All for One or One for All? A Comparative Study of Grouped Data in Mixed-Effects Additive Bayesian Networks
An Efficient Quasi-Monte Carlo Algorithm for High Dimensional Numerical Integration
Joint Discrete Approximation by Shifts of Hurwitz Zeta-Function: The Case of Short Intervals
First-Order Axiom Systems ε_d and ε_da Extending Tarski’s ε₂ with Distance and Angle Function Symbols for Quantitative Euclidean Geometry
Transient Waves in Linear Dispersive Media with Dissipation: An Approach Based on the Steepest Descent Path

Journal Description

Mathematics

Mathematics is a peer-reviewed, open access journal which provides an advanced forum for studies related to mathematics, and is published semimonthly online by MDPI. The Committee on Mathematical Logic of the Chinese Mathematical Society (CML-CMS), European Society for Fuzzy Logic and Technology (EUSFLAT) and International Society for the Study of Information (IS4SI) are affiliated with Mathematics and their members receive a discount on article processing charges.

Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
High Visibility: indexed within Scopus, SCIE (Web of Science), RePEc, and other databases.
Journal Rank: JCR - Q1 (Mathematics) / CiteScore - Q1 (General Mathematics)
Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 17.3 days after submission; acceptance to publication is undertaken in 2.8 days (median values for papers published in this journal in the second half of 2025).
Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Sections: published in 17 topical sections.
Companion journals for Mathematics include: Foundations, Analytics, International Journal of Topology, Geometry and Logics.
Journal Cluster of Mathematics and Its Applications: AppliedMath, Axioms, Computation, Fractal and Fractional, Geometry, International Journal of Topology, Logics, Mathematics and Symmetry.

Impact Factor: 2.2 (2024); 5-Year Impact Factor: 2.0 (2024)

Imprint Information Journal Flyer Open Access ISSN: 2227-7390

Latest Articles

17 pages, 7284 KB

Open AccessArticle

Dynamics and Solution Behavior of the Variable-Order Fractional Newton–Leipnik System

by Rania Saadeh, Nidal E. Taha, Mohamed Hafez, Ghozail Sh. Al-Mutairi and Manahil A. M. Ashmaig

Mathematics 2026, 14(2), 312; https://doi.org/10.3390/math14020312 - 16 Jan 2026

Abstract

This paper considers the solution behavior and dynamical properties of the variable-order fractional Newton–Leipnik system defined via Liouville–Caputo derivatives of variable order. In contrast to integer-order models, the presence of variable-order fractional operators in the Newton–Leipnik structure enriches the model by providing memory-dependent effects that vary with time; hence, it is capable of a broader and more flexible range of nonlinear responses. Numerical simulations have been conducted to study how different order functions influence the trajectory and qualitative dynamics: clear transitions in oscillatory patterns have been identified by phase portraits, time-series profiles, and three-dimensional state evolution. The work goes further by considering the development of bifurcations and chaotic regimes and stability shifts and confirms the occurrence of several phenomena unattainable in fixed-order and/or integer-order formulations. Analysis of Lyapunov exponents confirms strong sensitivity to the initial conditions and further details how the memory effects either reinforce or prevent chaotic oscillations according to the type of order function. The results, in fact, show that the variable-order fractional Newton–Leipnik framework allows for more expressive and realistic modeling of complex nonlinear phenomena and points out the crucial role played by evolving memory in controlling how the system moves between periodic, quasi-periodic, and chaotic states. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

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19 pages, 1973 KB

Open AccessArticle

Continuous Smartphone Authentication via Multimodal Biometrics and Optimized Ensemble Learning

