Mathematics

21 pages, 4593 KB

Open AccessFeature PaperArticle

Adaptive-Scale Color Offset and Error-Guided Gaussian Reallocation for Multi-Scale 3D Gaussian Splatting

by Hyeonbin Park, Yooho Lee, Dongho Lee and Dongsan Jun

Mathematics 2026, 14(15), 2713; https://doi.org/10.3390/math14152713 - 30 Jul 2026

Multi-scale rendering exposes scale-dependent reconstruction errors in 3D Gaussian splatting (3DGS) when the rendering resolution differs from the training resolution. Existing anti-aliasing and multi-scale optimization methods reduce these artifacts, but each Gaussian usually keeps a fixed color representation and a fixed spatial allocation [...] Read more.

Multi-scale rendering exposes scale-dependent reconstruction errors in 3D Gaussian splatting (3DGS) when the rendering resolution differs from the training resolution. Existing anti-aliasing and multi-scale optimization methods reduce these artifacts, but each Gaussian usually keeps a fixed color representation and a fixed spatial allocation after densification. This fixed representation is restrictive at coarse scales because one rendered pixel combines multiple Gaussian contributions into an average color that can differ from the color assigned to each Gaussian. In addition, Gaussians allocated by fine-scale reconstruction criteria may provide insufficient coverage for regions with persistent coarse-scale errors. This paper proposes adaptive-scale color offset and error-guided Gaussian reallocation for multi-scale 3DGS. The adaptive-scale color offset applies a scale-specific correction to the shared base color of each Gaussian while keeping its geometry unchanged. Error-guided Gaussian reallocation shifts low-contribution Gaussians toward high-error regions under a fixed primitive budget and preserves Gaussians that support coarse-scale rendering. Experimental results show that the proposed method improves multi-scale rendering quality across rendering scales on Mip-NeRF 360, Tanks and Temples, and Deep Blending, with larger gains at coarse scales. These results demonstrate that adapting Gaussian color and primitive allocation together improves scale-consistent 3DGS rendering without modifying the inference-stage rendering process. Full article

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14 pages, 348 KB

Open AccessArticle

Raw Moments of Asset Returns Under a Non-Homogeneous Poisson Bid Arrival Process with Beta-Distributed Percentage Bids

by Steve Drekic

Mathematics 2026, 14(15), 2712; https://doi.org/10.3390/math14152712 - 30 Jul 2026

Abstract

Suppose that bids, or offers, to purchase an asset for sale occur according to a non-homogeneous Poisson process. Bid amounts, measured in percentages, are assumed to be independent and identically distributed random variables. Faced with the pressure of having to sell the asset [...] Read more.

Suppose that bids, or offers, to purchase an asset for sale occur according to a non-homogeneous Poisson process. Bid amounts, measured in percentages, are assumed to be independent and identically distributed random variables. Faced with the pressure of having to sell the asset by a certain time period (or else risk getting nothing at all for it), this article analyzes the probability distribution of the obtained return on the asset under a specific type of selling strategy, advancing the initial results found in Karlin’s seminal paper from 1962. In particular, the main result derived is an explicit expression for the raw moments of the return on the asset. Such moments are widely used in probability theory and stochastic modeling, since, in most instances, having all the raw moments fully determines the underlying probability distribution. The special case when bid amounts have a beta distribution with integer shape parameters is examined in depth. Numerical results are presented and discussed for a variety of different selling strategies, time-varying offer rates, and bid amount distributions. Full article

(This article belongs to the Special Issue Advances in Mathematical Optimization in Operational Research)

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26 pages, 3160 KB

Open AccessArticle

Complex Krawtchouk and Hahn Polynomial Operators for Localized and Numerically Stable Discrete Image Enhancement

by Hasan Bayram, Sibel Yalçın and Alina Alb Lupaş

Mathematics 2026, 14(15), 2711; https://doi.org/10.3390/math14152711 - 30 Jul 2026

Abstract

The Krawtchouk and Hahn families are discrete orthogonal polynomials defined on the integer pixel grid, yet as polynomials, they are entire functions of a complex variable. Adopting this complex variable and complex-valued viewpoint, we develop an image enhancement framework that connects discrete orthogonal [...] Read more.

