wheel-side planetary reducer, considering the tooth wear effect. The tooth wear model based on the Archard adhesion wear theory is established, and the impact of tooth wear on meshing stiffness and piecewise-linear backlash of the planetary gear system is discussed. Then, the torsional vibration model and dimensionless differential equations considering tooth wear for the wheel-side planetary reducer are established, in which meshing excitations include time-varying mesh stiffness (TVMS), piecewise-linear backlash, and transmission error. The dynamic responses are numerically solved using the fourth-order Runge–Kutta method. On this basis, the nonlinear dynamics, such as the bifurcation and chaos properties of the wheel-side planetary reducer with tooth wear, are analyzed. Simulation results demonstrate that the existence of tooth wear reduces meshing stiffness and increases backlash. The reduction in the meshing stiffness changes the bifurcation path and chaotic amplitude of the system, inducing chaotic phenomena more easily. The increase in the gear backlash causes a higher amplitude of the relative displacement and more severe vibration. Full article

(This article belongs to the Section C2: Dynamical Systems)

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16 pages, 2791 KB

Open AccessArticle

Adaptive Penalized Regression for High-Efficiency Estimation in Correlated Predictor Settings: A Data-Driven Shrinkage Approach

by Muhammad Shakir Khan and Amirah Saeed Alharthi

Mathematics 2025, 13(17), 2884; https://doi.org/10.3390/math13172884 - 6 Sep 2025

Viewed by 348

Abstract

Penalized regression estimators have become widely adopted alternatives to ordinary least squares while analyzing collinear data, despite introducing some bias. However, existing penalized methods lack universal superiority across diverse data conditions. To address this limitation, we propose a novel adaptive ridge estimator that [...] Read more.

Penalized regression estimators have become widely adopted alternatives to ordinary least squares while analyzing collinear data, despite introducing some bias. However, existing penalized methods lack universal superiority across diverse data conditions. To address this limitation, we propose a novel adaptive ridge estimator that automatically adjusts its penalty structure based on key data characteristics: (1) the degree of predictor collinearity, (2) error variance, and (3) model dimensionality. Through comprehensive Monte Carlo simulations and real-world applications, we evaluate the estimator’s performance using mean squared error (MSE) as our primary criterion. Our results demonstrate that the proposed method consistently outperforms existing approaches across all considered scenarios, with particularly strong performance in challenging high-collinearity settings. The real-data applications further confirm the estimator’s practical utility and robustness. Full article

(This article belongs to the Special Issue Statistical Machine Learning: Models and Its Applications)

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18 pages, 355 KB

Open AccessArticle

Splitting-Based Regenerations for Accelerated Simulation of Queues

by Irina Peshkova, Evsey Morozov and Michele Pagano

Mathematics 2025, 13(17), 2883; https://doi.org/10.3390/math13172883 - 6 Sep 2025

Viewed by 317

Abstract

In this paper, we address the problem of increasing the number of regenerations in the simulation of the workload process in a single-server queueing system. To this end, we extend the splitting technique developed for the Markov workload process in the M/M/1 queue [...] Read more.

In this paper, we address the problem of increasing the number of regenerations in the simulation of the workload process in a single-server queueing system. To this end, we extend the splitting technique developed for the Markov workload process in the M/M/1 queue to the more general GI/M/1 queueing systems. This approach is based on a minorization condition for the transition kernel of the workload process, which is a Markov chain defined by the Lindley recursion. The proposed method increases the number of regenerations during the simulation and potentially reduces the time required to estimate stationary performance metrics with a given level of precision. Full article

(This article belongs to the Special Issue Recent Research in Queuing Theory and Stochastic Models, 2nd Edition)

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20 pages, 3410 KB

Open AccessArticle

Impact of Polar Ice Layers on the Corrosion-Related Static Electric Field of a Submerged Underwater Vehicle

by Mingjie Qiu, Mingyong Hu, Yuhong Li, Dingfeng Yu and Cong Chen

Mathematics 2025, 13(17), 2882; https://doi.org/10.3390/math13172882 - 6 Sep 2025

Viewed by 386

Abstract

The influence of polar ice-covered environments on the corrosion-related static electric field (CRSE) of underwater vehicles is critical for understanding and applying the characteristics of underwater electric fields in polar regions. This study utilizes a combined methodology involving COMSOL Multiphysics 6.1 simulations and [...] Read more.