by Chia-Sheng Cheng, Ko-Chien Chang, Hsing-Chung Chen and Chao-Lung Chou

Mathematics 2026, 14(2), 311; https://doi.org/10.3390/math14020311 - 15 Jan 2026

Abstract

The ubiquity of smartphones has transformed them into primary repositories of sensitive data; however, traditional one-time authentication mechanisms create a critical trust gap by failing to verify identity post-unlock. Our aim is to mitigate these vulnerabilities and align with the Zero Trust Architecture (ZTA) framework and philosophy of “never trust, always verify,” as formally defined by the National Institute of Standards and Technology (NIST) in Special Publication 800-207. This study introduces a robust continuous authentication (CA) framework leveraging multimodal behavioral biometrics. A dedicated application was developed to synchronously capture touch, sliding, and inertial sensor telemetry. For feature modeling, a heterogeneous deep learning pipeline was employed to capture modality-specific characteristics, utilizing Convolutional Neural Networks (CNNs) for sensor data, Long Short-Term Memory (LSTM) networks for curvilinear sliding, and Gated Recurrent Units (GRUs) for discrete touch. To resolve performance degradation caused by class imbalance in Zero Trust environments, a Grid Search Optimization (GSO) strategy was applied to optimize a weighted voting ensemble, identifying the global optimum for decision thresholds and modality weights. Empirical validation on a dataset of 35,519 samples from 15 subjects demonstrates that the optimized ensemble achieves a peak accuracy of 99.23%. Sensor kinematics emerged as the primary biometric signature, followed by touch and sliding features. This framework enables high-precision, non-intrusive continuous verification, bridging the critical security gap in contemporary mobile architectures. Full article

(This article belongs to the Special Issue Emerging Applications of Artificial Intelligence Algorithms in Computer and Network Security)

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11 pages, 857 KB

Open AccessFeature PaperArticle

On the Characterization of Classes of Floorplans by Pattern-Avoiding Permutation Matrices

by Andrea Frosini, Elisa Pergola and Simone Rinaldi

Mathematics 2026, 14(2), 310; https://doi.org/10.3390/math14020310 - 15 Jan 2026

Abstract

Let R be an axis-aligned rectangle. We define a floorplan as a partition of R into rectangular regions (rooms) such that each vertex is shared by at most three rooms. Following the approach of Nakano et al.,we also assume the presence of a set of points that impose constraints on the walls passing through them, allowing only horizontal or vertical segments. These constraints can be encoded by a permutation matrix whose entries are labeled H and V, which we refer to as a pattern matrix. In this work, we characterize the well-known classes of guillotine, diagonal, and diagonal–guillotine floorplans in terms of the presence of specific families of pattern matrices. In this way, we translate a purely geometric characterization into a combinatorial one. Full article

(This article belongs to the Section B: Geometry and Topology)

25 pages, 1857 KB

Open AccessArticle

Exponentially Clustered Synchronization of a Stochastic Complex Network with Reaction–Diffusion Terms and Time Delays via a Pinning Boundary Control

by Binglong Lu and Mei Liu

Mathematics 2026, 14(2), 309; https://doi.org/10.3390/math14020309 - 15 Jan 2026

Abstract

A pinning boundary control strategy that can achieve the exponentially clustered synchronization of a specific class of complex networks is developed. Firstly, the studied model captures the essential features of networks, including spatial dependence, stochastic switching, noise perturbation, and time delays. Secondly, the proposed control algorithm can save the implementation cost and overcome environmental constraint by acting on the boundary of a few nodes. Thirdly, an average state related to the directed topology of the nodes in the same cluster is calculated as the target network. Finally, nonlinear simulations show that the proposed controller can solve the cluster synchronization of a directed coupled reaction–diffusion neural network with Markovian switching, stochastic noise and time delay. Full article

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15 pages, 4450 KB

Open AccessArticle

Eigenvalues of the Operator Describing Magnetohydrodynamic Problems in Outer Parts of Galaxies

by Evgeny Mikhailov and Tatiana Khasaeva

Mathematics 2026, 14(2), 308; https://doi.org/10.3390/math14020308 - 15 Jan 2026

Abstract

The magnetic field generation studies in astronomy lead to a number of interesting problems in mathematical physics. In the dynamo theory, the problem is reduced to a system of parabolic equations for the field components. Assuming that the field grows exponentially, we obtain an eigenvalue problem for the corresponding elliptic operator. The possibility of the field generation and behaviour of the system is characterized by the spectra of the operator. If all eigenvalues lie in the left half of the complex plane, the perturbations will decay. On the other hand, if some of the eigenvalues have positive real parts, the large-scale structures of the field can be generated. From the astrophysical point of view, galactic magnetic fields are very important to study. One of the main problems is connected with the peripheral regions, where the properties of the medium complicate the operator structure. We can use the perturbation theory to find the eigenvalues. However, the problem can be solved analytically by considering some specific approximations. We can find the spectra using numerical approaches in the case of the conditions that are close to the real ones. In this paper, we solve eigenvalue problems for different operators which are connected with magnetohydrodynamic processes in outer parts of galaxies. Full article

