Mathematics

46 pages, 7426 KB

Open AccessArticle

How Supply-Side Policies Influence High-Quality Technology Diffusion: A Complex Network Simulation Based on Evolutionary Game Theory

by Lin Zhang, Jiakun Wu and Xianwei Liu

Mathematics 2026, 14(14), 2616; https://doi.org/10.3390/math14142616 (registering DOI) - 18 Jul 2026

Abstract

To examine the mechanisms through which supply-side policy instruments influence the early diffusion of high-quality technologies in technology markets, this study constructs a bilateral evolutionary model incorporating finitely rational technology suppliers and demanders based on evolutionary game theory and a bipartite small-world network. [...] Read more.

To examine the mechanisms through which supply-side policy instruments influence the early diffusion of high-quality technologies in technology markets, this study constructs a bilateral evolutionary model incorporating finitely rational technology suppliers and demanders based on evolutionary game theory and a bipartite small-world network. Through multi-agent simulations, the study analyzes the effects of government R&D funding, tax relief and technology transaction subsidies on the diffusion of high-quality technologies and the process of supply-demand strategy adaptation, with the aim of characterizing the role of public fiscal funds in screening for effective technology supply. The study reached the following main conclusions: First, under the baseline scenario, high-quality technologies exhibit a strong tendency toward endogenous diffusion, and supply-side policies primarily serve to accelerate marginal growth in the early stages rather than fundamentally altering the long-term convergence trend. Among these policies, R&D funding, as an ex ante incentive tool, has the most direct impact on early-stage diffusion; however, high-intensity funding leads to a decline in fiscal efficiency; Second, the independent effects of ex post incentive tools like tax relief and technology transaction subsidies are relatively moderate, with the former exhibiting high fiscal efficiency at low to medium intensities, and the latter exerting a moderate incentive effect by increasing the returns on successful transactions, though both suffer from diminishing marginal returns; Third, the supply-demand strategy fit index can be used to help characterize the supply-demand coordination process, but it cannot be directly interpreted as an indicator of technology quality or diffusion quality; Finally, robustness tests indicate that the main conclusions remain stable under perturbations to network topology, initial conditions, and Fermi noise parameters. Overall, the design of supply-side policies should simultaneously consider the timing of policy implementation, market-matching mechanisms, and fiscal cost-effectiveness. Full article

(This article belongs to the Special Issue Dynamic Analysis and Decision-Making in Complex Networks, 2nd Edition)

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24 pages, 61324 KB

Open AccessArticle

Target Detection for Traffic Flow in Low-Altitude Unmanned Aerial Vehicle Scenarios

by Tian Luan, Fan Yang, Huanxia Wei and Weijun Pan

Mathematics 2026, 14(14), 2615; https://doi.org/10.3390/math14142615 (registering DOI) - 18 Jul 2026

Abstract

Low-altitude unmanned aerial vehicle (UAV)-based traffic object detection is challenged by substantial scale variations from aerial perspectives, the extremely small pixel proportions of distant traffic participants, complex road background interference, unstable illumination, and severe occlusion in dense traffic scenes. To address these problems, [...] Read more.

Low-altitude unmanned aerial vehicle (UAV)-based traffic object detection is challenged by substantial scale variations from aerial perspectives, the extremely small pixel proportions of distant traffic participants, complex road background interference, unstable illumination, and severe occlusion in dense traffic scenes. To address these problems, this paper proposes ACP2-YOLO, an improved YOLO11-based detection framework for low-altitude UAV traffic scenarios, with the goal of enhancing the detection of vehicles, pedestrians, and non-motorized traffic participants. The proposed framework introduces two key improvements. First, a lightweight hybrid ACmix module that integrates convolution and self-attention is embedded into the network, enabling the model to jointly capture local detailed features and global contextual dependencies and thereby strengthen feature representation under complex backgrounds. Second, a P2 small-object detection layer is added to the original three-scale detection structure of YOLO11 to construct a four-scale P2–P5 feature pyramid. By allowing shallow high-resolution features to directly participate in object prediction, this design effectively reduces spatial information loss caused by deep downsampling and improves small-object perception. Experiments on the VisDrone2019 dataset show that the improved model achieves 53.1% Precision, 41.1% Recall, 42.9% mAP@50, and 26.3% mAP@50–95, outperforming the baseline YOLO11 by 4.2, 4.2, 5.0, and 3.6 percentage points, respectively. Comparisons with mainstream YOLO-series detectors further demonstrate its superior overall accuracy, small-object detection capability, and adaptability to complex scenes, indicating its potential for UAV-based traffic monitoring, road safety inspection, and intelligent transportation perception. Full article

(This article belongs to the Special Issue Optimization and Machine Learning-Based Methods in Air Traffic Management and Aeronautical Domains)

27 pages, 3834 KB

Open AccessArticle

A Framework for Queueing Network Construction Based on Business Process Structural Properties

by Konstantin Terentyev, Julia Babina, Alexey Ermolaev and Irina Kochetkova

Mathematics 2026, 14(14), 2614; https://doi.org/10.3390/math14142614 (registering DOI) - 18 Jul 2026

Abstract

Business analytics increasingly relies on analytical performance models to support real-time, data-driven decision-making in multi-stage service processes. Discrete-event simulation and process mining offer valuable insights but face scalability and explainability limits when closed-form performance indicators are needed for optimization. This paper develops a [...] Read more.

