Mathematics

32 pages, 4720 KB

Open AccessArticle

Explainability-Guided Transformer Models for Hourly Cryptocurrency Forecasting: A Comparative Study with SHAP-Based Feature Refinement

by Zeynep Hilal Kilimci and Erçin Dinçer

Mathematics 2026, 14(18), 3402; https://doi.org/10.3390/math14183402 (registering DOI) - 19 Sep 2026

Abstract

Accurate cryptocurrency price forecasting represents an important yet challenging problem in financial time-series analysis due to the highly volatile, nonlinear, and noise-sensitive nature of digital asset markets. Although transformer-based architectures have recently demonstrated strong capabilities in temporal sequence modeling, their behavior under high-frequency [...] Read more.

Accurate cryptocurrency price forecasting represents an important yet challenging problem in financial time-series analysis due to the highly volatile, nonlinear, and noise-sensitive nature of digital asset markets. Although transformer-based architectures have recently demonstrated strong capabilities in temporal sequence modeling, their behavior under high-frequency cryptocurrency dynamics and the role of explainability-guided feature refinement remain insufficiently explored. To address this gap, this study presents a comprehensive transformer-based forecasting framework for hourly cryptocurrency price prediction and investigates the impact of explainability-guided feature optimization on forecasting performance, robustness, and interpretability. Five transformer architectures—Vanilla Transformer, Informer, Autoformer, Reformer, and Temporal Fusion Transformer (TFT)—are systematically evaluated across five major cryptocurrency assets: Bitcoin (BTC), Ethereum (ETH), Solana (SOL), Dogecoin (DOGE), and Ripple (XRP). The experimental framework employs Open, High, Low, Close, and Volume (OHLCV) data together with a broad set of engineered technical indicators and evaluates model performance using Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), Mean Squared Error (MSE), Mean Absolute Percentage Error (MAPE), and the coefficient of determination (R²). To improve interpretability and reduce feature redundancy, SHAP (SHapley Additive exPlanations) and LIME (Local Interpretable Model-agnostic Explanations) are integrated directly into the forecasting pipeline. Based on the resulting explanations, asset-specific feature subsets are constructed, and all models are subsequently retrained using the refined feature representations. The results show that explainability-guided feature refinement provides compact, model-aware, and interpretable feature subsets with competitive forecasting performance; however, its effect on prediction accuracy is dependent on the cryptocurrency asset, transformer architecture, retained feature subset size, and market conditions. Additional robustness, sensitivity, alternative feature-selection, and statistical significance analyses indicate that the SHAP–LIME Top-15 subset should be interpreted as a conservative dimensionality-reduction strategy rather than a universally optimal feature-selection rule. The findings further reveal that transformer architectures incorporating sparse attention, decomposition mechanisms, or gating structures generally provide stronger performance than the Vanilla Transformer under highly volatile hourly market conditions. Overall, the proposed framework demonstrates that combining transformer-based forecasting with explainability-guided feature refinement can support interpretable and parsimonious high-frequency financial time-series modeling, while highlighting the importance of evaluating robustness, feature-selection sensitivity, and statistical variability alongside average forecasting errors. Full article

(This article belongs to the Special Issue Advances in Machine Learning Applied to Financial Economics)

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26 pages, 2567 KB

Open AccessArticle

A Recentered-Domain Yau–Yau Filter with Reduced-Order FKE Propagation for Nonlinear State Estimation

by Lei Ma, Yuzhong Hu and Xiaoming John Zhang

Mathematics 2026, 14(18), 3401; https://doi.org/10.3390/math14183401 (registering DOI) - 19 Sep 2026

Abstract

Nonlinear filtering can be formulated as the propagation and update of conditional probability densities, but direct numerical propagation of the associated Forward Kolmogorov equation (FKE) over a large fixed domain is computationally expensive. This paper proposes a Recentered-Domain Yau–Yau Filter (RD-YYF) with reduced-order [...] Read more.

Nonlinear filtering can be formulated as the propagation and update of conditional probability densities, but direct numerical propagation of the associated Forward Kolmogorov equation (FKE) over a large fixed domain is computationally expensive. This paper proposes a Recentered-Domain Yau–Yau Filter (RD-YYF) with reduced-order FKE propagation for nonlinear state estimation. The method solves the FKE on a fixed-size local computational window centered at the latest state estimate, thereby concentrating numerical resolution near the dominant posterior density. In the offline stage, physics-informed neural networks (PINNs) generate FKE solution snapshots, principal component analysis constructs a low-dimensional representation of density evolution, and a lightweight residual surrogate maps initial-condition coefficients and the domain center to terminal-solution coefficients. In the online stage, the pretrained surrogate performs per-timestep density prediction within the recentered window, followed by observation update and state estimation. Numerical experiments on two geometrically constrained target-tracking models show that RD-YYF achieves lower tracking errors than the extended Kalman filter and particle filter under matched online evaluation conditions. A fixed-domain ablation further shows that recentering improves density approximation in high-probability regions and reduces offline PINN training epochs. These results indicate that recentered-domain reduced-order FKE propagation is a practical computational strategy for nonlinear density-based filtering. Full article