The influence of polar ice-covered environments on the corrosion-related static electric field (CRSE) of underwater vehicles is critical for understanding and applying the characteristics of underwater electric fields in polar regions. This study utilizes a combined methodology involving COMSOL Multiphysics 6.1 simulations and laboratory-simulated experiments to systematically investigate the distribution characteristics of underwater vehicle electric fields under ice-covered conditions. By comparing the electric field distributions in scenarios with and without ice coverage, this study clarifies the effect of ice presence on the behavior of underwater electric fields. The simulation results demonstrate that the existence of ice layers enhances both the electric potential and field strength, with the degree of influence depending on the ice layer conductivity, thickness, and proximity of the measurement points to the ice layer. The accumulation of error analysis and laboratory experiments corroborates the reliability of the simulation results, demonstrating that ice layers enhance electric field signals by modifying the conductive properties of the surrounding medium, whereas the overall spatial distribution characteristics remain largely consistent. These findings offer a theoretical and technical basis for the optimization of stealth strategies in polar underwater vehicles and contribute to the advancement of electric field detection technologies. Full article

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18 pages, 1084 KB

Open AccessArticle

Tractor and Semitrailer Scheduling with Time Windows in Highway Ports with Unbalanced Demand Under Network Conditions

by Hongxia Guo, Fengjun Wang, Yuyan He and Yuyang Zhou

Mathematics 2025, 13(17), 2881; https://doi.org/10.3390/math13172881 - 6 Sep 2025

Viewed by 454

Abstract

To address the challenges of unbalanced demand and high operational costs in highway port logistics, this study investigates the scheduling of tractors and semitrailers under time window constraints in a networked environment, where geographically distributed ports are interconnected by fixed routes, and tractors [...] Read more.

To address the challenges of unbalanced demand and high operational costs in highway port logistics, this study investigates the scheduling of tractors and semitrailers under time window constraints in a networked environment, where geographically distributed ports are interconnected by fixed routes, and tractors dynamically transport semitrailers between ports to balance asymmetric demands. A mathematical optimization model is developed, incorporating multiple car yards, diverse transport demands, and temporal constraints. To solve the model efficiently, an Adaptive Large Neighborhood Search (ALNS) algorithm is proposed and benchmarked against an improved Ant Colony System (IACS). Simulation results show that, compared to traditional scheduling methods, the proposed approach reduces the number of required tractors by up to 61% and operational costs by up to 21%, depending on tractor working hours. The tractor-to-semitrailer ratio improves from 1.00:1.10 to 1.00:2.59, demonstrating the enhanced resource utilization enabled by the ALNS algorithm. These findings offer practical guidance for optimizing tractor and semitrailer configurations in highway port operations under varying conditions. Full article

(This article belongs to the Special Issue AI-Based and Data-Driven Modeling and Control: Mathematical Methods and Industrial Applications)

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52 pages, 3053 KB

Open AccessArticle

Orthonormal Right-Handed Frames on the Two-Sphere and Solutions to Maxwell’s Equations via de Broglie Waves

by David Carfì

Mathematics 2025, 13(17), 2880; https://doi.org/10.3390/math13172880 - 5 Sep 2025

Viewed by 276

Abstract

This paper explores some frame bundles and physical implications of Killing vector fields on the two-sphere

S^{2}

, culminating in a novel application to Maxwell’s equations in free space. Initially, we investigate the Killing vector fields on

S^{2}

(represented by the [...] Read more.

This paper explores some frame bundles and physical implications of Killing vector fields on the two-sphere

S^{2}

, culminating in a novel application to Maxwell’s equations in free space. Initially, we investigate the Killing vector fields on

S^{2}

(represented by the unit sphere of

R^{3}

), which generate the isometries of the sphere under the rotation group

S O (3)

. These fields, realized as functions

K_{v} : S^{2} \to R^{3}

, defined by

K_{v} (q) = v \times q

for a fixed

v \in R^{3}

and any

q \in S^{2}

, generate a three-dimensional Lie algebra isomorphic to

so (3)

. We establish an isomorphism

K : R^{3} \to K (S^{2})

, mapping vectors

v = a u

(with

u \in S^{2}

) to scaled Killing vector fields

a K_{u}

, and analyze its relationship with

S O (3)

through the exponential map. Subsequently, at a fixed point

e \in S^{2}

, we construct a smooth orthonormal right-handed tangent frame

f_{e} : S^{2} \ {e, - e} \to T {(S^{2})}^{2}

, defined as

f_{e} (u) = ({\hat{K}}_{e} (u), u \times {\hat{K}}_{e} (u))

, where

{\hat{K}}_{e}

is the unit vector field of the Killing field

K_{e}

. We verify its smoothness, orthonormality, and right-handedness. We further prove that any smooth orthonormal right-handed frame on