Business analytics increasingly relies on analytical performance models to support real-time, data-driven decision-making in multi-stage service processes. Discrete-event simulation and process mining offer valuable insights but face scalability and explainability limits when closed-form performance indicators are needed for optimization. This paper develops a systematic procedure for constructing BCMP queueing-network models directly from the structural properties of a business process formalized in BPMN. The framework introduces a six-level state-space hierarchy (V0–V5) that ranges from a full path encoding to a single system-level counter, and three allocation strategies (D1–D3) that govern how the actions of an organizational unit are distributed among its workers. A product-form stationary distribution is derived for the resulting open network with FCFS and infinite-server nodes, yielding closed-form expressions for the mean sojourn time at a node and for the mean process flow time. We validate the framework on the university admissions campaign at RUDN University (2023 cycle, more than 85,000 applications). Numerical experiments show that the baseline mean flow time of about 174 min can be reduced to 111 min—a 33% improvement—by only a 3% increase in total staffing, allocated by greedy coordinate descent. The closed-form model agrees with discrete-event simulation under exponential service times to within 1%, and its analytical tractability enables real-time staffing recommendations that simulation alone cannot deliver. Full article

(This article belongs to the Special Issue Business Analytics and Decision-Making: Models, Algorithms and Applications)

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35 pages, 5673 KB

Open AccessArticle

Complete Optimal Cost Design for RIFs, Assuming That the Contact Area Is Partially Compressed with Eccentric Column

by Arnulfo Luévanos-Rojas, Griselda Santiago-Hurtado, Víctor Manuel Moreno-Landeros, Eyran Roberto Díaz-Gurrola, Rajeswari Narayanasamy, Facundo Cortés-Martínez, Aldo Emelio Landa-Gómez, María Dolores Arriaga-Pons, María Teresa Mora-Cabral and Francisco Luis Camporredondo-Reyes

Mathematics 2026, 14(14), 2613; https://doi.org/10.3390/math14142613 (registering DOI) - 18 Jul 2026

Abstract

This article presents a complete cost-optimized design for rectangular isolated footings (RIFs) under biaxial bending with a partially compressed contact area (PCCA) and an eccentric column, i.e., part of the base is in contact with the soil (generating compression), and the other part [...] Read more.

This article presents a complete cost-optimized design for rectangular isolated footings (RIFs) under biaxial bending with a partially compressed contact area (PCCA) and an eccentric column, i.e., part of the base is in contact with the soil (generating compression), and the other part is not in contact with the soil (generating neither compression nor tension), and the soil pressure behaves linearly. The proposed model is presented by integration to find the moments in the critical sections, the bending shears in the critical sections and the punching shear acting on the RIF, and these are compared with those that must be resisted according to the standards (ACI 318-19) Some articles show complete designs for RIFs with a fully compressed contact area (FCCA) and an eccentric column, and other articles show only the minimum area (A_min) for RIFs with a PCCA and an eccentric column. The main contributions of this article are as follows: (1) the A_min is determined from the loads and moments, assuming that the loads and moments it resists must be greater than or equal to those acting on the RIF; (2) the minimum cost (C_min) is determined. Three numerical studies are described to demonstrate the advantages of this study over other models. The results indicate that, according to the studies presented in this article, savings of up to 45.99% for A_min and 93.62% for C_min can be achieved using the model proposed in this article (MPA); this occurs when the pressure acting on the foundation does not reach the maximum available soil pressure (p_max). Therefore, this paper could be of great use to engineers dedicated to the construction of foundations. Full article

(This article belongs to the Special Issue Modeling and Control in Vibrational and Structural Dynamics)

26 pages, 1392 KB

Open AccessArticle

Polar Orders on Lattices of Real Intervals

by Benedetta Amicizia, Maria Letizia Guerra and Luciano Stefanini

Mathematics 2026, 14(14), 2612; https://doi.org/10.3390/math14142612 (registering DOI) - 18 Jul 2026

Abstract

Two classes of partial orders are considered and analyzed for the space

K_{C}

of compact real intervals. The first generalizes the well-known Lower-Upper (LU) partial order, based on the relative position of the intervals’ elements on the real line; the second extends [...] Read more.

