Mathematics

20 pages, 344 KB

Open AccessArticle

On Exact Totient Recovery in Semiprimes via Square-Root Proximity

by Abdinabi Mukhamadiyev, Ugiloy Akhadova, Ilkhom Boykuziev, Bakhtiyor Abdurakhimov, Ergashevich Halimjon Khujamatov and Razvan Craciunescu

Mathematics 2026, 14(10), 1784; https://doi.org/10.3390/math14101784 - 21 May 2026

Viewed by 386

Abstract

This paper studies structural properties of semiprimes

N = p q

in computational number theory, focusing on cases where the prime factors are close. We analyze the relationship between

\sqrt{N}

and

\sqrt{φ (N)}

and show that, under a bounded prime gap [...] Read more.

This paper studies structural properties of semiprimes

N = p q

in computational number theory, focusing on cases where the prime factors are close. We analyze the relationship between

\sqrt{N}

and

\sqrt{φ (N)}

and show that, under a bounded prime gap condition, these quantities exhibit strong proximity. Specifically, assuming

| p - q | \leq 2^{l / 4}

for an l-bit semiprime, we prove that the Euler totient function admits the exact representation

φ (N) = N - 1 - 2 ⌊ \sqrt{N} ⌋

. Based on this result, we develop an interval-based method for reconstructing

φ (N)

within a narrow neighborhood derived from square-root bounds, followed by a discriminant-based refinement step for recovering the prime factors. Experimental evaluation on large semiprimes, including RSA-type moduli of 4095 and 4096 bits, shows that the method operates efficiently under the stated structural condition using only elementary integer arithmetic. These results provide a theoretical characterization of semiprimes with small prime gaps and offer a framework for identifying structurally weak RSA moduli. This method, given its high efficiency when the prime factors are close to each other, can be regarded as an alternative to Fermat’s factorization method. In particular, for semiprime integers with a small prime gap (i.e.,

| p - q |

is small), the proposed approach exploits structural properties based on the proximity of square roots, thereby significantly accelerating the factorization process. Consequently, it not only aligns with the theoretical foundation of Fermat’s method but, under certain conditions, may also achieve comparable or even superior practical performance. Full article

(This article belongs to the Section E: Applied Mathematics)

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39 pages, 909 KB

Open AccessFeature PaperArticle

Projective Solutions Methods Automatically Satisfying the Stokes, Oseen and Brinkman Equations

by Chein-Shan Liu, Tai-Wen Hsu and Chia-Cheng Tsai

Mathematics 2026, 14(10), 1783; https://doi.org/10.3390/math14101783 - 21 May 2026

Cited by 1 | Viewed by 308

Abstract

The new projective solutions methods (PSMs) for solving the Stokes, Oseen, and Brinkman flow problems are presented in this paper. They automatically satisfy the governing equations and are therefore Trefftz-type methods. Utilizing the third-order formulation and three-dimensional analytic functions, we derive a meshless [...] Read more.

The new projective solutions methods (PSMs) for solving the Stokes, Oseen, and Brinkman flow problems are presented in this paper. They automatically satisfy the governing equations and are therefore Trefftz-type methods. Utilizing the third-order formulation and three-dimensional analytic functions, we derive a meshless Trefftz-type method to solve three-dimensional Stokes flow problems. The Oseen and Brinkman equations are transformed into four coupled third-order/first-order partial differential equations. The projective-type particular solution (PTPS) is obtained via a projective function in terms of the projective variable; the third-order ordinary differential equations (ODEs) with constant coefficients are derived to determine the projective functions. The Trefftz-type PSM is extremely accurate, because the governing equations (including the incompressibility condition) are implemented automatically. For the Brinkman equations, the general solutions of velocity and pressure are presented by using the Helmholtz function and a harmonic function, whose corresponding Trefftz-type numerical method is developed. Upon comparison with the method of fundamental solutions (MFS), the new methods exhibit some advantages, including lower condition numbers, faster convergence, and better accuracy. We also apply the Trefftz-type PSM to solve the exterior problem of the Stokes equations, where the velocity tends to zero at infinity. Full article

(This article belongs to the Section E: Applied Mathematics)

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28 pages, 3294 KB

Open AccessArticle

Optimization of Material Permeability Analysis Algorithm for 3D Raster Structures Using Graph-Based and Morphological Approaches

by Jan Mrógala, Martin Kotyrba, Eva Volná, Hashim Habiballa and Alexej Kolcun

Mathematics 2026, 14(10), 1782; https://doi.org/10.3390/math14101782 - 21 May 2026

Viewed by 310

Abstract

Quantitative characterization of permeability in porous media represents a central problem in filtration theory, geosciences, and materials engineering. Standard numerical approaches, including finite element methods and Lattice Boltzmann simulations, typically require extensive domain-specific expertise together with specialized computational software. This motivates the development [...] Read more.

Quantitative characterization of permeability in porous media represents a central problem in filtration theory, geosciences, and materials engineering. Standard numerical approaches, including finite element methods and Lattice Boltzmann simulations, typically require extensive domain-specific expertise together with specialized computational software. This motivates the development of computationally simpler and more accessible geometric approaches applicable directly to binary volumetric data. We introduce a novel algorithmic framework for the analysis of porous structures that reformulates permeability-related characterization in terms of discrete geometry and graph-based computation. The method combines parallel raster-grid and graph representations of a binarized three-dimensional CT image. The principal transport-limiting feature of the pore network, interpreted as the minimal constriction governing connectivity, is identified through iterative morphological dilation coupled with a three-dimensional scanline seed-fill procedure. In addition, a dichotomous bisection strategy is proposed to accelerate the determination of the critical bottleneck scale. The proposed methodology was evaluated on five volumetric datasets of size 100 × 100 × 100 voxels obtained from CT-derived porous structures. Experimental results demonstrate that dilation- and erosion-based formulations yield equivalent estimates of the bottleneck parameter in four of the five investigated samples. Furthermore, incorporation of the bisection optimization reduces computational time in three-dimensional experiments by approximately 50% relative to sequential iteration. The presented approach provides a computationally efficient and fully open-source alternative to conventional physics-based permeability solvers for binary porous media. The resulting bottleneck parameter b should be interpreted as a discrete geometric invariant characterizing the pore-network connectivity and minimal transport cross-section. It is not intended to replace the absolute permeability coefficient K appearing in Darcy’s law, but rather to serve as an independent structural descriptor suitable for comparative and topological analysis of porous systems. Full article

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

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22 pages, 3548 KB

Open AccessArticle

Adaptive Fixed-Time Prescribed Performance Command-Filtered Control for Nonlinear Systems with Unknown Control Gains and Actuator Faults

by Hadil Alhazmi, Mohamed Kharrat, Asma Al-Jaser and Paolo Mercorelli

Mathematics 2026, 14(10), 1781; https://doi.org/10.3390/math14101781 - 21 May 2026

Cited by 1 | Viewed by 451

Abstract

This paper investigates the adaptive prescribed performance fixed-time control problem for uncertain strict-feedback nonlinear systems in the presence of unknown control coefficients and actuator faults. A switching-based control strategy is developed to address the uncertainty in control coefficients, where adaptive parameters are adjusted [...] Read more.