The Krawtchouk and Hahn families are discrete orthogonal polynomials defined on the integer pixel grid, yet as polynomials, they are entire functions of a complex variable. Adopting this complex variable and complex-valued viewpoint, we develop an image enhancement framework that connects discrete orthogonal polynomial theory with geometric function theory. Two operators are introduced. The Krawtchouk operator exploits the binomial weight, for which the parameter p concentrates the basis around a selectable tonal level

x = p N

, producing a localized contrast enhancement steerable toward shadows, midtones, or highlights. The Hahn operator uses the two-parameter Hahn polynomials, the discrete analogue of the Jacobi family, in which

(α, β)

give asymmetric control of dark and light bands. Each operator is the real restriction of a holomorphic near identity map

F (z) = z + \sum_{k} c_{k} {\tilde{ϕ}}_{k} (z)

, with the intensity entering the discrete basis through

x = N I

, realized as a monotone 256-entry lookup table. We prove a bounded deviation estimate

| F - id | \leq \sum_{k} | c_{k} |

on the intensity segment

[0, 1]

and a positive slope condition that, via the Noshiro–Warschawski criterion, is a univalence condition for the analytic transfer map, ruling out intensity order reversal and oscillatory folding. Because Hahn polynomials lose orthogonality at high order in naive arithmetic, we show that a three-term recurrence with log-gamma weights preserves orthonormality to within about

10^{- 10}

on the full 8-bit grid, where single precision computation fails, and that the same scheme remains at the double-precision roundoff level on 10-, 12-, and 16-bit grids (

N = 1023, 4095, 65, 535

). The operators cost

O (m L)

table construction plus one lookup per pixel (about 3 ms for a

1024 \times 1024

color image), and a histogram-based differential entropy criterion selects the focus parameters automatically. Experiments on imagery of fine art, wildlife, archaeology, and architecture show that the Krawtchouk operator yields stronger localized contrast compared to seven classical methods, while the Hahn operator attains higher PSNR/SSIM, both as deterministic fast slope-controlled transforms. A diffusion MRI example further demonstrates that matching the focus parameter to the tonal mass adapts the same operators to dark-dominated medical scan imagery. Full article

(This article belongs to the Section C: Mathematical Analysis)

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42 pages, 1574 KB

Open AccessArticle

A Comparative Study of Structure-Preserving Representations for Variable-Turbine-Number Wind Farm Layout Optimization

by Haichuan Yang, Yifei Yang, Shanxian Lin, Qiong Fu, Guodong Cui, Ancai Zhang and Yuichi Nagata

Mathematics 2026, 14(15), 2710; https://doi.org/10.3390/math14152710 - 29 Jul 2026

Abstract

Wind farm layout optimization is commonly studied under a fixed number of turbines, although turbine quantity is closely coupled with wake effects and power generation efficiency. This study formulates the problem as a variable-turbine-number optimization task and systematically compares three representative variable-length approaches: [...] Read more.

Wind farm layout optimization is commonly studied under a fixed number of turbines, although turbine quantity is closely coupled with wake effects and power generation efficiency. This study formulates the problem as a variable-turbine-number optimization task and systematically compares three representative variable-length approaches: dimension activation/deactivation, representation mapping, and variable-length evolutionary representation. The study also investigates structure-preserving search strategies based on binary occupancy representations. The findings suggest that, for variable-turbine-number wind farm layout optimization, the effective preservation and exploitation of spatial layout structures are more critical than merely handling dimensional changes. From a fitness-landscape perspective, structure-preserving representations appear to create more coherent neighborhoods and more accessible search topologies. These properties facilitate transitions across heterogeneous turbine-number subspaces and improve search performance. Full article

(This article belongs to the Special Issue Ensemble Evolutionary Algorithms and Machine Learning for Solving Complex Optimization and Scheduling Problems, 2nd Edition)

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60 pages, 1692 KB

Open AccessReview

A Review of Language-Guided Dexterous Grasping with Human-in-the-Loop Adaptation

by Eugene Jeong and Soo-Chul Lim

Mathematics 2026, 14(15), 2709; https://doi.org/10.3390/math14152709 - 29 Jul 2026

Abstract

Language-guided dexterous grasping aims to translate natural-language instructions and multimodal observations into multi-fingered grasps that are physically admissible, semantically aligned, functionally suitable, user-adaptive, and robust under execution uncertainty. Existing research remains fragmented across language-conditioned manipulation, dexterous grasp generation, human-in-the-loop learning, and simulation-based validation. [...] Read more.