Two classes of partial orders are considered and analyzed for the space

K_{C}

of compact real intervals. The first generalizes the well-known Lower-Upper (LU) partial order, based on the relative position of the intervals’ elements on the real line; the second extends the standard partial order induced by set inclusion. Using mid-point representation, non-empty intervals are associated to points in the ordinary upper half-plane; the two classes of order relations are characterized in terms of two real parameters, appropriately combined to linearly partition the half-plane into four disjoint sliced subsets, which separate comparable from incomparable intervals. We show that pairs of order relations obtained with the same parameters (denote these orders by ⪷ and ⫅, respectively) have an interesting polarity property which allows one to formalize a general setting for analyzing questions of Interval Analysis related to different forms of comparability. In particular, the obtained lattice structures

(K_{C}, ⪷)

,

(K_{C}, ⫅)

are detailed and the polarity property is proved for them: if two intervals are incomparable with respect to ⪷ (or ⫅), they are necessarily comparable with respect to ⫅ (or ⪷, respectively). Finally, for any pair of orders

(⪷, ⫅)

with the polarity property, we prove that two total (linear) orders

⪷^{T}

and

⫅^{T}

are (uniquely) constructed in

K_{C}

which completely remove incomparable cases. The presentation is supported by several illustrative pictures and some examples. Full article

(This article belongs to the Special Issue New Perspectives in Interval Analysis and Fuzzy-Valued Functions)

14 pages, 710 KB

Open AccessArticle

Improvements for Inventory Models with Generalized Interarrival Times

by Daniel Yi-Fong Lin

Mathematics 2026, 14(14), 2611; https://doi.org/10.3390/math14142611 (registering DOI) - 18 Jul 2026

Abstract

Single-machine, single-product inventory models with generalized interarrival times have lacked a fully rigorous and computationally efficient optimization framework because a sign error in earlier derivations produced ad hoc feasibility restrictions and overly broad search domains. This study presents a corrected derivation that proves [...] Read more.

Single-machine, single-product inventory models with generalized interarrival times have lacked a fully rigorous and computationally efficient optimization framework because a sign error in earlier derivations produced ad hoc feasibility restrictions and overly broad search domains. This study presents a corrected derivation that proves the strict convexity of the minimum-cost objective and establishes the existence and uniqueness of an interior optimum without auxiliary conditions, therefore consolidating previously fragmented results into a single theorem. Building on these structural properties, the maximum-profit formulation is reduced to a one-dimensional program with natural finite bounds that tightly bracket the optimizer, replacing earlier paired bounds defined on an effectively unbounded domain. Numerical results for a canonical benchmark show that the tightened admissible interval recovers the same optimum with fewer function evaluations, thereby improving computational efficiency and implementation robustness. The paper therefore contributes a corrected optimality theory, a problem-native one-dimensional reformulation of the profit model, and a more reproducible computational procedure for capacity-constrained single-machine systems under generalized interarrival times. Full article

21 pages, 1907 KB

Open AccessArticle

Top-Weighted Structural Analysis of Ordered List Averaging for Medical Knowledge Extraction

by Ricardo Sánchez-de-Madariaga, Mario Pascual Carrasco and Adolfo Muñoz Carrero

Mathematics 2026, 14(14), 2610; https://doi.org/10.3390/math14142610 (registering DOI) - 18 Jul 2026

Abstract

Ordered List Averaging (OLA) was previously introduced as a methodology for evaluating machine learning algorithms according to their ability to prioritize medically relevant feature subsets. The original methodology demonstrated, through Friedman–Nemenyi statistical analysis applied to probability distributions of related datasets, that OLA can [...] Read more.

Ordered List Averaging (OLA) was previously introduced as a methodology for evaluating machine learning algorithms according to their ability to prioritize medically relevant feature subsets. The original methodology demonstrated, through Friedman–Nemenyi statistical analysis applied to probability distributions of related datasets, that OLA can identify statistically supported machine learning algorithms for medical knowledge extraction. However, the internal structural organization of OLA-generated rankings has not previously been investigated. This work presents a structural analysis framework for OLA based on top-weighted ranking-agreement measures. Rather than proposing a new machine learning model-selection algorithm, the objective is to determine whether the ordering induced by OLA scores exhibits an intrinsic structural organization that provides complementary evidence supporting the interpretation of OLA rankings. To this end, ranking signatures generated by OLA are compared using Kendall distance, while the agreement between score-based and structure-based rankings is evaluated through Top-k overlap and Rank-Biased Overlap (RBO). Experiments conducted on nine biomedical datasets show that datasets previously exhibiting statistically validated OLA behavior consistently present stronger top-weighted structural agreement than datasets with weaker statistical support. Furthermore, the observed structural patterns remain stable across different values of the RBO persistence parameter and are robust to variations in dataset size. The results indicate that OLA rankings possess a meaningful and reproducible structural organization beyond their numerical scores. Consequently, top-weighted structural agreement provides complementary evidence supporting the interpretation of OLA rankings and strengthens the practical interpretation of OLA-based algorithm selection, particularly in situations where only a single working dataset is available and statistical validation through Friedman–Nemenyi cannot be performed. Full article

(This article belongs to the Special Issue Data Sciences and Artificial Intelligence Optimization Methods)

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16 pages, 1305 KB

Open AccessArticle

GA-Optimized Feature Weighting for Fuzzy C-Means Classification

by Zhiwen Li, Xinghua Wang, Yongtao Li and Yubin Zhong

Mathematics 2026, 14(14), 2609; https://doi.org/10.3390/math14142609 (registering DOI) - 18 Jul 2026

Abstract

The problem of insufficient classification accuracy in fuzzy clustering algorithms for multidimensional data is addressed in this paper. To tackle this issue, an improved genetic algorithm (GA)-based fuzzy controller is proposed, which combines the advantages of an improved genetic algorithm and a fuzzy [...] Read more.