This paper investigates the adaptive prescribed performance fixed-time control problem for uncertain strict-feedback nonlinear systems in the presence of unknown control coefficients and actuator faults. A switching-based control strategy is developed to address the uncertainty in control coefficients, where adaptive parameters are adjusted online according to design requirements. To regulate the transient and steady-state behavior, a fixed-time prescribed performance function is incorporated into the control design, ensuring that the tracking error evolves within predefined bounds. The command filter technique is employed to simplify the backstepping procedure and avoid the issue of complexity growth, while filter-induced errors are compensated using auxiliary signals. Rigorous Lyapunov analysis establishes that all closed-loop signals remain bounded and that the tracking error converges to a small neighborhood of zero within a fixed time, independent of initial conditions. The effectiveness of the proposed method is demonstrated through numerical simulations and a practical example. Full article

(This article belongs to the Special Issue Mathematics and Applications, 2nd Edition)

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18 pages, 420 KB

Open AccessFeature PaperArticle

On Rationality of Fields of Invariants for Abelian Groups of Odd Order

by Ivo Michailov Michailov

Mathematics 2026, 14(10), 1780; https://doi.org/10.3390/math14101780 - 21 May 2026

Viewed by 322

Abstract

In this paper, we investigate Noether’s problem concerning the rationality of the field of invariants

k (G)

for finite abelian groups G of odd order. We establish necessary and sufficient conditions for the rationality of the extension

k (G)

[...] Read more.

In this paper, we investigate Noether’s problem concerning the rationality of the field of invariants

k (G)

for finite abelian groups G of odd order. We establish necessary and sufficient conditions for the rationality of the extension

k (G)

over an arbitrary field k of characteristic 0, providing a more flexible alternative to the classical requirement that all cyclic primary components be rational. Specifically, we present a criterion for elementary abelian q-groups

C_{q}^{n}

in terms of the properties of certain norm maps (Theorem 4) and generalize these results to provide necessary and sufficient conditions for arbitrary abelian groups of odd order (Theorem 5). Furthermore, we provide a computational implementation of these criteria in PARI/GP and offer a concrete arithmetic classification of rational invariant fields over

Q

for groups of various odd orders. Full article

(This article belongs to the Special Issue Advanced Researches in Algebraic Geometry)

3 pages, 133 KB

Open AccessEditorial

Preface to the Special Issue “Advanced Theories and Novel Methods for Nonlinear Analysis, Optimization and Applications”

by Wei-Shih Du and Yousuke Araya

Mathematics 2026, 14(10), 1779; https://doi.org/10.3390/math14101779 - 21 May 2026

Viewed by 228

Abstract

After more than a century of unremitting efforts by scholars, nonlinear analysis has found widespread and important applications in many fields that are at the core of many branches of pure and applied mathematics, including functional analysis, fixed point theory, nonlinear ordinary and [...] Read more.

After more than a century of unremitting efforts by scholars, nonlinear analysis has found widespread and important applications in many fields that are at the core of many branches of pure and applied mathematics, including functional analysis, fixed point theory, nonlinear ordinary and partial differential equations, variational analysis, dynamical system theory, control theory, convex analysis, nonsmooth analysis, critical point theory, nonlinear optimization, fractional calculus and its applications, probability and statistics, mathematical economics, data mining, signal processing, biological engineering, electronic networks, electromagnetic theory, and so forth [...] Full article

(This article belongs to the Special Issue Advanced Theories and Novel Methods for Nonlinear Analysis, Optimization and Applications)

28 pages, 4319 KB

Open AccessArticle

Reliability-Based Multi-Objective Design of an FOPID Controller for Solar Furnaces Under Stochastic Parameter Uncertainties

by Mohamed Nejlaoui and Abdullah Alghafis

Mathematics 2026, 14(10), 1778; https://doi.org/10.3390/math14101778 - 21 May 2026

Viewed by 378

Abstract

Reliable solar energy harvesting demands advanced control strategies capable of maintaining thermal precision despite inherent environmental unpredictability. This research addresses the critical challenge of temperature regulation in the solar furnace system, which is hindered by severe non-linearities and stochastic environmental uncertainties. The study [...] Read more.

Reliable solar energy harvesting demands advanced control strategies capable of maintaining thermal precision despite inherent environmental unpredictability. This research addresses the critical challenge of temperature regulation in the solar furnace system, which is hindered by severe non-linearities and stochastic environmental uncertainties. The study aims to transition Fractional-Order PID (FOPID) control from theoretical design to reliable industrial application by accounting for the Uncertain Design Vector (UDV) during the tuning phase. A Reliability-Based Design Optimization (RBDO) framework is proposed, utilizing a hybrid Multi-Objective Imperialist Competitive Algorithm (MOICA) integrated with Monte Carlo Analysis (MCAR). This approach simultaneously optimizes the Maximum Sensitivity (Ms), the integral of Time-weighted Absolute Error (ITAE) and their sensitivities, while ensuring physical realizability through the FOPID structure. Crucially, the simulation results demonstrate that the RBDO-tuned FOPID design achieves optimal performance levels comparable to deterministic methods while significantly reducing the overall system sensitivity by 35% to 55% compared to both deterministic and literature-based methods (GA-FOPID and PSO-FOPID). The study concludes that integrating probabilistic reliability into multi-objective metaheuristics provides a robust control strategy for high-temperature solar facilities, effectively mitigating the performance degradation caused by real-world parameter fluctuations and ensuring consistent operational stability. Full article

(This article belongs to the Section E: Applied Mathematics)

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22 pages, 716 KB

Open AccessArticle

Bridging Markov Chain Monte Carlo Techniques and Tierney–Kadane Approximations for Progressively Censored Garhy Reliability Models: Simulation Insights and a Medical Application

by Abdullah H. Alenezy, Anis Ben Ghorbal, Khudhayr A. Rashedi and Ghareeb A. Marei

Mathematics 2026, 14(10), 1777; https://doi.org/10.3390/math14101777 - 21 May 2026

Cited by 2 | Viewed by 333

Abstract

This paper investigates the estimation of the stress–strength reliability parameter

R = P (Y < X)

when both stress and strength follow independent Garhy distributions under progressive Type-II censoring schemes. A closed-form expression for $R$ is explicitly derived, enabling effective [...] Read more.