(This article belongs to the Special Issue Computational Mathematics Methods and Applications in Engineering Science)

33 pages, 2722 KB

Open AccessArticle

Multi-Agent Cooperative Navigation Algorithm Based on Trajectory Optimization with Global Structural Constraints and Attention-Enhanced Graph Neural Networks

by Bin Zhao, Ruohuai Sun, Zhenyu Liu and Xiang Li

Mathematics 2026, 14(18), 3400; https://doi.org/10.3390/math14183400 (registering DOI) - 19 Sep 2026

Abstract

To address the challenges of formation maintenance, collision avoidance, and cooperative navigation for multi-agent systems in shared environments, a spatio-temporal trajectory optimization method under global structural constraints and A attention-enhanced graph neural network (AE-GNN)-based cooperative navigation strategy are proposed. First, a differentiable formation [...] Read more.

To address the challenges of formation maintenance, collision avoidance, and cooperative navigation for multi-agent systems in shared environments, a spatio-temporal trajectory optimization method under global structural constraints and A attention-enhanced graph neural network (AE-GNN)-based cooperative navigation strategy are proposed. First, a differentiable formation similarity metric based on graph Laplacian theory is constructed to characterize the intrinsic geometric structure of formations, which is invariant to translation, rotation, and scale transformations. Based on this metric, a distributed trajectory optimization framework is developed by jointly considering formation maintenance, obstacle avoidance, inter-agent collision avoidance, and trajectory smoothness. Furthermore, an adaptive formation constraint adjustment mechanism based on environmental passage margin is introduced, allowing agents to relax rigid formation constraints in narrow passages and gradually recover the desired formation after obstacle avoidance. To improve cooperative decision-making in densely cluttered environments, an AE-GNN navigation method is proposed, where an attention mechanism dynamically assigns higher weights to critical neighboring agents during feature aggregation, enhancing cooperative information interaction and collision avoidance capability. Simulation and physical experiments validate the effectiveness of both proposed methods. The trajectory optimization method achieves a success rate of 97.5% in dense-obstacle formation-navigation simulations, while AE-GNN demonstrates effective cooperative navigation and collision avoidance in multi-agent scenarios. Full article

(This article belongs to the Special Issue Learning-Based Control of Networked Systems)

29 pages, 5663 KB

Open AccessArticle

A Brinkman-Penalized Finite Element Method for the Boussinesq Equations with Immersed Rigid Obstacles Under Uncertain Obstacle Motion

by Zhadra Zhaxylykova, Nurlana Alimbekova, Farida Amenova and Nurlan Temirbekov

Mathematics 2026, 14(18), 3399; https://doi.org/10.3390/math14183399 (registering DOI) - 19 Sep 2026

Abstract

The Boussinesq equations are widely used to model incompressible thermally driven flows, but their numerical simulation becomes more challenging in domains containing fixed or moving rigid obstacles, particularly when the prescribed obstacle motion is uncertain. In this work, we develop a Brinkman-type fictitious-domain [...] Read more.

The Boussinesq equations are widely used to model incompressible thermally driven flows, but their numerical simulation becomes more challenging in domains containing fixed or moving rigid obstacles, particularly when the prescribed obstacle motion is uncertain. In this work, we develop a Brinkman-type fictitious-domain formulation for the incompressible Boussinesq system on a fixed computational domain. Brinkman penalization is used to impose the prescribed solid velocity, while a thermal penalty term enforces the temperature inside the immersed bodies. The resulting problem is discretized by a finite element method using skew-symmetric convective forms and grad-div stabilization. The uncertainty in the obstacle motion is introduced through a random oscillation amplitude and treated by a non-intrusive stochastic collocation method based on Gauss–Legendre quadrature. Stability and convergence of the fully discrete scheme are established. Numerical experiments for moving heated and hot–cold obstacles demonstrate the influence of the uncertain oscillation amplitude on the Nusselt number, kinetic energy, mean vorticity, and the spatial distributions of the mean and standard deviation of the solution fields. The results indicate that the proposed method provides a stable numerical framework for simulating incompressible thermally coupled flows with moving immersed obstacles under uncertainty in the prescribed obstacle motion. Full article

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

31 pages, 4284 KB

Open AccessArticle

Stability and Bifurcation Structure of a Discrete-Time Chemostat Model with Nutrient Recycling

by Saad Jamhan Aldosari

Mathematics 2026, 14(18), 3398; https://doi.org/10.3390/math14183398 (registering DOI) - 19 Sep 2026

Abstract

This work aims to study the dynamic behavior of a discrete-time chemostat model involving the effect of nutrient recycling, which is a common biological phenomenon usually left out in conventional models. Based on a continuous time model involving the interrelationship between bacterial population [...] Read more.