Language-guided dexterous grasping aims to translate natural-language instructions and multimodal observations into multi-fingered grasps that are physically admissible, semantically aligned, functionally suitable, user-adaptive, and robust under execution uncertainty. Existing research remains fragmented across language-conditioned manipulation, dexterous grasp generation, human-in-the-loop learning, and simulation-based validation. This structured scoping review synthesizes these areas through a formulation-oriented analytical lens and organizes prior work around the main decisions involved in language-guided dexterous grasping: candidate generation, admissibility verification, multi-objective evaluation, human-feedback-based adaptation, and simulation-based or digital twin-based validation. We compare language, vision–language, and vision–language–action models, optimization-based and generative grasp methods, feedback mechanisms, and validation strategies in terms of their representations, assumptions, data requirements, evaluation settings, and failure modes. The synthesis indicates that no single paradigm currently provides reliable end-to-end language-guided dexterous grasping: semantic reasoning lacks contact-level guarantees, physically grounded generators provide limited task and user conditioning, human feedback introduces scalability costs, and simulation remains vulnerable to model mismatch. Finally, we identify benchmark requirements and research directions for language-conditioned datasets, semantic alignment, functional suitability, tactile integration, low-burden feedback, execution-risk estimation, and calibrated sim-to-real validation. Full article

(This article belongs to the Special Issue AI-Driven Computational Methods: Theories, Algorithms and Applications)

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12 pages, 777 KB

Open AccessArticle

The Least Distance Eigenvalue of Complement Graphs with Two Pendant Vertices

by Jinfeng Zhang, Xu Chen, Jiawen Deng, Shuo Li, Guodong Li and Liyan Hua

Mathematics 2026, 14(15), 2708; https://doi.org/10.3390/math14152708 - 29 Jul 2026

Abstract

Let G be a connected simple graph with the vertex set

V (G) = {v_{1}, v_{2}, \dots, v_{n}}

, where the distance

d_{G} (v_{i}, v_{j})

is the [...] Read more.

Let G be a connected simple graph with the vertex set

V (G) = {v_{1}, v_{2}, \dots, v_{n}}

, where the distance

d_{G} (v_{i}, v_{j})

is the length of a shortest path between

v_{i}

and

v_{j}

, and the distance matrix

D (G) = (d_{i j}) n \times n

is defined by

d_{i j} = d_{G} (v_{i}, v_{j})

. Being real symmetric and non-negative,

D (G)

has real eigenvalues

λ_{1} (G) \geq λ_{2} (G) \geq \dots \geq λ_{n} (G)

, with

λ_{n} (G)

referred to as the least distance eigenvalue of G. In this paper, we give the maximum value of the least distance eigenvalue, and determine the unique extremal graph with two pendant vertices whose complements attain the value. Full article

(This article belongs to the Special Issue Advances in Combinatorics, Discrete Mathematics and Graph Theory, 2nd Edition)

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26 pages, 532 KB

Open AccessArticle

On the Optimality of

k = \sqrt{n}

in k-Nearest Neighbor Classification: Sub-Optimality Rates, Dimension-Aware Selection, and Hassanat Distance Comparison

by Ahmad B. Hassanat, Anas A. Alkasasbeh, Esra’a Alkafaween, Omar Lasassmeh, Khalid Almohammadi, Malek Alrashidi, Abdulkareem Alzahrani, Maha Alamri and Ahmad S. Tarawneh

Mathematics 2026, 14(15), 2707; https://doi.org/10.3390/math14152707 - 29 Jul 2026

Abstract

The k-nearest neighbor (KNN) algorithm remains one of the most fundamental and widely-used methods in machine learning. A common rule of thumb sets the number of neighbors as

k = \sqrt{n}

, where n is the size of the training set. Despite [...] Read more.

The k-nearest neighbor (KNN) algorithm remains one of the most fundamental and widely-used methods in machine learning. A common rule of thumb sets the number of neighbors as

k = \sqrt{n}

, where n is the size of the training set. Despite its widespread adoption, the theoretical justification for this choice has remained obscure. We provide a comprehensive rate-based analysis. First, we derive the minimax-optimal exponent

β^{*} = 4 / (d + 4)

for

k = n^{β}

under standard Hölder-smoothness assumptions, recovering as a special case Theorem 1: under these second-order Hölder-smoothness assumptions,

k = \sqrt{n}

attains the asymptotic minimax-optimal exponent if and only if the feature space has dimensionality

d = 4

; this concerns the asymptotic rate exponent, not the finite-sample optimal value of k. Second, Theorem 2 quantifies the sub-optimality of any fixed