The problem of insufficient classification accuracy in fuzzy clustering algorithms for multidimensional data is addressed in this paper. To tackle this issue, an improved genetic algorithm (GA)-based fuzzy controller is proposed, which combines the advantages of an improved genetic algorithm and a fuzzy C-means (FCM) clustering algorithm. The population initialization is performed using the Tent chaotic map, while the best individual retention strategy and last elimination selection operator are employed to adjust the population structure. Furthermore, an elitist crossover operator, an adaptive trial mutation operator, and a nonlinear convergence factor are introduced to mitigate the risk of falling into local optima. The controller algorithm integrates the intermediate parameters of FCM clustering into the fitness function of the genetic algorithm, and effectively improves the classification accuracy by screening the optimal feature subset and then fuzzy clustering. The experimental results on the Pistachio, WDBC, and Wine datasets show that the proposed method achieves competitive classification accuracy compared with other FCM-based feature-weighting optimization methods. Full article

(This article belongs to the Special Issue Advances in Fuzzy Intelligence and Non-Classical Logical Computing)

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17 pages, 2217 KB

Open AccessArticle

Propagation Behaviors of Rayleigh Waves in Partially Saturated Poroviscoelastic Half-Space

by Xinzhu Ma and Tengyu Ma

Mathematics 2026, 14(14), 2608; https://doi.org/10.3390/math14142608 (registering DOI) - 17 Jul 2026

Abstract

This study investigates the propagation characteristics of Rayleigh waves in partially saturated poroviscoelastic half-spaces based on fundamental theories of elastic wave propagation. Firstly, a poroviscoelastic model tailored to unsaturated porous media is developed, integrating constitutive relations and dynamic equilibrium equations with fractional derivatives [...] Read more.

This study investigates the propagation characteristics of Rayleigh waves in partially saturated poroviscoelastic half-spaces based on fundamental theories of elastic wave propagation. Firstly, a poroviscoelastic model tailored to unsaturated porous media is developed, integrating constitutive relations and dynamic equilibrium equations with fractional derivatives to effectively capture the complex, history-dependent viscoelastic behavior of materials. Secondly, the potential function method is employed to derive the dispersion equation for Rayleigh waves in the proposed medium. Lastly, numerical simulations are conducted to analyze the effects of liquid saturation and fractional order parameters on Rayleigh wave velocity, with results presented through graphical illustrations. The analysis reveals that both liquid saturation and fractional order significantly influence the propagation features of Rayleigh waves. This work provides valuable theoretical support for seismic exploration and related engineering applications involving unsaturated porous formations. Full article

(This article belongs to the Special Issue Advances in Fractional Calculus for Modeling and Applications)

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39 pages, 1344 KB

Open AccessArticle

A Hybrid Multilayer Dynamic Modelling Framework for Path-Dependent Stochastic Systems: Application to EU Digital Public Services

by Oana-Ramona Lobonț, Andrei Trip, Florina Stanciu, Cristina Criste, Iuliana Militaru and Daniel Brîndescu-Olariu

Mathematics 2026, 14(14), 2607; https://doi.org/10.3390/math14142607 (registering DOI) - 17 Jul 2026

Abstract

Path-dependent dynamic systems are often analysed through separate metric, stochastic, econometric or nonlinear methods, while their integration within a unified architecture remains limited. This study develops and validates a Hybrid Dynamic Modelling Framework (HDMF) for path-dependent stochastic systems, using EU-27 digital public service [...] Read more.

Path-dependent dynamic systems are often analysed through separate metric, stochastic, econometric or nonlinear methods, while their integration within a unified architecture remains limited. This study develops and validates a Hybrid Dynamic Modelling Framework (HDMF) for path-dependent stochastic systems, using EU-27 digital public service trajectories during 2017-2022 as an empirical application. The framework combines DTW-TimeSeriesKMeans clustering, Markov transition analysis, PVAR, SEM and MLP approximation. The findings answer the four research questions sequentially. First, EU digital public service development is represented by three distinct trajectory regimes rather than a single convergence path. Second, the Markov layer shows strong diagonal dominance, limited mobility and practical non-ergodicity over the finite observation horizon, indicating persistent state dependence. Third, the PVAR layer confirms dynamic stability and bounded feedback within the institutional–digital system. Fourth, the SEM and MLP layers identify governance as a coherent latent institutional structure and reveal nonlinear predictive heterogeneity, suggesting possible threshold-type behaviour. Overall, the findings support the HDMF as a coherent multilayer architecture for modelling complex path-dependent systems and demonstrate its empirical usefulness through the EU digital public service case. Full article