S^{2} \ {e, - e}

is either

f_{e}

or a rotation thereof by a smooth map

ρ : S^{2} \ {e, - e} \to S O (3)

, reflecting the triviality of the frame bundle over the parallelizable domain. The paper then pivots to an innovative application, constructing solutions to Maxwell’s equations in free space by combining spherical symmetries with quantum mechanical de Broglie waves in tempered distribution wave space. The deeper scientific significance lies in bringing together differential geometry (via

S O (3)

symmetries), quantum mechanics (de Broglie waves in Schwartz distribution theory), and electromagnetism (Maxwell’s solutions in Schwartz tempered complex fields on Minkowski space-time), in order to offer a unifying perspective on Maxwell’s electromagnetism and Schrödinger’s picture in relativistic quantum mechanics. Full article

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23 pages, 775 KB

Open AccessArticle

Belief-Based Model of Career Dropout Under Monopsonistic Employment and Noisy Evaluation

by Iñaki Aliende, Lorenzo Escot and Julio E. Sandubete

Mathematics 2025, 13(17), 2879; https://doi.org/10.3390/math13172879 - 5 Sep 2025

Viewed by 316

Abstract

This paper develops a belief-based dynamic optimisation framework to explain career continuation decisions in settings characterised by monopsonistic employment and asymmetric performance evaluation. Extending Holmström’s career concerns model, we consider agents who must decide whether to continue or exit their vocation based on [...] Read more.

This paper develops a belief-based dynamic optimisation framework to explain career continuation decisions in settings characterised by monopsonistic employment and asymmetric performance evaluation. Extending Holmström’s career concerns model, we consider agents who must decide whether to continue or exit their vocation based on subjective beliefs updated from noisy signals. Unlike the original framework, our model assumes a single institutional employer and limited feedback transparency, turning the agent’s decision into an optimal stopping problem governed by evolving belief thresholds. Analytical results demonstrate how greater signal noise, higher effort costs, and more attractive outside options raise the probability of exit. To validate the framework, we confront belief-based dropout decisions using original survey data from over 8000 football referees in Europe, showing that threats, unmet development expectations, and perceived stagnation significantly predict dropout. The results offer practical insights for institutions, such as sports federations, academic bodies, and civil services, on how to improve retention through increased transparency and better support structures. This study contributes to the literature by integrating optimal stopping theory and dynamic labor models in a novel context of constrained career environments. Full article

(This article belongs to the Special Issue Mathematical Economics and Its Applications)

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35 pages, 1992 KB

Open AccessArticle

Integrating Large Language Models into a Novel Intuitionistic Fuzzy PROBID Method for Multi-Criteria Decision-Making Problems

by Ferry Anhao, Amir Karbassi Yazdi, Yong Tan and Lanndon Ocampo

Mathematics 2025, 13(17), 2878; https://doi.org/10.3390/math13172878 - 5 Sep 2025

Viewed by 243

Abstract

As vision and mission statements embody the directions set forth by an organization, their connection to the Sustainable Development Goals (SDGs) must be made explicit to guide overall decision-making in taking strides toward the sustainability agenda. The semantic alignment of these strategic statements [...] Read more.

As vision and mission statements embody the directions set forth by an organization, their connection to the Sustainable Development Goals (SDGs) must be made explicit to guide overall decision-making in taking strides toward the sustainability agenda. The semantic alignment of these strategic statements with the SDGs is investigated in a previous study, although several limitations need further exploration. Thus, this study aims to advance two contributions: (1) utilizing the capabilities of LLMs (Large Language Models) in text semantic analysis and (2) integrating fuzziness into the problem domain by using a novel intuitionistic fuzzy set extension of the PROBID (Preference Ranking On the Basis of Ideal-average Distance) method. First, a systematic approach evaluates the semantic alignment of organizational strategic statements with the SDGs by leveraging the use of LLMs in semantic similarity and relatedness tasks. Second, viewing it as a multi-criteria decision-making (MCDM) problem and recognizing the limitations of LLMs, the evaluations are represented as intuitionistic fuzzy sets (IFSs), which prompted the development of an IF extension of the PROBID method. The proposed IF-PROBID method was then deployed to evaluate the 47 top Philippine corporations. Utilizing ChatGPT 3.5, 7990 prompts with repetitions generated the membership, non-membership, and hesitance scores for each evaluation. Also, we developed a cohort-dependent SDG–vision–mission matrix that categorizes corporations into four distinct classifications. Findings suggest that “highly-aligned” corporations belong to the private and technology sectors, with some in the industrial and real estate sectors. Meanwhile, “weakly-aligned” corporations come from the manufacturing and private sectors. In addition, case-specific insights are presented in this work. The comparative analysis yields a high agreement between the results and those generated by other IF-MCDM extensions. This paper is the first to demonstrate two methodological advances: (1) the integration of LLMs in MCDM problems and (2) the development of the IF-PROBID method that handles the resulting inherently imprecise evaluations. Full article