This paper investigates the estimation of the stress–strength reliability parameter

R = P (Y < X)

when both stress and strength follow independent Garhy distributions under progressive Type-II censoring schemes. A closed-form expression for $R$ is explicitly derived, enabling effective and precise calculation without numerical integration. The Garhy distribution, a flexible one-parameter lifetime model with an increasing hazard function, is confirmed by full-scale goodness-of-fit diagnostics. A Bayesian estimation model is trained on non-informative priors (normal and extended Jeffreys priors) under squared error loss. The posterior expectations are analytically intractable; we adopt two complementary methods of computation: (i) Markov Chain Monte Carlo (MCMC) using the Metropolis–Hastings algorithm and (ii) the Tierney–Kadane (TK) approximation, which provides extremely precise analytical estimates with significantly reduced computational burden. Monte Carlo simulations are large-scale and compare the proposed estimators under different censoring schemes, sample sizes, and parameter configurations in terms of bias and mean squared error (MSE). The methodology is further applied to a real medical dataset comprising kidney dialysis patient survival times, demonstrating its practical relevance in clinical reliability assessment. Results consistently indicate that Bayesian methods, particularly with the extended Jeffreys prior, outperform classical MLEs in terms of stability and accuracy, especially under heavy censoring. Moreover, the TK approximation yields estimates virtually identical to MCMC while requiring only a fraction of the computational effort. We further extend the TK framework to approximate the posterior variance of R and the expected log-likelihood, providing a fully analytical alternative to MCMC for comprehensive Bayesian inference. Full article

(This article belongs to the Special Issue Reliability Estimation and Mathematical Statistics, 2nd Edition)

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24 pages, 1304 KB

Open AccessArticle

A Causally Constrained Framework Coupling Causal Discovery and SEIR Mechanisms for Interpretable Epidemic Modeling

by Rui Zhu, Yijiang Zhao, Zhixiong Fang and Yizhi Liu

Mathematics 2026, 14(10), 1776; https://doi.org/10.3390/math14101776 - 21 May 2026

Viewed by 373

Abstract

Infectious disease transmission is a complex dynamic process governed by intrinsic causal mechanisms rather than simple statistical correlations. Although deep learning paradigms have demonstrated powerful nonlinear representation capabilities, their “black-box” and purely data-driven nature often lead to a severe lack of causal consistency [...] Read more.

Infectious disease transmission is a complex dynamic process governed by intrinsic causal mechanisms rather than simple statistical correlations. Although deep learning paradigms have demonstrated powerful nonlinear representation capabilities, their “black-box” and purely data-driven nature often lead to a severe lack of causal consistency and logical transparency. To bridge this gap, this paper proposes CCSANet (Causally Constrained SEIR-Aware Network), an interpretable forecasting framework that seamlessly embeds epidemiological priors directly into the neural architecture. The model integrates SEIR dynamics into a temporal causal discovery framework, utilizing a mechanism-aware prior loss to guide a CausalFormer in learning a global temporal causal graph from multi-source heterogeneous data. This ensures that the identified relationships strictly adhere to the fundamental evolutionary logic of contagion. Subsequently, the extracted causal subgraphs are encoded as structural priors within a Causal-SCI-Block via a specialized masking mechanism, effectively forcing information to propagate exclusively along epidemiologically legitimate pathways. To ensure deep alignment between neural representations and physical reality, a causal strength alignment loss is introduced to synchronize the network’s attention weights with actual transmission intensities. Experimental evaluations on real-world multi-city datasets demonstrate that this integrated approach significantly outperforms baselines such as LSTM, Informer, and its predecessor, ESASNet. Under a 7-day sliding window configuration, the model maintains a Coefficient of Determination

R^{2}

stably above 0.97, achieving an accuracy improvement of 5.5% to 6.2% and an 8% to 10% reduction in SMAPE, thereby demonstrating that coupling causal discovery with SEIR constraints substantially enhances both predictive precision and physical interpretability. Full article

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13 pages, 322 KB

Open AccessArticle

Existence and Blow-Up of Compressible Spherically Symmetric Euler Equations with Vacuum Free Boundary

by Lijun Zhang, Junmei Shi and Chaudry Masood Khalique

Mathematics 2026, 14(10), 1775; https://doi.org/10.3390/math14101775 - 21 May 2026

Viewed by 397

Abstract

This paper studies the compressible spherically symmetric Euler equations with a vacuum free boundary, a fundamental model for astrophysical gas dynamics. We rigorously resolve an open problem by proving that nontrivial homogeneous linear velocity solutions exist if, and only if, the adiabatic exponent [...] Read more.

This paper studies the compressible spherically symmetric Euler equations with a vacuum free boundary, a fundamental model for astrophysical gas dynamics. We rigorously resolve an open problem by proving that nontrivial homogeneous linear velocity solutions exist if, and only if, the adiabatic exponent

γ = 4 / 3

, the critical value for monatomic gases and radiative stellar atmospheres. Using qualitative analysis of the reduced planar dynamical system, we characterize the flow’s global existence and finite-time blow-up behavior, establish a sharp existence threshold, and derive an explicit upper bound for the blow-up time. Quantitative energy estimates via Bernoulli’s head verify the physical consistency of solutions in both regimes. Our results complete the classification of self-similar solutions in this class, laying a rigorous theoretical foundation for planetary atmospheric gas flows and providing a practical criterion for predicting blow-up. Full article

(This article belongs to the Special Issue Computational Mechanics and Applied Mathematics, 2nd Edition)

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18 pages, 4476 KB

Open AccessArticle

High-Efficiency Lightweight Quantum Key Agreement Scheme Based on Bell State Entanglement

by Chunyu Zhang, Yanbing Liu, Yinghua Jiang and Sen Zheng

Mathematics 2026, 14(10), 1774; https://doi.org/10.3390/math14101774 - 21 May 2026

Viewed by 518

Abstract

To address the low qubit efficiency and high user-side operational complexity in existing quantum key agreement schemes, this paper proposes a high-efficiency and lightweight quantum key agreement scheme based on Bell states. The scheme is constructed upon a time-reversed EPR architecture, in which [...] Read more.

To address the low qubit efficiency and high user-side operational complexity in existing quantum key agreement schemes, this paper proposes a high-efficiency and lightweight quantum key agreement scheme based on Bell states. The scheme is constructed upon a time-reversed EPR architecture, in which a quantum server performs entangled-state preparation and Bell state measurement. Furthermore, a bidirectional decoy photon mechanism is incorporated into the architecture to achieve eavesdropping detection. By exploiting the completeness and orthogonality of the Bell basis, the scheme introduces an encoding mechanism based on local Pauli operations, enabling a single Bell state to carry 2 bits of key information and thereby realizing dense coding, which improves qubit efficiency. Meanwhile, users are only required to perform single-qubit operations, which reduces the quantum operational requirements on the user side. Based on the properties of Bell states, this paper derives the mapping relationship between local Pauli operations and Bell state measurement outcomes. Experimental results on the SpinQ Gemini quantum computing platform are consistent with the theoretical analysis, verifying the feasibility of the proposed scheme. In addition, security analysis shows that, owing to the bidirectional decoy photon mechanism, the scheme can resist various quantum attacks. The proposed scheme combines high efficiency with a lightweight implementation, reducing quantum hardware requirements on the user side and network deployment costs, thereby providing a cost-effective solution for practical quantum key agreement. Full article

(This article belongs to the Topic Future-Proofing Digital Trust: From Classical to Quantum-Safe Security)

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20 pages, 1418 KB

Open AccessArticle

A Multimodal Fake News Detection Method Based on Contrastive Learning and Variational Autoencoder

by Baowen Wu, Ruijiao Hu, Jilin Wang, Xin Sui, Jiaxing Sun, Jie Liu and Youli Qu

Mathematics 2026, 14(10), 1773; https://doi.org/10.3390/math14101773 - 21 May 2026

Viewed by 598

Abstract

Fake news often exhibits pronounced bias and misleading content. To foster a harmonious information environment, there is an urgent need for rapid fake news identification. Fake news detection can assess news authenticity by analyzing multidimensional information such as text, images, and comments. This [...] Read more.