(This article belongs to the Special Issue Advanced Research in Mathematical Economics and Financial Modelling, 2nd Edition)

33 pages, 3147 KB

Open AccessArticle

Constrained Robust Synthesis of Fixed Positive Input Shapers for a Trolley–Pendulum Crane System Under Parametric Uncertainty

by Rosen Mitrev and Dragan Marinkovic

Mathematics 2026, 14(14), 2606; https://doi.org/10.3390/math14142606 (registering DOI) - 17 Jul 2026

Abstract

This paper presents a constrained robust procedure for synthesizing fixed positive input shapers for a trolley–pendulum crane that accounts for motor dynamics, actuator lag, and PD trajectory tracking. Rope length L and payload mass m are treated as uncertain parameters within a prescribed [...] Read more.

This paper presents a constrained robust procedure for synthesizing fixed positive input shapers for a trolley–pendulum crane that accounts for motor dynamics, actuator lag, and PD trajectory tracking. Rope length L and payload mass m are treated as uncertain parameters within a prescribed operating domain, while remaining constant during each maneuver. The method searches offline for a single impulse sequence that can be used across the entire (L,m) domain without online retuning. The impulse amplitudes and delays are obtained from a linearized closed-loop electromechanical model. Mean and worst-case peak payload sway are evaluated during synthesis, while residual RMS sway, terminal-position error, peak motor current, and peak actuator voltage are imposed as feasibility constraints. The resulting shapers are tested on a dense nonlinear validation grid and compared with nominal classical shapers, frequency-robust references, and pointwise-retuned benchmarks. The synthesized four-impulse robust shaper yields the lowest worst-case peak sway among the tested fixed shapers and satisfies all imposed feasibility limits over the validation domain. A frequency-robust reference retains a lower mean peak sway, a shorter shaping horizon, and lower drive demand. The proposed shaper is better suited to applications in which limiting the maximum payload excursion is the primary design objective. Full article

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

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22 pages, 1547 KB

Open AccessArticle

Image Reconstruction by Frequency Extrapolation and Deep Learning in Three-Layer Medium

by Chien-Ching Chiu, Po-Hsiang Chen, Guan-Jang Li and Eng Hock Lim

Mathematics 2026, 14(14), 2605; https://doi.org/10.3390/math14142605 (registering DOI) - 17 Jul 2026

Abstract

This paper proposes a novel multi-frequency extended Deep Learning (DL) model for electromagnetic image reconstruction under Transverse Magnetic (TM) wave incidence in layered media, inspired by conventional microwave imaging techniques that combine nonlinear inversion algorithms with neural networks to improve reconstruction performance. The [...] Read more.

This paper proposes a novel multi-frequency extended Deep Learning (DL) model for electromagnetic image reconstruction under Transverse Magnetic (TM) wave incidence in layered media, inspired by conventional microwave imaging techniques that combine nonlinear inversion algorithms with neural networks to improve reconstruction performance. The proposed framework adopts a two-stage neural network architecture. In the first stage, a Deep Residual Convolutional Neural Network (DRCNN) is employed to extrapolate multi-frequency scattered fields from single-frequency input data, thereby enriching the frequency-dependent scattering information available for reconstruction. Subsequently, the extrapolated multi-frequency scattered fields are fed into a Deep Convolutional Encoder–Decoder (DCED) network to reconstruct an accurate dielectric constant distribution within the imaging domain. To validate the effectiveness of the proposed approach, two representative comparison methods are considered: (1) a hybrid framework combining the Back-Propagation Scheme (BPS) with a Convolutional Neural Network (CNN), and (2) a framework integrating the Dominant Current Scheme (DCS) with a CNN. In both approaches, conventional inversion algorithms are first utilized to generate coarse initial reconstructions, which are subsequently refined by the neural network. Numerical simulations and experimental results show that the proposed multi-frequency extension model achieves lower reconstruction error and higher structural similarity than the reference methods. These results confirm the effectiveness and potential of the proposed framework for advanced electromagnetic imaging applications. Full article

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

22 pages, 1522 KB

Open AccessArticle

Bird Strike Risk in Aviation: A Calibrated Threshold-Optimized Machine Learning Decision Support

by Luís F. F. M. Santos, Victor Tavares, Flávio Lázaro, Rui Melicio and Duarte Valério

Mathematics 2026, 14(14), 2604; https://doi.org/10.3390/math14142604 (registering DOI) - 17 Jul 2026

Abstract

Bird–aircraft strikes are rare but high-impact events that pose persistent safety and operational challenges in aviation. Existing approaches often focus on predictive accuracy, with limited integration of probability calibration, rare-event precision, and decision-making under operational constraints. This paper proposes a calibrated threshold-optimized decision-support [...] Read more.