(This article belongs to the Special Issue Mathematical Physics and Applied Mathematics: Latest Advances and Prospects)

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27 pages, 2227 KB

Open AccessArticle

Application of a Reinforcement Learning-Based Improved Genetic Algorithm in Flexible Job-Shop Scheduling Problems

by Guoli Zhao, Jiansha Lu, Gangqiang Liu, Weini Weng and Ning Wang

Mathematics 2026, 14(2), 307; https://doi.org/10.3390/math14020307 - 15 Jan 2026

Abstract

This paper addresses the limitations of genetic algorithms in solving the Flexible Job-Shop Scheduling Problem (FJSP) including slow convergence, susceptibility to local optima, and sensitivity to parameter settings. The paper proposes an Improved Genetic Algorithm based on Reinforcement Learning (IGARL). First, a hybrid population selection mechanism that combines the Queen Bee Mating Flight (QBMF) strategy with the Tournament Selection (TS) method is introduced. This mechanism significantly accelerates convergence by optimizing the population structure. Second, a dynamic population update strategy based on tunnel vision, termed the Solution Space Diversity Awakening (SSDA) strategy, is developed. When the population becomes trapped in local optima, this strategy intelligently triggers random perturbations and introduces high-potential individuals to enhance the algorithm’s ability to escape local optima and promote population diversity. Third, a novel multi-Q-table reinforcement learning framework is embedded within the iterative process to dynamically adjust key genetic algorithm parameters (such as selection, mutation, and crossover rates) and enable multi-dimensional performance evaluation, thereby effectively guiding the search toward better solutions. Experimental results demonstrate that the IGARL algorithm achieves a 10% to 60% improvement in convergence speed on Brandimarte benchmark instances, with solution quality significantly surpassing that of the basic genetic algorithm. Moreover, the fluctuation of the average optimal solution remains within 20%, indicating strong stability and robustness. Full article

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

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18 pages, 604 KB

Open AccessArticle

Making Chaos Out of COVID-19 Testing

by Bo Deng, Jorge Duarte, Cristina Januário and Chayu Yang

Mathematics 2026, 14(2), 306; https://doi.org/10.3390/math14020306 - 15 Jan 2026

Abstract

Mathematical models for infectious diseases, particularly autonomous ODE models, are generally known to possess simple dynamics, often converging to stable disease-free or endemic equilibria. This paper investigates the dynamic consequences of a crucial, yet often overlooked, component of pandemic response: the saturation of public health testing. We extend the standard SIR model to include compartments for ‘Confirmed’ (C) and ‘Monitored’ (M) individuals, resulting in a new SICMR model. By fitting the model to U.S. COVID-19 pandemic data (specifically the Omicron wave of late 2021), we demonstrate that capacity constraints in testing destabilize the testing-free endemic equilibrium (

E_{1}

). This equilibrium becomes an unstable saddle-focus. The instability is driven by a sociological feedback loop, where the rise in confirmed cases drive testing effort, modeled by a nonlinear Holling Type II functional response. We explicitly verify that the eigenvalues for the best-fit model satisfy the Shilnikov condition (

λ_{u} > λ_{s}

), demonstrating the system possesses the necessary ingredients for complex, chaotic-like dynamics. Furthermore, we employ Stochastic Differential Equations (SDEs) to show that intrinsic noise interacts with this instability to generate ’noise-induced bursting,’ replicating the complex wave-like patterns observed in empirical data. Our results suggest that public health interventions, such as testing, are not merely passive controls but active dynamical variables that can fundamentally alter the qualitative stability of an epidemic. Full article

(This article belongs to the Special Issue Applications of Differential Equations and Mathematical Modelling in Mathematical Biology)