Fake news often exhibits pronounced bias and misleading content. To foster a harmonious information environment, there is an urgent need for rapid fake news identification. Fake news detection can assess news authenticity by analyzing multidimensional information such as text, images, and comments. This automated approach significantly reduces human and material resource costs. However, existing detection methods often focus on extracting textual features, employing coarse-grained fusion techniques when integrating multi-modal information, and neglecting the inherent correlations between different modalities. Meanwhile, these methods rely on static network structures and fixed feature weighting strategies, lacking targeted neural network optimization and adaptive learning mechanisms, which results in insufficient interpretability and limited generalization performance across most detection approaches. To address these challenges, from the perspective of neural network optimization and regularization enhancement, this paper proposes a multi-modal fake news detection method based on contrastive learning and variational autoencoders. Firstly, we design a dual-contrastive learning loss function as a specialized regularization strategy for multimodal neural networks. By learning features through comparing similar and dissimilar samples, it more effectively captures correlations across multimodal data, optimizing the feature distribution and enhancing the model’s generalization capability via contrastive regularization. Second, it introduces a variational autoencoder to realize adaptive learning and dynamic weight optimization assigned to unimodal and multimodal features during decision-making. This adaptive mechanism enables the model to distinguish the relative importance of different modal information, optimizing the decision-making process of the multimodal neural network and thereby improving detection accuracy. Experiments conducted on the public Chinese dataset Weibo and English dataset Twitter demonstrate that the proposed optimized network architecture outperforms other multimodal methods by 3% to 8% in terms of detection accuracy, validating the superiority of this neural network optimization-based approach for multimodal fake news detection tasks. Full article

(This article belongs to the Special Issue Recent Advances of Neural Network Optimization and Algorithms in Deep Learning)

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15 pages, 582 KB

Open AccessArticle

Bayesian Estimation for α-Mixture Survival Models

by Feng Luan, Duchwan Ryu, Zhexuan Yang and Devrim Bilgili

Mathematics 2026, 14(10), 1772; https://doi.org/10.3390/math14101772 - 21 May 2026

Viewed by 276

Abstract

Heterogeneity in survival data poses substantial challenges for identifying appropriate mixture structures. The

α

-mixture family provides a flexible class of survival models that generalizes standard mixture formulations through a continuous weighting parameter, allowing it to balance failure rates and distributional shapes. Despite [...] Read more.

Heterogeneity in survival data poses substantial challenges for identifying appropriate mixture structures. The

α

-mixture family provides a flexible class of survival models that generalizes standard mixture formulations through a continuous weighting parameter, allowing it to balance failure rates and distributional shapes. Despite its theoretical appeal, the Bayesian inference for

α

-mixture survival models has received limited attention. In this paper, we develop a Bayesian framework for inference for

α

-mixture survival models, with a particular emphasis on estimation and structural identification. The posterior inference is conducted using Markov chain Monte Carlo methods, and simulation studies demonstrate accurate recovery of model parameters across a range of heterogeneous survival settings. The posterior distribution of the mixing parameter

α

offers a principled mechanism for model selection by identifying the mixture structure most consistent with the observed data. Applications to real-world datasets illustrate the interpretability and practical utility of the proposed approach in survival analysis. Full article

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28 pages, 6252 KB

Open AccessSystematic Review

Machine Learning-Enabled Robust Optimization for Green Vehicle Routing Problems: A Systematic Literature Review

by Wibi Anto, Herlina Napitupulu, Diah Chaerani and Adibah Shuib

Mathematics 2026, 14(10), 1771; https://doi.org/10.3390/math14101771 - 21 May 2026

Viewed by 832

Abstract

This systematic literature review (SLR) synthesizes current research on integrating machine learning (ML) into robust optimization (RO) frameworks for solving Green Vehicle Routing Problems (Green-VRP) under uncertainty. The key contributions include utilizing the EmbedSLR 2.0 framework for objective screening, establishing a functional ML [...] Read more.

This systematic literature review (SLR) synthesizes current research on integrating machine learning (ML) into robust optimization (RO) frameworks for solving Green Vehicle Routing Problems (Green-VRP) under uncertainty. The key contributions include utilizing the EmbedSLR 2.0 framework for objective screening, establishing a functional ML role taxonomy, and mapping uncertainty sets to computational tractability. Following PRISMA guidelines, searches across Scopus, Sage, and Dimensions identified 82 eligible studies validated through a three-point quality assessment scale. Bibliometric analysis indicates that the VRP has evolved into an interdisciplinary field that combines the power of rigorous RO with the integration capabilities of ML to achieve sustainability and resilience goals. Based on the results of the literature review, it was found that ML plays four crucial functional roles: as an end-to-end problem solver, a tool for predicting input parameters, a guide for search subroutines, and a mechanism for constructing more precise uncertainty sets. Various frameworks such as Adjustable Robust Optimization (ARO), Distributionally Robust Optimization (DRO), and Data-Driven Robust Optimization (DDRO) have been reported in various studies to offer improved cost efficiency and robustness compared to conventional static RO models by utilizing data more dynamically to reduce the level of conservatism. The integration of these environmental factors is carried out through emission and energy consumption parameters, which systematically give rise to operational trade-offs. This SLR has several limitations, including database and language limitations, the absence of cross-reference validation in EmbedSLR 2.0, and limitations in quality assessment. This publication is funded by the Universitas Padjadjaran through the LPDP on behalf of the Indonesian Ministry of Higher Education, Science and Technology and managed under the EQUITY Program (Contract No. 4303/B3/DT.03.08/2025 and 3927/UN6.RKT/HK.07.00/2025), as well as the Universitas Padjadjaran Research Grant under Research Grant for Graduate Students (Hibah Riset Melibatkan Mahasiswa Pascasarjana - RMMP) with contract number 5598/UN6.3.1/PT.00/2025. This systematic review was registered on the Open Science Framework (OSF) on 8 May 2026. Full article

(This article belongs to the Section D2: Operations Research and Fuzzy Decision Making)

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18 pages, 1326 KB

Open AccessArticle

Distributed Generalized Nash Equilibrium Seeking for Constrained Population Games via Consensus-Based Revision Protocols

by Jiajia Liu, Xuelei Fu and Ning Jiang

Mathematics 2026, 14(10), 1770; https://doi.org/10.3390/math14101770 - 21 May 2026

Viewed by 375

Abstract

This study addresses distributed decision-making in multi-agent systems under shared constraints. Existing methods often fail to guarantee strict constraint satisfaction or require sensitive parameter tuning. We propose a novel algorithm that integrates a revision protocol with a consensus mechanism. The key innovation is [...] Read more.