Bird–aircraft strikes are rare but high-impact events that pose persistent safety and operational challenges in aviation. Existing approaches often focus on predictive accuracy, with limited integration of probability calibration, rare-event precision, and decision-making under operational constraints. This paper proposes a calibrated threshold-optimized decision-support framework that anticipates short-term bird strike risk and translates probabilistic forecasts into constrained intervention plans. The framework combines (i) a calibrated occurrence model for short-term strike probability, (ii) a conditional intensity model for expected strike magnitude, and (iii) a constrained planning module that selects a limited number of high-risk time windows per day. The approach is evaluated using EASA bird strike data (2019–2024), enriched with environmental, meteorological, astronomical, tide, and traffic-proxy features. After this stricter temporal reconstruction, the calibrated occurrence model reaches ROC-AUC

\approx 0.636

and PR-AUC

\approx 0.050

across forward chain folds, while the OOF aggregate gives ROC-AUC

= 0.648

and PR-AUC

= 0.047

against a forward label prevalence of

0.0246

. Under a strict daily alarm budget (

K = 2

), OOF operational Precision@K is

0.063

and Recall@K is

0.075

, with a median credited lead time of 90 min. The results highlight the importance of calibrated probabilities and decision-oriented evaluation, showing that effective rare-event management requires not only prediction, but also integration with operational planning. Full article

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45 pages, 3792 KB

Open AccessArticle

Multi-Objective Live-Streaming Mode Selection and Preservation Investment in Fresh Agricultural Product Supply Chains: A Game-Theoretic Analysis

by Lanying Liu and Songping Zhu

Mathematics 2026, 14(14), 2603; https://doi.org/10.3390/math14142603 (registering DOI) - 17 Jul 2026

Abstract

Fresh agricultural product live-streaming involves multi-objective operational trade-offs rather than a simple channel choice problem. Farmers, influencers, and platforms must balance demand expansion and freshness assurance against pricing pressure, selling effort cost, preservation cost, revenue sharing, and platform fees. We examine a fresh [...] Read more.

Fresh agricultural product live-streaming involves multi-objective operational trade-offs rather than a simple channel choice problem. Farmers, influencers, and platforms must balance demand expansion and freshness assurance against pricing pressure, selling effort cost, preservation cost, revenue sharing, and platform fees. We examine a fresh agricultural product live-streaming supply chain involving a farmer, an influencer, and a platform under three modes: farmer self-streaming (FS), influencer streaming (IS), and farmer–influencer hybrid streaming (FIS). We develop game-theoretic models to jointly analyze pricing, selling effort, preservation effort, traffic spillover, and payoff allocation, and derive equilibrium decisions through analytical optimization and backward induction. The results show that the FS and IS modes follow different value-conversion logics: the FS mode relies more on farmer-channel information value, while the IS mode relies more on influencer-channel traffic value. The FIS mode combines these two value sources, but its advantage is not merely due to channel addition. A no-spillover comparison shows that the FIS mode creates value through channel addition and traffic-spillover-based synergy. The results also reveal threshold-like amplification in freshness-value realization within the modeled parameter environment: preservation investment becomes more valuable when preservation efficiency, consumer freshness sensitivity, demand scale, and channel complementarity are relatively strong. These findings guide live-streaming mode selection, traffic collaboration, and preservation investment. Full article

(This article belongs to the Special Issue Multi-Objective Optimization: Mathematical Methods, Algorithms, and Applications)

25 pages, 1272 KB

Open AccessArticle

Time-Variant Reliability Analysis via Copula Failure Correlation and Advanced Kriging

by Debiao Meng, Huaxi Wu, Muhammad Umar Khan, Juan Huang, Lu-Kai Song and Shiyuan Yang

Mathematics 2026, 14(14), 2602; https://doi.org/10.3390/math14142602 (registering DOI) - 17 Jul 2026

Abstract

In practical engineering, mechanical structures are affected by time-dependent uncertainties and by correlated failure modes. This study proposes an enhanced time-variant reliability analysis framework by coupling a mixed Archimedean Copula model with an adaptive Kriging model driven by the maximum expected prediction error [...] Read more.

In practical engineering, mechanical structures are affected by time-dependent uncertainties and by correlated failure modes. This study proposes an enhanced time-variant reliability analysis framework by coupling a mixed Archimedean Copula model with an adaptive Kriging model driven by the maximum expected prediction error (MEPE) learning function. The mixed Copula combines Gumbel, Clayton and Frank components, so that upper-tail, lower-tail and nearly symmetric dependence can be represented in one model. The MEPE function combines Kriging prediction variance and leave-one-out cross-validation error through a dynamic balance factor, which guides early global exploration and later refinement near the most probable point trajectory. For the planar three-bar structure, the mixed-Copula failure probability at t = 7 is 0.4078, close to the direct-MCS benchmark 0.4059, with a relative error of 0.47%; the number of original limit-state evaluations decreases from 1.00 × 10⁶ to 326, corresponding to a 99.97% reduction. For the two-rod parallel structure, the mixed-Copula failure probability at T = 2 is 0.0987, close to the direct-MCS benchmark 0.0992, with a relative error of 0.50%; the number of original limit-state evaluations decreases to 284. The calculated upper- and lower-tail dependence coefficients of the mixed Copula are 0.1345 and 0.2985 for the planar three-bar structure and 0.0003 and 0.3709 for the two-rod parallel structure. These quantitative results show that the proposed framework can describe nonlinear failure dependence more flexibly than a single Copula while retaining high computational efficiency. Full article