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18 pages, 295 KB

Open AccessArticle

Characterizations of Pseudo-Symmetric Space–Times in Gray’s Subspaces and f(R)-Gravity Vacuum Solutions

by Awatif Al-Jedani, Sameh Shenawy, Uday Chand De and Abdallah Abdelhameed Syied

Mathematics 2026, 14(2), 305; https://doi.org/10.3390/math14020305 - 15 Jan 2026

Abstract

This paper investigates pseudo-symmetric space–times within two interrelated frameworks: vacuum

f (R)

-gravity and Gray’s seven canonical decomposition subspaces. First, it is established that any conformally flat pseudo-symmetric space–time satisfying the vacuum field equations of

f (R)

-gravity necessarily [...] Read more.

This paper investigates pseudo-symmetric space–times within two interrelated frameworks: vacuum

f (R)

-gravity and Gray’s seven canonical decomposition subspaces. First, it is established that any conformally flat pseudo-symmetric space–time satisfying the vacuum field equations of

f (R)

-gravity necessarily corresponds to a perfect fluid. Subsequently, a detailed analysis of Gray’s subspaces reveals the following structural results: In the trivial and

𝒜

subspaces, pseudo-symmetric space–times are Ricci-simple and Weyl-harmonic, and thus are necessarily generalized Robertson–Walker space–times. In the

B

and

𝒜 \oplus B

subspaces, the associated time-like vector field

ξ^{l}

is shown to be an eigenvector of the Ricci tensor with the eigenvalue

- R / 2

. Furthermore, for a perfect fluid pseudo-symmetric space–time obeying

f (R)

-gravity and belonging to the trivial,

𝒜

B

, or

𝒜 \oplus B

subspaces, the isotropic pressure p and energy density

σ

are proven to be constants. Additionally, it is demonstrated that Gray’s

I

subspace reduces to the

B

subspace in the pseudo-symmetric setting. Finally, under specific geometric conditions, pseudo-symmetric space–times in the

I \oplus 𝒜

and

I \oplus B

subspaces are also shown to admit perfect fluid representations. These results collectively clarify the geometric and physical constraints imposed by pseudo-symmetry within

f (R)

-gravity and Gray’s classification scheme. Full article

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

29 pages, 16318 KB

Open AccessArticle

A Novel Algorithm for Determining the Window Size in Power Load Prediction

by Haobin Liang, Zefang Song, Yiran Liu and Yiwei Huang

Mathematics 2026, 14(2), 304; https://doi.org/10.3390/math14020304 - 15 Jan 2026

Abstract

The sliding window method is a commonly used data processing in time series forecasting tasks, and determining the appropriate window size is a crucial step in constructing predictive models. However, the current setting of window size parameters is often based on empirical knowledge, making the scientific determination of the optimal sliding window size highly significant. This paper proposes an algorithm for optimizing window size based on sample entropy, which is applicable not only to the original undecomposed sequences but also effectively to the decomposed sequences. The proposed algorithm has been validated using the open-source Elia grid data across multiple model architectures, including recurrent (GRU/LSTM) and attention-based (Transformer) networks. Experimental results demonstrate that the algorithm effectively determines an optimal window size of 106. The optimized window consistently leads to superior prediction performance, with the CEEMD-GRU model achieving a MAPE of 0.256, RMSE of 22.529, and MAE of 18.186—representing reductions of over 5% compared to the undecomposed benchmark. Furthermore, the enhancement is more significant for decomposed sequences, and the algorithm’s efficacy is validated across different neural network architectures (e.g., LSTM, GRU, Transformer), confirming its practical utility and generalizability. Full article

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

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8 pages, 248 KB

Open AccessFeature PaperArticle

Fermi Sea Topology and Boundary Geometry for Free Particles in One- and Two-Dimensional Lattices

by Guillermo R. Zemba

Mathematics 2026, 14(2), 303; https://doi.org/10.3390/math14020303 - 15 Jan 2026

Abstract

Free gasses of spinless fermions moving on a lattice-symmetric geometric background are considered. Their topological properties at zero temperature can be used to classify their Fermi seas and associated boundaries. The flat orbifolds

R^{d} / Γ

, where

Γ

is the crystallographic [...] Read more.