This work aims to study the dynamic behavior of a discrete-time chemostat model involving the effect of nutrient recycling, which is a common biological phenomenon usually left out in conventional models. Based on a continuous time model involving the interrelationship between bacterial population and nutrient concentration, a new system with nutrient recycling term is obtained by introducing the process of nutrient recycling from biomass. This model is subsequently transformed to a discrete form using the forward Euler approximation scheme. Existence and stability of all possible equilibria are examined using the characteristic equation. Necessary conditions for stability and bifurcations, including transcritical and flip types, have been derived. It turns out that the considered model cannot experience Neimark–Sacker bifurcation. The results indicate that the recycling parameter plays an important role in changing stability regions and can generate complex dynamics, including oscillations and chaos. Numerical simulations are used to verify the analytical results and to show transitions between different types of system behavior. The results imply that the relationship between nutrient input, bacterial proliferation, and the recycling process has an essential influence on the dynamics and improves the comprehension of discrete ecological models. Full article

(This article belongs to the Special Issue On Stability, Bifurcation, and Chaos in Dynamical Systems: Theory and Applications)

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31 pages, 1166 KB

Open AccessArticle

Aggregation Semantics for Prioritization in Cooperative Automated Decision-Making

by Pavel Novoa-Hernández, Mariia Godz and David A. Pelta

Mathematics 2026, 14(18), 3397; https://doi.org/10.3390/math14183397 (registering DOI) - 19 Sep 2026

Abstract

Cooperative automated decision-making systems can combine predictive assessments with contextual knowledge to prioritize alternatives under resource constraints. The mechanism used to integrate these sources can influence whether contextual requirements are preserved or can be offset by predictive evidence. This study examines this issue [...] Read more.

Cooperative automated decision-making systems can combine predictive assessments with contextual knowledge to prioritize alternatives under resource constraints. The mechanism used to integrate these sources can influence whether contextual requirements are preserved or can be offset by predictive evidence. This study examines this issue in CADEMAS-ML, a recently proposed framework for cooperative automated decision-making that separates predictive and contextual decision pathways. Three aggregation regimes are analyzed: linear, weighted geometric, and minimum aggregation. The study combines theoretical analysis, controlled Monte Carlo experiments, and an employee-attrition case study. The theoretical analysis characterizes the conditions under which contextual vetoes are preserved during integration. The simulation study evaluates policy compliance, sensitivity to predictive uncertainty, ranking stability, and changes in the Top-K intervention set. The case study applies the same aggregation mechanisms to a reproducible CADEMAS-ML workflow based on predictive models and fuzzy contextual rules. The results show that aggregation semantics can affect policy compliance and prioritization outcomes. Under the considered conditions, zero-absorbing operators preserve contextual vetoes, whereas linear aggregation can allow predictive evidence to compensate for a contextual veto. This effect depends on the relative weighting of the two sources and on the characteristics of the decision population. Predictive uncertainty can also modify rankings and intervention sets, with its effects depending on the aggregation regime and intervention capacity. In the employee-attrition case study, changing only the aggregation mechanism produces different intervention sets from the same predictive and contextual assessments. These findings support treating aggregation as an explicit design choice in cooperative automated decision-making, since its semantics influence how contextual requirements are translated into automated prioritization. Full article

(This article belongs to the Special Issue Advanced Fuzzy Optimization and Decision Making)

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30 pages, 11017 KB

Open AccessArticle

Chaotic-Saddle-Organized Hidden Bursting Oscillations in a 4D Slow–Fast System with No Equilibria

by Shaolong Li, Haibo Jiang, Weipeng Lyu, Liping Zhang, Lizhou Zhuang, Juanjuan Huang, Bo Liang and Zhenyang Chen

Mathematics 2026, 14(18), 3396; https://doi.org/10.3390/math14183396 (registering DOI) - 19 Sep 2026

Abstract

Bursting oscillations, characterized by alternating quiescent and spiking states, are a fundamental class of nonlinear dynamics that play a crucial role in diverse scientific fields. Therefore, understanding the mechanisms that generate bursting is of both theoretical and practical importance. Bursting is typically studied [...] Read more.