18 pages, 4316 KB

Open AccessArticle

On

(2, 3)

-Knight’s Tours on Rectangular Chessboards

by Nur Narimah Mohd Yusof, Siew Hui Ong and Kok Bin Wong

Mathematics 2026, 14(14), 2601; https://doi.org/10.3390/math14142601 (registering DOI) - 17 Jul 2026

Abstract

A generalized

(a, b)

-knight’smove is an outcome when the knight moves on a rectangular chessboard a cells in either the horizontal or vertical direction, followed by a perpendicular move of b cells. An

(a, b)

-knight’s [...] Read more.

A generalized

(a, b)

-knight’smove is an outcome when the knight moves on a rectangular chessboard a cells in either the horizontal or vertical direction, followed by a perpendicular move of b cells. An

(a, b)

-knight’s tour on a chessboard is a sequence of

(a, b)

-knight’s moves such that the knight visits every cell of the chessboard exactly once. Such a tour is said to be closed if the final position of the knight coincides with its initial position; otherwise, the tour is said to be open. In this paper, we investigate the existence of open

(2, 3)

-knight’s tours on an

m \times n

rectangular chessboard. In particular, we establish sufficient and necessary conditions under which a

5 p \times n

rectangular chessboard admits an open

(2, 3)

-knight’s tour. The results presented here provide a complete characterization for this class of rectangular chessboards. Full article

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42 pages, 1119 KB

Open AccessArticle

Analytical Formulation and Equilibrium Structure of a 26-State Nonlinear Dynamical System for DFIG

by Abdullah Alassaf and Ibrahim Alsaleh

Mathematics 2026, 14(14), 2600; https://doi.org/10.3390/math14142600 (registering DOI) - 17 Jul 2026

Abstract

We formulate and analyze a 26-dimensional nonlinear dynamical system governing a doubly-fed induction generator (DFIG) wind energy conversion system coupled to an infinite bus through a dynamic transmission line. Seven interacting subsystems—aerodynamics, a two-mass drivetrain, a fourth-order machine, rotor- and grid-side converter controllers, [...] Read more.

We formulate and analyze a 26-dimensional nonlinear dynamical system governing a doubly-fed induction generator (DFIG) wind energy conversion system coupled to an infinite bus through a dynamic transmission line. Seven interacting subsystems—aerodynamics, a two-mass drivetrain, a fourth-order machine, rotor- and grid-side converter controllers, a phase-locked loop, and a pitch regulator—are assembled into a single vector field

\dot{x} = f (x, u)

on

R^{26}

, derived in dimensionless coordinates. Strict positivity of the determinant

Δ = L_{s} L_{r} - L_{m}^{2} = σ L_{s} L_{r}

for every physically admissible machine renders the flux–current map invertible, so the right-hand side is well defined; the nodal Kirchhoff constraint forms a semi-explicit differential-algebraic relation that we eliminate to obtain an explicit ordinary differential equation. The central contribution is a constructive scheme for the equilibria: the 26 stationarity conditions

f (x^{★}, u) = 0

are solved by an iterative voltage-matching procedure converging to a residual below

10^{- 11}

per unit—essentially machine precision—which removes the spurious start-up transients common in reported simulations. Analytically chosen feedback gains induce a hierarchy of well-separated time scales, placing the closed loop in the multiple-time-scale class; the separation is made quantitative through explicit small parameters

ε_{i}

formed from the ratios of subsystem time constants. Numerical integration of a GE

3.6

MW configuration confirms the construction: under stationary forcing, the rotor speed stays within

1.32 \times 10^{- 5}

pu of the equilibrium, and under a large-amplitude wind program (

11 \to 14 \to 9

m/s) spanning the full operating envelope, it is regulated to within

0.065 %

, while the DC-link voltage deviation remains below

2.4 \times 10^{- 5}

pu and the power balance closes with residual below

10^{- 3}

pu, the

\approx 2 %

mechanical–electrical gap being the modeled losses. Linearization about the computed equilibrium yields a Jacobian whose spectrum lies entirely in the open left half-plane, establishing local asymptotic stability and exposing the individual electromagnetic, torsional, and control modes. The model furnishes a rigorously initialized, analytically transparent basis for linearization, spectral stability analysis, and bifurcation study. Its practical value is that a consistent equilibrium and a certified spectrum remove the start-up transients and undocumented tuning that otherwise let initialization artifacts masquerade as genuine dynamics, so that the model can serve as a trustworthy building block for weak-grid and wind-farm stability studies. Full article

(This article belongs to the Topic Power System Modeling and Control, 3rd Edition)