R^{d} / Γ

, where

Γ

is the crystallographic group of symmetry in d-dimensional momentum space, are used to accomplish this task. Two topological classes exist for

d = 1

: an interval, which is identified as a conductor, and a circumference, which corresponds to an insulator. The number of topological classes increases to 17 for

d = 2

: 8 have the topology of a disk, that are generally recognized as conductors, and 4 correspond to a two-sphere, matching insulators. Both sets eventually contain a finite number of conical singularities and reflection corners at the boundaries. The remaining cases in the listing relate to conductors (annulus, Möbius strip) and insulators (two-torus, real projective plane, Klein bottle). Examples that fall under this list are given, along with physical interpretations of the singularities. It is anticipated that the findings of this classification will be robust under perturbative interactions due to its topological character. Full article

(This article belongs to the Special Issue Effective Field Theories for Condensed Matter and Statistical Systems)

24 pages, 1542 KB

Open AccessArticle

Joint Ordering Optimization for a Two-Echelon Pharmaceutical Supply Chain Considering Shelf Life and a Transshipment Mechanism

by Shiju Li, Ruizhi Ouyang, Li Guo, Hongjie Lan, Tingting Wang and Kaiye Gao

Mathematics 2026, 14(2), 302; https://doi.org/10.3390/math14020302 - 14 Jan 2026

Abstract

Pharmaceutical supply chains face high inventory and stockout risks because of short product shelf lives and volatile demand. To enhance coordination efficiency and reduce drug waste, this study examines a two-echelon supply chain comprising a manufacturer and multiple medical institutions. We built a joint ordering and transshipment optimization model that simultaneously incorporates shelf-life constraints, the first-in–first-out (FIFO) policy, inventory capacity limits, and peer-level transshipment. Under deterministic and stochastic demand, we solved the model using Bayesian optimization and Monte Carlo simulation. The results show that moderate inventory transshipment effectively mitigates risk from demand uncertainty and increases total supply-chain profit; under stochastic demand, the optimal strategy relies more heavily on coordinated transshipment to reduce excess inventory and near-expiry waste. Full article

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25 pages, 361 KB

Open AccessArticle

Killing and 2-Killing 1-Forms on Poisson Doubly Warped Product Manifolds

by Bang-Yen Chen, Majid Ali Choudhary, Foued Aloui and Ibrahim Al-Dayel

Mathematics 2026, 14(2), 301; https://doi.org/10.3390/math14020301 - 14 Jan 2026

Abstract

In this paper we investigate Killing and 2-Killing 1-forms on a Poisson doubly warped product manifold (

PDWPM

). We establish a connection between the property of 1-form on a

PDWPM

and its components on the factor manifolds to be Killing or 2-Killing. [...] Read more.

In this paper we investigate Killing and 2-Killing 1-forms on a Poisson doubly warped product manifold (

PDWPM

). We establish a connection between the property of 1-form on a

PDWPM

and its components on the factor manifolds to be Killing or 2-Killing. We also demonstrate that a Killing 1-form on a

PDWPM

generates a contravariant Ricci soliton factor manifold if and only if it is a contravariant Einstein manifold. Additionally, we provide necessary and sufficient conditions for a

PDWPM

to be a Poisson singly warped product manifold (

PSWPM

). Finally, we present examples of Killing and 2-Killing 1-forms on

PDWPM s

with one-dimensional, and two-dimensional base spaces. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications, 2nd Edition)

25 pages, 951 KB

Open AccessReview

Statistical Hypothesis Testing: A Comprehensive Review of Theory, Methods, and Applications

by Vesna Rajić

Mathematics 2026, 14(2), 300; https://doi.org/10.3390/math14020300 - 14 Jan 2026

Abstract

Statistical hypothesis testing is a foundational concept in inferential statistics, providing formal mechanisms for evaluating evidence and making decisions. This review article provides a comprehensive overview of the principles of hypothesis testing, including classical frameworks such as the Neyman–Pearson paradigm and Fisher’s significance testing, as well as modern perspectives including Bayesian approaches, robust testing, and high-dimensional inference. The article synthesizes key theoretical underpinnings, outlines commonly used statistical tests, and discusses methodological challenges and criticisms. Applications across various fields—ranging from medicine and economics to machine learning and psychology—are reviewed to demonstrate the versatility and evolving role of hypothesis testing in contemporary research. Finally, current debates and future research directions are highlighted, emphasizing the need for more robust, transparent, and reproducible statistical practices. Full article