Bursting oscillations, characterized by alternating quiescent and spiking states, are a fundamental class of nonlinear dynamics that play a crucial role in diverse scientific fields. Therefore, understanding the mechanisms that generate bursting is of both theoretical and practical importance. Bursting is typically studied as a class of self-excited or forced oscillations in classical slow–fast dynamics, with analysis relying on stable invariant sets and their bifurcations. In contrast to these cases, we extend the analysis to hidden bursting in a 4D slow–fast system with no equilibria using classical slow–fast decomposition. Specifically, we show how chaotic saddle dynamics in the fast subsystem organize the bursting mechanism. Bifurcation and basin analyses identify two symmetric chaotic saddle ranges between interior crises and the folds of limit cycles. When quiescent-to-spiking transitions fall within these ranges, chaotic saddles render the full system highly sensitive to parameters, acting as the organizing center for hidden bursting. By tuning the slow adaptation strength, we uncover three transition-path modes: lift-captured double-reversal (LCDR), lift-escape single-reversal (LESR), and direct-captured no-reversal (DCNR). These modes and their combinations classify the observed bursting patterns. These findings show that non-attracting invariant sets can affect bursting dynamics, offering a new perspective on bursting dynamics. Full article

(This article belongs to the Special Issue Bifurcation Theory and Qualitative Analysis of Dynamical Systems)

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34 pages, 6452 KB

Open AccessArticle

Production-Profile Interpretation for Hydraulically Fractured Horizontal Gas Wells Using DTS Temperature Responses and Dual-Objective Inversion

by Zhe Zhang, Qinfeng Su, Yi Yang and Kuan Sun

Mathematics 2026, 14(18), 3395; https://doi.org/10.3390/math14183395 (registering DOI) - 18 Sep 2026

Abstract

Production-profile interpretation based on distributed temperature sensing (DTS) is an important method for the dynamic evaluation of hydraulically fractured horizontal gas wells. Because downhole temperature is jointly controlled by reservoir flow, variable-mass wellbore flow, and wellbore–formation heat transfer, the relationship between the temperature [...] Read more.

Production-profile interpretation based on distributed temperature sensing (DTS) is an important method for the dynamic evaluation of hydraulically fractured horizontal gas wells. Because downhole temperature is jointly controlled by reservoir flow, variable-mass wellbore flow, and wellbore–formation heat transfer, the relationship between the temperature response and the production rate of an individual cluster is nonlinear, making inversion based solely on temperature fitting prone to non-uniqueness. In this study, a temperature forward model incorporating reservoir flow, wellbore flow, and heat-transfer processes was developed. Orthogonal experiments, sensitivity analysis, and error-surface analysis were combined to screen the key inversion parameters, and a dual-objective inversion function was constructed using the DTS temperature error and the relative total wellhead gas-production error. On this basis, the covariance matrix adaptation evolution strategy (CMA-ES) was used to jointly invert effective fracture half-lengths and the production profile. This framework uses the total wellhead gas production as an additional constraint during the search, allowing parameter combinations with similar temperature responses but different production responses to be distinguished. Controlled synthetic tests were further performed to assess inversion robustness under different initialization and model settings, followed by application to a field-case well. The analyses showed that the production constraint reduces the admissible low-error solution space but does not establish uniqueness. Results from the case well showed good agreement between the simulated temperature profile and the measured DTS profile, with temperature-drop errors of less than 0.05 °C near the potential producing clusters. The inverted total gas-production rate was 26,649.67 m³/d, with a relative error of 0.6492% compared with the measured wellhead value. These errors characterize the temperature fitting and whole-well production closure achieved for the present field case and are not used here as a quantitative measure of improvement over previous methods. The inversion results revealed a distinctly heterogeneous distribution of production contributions among the fracturing clusters. This method provides a model-constrained analytical approach for production-profile interpretation in hydraulically fractured horizontal gas wells without production logging tool (PLT) data. Without independent cluster-level flow measurements, the resulting production profile should be interpreted as a model-constrained estimate rather than a uniquely determined true profile. Full article

(This article belongs to the Special Issue Inverse Problems and Numerical Computation in Mathematical Physics)

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31 pages, 1198 KB

Open AccessArticle

A Composite Evaluation Framework for Dimensionality Reduction Methods in Industry 4.0 Data

by Özlem Gürünlü Alma, Ali Valiyev and Reza Arabi Belaghi

Mathematics 2026, 14(18), 3394; https://doi.org/10.3390/math14183394 (registering DOI) - 18 Sep 2026

Abstract

Dimensionality reduction is essential in industrial analytics, as high-dimensional process data often contain redundant and correlated variables that increase computational burden and model complexity. This study proposes a weighted composite evaluation framework (S_wc) that jointly considers predictive performance, computational execution time, [...] Read more.