This study addresses distributed decision-making in multi-agent systems under shared constraints. Existing methods often fail to guarantee strict constraint satisfaction or require sensitive parameter tuning. We propose a novel algorithm that integrates a revision protocol with a consensus mechanism. The key innovation is a built-in, parameter-free constraint-checking function within the revision protocol, which automatically halts infeasible strategy updates. This approach enables agents using only local neighbor communication to seek a Generalized Nash Equilibrium (GNE). Theoretical analysis proves that the algorithm converges exponentially. Extensive simulations demonstrate its superiority: it achieves faster convergence and ensures strict per-iteration constraint satisfaction, significantly outperforming traditional gradient descent and penalty-based methods across various network topologies. Full article

(This article belongs to the Special Issue Optimization Theory, Algorithms and Applications)

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24 pages, 467 KB

Open AccessFeature PaperArticle

Atomic Contrastive Verification: Fine-Grained Fact-Checking via Claim Decomposition and Knowledge Graph-Grounded Contrastive Reasoning

by Hyeong-Geun Kim, Tea-Sung Jun and Taeseon Lee

Mathematics 2026, 14(10), 1769; https://doi.org/10.3390/math14101769 - 21 May 2026

Viewed by 893

Abstract

Large language models (LLMs) frequently produce text that is fluent yet factually inconsistent with source documents. Detecting such inconsistency remains challenging, particularly when errors involve subtle entity substitutions, temporal distortions, or relational misattributions embedded within lengthy outputs. We propose Atomic Contrastive Verification (ACV), [...] Read more.

Large language models (LLMs) frequently produce text that is fluent yet factually inconsistent with source documents. Detecting such inconsistency remains challenging, particularly when errors involve subtle entity substitutions, temporal distortions, or relational misattributions embedded within lengthy outputs. We propose Atomic Contrastive Verification (ACV), a training-free, graph-grounded fact-checking framework that decomposes both generated claims and source documents into atomic claims—minimal, self-contained factual units—and performs structured contrastive reasoning over each unit independently. For each atomic claim, ACV extracts a knowledge graph triple and generates contrastive claim variants through a multi-type perturbation taxonomy covering entity, relation, temporal, and quantitative dimensions. A novel Knowledge-Weighted Contrastive MMR mechanism, integrating graph-structural centrality and NLI-based logical diversity, selects the most discriminative subset of variants. Each selected variant is then pairwise compared against the claim; the resulting comparison responses are summarized to produce a per-claim verdict, and per-claim verdicts are aggregated into a document-level judgment. Experiments on the LLM-AggreFact benchmark (eleven subsets) demonstrate that ACV achieves competitive or superior performance compared to both specialized fine-tuned fact-checkers and large-scale LLMs. Beyond accuracy, ACV provides interpretable, claim-level error localization that existing methods cannot offer. Full article

(This article belongs to the Section E: Applied Mathematics)

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17 pages, 2094 KB

Open AccessArticle

Physics-Guided Graph Convolutional Network for Ship Structural Failure Mode Classification

by Shengpeng Li, Yi Xu, Hanxi Cao, Pengyu Wei, Ruonan Zhang and Zhikui Zhu

Mathematics 2026, 14(10), 1768; https://doi.org/10.3390/math14101768 - 21 May 2026

Viewed by 395

Abstract

Ship structural failure mode classification still relies heavily on subjective expert judgment, which is time-consuming and may introduce uncertainty in safety assessment. Although deep learning provides a promising avenue for automation, many existing learning approaches rely on 2D image representations and may therefore [...] Read more.

Ship structural failure mode classification still relies heavily on subjective expert judgment, which is time-consuming and may introduce uncertainty in safety assessment. Although deep learning provides a promising avenue for automation, many existing learning approaches rely on 2D image representations and may therefore suffer from geometric occlusion and information loss when projecting complex 3D stiffened structures. To address these challenges, we propose a Physics-Guided Graph Convolutional Network (PGGCN) for failure mode classification. Specifically, our method models finite-element (FE) meshes directly as graphs, preserving the holistic topology and displacement-field fidelity without viewpoint dependency. We further incorporate domain knowledge through a hybrid strategy: a Deep Graph Convolutional Network (DeepGCN) first detects local component buckling states such as plate or web buckling, and a logic matrix derived from classical failure definitions subsequently determines panel-level failure modes. To enable systematic evaluation, we construct a dataset spanning diverse stiffened-panel geometries via Latin Hypercube Sampling. Progressive analysis states from each loading case are organized into task-specific graph samples for supervised learning. Experiments on the test set achieve accuracies of 95.48% and 91.42% for plate- and web-buckling classification, respectively, and 89.56% for panel-level failure mode discrimination. These results demonstrate that the proposed method provides an interpretable framework for automated failure mode classification from FE meshes in ship stiffened panels. Full article

(This article belongs to the Special Issue Computational Intelligence for Multimodal Data Modeling and Perception)

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22 pages, 796 KB

Open AccessEditor’s ChoiceArticle

Multi-View Clustering via Projection-Enhanced Bipartite Graph Learning and Consensus Fusion

by Xun Liu, Qing-Wen Wang and Jiang-Feng Chen

Mathematics 2026, 14(10), 1767; https://doi.org/10.3390/math14101767 - 21 May 2026

Viewed by 365

Abstract

Anchor-based bipartite graph methods provide scalable solutions for multi-view clustering, but most of them construct graphs in the original feature space, where high dimensionality distorts the proximity between samples and anchors and degrades graph quality. In addition, the K-means step commonly used to [...] Read more.

Anchor-based bipartite graph methods provide scalable solutions for multi-view clustering, but most of them construct graphs in the original feature space, where high dimensionality distorts the proximity between samples and anchors and degrades graph quality. In addition, the K-means step commonly used to discretize spectral embeddings may produce different cluster assignments across random seeds. To address these limitations, this paper proposes projection-enhanced bipartite graph learning (PEBGL), which first projects each view onto a compact PCA subspace and then jointly performs bipartite graph construction, consensus graph fusion with adaptive view weighting, spectral embedding, and discrete label assignment within an alternating optimization framework. Most subproblems admit closed-form or efficient projection-based updates, and the final labels are obtained by connected-component detection on the learned consensus graph, reducing the dependence on K-means post-processing. Experiments on six benchmark datasets demonstrate that PEBGL achieves competitive clustering performance against recent graph-based and bipartite graph-based methods. These results validate the effectiveness of the proposed framework. Full article

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19 pages, 287 KB

Open AccessArticle

A Generalized Nonlinear Bagley–Torvik Equation in Distributions

by Chenkuan Li, Ehsan Pourhadi and Alison Gray

Mathematics 2026, 14(10), 1766; https://doi.org/10.3390/math14101766 - 21 May 2026

Viewed by 508

Abstract

This paper investigates the fractional calculus of distributions supported on

R^{+}

in the sense of L. Schwartz, based on distributional convolutions. We further study a generalized Bagley–Torvik equation involving an arbitrary number of fractional derivative terms with orders in the interval [...] Read more.