β

as

R_{n} (n^{β}) = Θ (n^{- r (β, d)})

with

r (β, d) = min {β, 4 (1 - β) / d}

, yielding an asymmetric penalty for the classical rule when

d \neq 4

that we make precise in Corollary 1. The predicted rate is empirically verified across

d \in {2, \dots, 20}

on controlled synthetic data. On 48 datasets from the OpenML-CC18 benchmark suite, the dimension-aware rule

k = ⌊ n^{4 / (d + 4)} ⌋

outperforms the classical

\sqrt{n}

rule in 32 of 48 head-to-head comparisons (paired Wilcoxon

p = 4.6 \times 10^{- 4}

, mean accuracy gain

+ 2.5

percentage points), demonstrating that the theoretical improvement translates to a practical one. We further test the Hassanat distance metric against Euclidean across all KNN variants on the same 48 datasets, finding that Hassanat outperforms Euclidean in five of six configurations (paired Wilcoxon

p < 0.05

), with the largest gains on unstandardized data. Cross-validation remains the strongest k-selection strategy when computationally feasible, and the theoretical results provide a principled non-cross-validated alternative. Full article

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

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40 pages, 2262 KB

Open AccessArticle

Zealots and Preachers: A Heterogeneous Node–Edge Diffusion Model on Networks

by Ricardo Gimeno and Ruth Mateos de Cabo

Mathematics 2026, 14(15), 2706; https://doi.org/10.3390/math14152706 - 29 Jul 2026

Abstract

We introduce the Zealot–Preacher model, a generalized diffusion framework on networks with simultaneous heterogeneity at the node and edge levels. Standard diffusion models typically assume homogeneous susceptibility across nodes and homogeneous transmission strength across edges. We relax both assumptions by assigning to each [...] Read more.

We introduce the Zealot–Preacher model, a generalized diffusion framework on networks with simultaneous heterogeneity at the node and edge levels. Standard diffusion models typically assume homogeneous susceptibility across nodes and homogeneous transmission strength across edges. We relax both assumptions by assigning to each node a latent receptiveness parameter and to each edge a latent conductiveness parameter governing influence propagation. The framework accommodates both binary contagion and continuous consensus dynamics and nests standard homogeneous diffusion as a special case. We characterize the resulting weighted consensus process and show that low-receptiveness nodes disproportionately influence long-run network outcomes. To estimate the model in large networks, we develop a hierarchical latent-variable approach in which receptiveness and conductiveness follow Beta distributions, with hyperparameters estimated by maximum likelihood and individual effects recovered through Bayesian updating. An application to U.S. interlocking director networks shows that diffusion of board gender diversity is constrained primarily by widespread node-level resistance rather than by the absence of conductive transmission channels. Female shared directors exhibit higher average conductiveness than male shared directors, suggesting gendered heterogeneity in diffusion channels. Full article

(This article belongs to the Special Issue Complex Systems and Networks)

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19 pages, 3328 KB

Open AccessArticle

Algebraic Elliptical Extent Estimation for Robust Extended Object Tracking from Partial Contour Observations

by In Ho Lee

Mathematics 2026, 14(15), 2705; https://doi.org/10.3390/math14152705 - 29 Jul 2026

Abstract

Extended object tracking (EOT) has emerged as a critical challenge in modern surveillance and autonomous systems, where high-resolution sensors yield multiple measurements per target in a single scan. This task necessitates the simultaneous estimation of an object’s spatial extent and kinematic states by [...] Read more.

Extended object tracking (EOT) has emerged as a critical challenge in modern surveillance and autonomous systems, where high-resolution sensors yield multiple measurements per target in a single scan. This task necessitates the simultaneous estimation of an object’s spatial extent and kinematic states by systematically assimilating distributed detections. In practical scenarios, however, sensor visibility constraints often restrict measurements to partial contours rather than the full object geometry, posing a significant hurdle for accurate state estimation. This paper proposes a robust EOT algorithm that addresses these challenges by extracting elliptical features—including size, orientation, and noise characteristics—in an affine-invariant manner from incomplete measurement sets. By formulating the ellipse-fitting problem through an algebraic solution, the proposed method effectively reconstructs the target’s morphology even under severe self-occlusion. The tracking framework is established as a Bayesian inference problem, utilizing a standard Kalman filter as a recursive estimator to ensure both computational efficiency and numerical robustness. In Monte Carlo simulations in which measurements are restricted to the sensor-facing contour, the proposed method reduces the average extent estimation error, quantified by the Gaussian Wasserstein distance, to 9.15 m, compared with 64.84–70.62 m for three state-of-the-art baselines, while retaining real-time feasibility with an average processing time of 0.94 ms per scan. Experiments on real-world LiDAR vehicle-tracking data further confirm consistent and accurate extent estimation under partial observability and complex maneuvers. Full article