Dimensionality reduction is essential in industrial analytics, as high-dimensional process data often contain redundant and correlated variables that increase computational burden and model complexity. This study proposes a weighted composite evaluation framework (S_wc) that jointly considers predictive performance, computational execution time, and representation dimensionality when comparing feature selection (FS) and feature extraction (FE) methods. The framework was evaluated on the Tennessee Eastman Process benchmark using seven fault classes, 52 process variables, a common XGBoost classifier, and 10 independently seeded stratified repetitions. Nine FS methods and four FE methods were benchmarked across four operational weighting scenarios: Balanced, Predictive Focus, Efficiency Focus, and Compactness Focus. Among the FE techniques, linear discriminant analysis (LDA) consistently achieved the highest composite score across all scenarios (mean rank = 1.00) due to its high-class separability and minimal computational overhead. For FS methods, forward selection and XGBoost-based selection jointly achieved the leading overall average rank (mean rank = 1.50), with XGBoost leading under predictive- and compactness-focused scenarios, while forward selection led under Balanced and Efficiency Focus constraints. Nonparametric statistical analysis indicated significant overall differences among methods based on the Friedman test (p < 0.05), while no individual pairwise comparison remained significant after Holm correction. The results demonstrate that evaluating methods solely by predictive performance provides an incomplete assessment. The proposed framework serves as a transparent multicriteria decision support mechanism for selecting dimensionality reduction strategies aligned with specific Industry 4.0 operational priorities. Full article

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16 pages, 404 KB

Open AccessFeature PaperArticle

Asian Option Pricing Under a Two-Factor Stochastic Volatility with a Stochastic Long-Term Mean

by Junkee Jeon and Geonwoo Kim

Mathematics 2026, 14(18), 3393; https://doi.org/10.3390/math14183393 (registering DOI) - 18 Sep 2026

Abstract

This paper studies the valuation of continuously monitored geometric Asian options under a two-factor stochastic volatility model with stochastic long-term variance levels. We derive the joint characteristic function required for continuous geometric averaging and obtain analytical pricing formulas for fixed-strike call and put [...] Read more.

This paper studies the valuation of continuously monitored geometric Asian options under a two-factor stochastic volatility model with stochastic long-term variance levels. We derive the joint characteristic function required for continuous geometric averaging and obtain analytical pricing formulas for fixed-strike call and put options through Fourier inversion. The analytical formulas are verified by comparison with Monte Carlo simulation. Numerical experiments show that option values are particularly sensitive to the initial variance levels, while changes in the trends of the stochastic long-term means and leverage correlations also affect prices across strikes. The proposed approach extends double Heston Asian option pricing while maintaining computational tractability, and it provides a flexible method for examining the impact of time-varying long-term volatility expectations. Full article

(This article belongs to the Special Issue Advanced Methods, Modeling and Optimization for Financial Engineering and Risk Management)

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40 pages, 3924 KB

Open AccessArticle

Structural Embedding Representation Method (SERM): A Masked Self-Supervised Framework for Structural Embedding Learning in Complex Networks

by Jingjing Chen, Haipeng Peng, Lu Li, Lixiang Li and Cuicui Wang

Mathematics 2026, 14(18), 3392; https://doi.org/10.3390/math14183392 (registering DOI) - 18 Sep 2026

Abstract

Complex networks are widely used to represent interactions in communication systems, biological systems, and large-scale infrastructure. Learning informative network-level representations is important for network analysis, structural similarity retrieval, and characterization of network properties. Existing representation learning methods, however, may have limited ability to [...] Read more.

Complex networks are widely used to represent interactions in communication systems, biological systems, and large-scale infrastructure. Learning informative network-level representations is important for network analysis, structural similarity retrieval, and characterization of network properties. Existing representation learning methods, however, may have limited ability to jointly capture multi-hop topological dependencies and node attribute information, particularly when network observations are incomplete. This work presents SERM, a masked self-supervised framework for network-level structural representation learning. SERM integrates topology-aware graph propagation with masked self-supervised reconstruction. Node-level structural attributes are propagated according to the normalized network topology, enabling attribute information and multi-hop neighborhood context to interact directly within the encoder. Masked reconstruction then provides the self-supervised learning objective by requiring the model to recover hidden node attributes from the available attributes and graph context. During training, a subset of node attributes is masked and reconstructed from the available attributes and graph context, encouraging the encoder to exploit information from both sources without requiring external labels. The resulting node representations are aggregated into fixed-dimensional network-level embeddings for downstream analysis. Experiments on complex-network datasets evaluate the learned representations through network retrieval, structural property prediction, cross-dataset evaluation, robustness analysis, and ablation studies. The results provide empirical evidence that the proposed framework captures useful structural information and remains applicable across several network-level evaluation settings. Full article