(This article belongs to the Section D1: Probability and Statistics)

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31 pages, 465 KB

Open AccessArticle

A Multi-Stage NLP Framework for Knowledge Discovery from Crop Disease Research Literature

by Jantima Polpinij, Manasawee Kaenampornpan, Christopher S. G. Khoo, Wei-Ning Cheng and Bancha Luaphol

Mathematics 2026, 14(2), 299; https://doi.org/10.3390/math14020299 - 14 Jan 2026

Abstract

Extracting and organizing knowledge from the agricultural crop disease research literature are challenging tasks because of the heterogeneous terminologies, complicated symptom descriptions, and unstructured nature of scientific documents. In this study, we developed a multi-stage natural language processing (NLP) pipeline to automate knowledge extraction, organization, and integration from the agricultural research literature into a domain-consistent crop disease knowledge graph. The model combines transformer-based sentence embeddings with variational deep clustering to extract topics, which are further refined via facet-aware relevance scoring for sentence selection to be included in the summary. Lexicon-guided named entity recognition helps in the precise identification and normalization of terms for crops, diseases, symptoms, etc. Relation extraction based on a combination of lexical, semantic, and contextual features leads to the meaningful generation of triplets for the knowledge graph. The experimental results show that the method yielded consistently good results at each stage of the knowledge extraction process. Among the combinations of embedding and deep clustering methods, SciBERT + VaDE achieved the best clustering results. The extraction of representative sentences for disease symptoms, control/treatment, and prevention obtained high F1-scores of around 0.8. The resulting knowledge graph has high node coverage and high relation completeness, as well as high precision and recall in triplet generation. The multi-stage NLP pipeline effectively converts unstructured agricultural research texts into a coherent and semantically rich knowledge graph, providing a basis for further research in crop disease analysis, knowledge retrieval, and data-driven decision support in agricultural informatics. Full article

(This article belongs to the Special Issue Research on Machine Learning, Data Mining, Natural Language Processes, and Optimization Methods)

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21 pages, 6222 KB

Open AccessFeature PaperArticle

Weighted, Mixed ℓ_p Norm Regularization for Gaussian Noise-Based Denoising Method Extension

by Yuanmin Wang and Jinsong Leng

Mathematics 2026, 14(2), 298; https://doi.org/10.3390/math14020298 - 14 Jan 2026

Abstract

Many denoising methods model noise as Gaussian noise. However, the realistic noise captured by camera devices does not satisfy Gaussian distribution. Hence, those methods do not perform well when being applied to real-world image denoising tasks. In this work, we indicate that the spatial correlation in noise and the variation of noise intensity are the main factors that impact the performance of Gaussian noise-based methods, and accordingly propose an extension of the method based on the weighted, mixed non-convex

ℓ_{p}

norm. The proposed method first strengthens the intensity of the noise pattern in the original denoising result through the Guided Filter, then removes the over-amplified frequency in the local area by the proposed regularization term. We prove that the optimal solution can be achieved through the sub-gradient-based iterative optimization scheme, and further reduce the computational cost by optimizing the initial values. Numerical experiments show that the proposed extending method can balance well texture preservation and noise removal, and the PSNR of the extending method’s results are greatly improved, even outperforming the recently proposed realistic noise removal methods which also include deep learning based methods. Full article

(This article belongs to the Special Issue Mathematical Methods for Image Processing and Computer Vision)