(This article belongs to the Special Issue Advanced Filtering and Control Methods for Stochastic Systems)

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25 pages, 2222 KB

Open AccessArticle

Abstracting Business Resources for Multi-Collaboration: The WFC-Net Approach

by Xiaotong Chen, Tong Gu, Jincheng Pan and Linfu Sun

Mathematics 2026, 14(15), 2704; https://doi.org/10.3390/math14152704 - 28 Jul 2026

Abstract

The industrial chain forms a network of interconnected enterprises spanning from raw material procurement to final product distribution, enabling cost reduction, operational efficiency improvement, and enhanced market competitiveness. However, business resource conversion among enterprises often leads to information asymmetry, which impedes effective interconnection [...] Read more.

The industrial chain forms a network of interconnected enterprises spanning from raw material procurement to final product distribution, enabling cost reduction, operational efficiency improvement, and enhanced market competitiveness. However, business resource conversion among enterprises often leads to information asymmetry, which impedes effective interconnection and collaboration. To address this challenge, WFC-Net (Workflow-Control Petri Net with Resource Fusion), a model tailored for multi-collaboration scenarios among interconnected enterprises in the industrial chain, is proposed. The proposed approach integrates WF-Net workflow characteristics with C-Net dependency modeling and Petri net transitions to enable precise workflow control and effective business resource coordination. The resource attributes are incorporated as first-class net elements, and a receding-horizon optimization policy is employed to dynamically adapt to varying operational conditions. Comparative simulation experiments demonstrate that WFC-Net achieves superior performance across all metrics, with an 89.1% authorization success rate and 36.0% event completion rate, outperforming Petri Net, C-Net, and WF-Net baselines. The full-rank reachability matrix provides a necessary condition for linearized controllability, offering a theoretical foundation for stable authorization behavior under the receding-horizon policy. These results demonstrate that WFC-Net consistently achieves or approximates target business-side metrics while ensuring controllability and soundness. Full article

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29 pages, 103702 KB

Open AccessArticle

Epidemic Spreading and Control on Preferential Attachment Hypergraph with Community

by Jialin Bi and Ninghan Sun

Mathematics 2026, 14(15), 2703; https://doi.org/10.3390/math14152703 - 28 Jul 2026

Abstract

The study of epidemic spreading in complex networks is fundamental to understanding diffusion processes across natural and social systems. While traditional graphs capture only pairwise interactions, many real-world processes involve higher-order group interactions that can be naturally represented by hypergraphs. In this work, [...] Read more.

The study of epidemic spreading in complex networks is fundamental to understanding diffusion processes across natural and social systems. While traditional graphs capture only pairwise interactions, many real-world processes involve higher-order group interactions that can be naturally represented by hypergraphs. In this work, we propose a community-based preferential attachment hypergraph model with tunable modularity and a heavy-tailed degree distribution, reproducing key structural properties in real systems. Based on this model, we develop a hypergraph-based SAIR framework to describe epidemic dynamics with asymptomatic transmission. A mean-field approximation is derived and compared with classical mean-field, heterogeneous mean-field, and Monte Carlo simulations, demonstrating improved predictive accuracy for community hypergraphs. The results show that epidemic spreading is regulated by community structure, transmission probability, and initial conditions, giving rise to localized, heterogeneous, and global diffusion regimes. By introducing a cross-community hyperedge index, we reveal that community structure suppresses spreading primarily through the reduction in inter-community transmission pathways. These factors collectively determine the spreading radius, propagation speed, and epidemic peak. We further evaluate behavioral, hyperedge-based, and node-based intervention strategies. Overall, this study provides a quantitative framework for analyzing epidemic spreading and control on community-structured hypergraphs, with potential relevance to studies of information diffusion and risk propagation. Full article

(This article belongs to the Special Issue Advanced Research in Complex Networks and Social Dynamics)

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29 pages, 7315 KB

Open AccessArticle

CreditR1: Calibration-Aware Reinforcement Learning for Interpretable Corporate Credit Risk Assessment with Large Language Models

by Yuxuan Wu, Haowen Dai, Yiheng Zhang and Jinping Ma

Mathematics 2026, 14(15), 2702; https://doi.org/10.3390/math14152702 - 28 Jul 2026

Abstract

Probability-of-default (PD) estimation under the Basel and IFRS 9 frameworks requires probabilities that are both discriminative and well-calibrated. Large language models applied to credit assessment via prompting or supervised fine-tuning (SFT) yield poorly calibrated probabilities, while reinforcement learning with binary correctness rewards is [...] Read more.