(This article belongs to the Special Issue Mathematical Modeling and Learning-Based Control of Information Dynamics in Complex Networks and Communication Systems)

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8 pages, 219 KB

Open AccessArticle

Integral of an Alpha Unpredictable Function

by Marat Akhmet and Nurlan Tilessov

Mathematics 2026, 14(18), 3391; https://doi.org/10.3390/math14183391 (registering DOI) - 18 Sep 2026

Abstract

The present paper is devoted to the study of integral properties of alpha unpredictable functions. The problem of invariance of recurrence under integration is one of the most challenging and interesting in the theory of functions. The problem was solved for quasiperiodic, almost [...] Read more.

The present paper is devoted to the study of integral properties of alpha unpredictable functions. The problem of invariance of recurrence under integration is one of the most challenging and interesting in the theory of functions. The problem was solved for quasiperiodic, almost periodic, and Poisson stable functions. All of the solutions were subjected to the condition of bounded integrals. The alpha-unpredictable functions are the endpoint in the row of recurrent functions. The new class is the cross-border point from regularity to chaos in the dynamics presented through the elements of the row. In the research, we finalized the proof that any theoretical functional recurrence is invariant with respect to integration, provided the result is bounded. Full article

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

33 pages, 1093 KB

Open AccessReview

Methodological Reporting in Humanitarian and Disaster Simulation Under Uncertainty

by James Byrne and Paul Liston

Mathematics 2026, 14(18), 3390; https://doi.org/10.3390/math14183390 (registering DOI) - 18 Sep 2026

Abstract

Simulation and simulation-based optimisation support humanitarian logistics and disaster-management decisions under uncertainty, but the evidential basis of their recommendations depends on how clearly publications connect model specification, computational implementation, experimentation, evaluation and uncertainty analysis. This secondary methodological review reanalyses a fixed corpus of [...] Read more.

Simulation and simulation-based optimisation support humanitarian logistics and disaster-management decisions under uncertainty, but the evidential basis of their recommendations depends on how clearly publications connect model specification, computational implementation, experimentation, evaluation and uncertainty analysis. This secondary methodological review reanalyses a fixed corpus of 109 peer-reviewed journal studies spanning discrete-event simulation, agent-based modelling, system dynamics, hybrid simulation and simulation-optimisation. Indicators were organised across four methodological domains and analysed using indicator-specific denominators, publication-level co-occurrence configurations, descriptive comparisons and exploratory Fisher’s exact tests for the simulation-optimisation subgroup. Conceptual model structure and scenario design were commonly reported, whereas computational environments, statistical characterisation of stochastic outputs, joint reporting of calibration, verification and validation procedures, and input-model uncertainty were less consistently visible. Descriptively, simulation-optimisation studies reported replications and structured experimental designs more frequently than other simulation studies, but no subgroup association remained statistically significant after Holm adjustment. The principal reporting discontinuities lay between conceptual models, executable implementations, computational experiments and the decision claims derived from them. The review develops conditional reporting priorities that relate methodological evidence to model architecture, uncertainty conditions and intended humanitarian use. These priorities provide a structured basis for appraising simulation-supported decisions while recognising that reporting requirements vary across paradigms, experimental designs and operational contexts. Full article

(This article belongs to the Special Issue Simulation-Based Optimisation and Information Systems in Disaster Management)

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19 pages, 1613 KB

Open AccessArticle

Seq2-ResGCRN: Sequence-to-Sequence Residual Graph Convolutional Recurrent Network for Traffic Flow Prediction

by Wenyan Yan and Tao Liu

Mathematics 2026, 14(18), 3389; https://doi.org/10.3390/math14183389 (registering DOI) - 17 Sep 2026

Abstract

Accurate traffic flow prediction remains challenging due to the complex spatio-temporal dependencies inherent in road networks. Existing graph-based prediction models typically construct the graph structure from geographical adjacency. This limits spatial feature extraction to physically neighboring nodes. Consequently, potential correlations among non-adjacent nodes [...] Read more.