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22 pages, 4203 KB

Open AccessFeature PaperArticle

Consensus and Divergence in Explainable AI (XAI): Evaluating Global Feature-Ranking Consistency with Empirical Evidence from Solar Energy Forecasting

by Kay Thari Thinn and Waddah Saeed

Mathematics 2026, 14(2), 297; https://doi.org/10.3390/math14020297 - 14 Jan 2026

Abstract

The growing reliance on solar energy necessitates robust and interpretable forecasting models for stable grid management. Current research frequently employs Explainable AI (XAI) to glean insights from complex black-box models, yet the reliability and consistency of these explanations remain largely unvalidated. Inconsistent feature attributions can mislead grid operators by incorrectly identifying the dominant drivers of solar generation, thereby affecting operational planning, reserve allocation, and trust in AI-assisted decision-making. This study addresses this critical gap by conducting a systematic statistical evaluation of feature rankings generated by multiple XAI methods, including model-agnostic (SHAP, PDP, PFI, ALE) and model-specific (Split- and Gain-based) techniques, within a time-series regression context. Using a LightGBM model for one-day-ahead solar power forecasting across four sites in Calgary, Canada, we evaluate consensus and divergence using the Friedman test, Kendall’s W, and Spearman’s rank correlation. To ensure the generalizability of our findings, we further validate the results using a CatBoost model. Our results show a strong overall agreement across methods (Kendall’s W: 0.90–0.94), with no statistically significant difference in ranking (p > 0.05). However, pairwise analysis reveals that the “Split” method frequently diverges from other techniques, exhibiting lower correlation scores. These findings suggest that while XAI consensus is high, relying on a single method—particularly the split count—poses risks. We recommend employing multi-method XAI and using agreement as an explicit diagnostic to ensure transparent and reliable solar energy predictions. Full article

(This article belongs to the Special Issue Applied Mathematics in Artificial Intelligence: Methods, Algorithms, and Applications)

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29 pages, 1782 KB

Open AccessArticle

Reinforcement Learning-Guided NSGA-II Enhanced with Gray Relational Coefficient for Multi-Objective Optimization: Application to NASDAQ Portfolio Optimization

by Zhiyuan Wang, Qinxu Ding, Ding Ding, Siying Zhu, Jing Ren, Yue Wang and Chong Hui Tan

Mathematics 2026, 14(2), 296; https://doi.org/10.3390/math14020296 - 14 Jan 2026

Abstract

In modern financial markets, decision-makers increasingly rely on quantitative methods to navigate complex trade-offs among multiple, often conflicting objectives. This paper addresses constrained multi-objective optimization (MOO) with an application to portfolio optimization for minimizing risk and maximizing return. To this end, and to address existing gaps, we propose a novel reinforcement learning (RL)-guided non-dominated sorting genetic algorithm II (NSGA-II) enhanced with gray relational coefficients (GRC), termed RL-NSGA-II-GRC, which combines an RL agent controller and GRC-based selection to improve the convergence and diversity of the Pareto-optimal fronts. The agent adapts key evolutionary parameters online using population-level metrics of hypervolume, feasibility, and diversity, while the GRC-enhanced tournament operator ranks parents via a unified score simultaneously considering dominance rank, crowding distance, and geometric proximity to ideal reference. We evaluate the framework on the Kursawe and CONSTR benchmark problems and on a NASDAQ portfolio optimization application. On the benchmarks, RL-NSGA-II-GRC achieves convergence metric improvements of about 5.8% and 4.4% over the original NSGA-II, while preserving a well-distributed set of non-dominated solutions. In the portfolio application, the method produces a smooth and densely populated efficient frontier that supports the identification of the maximum Sharpe ratio portfolio (with annualized Sharpe ratio = 1.92), as well as utility-optimal portfolios for different risk-aversion levels. The main contributions of this work are three-fold: (1) we propose an RL-NSGA-II-GRC method that integrates an RL agent into the evolutionary framework to adaptively control key parameters using generational feedback; (2) we design a GRC-enhanced binary tournament selection operator that provides a comprehensive performance indicator to efficiently guide the search toward the Pareto-optimal front; (3) we demonstrate, on benchmark MOO problems and a NASDAQ portfolio case study, that the proposed method delivers improved convergence and well-populated efficient frontiers that support actionable investment insights. Full article

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

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12 pages, 259 KB

Open AccessArticle

Generalized Derivations in Rings and Their Applications to Banach Algebra

by Amal S. Alali, Emine Koç Sögütcü, Zeliha Bedir and Nadeem ur Rehman

Mathematics 2026, 14(2), 295; https://doi.org/10.3390/math14020295 - 13 Jan 2026

Abstract

Let

R

be a prime ring, and let F denote a generalized derivation associated with a derivation d of

R

. Consider I as a nonzero ideal of

R

, and let

m, n, k, l

be fixed positive integers. [...] Read more.