Probability-of-default (PD) estimation under the Basel and IFRS 9 frameworks requires probabilities that are both discriminative and well-calibrated. Large language models applied to credit assessment via prompting or supervised fine-tuning (SFT) yield poorly calibrated probabilities, while reinforcement learning with binary correctness rewards is structurally unsuitable for probability prediction under extreme class imbalance. We propose CreditR1, a three-stage framework: an SFT cold start on evidence-filtered reasoning chains; Group Relative Policy Optimization, guided by a composite verifiable reward combining Brier-score calibration—a strictly proper scoring rule—pairwise ranking, evidence anchoring, and format compliance; and an anti-contamination evaluation protocol. On Chinese A-share corporate credit data, CreditR1 matches gradient-boosted baselines in discrimination (AUC:

0.883 \pm 0.004

vs. 0.891 for XGBoost) while reducing expected calibration error by 24.2% versus isotonic-calibrated XGBoost (ECE: 0.047 vs. 0.062) and by 47.2% versus uncalibrated XGBoost (0.089). Because the test set contains only 119 default events, all comparisons carry firm-level bootstrap confidence intervals; the calibration advantage remains significant against every baseline after Holm–Bonferroni correction, including Platt, beta, and Bayesian-binning recalibrations. Ablations confirm each reward component is necessary. CreditR1 delivers calibrated PDs with evidence-grounded reasoning that supports internal model validation and human review; transferability beyond the Chinese A-share market remains an open empirical question. Full article

(This article belongs to the Special Issue Intelligent Financial Systems: Algorithms, Learning, and Decision Mechanisms)

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17 pages, 2160 KB

Open AccessArticle

Mahalanobis-Constrained Hard-Clustering Algorithm

by Vedran Novoselac

Mathematics 2026, 14(15), 2701; https://doi.org/10.3390/math14152701 - 28 Jul 2026

Abstract

The paper proposes a modification of the standard k-means algorithm that removes low-probability tail points of Gaussian components using the p-quantile of the chi-squared distribution as a trimming threshold based on the squared Mahalanobis distance. The procedure is evaluated on synthetic [...] Read more.

The paper proposes a modification of the standard k-means algorithm that removes low-probability tail points of Gaussian components using the p-quantile of the chi-squared distribution as a trimming threshold based on the squared Mahalanobis distance. The procedure is evaluated on synthetic noisy datasets and compared with well-known state-of-the-art clustering methods. Applications in circular image pattern recognition demonstrate effective detection of spherical patterns in highly noisy environments. Pattern detection is driven by an optimization using a constructed spherical clustering-validity index, and the method’s practical potential is confirmed by comparisons with state-of-the-art circular detection techniques. Full article

(This article belongs to the Special Issue Innovative Intelligent Systems in Computer Vision: Methodologies, Algorithms, and Applications)

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17 pages, 1036 KB

Open AccessArticle

Dual-Scale Grid-Based Adaptive Trajectory Planning for UAVs in Urban Low-Altitude Airspace

by Xin Zhang, Guang Cheng, Chao Wang, Yu Liu, Guanwang Jiang and Ziye Jia

Mathematics 2026, 14(15), 2700; https://doi.org/10.3390/math14152700 - 28 Jul 2026

Abstract

With the rapid growth of the low-altitude economy, the number of unmanned aerial vehicles (UAVs) has grown rapidly. It is challenging to plan substantial UAV trajectories in complex urban low-altitude airspace, considering the airspace capacity, inter-vehicle safety, and communication reliability. To deal with [...] Read more.