Accurate traffic flow prediction remains challenging due to the complex spatio-temporal dependencies inherent in road networks. Existing graph-based prediction models typically construct the graph structure from geographical adjacency. This limits spatial feature extraction to physically neighboring nodes. Consequently, potential correlations among non-adjacent nodes within the same region are overlooked. Moreover, these models generally treat the influence of neighboring nodes uniformly, whereas, in practice, such influences are inherently heterogeneous. On the temporal side, many approaches rely on recurrent units such as Long Short-Term Memory (LSTM) or Gated Recurrent Units (GRUs) to capture short-term dependencies, yet these architectures are prone to gradient vanishing or explosion, leading to training instability. To address these issues, this paper proposes Seq2-ResGCRN, an encoder–decoder framework based on residual graph convolutional recurrent networks with an attention mechanism. The model reconstructs the adjacency matrix via graph diffusion convolution to learn adaptive edge weights, thereby capturing heterogeneous spatial influences. Seq2-ResGCRN further integrates graph convolution into the GRU architecture and introduces residual connections to alleviate gradient degradation during training. This study conducts experiments on two open-source real-world datasets (i.e., PEMS04 and PEMS08) to evaluate the Seq2-ResGCRN. The raw data, originally collected at a sampling frequency of 30 s, are aggregated into 5 min time intervals. Each record comprises three features: traffic flow, traffic speed, and road occupancy. The experimental results demonstrate that our Seq2-ResGCRN outperforms state-of-the-art methods, achieving 1.6–2.3% and 2.1–5.5% relative improvements in MAE and RMSE, respectively. Seq2-ResGCRN effectively captures the spatio-temporal correlations of short-term traffic flow and achieves superior predictive performance. Full article

(This article belongs to the Special Issue Advanced Methods in Intelligent Transportation Systems, 2nd Edition)

28 pages, 682 KB

Open AccessArticle

Curved-Stern Internal Waves with Applied Pressure in a Finite-Depth Two-Layer Fluid

by Osama Ogilat

Mathematics 2026, 14(18), 3388; https://doi.org/10.3390/math14183388 (registering DOI) - 17 Sep 2026

Abstract

The wave train generated by a moving vessel remains a persistent challenge in ship hydrodynamics, particularly in stratified oceans where internal waves contribute significantly to drag. This research investigates steady interfacial wave generation by flow past a semi-infinite curved plate in a two-layer [...] Read more.

The wave train generated by a moving vessel remains a persistent challenge in ship hydrodynamics, particularly in stratified oceans where internal waves contribute significantly to drag. This research investigates steady interfacial wave generation by flow past a semi-infinite curved plate in a two-layer fluid where both layers are of finite depth. Previous studies were limited by flat-plate geometries and infinite-depth approximations, neglecting the critical role of applied pressure as a design variable. We address these limitations to facilitate the engineering goal of wave drag reduction through optimal hull and pressure configurations. Using a scalar Wiener–Hopf formulation, we construct an analytical solution where the kernel is factorized via Cauchy-type integrals, bypassing the technical constraints of traditional infinite-product expansions. We derive a closed-form far-field amplitude and a wave-free condition,

M (μ) = 0

. This condition is a single complex equation, i.e., two independent real constraints. A one-parameter hull design can therefore generally only minimise the downstream wave amplitude; exact cancellation at linear order generally requires a second, independent real design freedom, such as a jointly optimised applied pressure. We quantify both regimes explicitly. Our findings demonstrate that physical stern flow patterns exhibit spatial relaxation and advection scales essential for energy transfer. This result provides a mathematically sharp framework for wave drag minimisation, quantifying the explicit performance trade-off between hull curvature and dynamic pressure control in stratified environments, and we are careful throughout to distinguish the formal linear theory cancellation from what is achievable with a single physically realisable (real-valued) design parameter. Full article

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

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24 pages, 1651 KB

Open AccessArticle

Architecturally Constrained Monotone Functions as Lyapunov-Stable Performance Certificates: A Function-Space Approach

by Xingda Li, Jianqiang Zhang, Yiping Liu, Pengfei Zhang and Ling Tan

Mathematics 2026, 14(18), 3387; https://doi.org/10.3390/math14183387 (registering DOI) - 17 Sep 2026

Abstract

Prescribed performance control (PPC) guarantees that tracking errors evolve within user-defined envelopes, yet the design of the performance certificate

ρ (t)

Full article

20 pages, 461 KB

Open AccessArticle

Beyond Midpoint Equivalence: A Full Axiomatic Characterization of the Moore–Shapley Value Under Interval Uncertainty

by Osman Palancı

Mathematics 2026, 14(18), 3386; https://doi.org/10.3390/math14183386 (registering DOI) - 17 Sep 2026

Abstract

A cooperative interval game assigns a range of possible worths, rather than a single number, to each coalition. For the Shapley-like rule based on Moore subtraction, existing axioms determine the midpoint of each player’s allocated interval but need not determine its lower and [...] Read more.