This paper investigates the fractional calculus of distributions supported on

R^{+}

in the sense of L. Schwartz, based on distributional convolutions. We further study a generalized Bagley–Torvik equation involving an arbitrary number of fractional derivative terms with orders in the interval

(0, 2)

. The existence and uniqueness of solutions for its nonlinear form are established in a space of continuous functions by applying Banach’s contraction principle, the Leray–Schauder fixed-point theorem, inverse operators, and the multivariate Mittag–Leffler function. Finally, several examples are presented, in which the values of multivariate Mittag–Leffler functions are computed to illustrate the main results. Full article

(This article belongs to the Special Issue Advanced Theoretical and Numerical Methods for Fractional Differential and Difference Equations)

23 pages, 2336 KB

Open AccessArticle

Extended State Observer-Based Design of a Bilateral Dual-Kernel Fuzzy Control Algorithm

by Chuqiang Liu, Lujun Chen, Zhulin Wang and Qunpo Liu

Mathematics 2026, 14(10), 1765; https://doi.org/10.3390/math14101765 - 21 May 2026

Viewed by 730

Abstract

For nonlinear problems in robotic systems, such as parametric uncertainties and external disturbances, this paper proposes a control method based on bilateral dual-kernel fuzzy control. To address the issue that joint angular velocities cannot be directly measured, an extended state observer (ESO) is [...] Read more.

For nonlinear problems in robotic systems, such as parametric uncertainties and external disturbances, this paper proposes a control method based on bilateral dual-kernel fuzzy control. To address the issue that joint angular velocities cannot be directly measured, an extended state observer (ESO) is introduced to simultaneously estimate the joint positions, velocities, and system nonlinearities, thereby achieving effective reconstruction of the system states. In terms of controller design, a dual-kernel function is adopted instead of the conventional single-kernel function. By exploiting its enhanced feature representation capability and fast response characteristics, the proposed approach improves the system dynamic response speed and reduces the settling time. For nonlinear residuals, the bilateral parallel control strategy further improves the approximation accuracy of the control system. Multiple dual-kernel fuzzy sub-controllers are integrated in a bilateral parallel manner, and the weighting parameters of both the fuzzy system and the bilateral structure are updated in real time based on the approximation error. This enables accurate approximation and compensation of the residuals estimated by the extended state observer. The stability of the closed-loop system is rigorously proved based on Lyapunov theory. Finally, simulations on the MATLAB R2022b platform and experiments on a robotic experimental platform are conducted to verify that the proposed bilateral dual-kernel fuzzy controller achieves significantly improved control accuracy for a two-degree-of-freedom robotic manipulator system compared with conventional controllers, thereby demonstrating the effectiveness and superiority of the proposed algorithm. Full article

(This article belongs to the Special Issue Recent Advances in Adaptive Control Theory and Applications for Nonlinear Systems)

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29 pages, 821 KB

Open AccessArticle

Optimisation of Fuzzy Reverse Logistics Networks for Express Packaging Considering Recycling Rates

by Kun Wang

Mathematics 2026, 14(10), 1764; https://doi.org/10.3390/math14101764 - 20 May 2026

Viewed by 466

Abstract

The recycling and reuse of discarded express delivery cartons can yield environmental, economic, and social benefits. A key factor influencing the volume of express packaging collected is the uncertainty in the total amount of such packaging within the service range of each collection [...] Read more.

The recycling and reuse of discarded express delivery cartons can yield environmental, economic, and social benefits. A key factor influencing the volume of express packaging collected is the uncertainty in the total amount of such packaging within the service range of each collection point. Additional uncertainties include the costs associated with the construction of recycling stations, operational expenses, transportation costs, additional recycling fees, and government subsidies. To address the issue of express packaging recycling, a fuzzy integer programming model for the reverse logistics network of express packaging is constructed. The model aims to minimise the total network cost and maximise the total recycling rate while enabling decisions regarding the location of recycling facilities and the flow between facilities. Then, a memetic algorithm based on dynamic local search is designed. Several alternative solution approaches were considered to evaluate the proposed algorithm, including the precision optimization method (CPLEX) and a hybrid priority-based genetic algorithm. The results confirm the feasibility of the memetic algorithm. Finally, the applicability of this fuzzy programming model is analysed and validated by changing the confidence level. The case study results reveal quantifiable trade-offs: as the confidence level (α) increases from 0.75 to 0.90 under a fixed recycling rate threshold (ε = 80%), the total network cost rises approximately linearly, while the required number of recycling stations increases, with their average facility level upgrading accordingly. Variations in confidence levels and the degree of total recycling rate achievement can significantly influence the increase in target values. Moreover, the magnitude of this influence exhibits irregularity, indicating that changes in confidence levels entail a certain degree of risk. Full article

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12 pages, 448 KB

Open AccessArticle

Parity-Based Level-Set Approach to the Collatz Conjecture

by Selcuk Koyuncu, Thevasha Sathiyakumar, Praise Alayode, Christopher Ellis and Peyton Thomas

Mathematics 2026, 14(10), 1763; https://doi.org/10.3390/math14101763 - 20 May 2026

Viewed by 456

Abstract

The Collatz conjecture concerns the iteration of the map

f (n) = 3 n + 1

for odd n and

f (n) = n / 2

for even n. In this paper, we study the level sets [...] Read more.

The Collatz conjecture concerns the iteration of the map

f (n) = 3 n + 1

for odd n and

f (n) = n / 2

for even n. In this paper, we study the level sets

l_{x} = {n \in N ∣ L (n) = x}

, where

L (n)

denotes the Collatz length. Using the parity representation of Collatz trajectories, we partition each

l_{x}

according to the number of odd steps and analyze the corresponding means

μ_{x, k}

. Under a natural scaling assumption, these means satisfy an approximate geometric progression, so that

log μ_{x, k}

is approximately linear in k. Computations for

n \leq 100, 000

and

10 \leq x \leq 50

show highly stable regression parameters and near-perfect linear fits. Full article

(This article belongs to the Section E: Applied Mathematics)

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30 pages, 3632 KB

Open AccessArticle

Intermittent Control for Synchronization-like Behavior of State-Dependent Impulsive Neural Networks via Interval–Impulse Differential Inequality

by Yanshou Dong, Junfang Zhao, Jinqiu Li, Tingting Dai and Yan Han

Mathematics 2026, 14(10), 1762; https://doi.org/10.3390/math14101762 - 20 May 2026

Viewed by 341

Abstract

This paper investigates the synchronization problem for a class of master–slave neural networks with state-dependent impulses. Different from fixed-time impulsive systems, the impulsive instants considered here depend on the current states of the neural networks, which makes the synchronization analysis more complicated. In [...] Read more.