(This article belongs to the Special Issue Advances in Operations Research: Applications in MCDA, Fuzzy Systems, DEA and Optimization)

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12 pages, 1541 KB

Open AccessArticle

On the Autocorrelation and Stationarity of Multi-Scale Returns

by Carlos Manuel Rodríguez-Martínez, Héctor Francisco Coronel-Brizio, Horacio Tapia-McClung, Manuel Enríque Rodríguez-Achach and Alejandro Raúl Hernández-Montoya

Mathematics 2025, 13(17), 2877; https://doi.org/10.3390/math13172877 - 5 Sep 2025

Viewed by 242

Abstract

In this article, we conduct a statistical analysis of the autocorrelation functions (ACF) of multi-scale logarithmic returns computed over maximal monotonic uninterrupted trends (runs) in financial indices’ daily data. We analyze the Dow Jones Industrial Average (DJIA) and the Mexican IPC (Índice de [...] Read more.

In this article, we conduct a statistical analysis of the autocorrelation functions (ACF) of multi-scale logarithmic returns computed over maximal monotonic uninterrupted trends (runs) in financial indices’ daily data. We analyze the Dow Jones Industrial Average (DJIA) and the Mexican IPC (Índice de Precios y Cotizaciones) over a period from 30 October 1978 to 19 May 2025. We examine how deterministic alternation of signs shapes the ACF of multi-scale returns, and we evaluate covariance stationarity via formal tests (e.g., Augmented Dickey–Fuller and Phillips–Perron). We conclude that, despite the persistent long-memory oscillations in the ACF, multi-scale return series pass the stationarity tests, an outcome with interesting implications for econometric modeling of financial time series. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics for Economic Data Analytics: Models, Methods, and Applications)

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17 pages, 340 KB

Open AccessFeature PaperArticle

Efficient Direct Reconstruction of Bipartite (Multi)Graphs from Their Line Graphs Through a Characterization of Their Edges

by Drago Bokal and Janja Jerebic

Mathematics 2025, 13(17), 2876; https://doi.org/10.3390/math13172876 - 5 Sep 2025

Viewed by 220

Abstract

We study the line graphs of bipartite multigraphs, which naturally arise in combinatorics, game theory, and applications such as scheduling and motion planning. We introduce a new characterization of these graphs via valid partial assignments of the edges of the underlying bipartite multigraph [...] Read more.

We study the line graphs of bipartite multigraphs, which naturally arise in combinatorics, game theory, and applications such as scheduling and motion planning. We introduce a new characterization of these graphs via valid partial assignments of the edges of the underlying bipartite multigraph to the vertices of its line graph. We show that an empty assignment extends to a complete one precisely when the graph is a line graph of a bipartite multigraph. Based on this, we design an

O (Δ (G) | E (G) |)

algorithm that incrementally constructs such assignments. The algorithm also provides a data structure supporting efficient solutions to problems of maximum clique, maximum weighted clique, minimum clique cover, chromatic number, and independence number. For line graphs of bipartite simple graphs these problems become solvable in linear time, improving on previously known polynomial-time results. For general bipartite multigraphs, our method enhances the

O (| V (G) |^{3})

recognition algorithm of Peterson and builds on the results of Demaine et al., Hedetniemi, Cook et al., and Gurvich and Temkin. Full article

(This article belongs to the Special Issue New Perspectives of Graph Theory and Combinatorics)

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10 pages, 467 KB

Open AccessFeature PaperArticle

Local Splitting into Incoming and Outgoing Waves and the Integral Representation of Regular Scalar Waves

by Didier Felbacq and Emmanuel Rousseau

Mathematics 2025, 13(17), 2875; https://doi.org/10.3390/math13172875 - 5 Sep 2025

Viewed by 233

Abstract

The problem of the integral representation over a bounded surface of a regular field satisfying the Helmholtz equation in all space is investigated. This problem is equivalent to local splitting into an incoming field and an outgoing field. This splitting is not possible [...] Read more.

The problem of the integral representation over a bounded surface of a regular field satisfying the Helmholtz equation in all space is investigated. This problem is equivalent to local splitting into an incoming field and an outgoing field. This splitting is not possible in general. Full article

(This article belongs to the Special Issue Analytical Methods in Wave Scattering and Diffraction, 3rd Edition)

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