A cooperative interval game assigns a range of possible worths, rather than a single number, to each coalition. For the Shapley-like rule based on Moore subtraction, existing axioms determine the midpoint of each player’s allocated interval but need not determine its lower and upper endpoints. We close this identification gap on the full class of interval games. In midpoint–radius coordinates, the center is the ordinary Shapley value, whereas each Moore marginal adds the radii of its predecessor and successor coalitions. We introduce two requirements for this radius component. Total Moore exposure fixes the aggregate radius generated along random permutation chains, and boundary exposure balance divides each proper coalition’s contribution equally, in aggregate, between its members and nonmembers. The familiar center axioms together with these two conditions uniquely recover the Moore–Shapley rule, including both endpoints. Without boundary exposure balance, one insider-share parameter remains for each proper coalition size, yielding an

(n - 1)

-dimensional family. We also derive an aggregate-envelope identity and the sharp Hausdorff–Lipschitz constant, establish logical independence of the six axioms for

n \geq 3

, and give an exact

O (n 2^{n - 1})

algorithm together with an unbiased permutation-sampling alternative. A stylized, non-empirical four-partner storage example separates total exposure from its allocation across players. All results concern the original Moore-based rule rather than other interval solution concepts. Full article

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17 pages, 330 KB

Open AccessArticle

A Note on Explicit Representations of (B, C)-Inverses as Inverses Along Suitable Matrices

by Jelena Višnjić and Ivana Petrović

Mathematics 2026, 14(18), 3385; https://doi.org/10.3390/math14183385 (registering DOI) - 17 Sep 2026

Abstract

In this note, starting from the known relation between

(B, C)

-inverses and inverses along suitable matrices, we study explicit representations of

(B, C)

-inverses of complex matrices as inverses along suitable matrices. More precisely, for a [...] Read more.

In this note, starting from the known relation between

(B, C)

-inverses and inverses along suitable matrices, we study explicit representations of

(B, C)

-inverses of complex matrices as inverses along suitable matrices. More precisely, for a given

(B, C)

-invertible matrix, we investigate explicit choices of a matrix D for which

A^{(B, C)} = A^{‖ D}

holds. For the three natural choices

D = B

,

D = C

and

D = B C

, we give necessary and sufficient conditions under which

A^{(B, C)} = A^{‖ D}

. Also, several recently studied generalized inverses, admitting

(B, C)

-inverse representations, are represented as inverses along suitable matrices. Finally, using fixed full-rank factorizations of B and C, we explicitly parametrize all matrices D satisfying

A^{(B, C)} = A^{‖ D}

. Full article

(This article belongs to the Special Issue Recent Advances in Generalized Inverses and Matrix Theory)

36 pages, 1076 KB

Open AccessArticle

Unified Euler–Lagrange Framework for Dynamic Modeling of Wheeled Mobile Robots: Holonomic and Non-Holonomic Architectures

by Jesús Said Pantoja-García, Alejandro Rodríguez-Molina, Miguel Gabriel Villarreal-Cervantes, Mario Aldape-Pérez, Jacobo Sandoval-Gutiérrez and Daniel Librado Martínez-Vázquez

Mathematics 2026, 14(18), 3384; https://doi.org/10.3390/math14183384 (registering DOI) - 17 Sep 2026

Abstract

This paper presents a unified Euler–Lagrange methodology for deriving the dynamic models of wheeled mobile robots (WMRs) across heterogeneous kinematic architectures. The work extends the authors’ previously developed kinematic methodology to the dynamic level, handling both holonomic platforms, where free roller motion eliminates [...] Read more.

This paper presents a unified Euler–Lagrange methodology for deriving the dynamic models of wheeled mobile robots (WMRs) across heterogeneous kinematic architectures. The work extends the authors’ previously developed kinematic methodology to the dynamic level, handling both holonomic platforms, where free roller motion eliminates kinematic constraints, and non-holonomic platforms, where pure-rolling conditions impose non-integrable velocity constraints. Through null-space-based reduction, constraint forces are eliminated, yielding control-affine state-space models that preserve structural properties essential for control synthesis, including inertia matrix symmetry, positive definiteness, and skew symmetry of the Coriolis matrix. The methodology is applied to four canonical architectures: differential drive, Ackermann steering, three-wheeled omnidirectional, and four-wheeled Mecanum drive. Distinct dynamic features are observed across topologies with direct implications for controller design. To confirm the correctness and practical utility of the derived models, trajectory tracking experiments are conducted in both ideal simulation and a realistic ROS2/Gazebo environment, with a model-based controller and a model-free strategy. Both successfully track the reference trajectories across all four architectures, with the model-based controller consistently achieving lower tracking errors. Full article

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