This paper investigates the synchronization problem for a class of master–slave neural networks with state-dependent impulses. Different from fixed-time impulsive systems, the impulsive instants considered here depend on the current states of the neural networks, which makes the synchronization analysis more complicated. In particular, when both the master and slave systems possess their own state-dependent impulses, the corresponding impulsive instants are generally asynchronous, so the synchronization error evolves over an impulsive interval rather than undergoing only a single instantaneous jump. To address this difficulty, two easily verifiable conditions are first proposed to guarantee that each trajectory intersects every impulsive surface exactly once, thereby excluding the beating phenomenon. Then, an interval–impulse differential inequality is established to characterize the error evolution on non-impulsive subintervals and to handle the mismatch between the impulsive times of the master and slave systems. Based on this inequality, an intermittent controller activated only outside the impulsive interval is designed so that the controller does not destroy the intrinsic state-dependent impulsive rhythm of the master system. By combining Lyapunov analysis with matrix inequality techniques, verifiable criteria are derived for local exponential synchronization-like behavior of the considered neural networks. Here, synchronization-like refers to exponential decay of the synchronization error on the non-mismatched time intervals since the master and slave systems generally possess asynchronous state-dependent impulsive instants. Finally, numerical examples are presented to illustrate the effectiveness of the proposed conditions and control strategy. The simulation results show that the designed controller can effectively suppress synchronization error and that increasing the control gain can significantly accelerate the convergence process. Full article

(This article belongs to the Section E: Applied Mathematics)

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29 pages, 2786 KB

Open AccessArticle

Enhanced Transmission Loss and Modal Coupling in Dual-Membrane Flexible-Shell Cylindrical Waveguides: A Rigorous Mode-Matching–Galerkin Framework

by Mohammed Alkinidri

Mathematics 2026, 14(10), 1761; https://doi.org/10.3390/math14101761 - 20 May 2026

Viewed by 284

Abstract

This paper develops an analytical treatment of vibro-acoustic wave propagation in a cylindrical waveguide containing two clamped elastic membranes and a central flexible-shell segment. The acoustic field obeys the time-harmonic Helmholtz equation, the shell motion is described by Donnell–Mushtari thin-shell theory under axisymmetric [...] Read more.

This paper develops an analytical treatment of vibro-acoustic wave propagation in a cylindrical waveguide containing two clamped elastic membranes and a central flexible-shell segment. The acoustic field obeys the time-harmonic Helmholtz equation, the shell motion is described by Donnell–Mushtari thin-shell theory under axisymmetric loading, and the membrane response is governed by classical membrane theory and incorporated through a tailored Galerkin scheme. The resulting coupled fluid–structure boundary-value problem is solved by the Mode-Matching Method: the acoustic potentials are expanded in orthogonal radial eigenfunctions within each subregion, and continuity of pressure, normal velocity, and structural displacement are enforced at every interface. The mirror symmetry of the configuration is exploited by an exact decomposition into symmetric and anti-symmetric sub-problems, each of which reduces to a truncated linear algebraic system of dimension

4 N + 4

for the unknown modal amplitudes. Acoustic power-balance identities provide a quantitative consistency check on the numerical implementation and diagnose convergence with respect to the truncation order; structural damping is accommodated through complex-modulus substitutions for the shell and the membrane tension without altering the algebraic structure of the system. The numerical results demonstrate that the dual-membrane configuration delivers transmission-loss values exceeding

25 dB

across the low-frequency band relevant to HVAC and automotive applications, with a representative plateau near

13 dB

at the reference geometry, through resonance-driven modal coupling between the acoustic field and the compliant interfaces. Parametric studies identify the excitation frequency, the inner-membrane radius, the shell radius, and the chamber length as effective design parameters for tuning the attenuation. The formulation furnishes a unified and computationally efficient analytical tool for predicting and optimising noise attenuation in flexibly coupled cylindrical duct systems. Full article

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

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41 pages, 1014 KB

Open AccessArticle

Geometric Structure of Genomes Across the Tree of Life: Toward a Geometric Theory of Sequence Structure

by Valentin E. Brimkov and Reneta P. Barneva

Mathematics 2026, 14(10), 1760; https://doi.org/10.3390/math14101760 - 20 May 2026

Viewed by 355

Abstract

This work develops a geometric and statistical framework for analyzing the structure of biological sequences and explores its implications for understanding the emergence and evolution of life. Motivated by questions concerning the transition from prebiotic chemistry to living systems, the quantification of negentropy [...] Read more.

This work develops a geometric and statistical framework for analyzing the structure of biological sequences and explores its implications for understanding the emergence and evolution of life. Motivated by questions concerning the transition from prebiotic chemistry to living systems, the quantification of negentropy in organic matter, and the distinction between random and biologically viable sequences, we introduce mathematical descriptors that measure deviation from linearity and related geometric irregularities of self-replicating macromolecules. These descriptors reveal a pronounced geometric separation between biological DNA and random sequences, underscoring the non-random structural organization characteristic of living systems. Using these descriptors, we compare a broad range of species across the Tree of Life and examine how geometric complexity varies between primitive and more advanced organisms. We further investigate whether these measures provide a natural way to compare organismal complexity, characterize the structure of viable sequence space, and identify potential constraints on evolutionary trajectories. The framework also offers an initial perspective on how natural selection and stochastic mutations may jointly influence genomic organization. Finally, we outline speculative connections between increasing geometric irregularity and the emergence of biological complexity, suggesting that such geometric transitions may offer insight into the origins of life and the theoretical limits of evolutionary development. Full article

(This article belongs to the Section E3: Mathematical Biology)

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37 pages, 4975 KB

Open AccessFeature PaperArticle

Fuzzy Iterative Learning Contouring Control

by Thanh-Quan Ta and Shyh-Leh Chen

Mathematics 2026, 14(10), 1759; https://doi.org/10.3390/math14101759 - 20 May 2026

Viewed by 370

Abstract

Iterative learning contouring control (ILCC) improves contouring accuracy in multi-axis motion systems via the equivalent contour error formulation. However, its convergence strongly depends on the learning gain. Large gains may induce overly aggressive updates and local divergence, degrading performance, whereas small gains lead [...] Read more.

Iterative learning contouring control (ILCC) improves contouring accuracy in multi-axis motion systems via the equivalent contour error formulation. However, its convergence strongly depends on the learning gain. Large gains may induce overly aggressive updates and local divergence, degrading performance, whereas small gains lead to slow convergence. Moreover, contour error convergence is typically non-uniform along the trajectory, and local divergence may still occur despite global convergence, particularly near error saturation regions. To address these issues, a fuzzy inference mechanism is integrated into the online ILCC framework, yielding an online ILCC with fuzzy-regulated convergence parameters (online ILCCf), enabling adaptive regulation of the learning gain. Two regulation strategies are developed: (i) online ILCCf_i, an independent multi-parameter regulation scheme; and (ii) online ILCCf_u, a unified single-parameter regulation scheme. The fuzzy mechanism adaptively adjusts the convergence parameters online according to the instantaneous magnitude of the equivalent contour error. Experimental results on a six-axis industrial robot demonstrate fast convergence while maintaining satisfactory contouring performance. Among all comparison cases, online ILCCf_i achieves the best performance, reducing the RMS position error from

7.26 \times 10^{- 1}

mm to

5.93 \times 10^{- 2}

mm and the RMS orientation error from

6.95 \times 10^{- 4}

rad to

5.64 \times 10^{- 5}

rad, without oscillation or local divergence. Further simulations confirm robustness under model uncertainty and measurement noise. Full article

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41 pages, 21816 KB

Open AccessArticle

Geometric Interpretation of Frequency Domain Robustness Constraints and Closed-Loop Pole Locations

by Vesela Karlova-Sergieva

Mathematics 2026, 14(10), 1758; https://doi.org/10.3390/math14101758 - 20 May 2026

Viewed by 474

Abstract

Requirements for robustness and performance in the frequency domain in control theory are usually formulated as constraints on the modulus of complex functions describing the open-loop system, the sensitivity function, and the complementary sensitivity function. These constraints generate circular sets that can be [...] Read more.