Let

R

be a prime ring, and let F denote a generalized derivation associated with a derivation d of

R

. Consider I as a nonzero ideal of

R

, and let

m, n, k, l

be fixed positive integers. In this study, we explore the behavior of the generalized derivation F within the structures of both prime and semiprime rings that satisfy the functional identity

{[F (η), d (τ)]}_{m} = η^{n} {[η, τ]}_{l} η^{k}

\forall, η, τ \in I .

Furthermore, we extend this investigation to the framework of Banach algebras, analyzing how generalized derivations operate in such algebras. A comparative discussion is also presented to highlight the distinctions and similarities in the behavior of generalized derivations within Banach algebraic settings under the above structural condition. Full article

(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra, 2nd Edition)

14 pages, 1539 KB

Open AccessArticle

Optimal Control of Orbit Rendezvous with Low-Thrust on Near-Circular Orbits Using Pontryagin’s Maximum Principle

by Xiao Zhou, Hongbin Deng, Yaxuan Li and Yigao Gao

Mathematics 2026, 14(2), 294; https://doi.org/10.3390/math14020294 - 13 Jan 2026

Abstract

This paper investigates the optimal control problem of orbital rendezvous for spacecraft in near-circular orbits with a low-thrust propulsion system. Two optimality criteria are considered: time-optimal and motor-time-optimal control. A linearized mathematical model of relative motion between the active and passive spacecraft is employed, which is formulated in dimensionless variables that characterize secular, periodic, and lateral motion components of the relative motion. By applying Pontryagin’s Maximum Principle, the equations governing the optimal relative motion of the spacecraft are derived. To address the discontinuities associated with the bang–bang switching function inherent in the motor-time-optimal problem, and the lack of a suitable initial guess, a homotopy method is adopted, in which the solution to the rendezvous time-optimal problem is used as an initial guess and is gradually deformed into the motor-time-optimal control. Considering the errors introduced by the linearization of the relative motion model, the obtained control law is validated via numerical simulations based on the original nonlinear dynamics of the system. Simulation results demonstrate that the proposed trajectory optimization methodology achieves high success rates and rapid convergence, providing valuable theoretical support and practical guidance for mission scenarios with similar trajectory design requirements. Full article

(This article belongs to the Special Issue Optimization and Control for Unmanned Systems and Intelligent Logistics)

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22 pages, 1343 KB

Open AccessArticle

Stability Improvement of PMSG-Based Wind Energy System Using the Passivity-Based Non-Fragile Retarded Sampled Data Controller

by Thirumoorthy Ramasamy, Thiruvenkadam Srinivasan and In-Ho Ra

Mathematics 2026, 14(2), 293; https://doi.org/10.3390/math14020293 - 13 Jan 2026

Abstract

This work presents the design of passivity based non-fragile retarded sampled data control (NFRSDC) for the wind energy system using permanent magnet synchronous generator. At first, the proposed system is characterized in terms of non-linear dynamical equations, which is later expressed in terms of linear sub-systems via fuzzy membership functions using the Takagi–Sugeno fuzzy approach. After that, a more applicative NFRSDC is proposed along with the delay involved during signal transmission as well as randomly occurring controller gain perturbations (ROCGPs). Here, the ROCGPs are modeled accordingly using stochastic variable which obeys the certain Bernoulli distribution sequences. Folowing that, an appropriate Lyapunov–Krasovskii functionals are constructed to obtain the sufficient conditions in the form of linear matrix inequalities. These obtained conditions are then used to ensure the global asymptotic stability of the given system with the exogenous disturbances. Finally, numerical simulations are performed using MATLAB/Simulink and the obtained results have clearly demonstrated the efficacy of the proposed controller. Full article

(This article belongs to the Special Issue Applied Mathematics and Intelligent Control in Electrical Engineering)

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