42 pages, 7845 KB

Open AccessArticle

Parameter Recovery in Noisy Tri-Periodic Horadam Sequences: A Fourier-Embedded Recurrence-Regularized Neural Estimator (RRNE)

by Bahar Kuloğlu, Hasan Gökbaş, George A. Anastassiou, Metin Zontul and Seda Karateke

Mathematics 2026, 14(14), 2599; https://doi.org/10.3390/math14142599 (registering DOI) - 17 Jul 2026

Abstract

This study investigates tri-periodic Horadam-type recurrence systems from both algebraic and computational perspectives. Algebraically, by adapting standard discrete Floquet-type and monodromy matrix viewpoints to the tri-periodic setting, the sequence is analyzed through induced recurrence relations, a monodromy matrix formulation, Binet-type representations, and an [...] Read more.

This study investigates tri-periodic Horadam-type recurrence systems from both algebraic and computational perspectives. Algebraically, by adapting standard discrete Floquet-type and monodromy matrix viewpoints to the tri-periodic setting, the sequence is analyzed through induced recurrence relations, a monodromy matrix formulation, Binet-type representations, and an effective spectral radius

ρ_{eff}

. Computationally, the inverse parameter estimation problem under noisy observations is shown to be highly ill-conditioned. In particular, classical phase-wise Ordinary Least Squares (OLS) estimators may suffer from attenuation bias under an Errors-in-Variables mechanism, causing coefficient shrinkage and over-smoothed trajectory reconstruction in noisy recursive settings. To address this difficulty, we propose a Fourier-Embedded Recurrence-Regularized Neural Estimator (RRNE). By combining Fourier feature embeddings for period-three phase encoding, a Log-Cosh loss for robust residual control, a teacher-forced recurrence constraint, and hybrid Adam–L-BFGS optimization, the RRNE acts as a structure-aware inverse estimator. Numerical experiments across varying noise levels and asymmetry regimes show that the proposed framework provides stable parameter recovery and long-horizon reconstruction in low- and moderate-asymmetry regimes, while exposing severe instability under extreme phase-switching dynamics. The results further indicate that inverse-estimation collapse cannot be explained by the effective spectral radius alone; rather, it reflects the joint effect of coefficient asymmetry, spectral amplification, observation noise, and optimization stiffness. Overall, this study connects tri-periodic recurrence theory, monodromy-based spectral analysis, classical regression limitations, and recurrence-regularized neural inverse modeling. Full article

(This article belongs to the Special Issue AI, Machine Learning and Optimization)

39 pages, 3397 KB

Open AccessArticle

Bipolar Metric Spaces for Geometric Regularization in Distance-Based Learning

by Gizen Mutlu and Çiğdem İnan Acı

Mathematics 2026, 14(14), 2598; https://doi.org/10.3390/math14142598 (registering DOI) - 17 Jul 2026

Abstract

Distance-based machine learning algorithms are widely used due to their simplicity and flexibility. However, they often suffer from sensitivity to noise, high computational cost, and instability in high-dimensional and complex data spaces. To address these limitations, this study introduces a novel geometric learning [...] Read more.

Distance-based machine learning algorithms are widely used due to their simplicity and flexibility. However, they often suffer from sensitivity to noise, high computational cost, and instability in high-dimensional and complex data spaces. To address these limitations, this study introduces a novel geometric learning framework based on Bipolar Metric Spaces (BMSs). The proposed framework extends classical metric structures by incorporating a quadrilateral inequality constraint, enabling the modeling of both similarity and contrast relationships across data points. Within this framework, geometric regularization is achieved through a structural filter mechanism that filters out structurally inconsistent assignments and stabilizes decision boundaries. Importantly, the proposed approach is not limited to specific algorithms and can be integrated into a broad class of distance-based learning methods. To demonstrate its effectiveness, the framework is instantiated on K-Nearest Neighbor (KNN) and K-means algorithms, resulting in Bipolar K-Nearest Neighbor (BP-KNN) and Bipolar K-means (BP-K-means). Theoretical analysis shows that the proposed framework improves Lipschitz stability, reduces variance, and enhances generalization performance. The models were tested on six different datasets, including benchmark datasets (Iris, Wine, and Breast Cancer) and real-world datasets (Abalone, CDC Diabetes, and Malicious Website Features (MWS)). Theoretical complexity analysis, supported by Floating Point Operation (FLOP) quantification, reveals that the bipolar approach offers a more stable decision boundary and significantly faster convergence, corroborating the observed efficiency improvements. Notably, on the large-scale MWS dataset, BP-KNN achieved an 11.6-fold increase in inference speed and a 75% reduction in RAM usage compared with the standard KNN. These results demonstrate that BMS provides a unified and scalable geometric regularization framework for distance-based learning, offering a principled solution to the trade-off between accuracy, robustness, and computational efficiency. Full article

(This article belongs to the Special Issue Mathematical Optimization in Machine Learning, Computer Vision, and Data Mining, 2nd Edition)

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