Requirements for robustness and performance in the frequency domain in control theory are usually formulated as constraints on the modulus of complex functions describing the open-loop system, the sensitivity function, and the complementary sensitivity function. These constraints generate circular sets that can be interpreted as admissible or forbidden regions in the complex plane, particularly in the Nyquist diagram. In engineering practice, they are often treated as method-specific constructions, without clarifying the general geometric mechanism by which they arise. This study develops a geometric interpretation in which a broad class of frequency-domain robustness constraints is represented as level sets of analytic and fractional-linear functions. The resulting circular sets in the Nyquist plane are characterized in a unified manner and mapped to admissible regions in the complex s-plane through preimage transformations. The approach is formulated entirely using complex transfer functions, remaining within the classical frequency-domain framework, without state-space representations, linear matrix inequalities, or optimization methods. Classical robustness measures, including gain margin, phase margin, and constraints on sensitivity and complementary sensitivity, are shown to be special cases of the same geometric structure. The main insight of this work is that these apparently different robustness constraints arise from the same underlying geometric mechanism. This interpretation establishes a geometric link between frequency domain robustness constraints and the location of closed-loop poles, allowing a qualitative assessment of robustness and dynamic properties of control systems without introducing new stability criteria or design procedures. The resulting admissible regions provide a geometric interpretation of frequency domain robustness specifications in terms of pole locations in the s-plane. Full article

(This article belongs to the Special Issue Advances in Robust Control Theory and Its Applications)

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15 pages, 318 KB

Open AccessArticle

Relational Nonlinear Almost Contractions of Mukherjea Type and Applications to Boundary Value Problems

by Doaa Filali, Mohammed Zayed Alruwaytie, Abdulaziz Abbas Alshammari, Adel Alatawi, Fahad M. Alamrani and Faizan Ahmad Khan

Mathematics 2026, 14(10), 1757; https://doi.org/10.3390/math14101757 - 20 May 2026

Viewed by 391

Abstract

In this paper, we explore specific fixed-point outcomes under a Mukherjea-Type nonlinear description of almost contractions in a metric space with a locally ℏ-transitive binary relation. Numerous previous insights are expanded, developed, improved, and consolidated in the outcomes reported herein. To demonstrate [...] Read more.

In this paper, we explore specific fixed-point outcomes under a Mukherjea-Type nonlinear description of almost contractions in a metric space with a locally ℏ-transitive binary relation. Numerous previous insights are expanded, developed, improved, and consolidated in the outcomes reported herein. To demonstrate the reliability of our results, a couple of instances are supplied. Our outcomes are deployed to evaluate the accuracy of the unique solution of a boundary value problem. Full article

22 pages, 4316 KB

Open AccessArticle

Spatiotemporal Forecasting of Seismic Activity Trends Using Wiener Filtering and Artificial Neural Networks

by Pengfei Ren, Peijia Li, Xiaoyang Chen, Tingkai Gu, Xiaoyu Song, Cong Wang and Kai Yan

Mathematics 2026, 14(10), 1756; https://doi.org/10.3390/math14101756 - 20 May 2026

Viewed by 452

Abstract

Reliable forecasting of seismic activity trends is essential for regional seismic hazard analysis. Based on earthquake catalogs from 1500 to 2026, this study investigates the spatiotemporal evolution of seismic activity in the North-South Seismic Belt using a hybrid framework that integrates Wiener filtering [...] Read more.

Reliable forecasting of seismic activity trends is essential for regional seismic hazard analysis. Based on earthquake catalogs from 1500 to 2026, this study investigates the spatiotemporal evolution of seismic activity in the North-South Seismic Belt using a hybrid framework that integrates Wiener filtering and artificial neural networks. Seismic activity is modeled as a discrete-time stochastic process, and a time series of earthquakes with magnitudes ≥ 6.0 is constructed. Wiener filtering is applied to establish an optimal linear relationship between input and output under the minimum mean square error criterion, and multi-origin extrapolation is employed to predict earthquakes with magnitudes ≥ 7.0 over the next century. The results reveal several stable peaks or peak clusters that agree well with historical strong earthquakes, with prediction errors generally within approximately three years. Sensitivity analyses indicate that longer time series (∼500 years) and higher threshold magnitudes (≥6.0) enhance prediction stability, although the method shows limitations in spatial prediction. To address this issue, a 16–8–4 artificial neural network model is developed, and seismic sequence features are extracted using a sliding time window approach to perform both temporal and spatial forecasting. The artificial neural network achieves high accuracy in temporal prediction (maximum error ≈ 0.5) and outperforms Wiener filtering in spatial prediction, capturing the migration characteristics of seismic activity. The results further suggest that earthquakes with magnitudes ≥ 7.0 are more likely to occur within the latitude range of 30.5–33.0° N in the near future. Full article

(This article belongs to the Special Issue Advances of Mathematics and Artificial Intelligence in Engineering Applications)

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16 pages, 1770 KB

Open AccessArticle

A Hybrid AI Approach for Intelligent Group Buying and Digital Marketing Strategy Optimization Based on Machine Learning and Evolutionary Algorithms

by Zhansaya Abildaeva, Raissa Uskenbayeva, Zhuldyz Kalpeyeva, Aizhan Kassymova, Aigul Dauitbayeva and Adranova Asselkhan

Mathematics 2026, 14(10), 1755; https://doi.org/10.3390/math14101755 - 20 May 2026

Viewed by 454

Abstract

This study considers the digital transformation of Kazakhstan’s agro-industrial complex, which has created an urgent need for scientifically grounded methods that can optimize marketing strategies under conditions of resource limitations, production seasonality, and heterogeneous consumer behavior. This study proposes a hybrid decision-support framework [...] Read more.

This study considers the digital transformation of Kazakhstan’s agro-industrial complex, which has created an urgent need for scientifically grounded methods that can optimize marketing strategies under conditions of resource limitations, production seasonality, and heterogeneous consumer behavior. This study proposes a hybrid decision-support framework integrating a modified NSGA-III algorithm with machine learning techniques for optimizing digital marketing strategies in the agro-industrial complex of Kazakhstan. The model considers three objectives: maximizing channel efficiency and audience reach while minimizing marketing costs. Experimental results based on a dataset of N = 1200 observations demonstrate that the proposed approach improves the composite performance indicator by 12.4% compared to baseline single-objective optimization methods. Pareto front analysis reveals three distinct clusters of strategies, corresponding to (1) high-impact integrated digital TV strategies, (2) cost-efficient traditional channel strategies, and (3) high-risk high-return allocations. The clustering validity is confirmed by a silhouette score of 0.624, indicating strong separation between strategy groups. The results highlight the practical significance of adaptive budget allocation and demonstrate the effectiveness of combining evolutionary optimization with machine learning for decision support in complex marketing environments. Full article

(This article belongs to the Topic AI and Data-Driven Advancements in Industry 4.0, 2nd Edition)

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