With the rapid growth of the low-altitude economy, the number of unmanned aerial vehicles (UAVs) has grown rapidly. It is challenging to plan substantial UAV trajectories in complex urban low-altitude airspace, considering the airspace capacity, inter-vehicle safety, and communication reliability. To deal with this challenge, we propose a dual-scale grid-based trajectory planning approach that separates the global routing and local refinement. Specifically, we discretize the three-dimensional airspace into coarse macro-grids for capacity-constrained routing and high-quality communication-aided fine micro-grids for collision-free trajectory refinement. Both consider an altitude-dependent energy model. To handle the complex dual-scale grid trajectory planning of UAVs, we propose a priority-driven dual-grid Theta* with adaptive relaxation (DGTAR) to balance the global planning efficiency and local obstacle avoidance accuracy. First, we design a priority-driven capacity allocation mechanism to enforce safe separation among UAVs. Then, a combined planner is proposed, which integrates a Theta*-enhanced global search with a sampling-based refinement algorithm, invoking on-demand boundary relaxation to ensure the feasibility. Simulation results reveal that the proposed method DGTAR achieves reductions in many aspects compared with benchmark mechanisms, while maintaining a high planning success rate in congested scenarios. Full article

(This article belongs to the Special Issue Computational Methods for Network Optimization and Security)

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18 pages, 333 KB

Open AccessArticle

High-Order Convergent Numerical Method for Pricing Asian Options Under Regime-Switching Model

by Taoshun He and Yong Chen

Mathematics 2026, 14(15), 2699; https://doi.org/10.3390/math14152699 - 28 Jul 2026

Abstract

In this paper, we study the fourth-order convergent finite difference method for solving a moving boundary partial differential equation (PDE) system of regime-switching Asian option pricing and conduct the convergence analysis in the discrete

L^{\infty}

-norm. First, the finite difference method is [...] Read more.

In this paper, we study the fourth-order convergent finite difference method for solving a moving boundary partial differential equation (PDE) system of regime-switching Asian option pricing and conduct the convergence analysis in the discrete

L^{\infty}

-norm. First, the finite difference method is proposed for solving the PDE system and combined with the Richardson extrapolation scheme for improving the computational accuracy. Then the fourth-order convergence rates in both time and space are rigorously proved in the discrete

L^{\infty}

-norm. Numerical examples are presented to demonstrate the convergence results and show the effectiveness of the studied scheme. Full article

(This article belongs to the Section E5: Financial Mathematics)

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28 pages, 721 KB

Open AccessArticle

Adaptive Wrapped Robust Canonical Correlation Analysis in High-Dimensional Data

by Hasan Bulut, Müjgan Zobu and Vedat Sağlam

Mathematics 2026, 14(15), 2698; https://doi.org/10.3390/math14152698 - 27 Jul 2026

Abstract

Classical canonical correlation analysis becomes numerically unstable when the number of variables is large relative to the sample size and is sensitive to contamination in observations or individual cells. This study develops an integrated robust and regularized procedure that combines bounded cellwise wrapping, [...] Read more.

Classical canonical correlation analysis becomes numerically unstable when the number of variables is large relative to the sample size and is sensitive to contamination in observations or individual cells. This study develops an integrated robust and regularized procedure that combines bounded cellwise wrapping, shrinkage estimation of the joint correlation matrix, and robust reweighting in a low-dimensional canonical score space. The resulting observation weights enter a second regularized canonical correlation fit, so the final estimator remains well defined when the combined number of variables exceeds the sample size. The simulation study shows that relative estimation accuracy depends on the signal strength, contamination mechanism, and dimensional configuration. The proposed estimator is competitive in several moderate-signal settings and has a clear computational advantage, whereas the minimum regularized covariance determinant plug-in estimator provides lower estimation error in many high-signal configurations. An additional ultra-high-dimensional experiment demonstrates numerical feasibility with modest memory use but also reveals substantial attenuation, identifying a limitation of the present dense estimator. The results therefore support a regime-dependent interpretation rather than a claim of uniform superiority. The complete reproducible simulation workflow is provided. Full article

(This article belongs to the Special Issue Advances in Multivariate Statistical Models: Approaches and Applications for Continuous Data)

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26 pages, 20245 KB

Open AccessArticle

A Method for 6-DOF Motion Measurement of Marine Floating Structures Based on Monocular Vision and Feature Point Tracking

by Chunyu Jiang, Hongda Shi, Chenyu Zhao, Qian Deng, Jian Li and Huihui Sun

Mathematics 2026, 14(15), 2697; https://doi.org/10.3390/math14152697 - 27 Jul 2026

Abstract

Accurate measurement of the 6-DOF motion responses of marine floating structures is essential for structural safety assessment and operational decision-making. To address the critical issues of integration drift in inertial navigation systems, susceptibility of GNSS to sea-surface multipath effects, and deployment complexity of [...] Read more.

Accurate measurement of the 6-DOF motion responses of marine floating structures is essential for structural safety assessment and operational decision-making. To address the critical issues of integration drift in inertial navigation systems, susceptibility of GNSS to sea-surface multipath effects, and deployment complexity of binocular vision systems, this paper proposed a 6-DOF motion measurement method for floating structures based on monocular vision and natural feature point tracking. This method eliminates the reliance on artificial cooperative targets and auxiliary sensors, instead utilizing the inherent surface textures of the floating structures as feature sources. Stable feature point tracking is achieved through multi-strategy cascaded detection and the pyramidal KLT optical flow algorithm. RANSAC geometric consistency verification is introduced to eliminate outlier matches, retaining only identical physical points between two consecutive frames for motion estimation. In-plane translations and

R Z

angle are extracted from the similarity transformation, while

R X

and

R Y

angles are estimated using principal component analysis of the covariance matrix of the feature point set. The depth-direction displacement is linearly mapped from variations in the scale factor. Subsequently, two series of physical model tests under different conditions were conducted to validate the measurement accuracy and robustness of the proposed method on different floating structures. The results demonstrate that the proposed method can accurately capture the motion attitudes of floating structures, maintaining a consistently high inlier ratio exceeding 80% in regular waves and averaging 85.2% in irregular waves, with a reprojection error of less than 0.05 pixels. The NRMSE for the primary motion directions are all below 10%, and the dominant frequency errors are essentially zero. It offers advantages such as low cost, easy deployment, and strong robustness, thereby providing valuable technical support for field monitoring of marine floating structures. Full article

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

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25 pages, 2184 KB

Open AccessArticle

Density-Based Outlier Detection of Atypical Timeout Request Contexts in EuroLeague Basketball

by José Miguel Contreras García, Elena Molina Portillo and Juan Manuel Fernández Luna

Mathematics 2026, 14(15), 2696; https://doi.org/10.3390/math14152696 - 27 Jul 2026

Abstract

Timeouts in elite basketball are scarce, time-constrained interventions requested under evolving pressure and contextual imbalance. Although previous clustering approaches have identified recurrent timeout profiles, less attention has been paid to structurally atypical timeout request contexts that do not conform to dominant patterns. This [...] Read more.

Timeouts in elite basketball are scarce, time-constrained interventions requested under evolving pressure and contextual imbalance. Although previous clustering approaches have identified recurrent timeout profiles, less attention has been paid to structurally atypical timeout request contexts that do not conform to dominant patterns. This study proposes a density-based outlier detection framework for EuroLeague basketball. Using play-by-play-derived data from 3743 games across 15 seasons, 25,031 timeout events were represented through differential and rate-adjusted indicators of scoring, shooting, free throws, rebounding, assists, turnovers, fouls, blocks, and score imbalance. Principal component analysis was used to construct a compact multivariate feature space. DBSCAN and OPTICS served as diagnostic density-based comparators, whereas HDBSCAN was used to derive observation-level outlier scores. Atypical contexts were defined using the highest outlier score events, with the top 10% group as the primary threshold. Flat density-based partitions did not support a compact timeout taxonomy, whereas HDBSCAN provided a stable and interpretable outlier score ranking. The most atypical timeouts were associated with larger score imbalances and rate-adjusted performance differentials, but an ablation analysis excluding score margin and score rate variables showed that the outlier ranking structure was not solely determined by direct scoreboard information. Full article

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

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24 pages, 2424 KB

Open AccessArticle

FraudDebate-Agent: A Multi-Agent LLM Framework with an Evidence-Based Debate Mechanism for Financial Statement Fraud Detection

by Xinran Yue, Jingyun Yang and Wenhe Liu

Mathematics 2026, 14(15), 2695; https://doi.org/10.3390/math14152695 - 27 Jul 2026

Abstract

Financial statement fraud inflicts large and recurring losses on capital markets, yet the dominant detection paradigm still relies on single, black-box classifiers (e.g., RUSBoost) trained on structured accounting ratios alone. Two limitations follow: (i) the rich, unstructured Management Discussion and Analysis (MD&A) narrative [...] Read more.

Financial statement fraud inflicts large and recurring losses on capital markets, yet the dominant detection paradigm still relies on single, black-box classifiers (e.g., RUSBoost) trained on structured accounting ratios alone. Two limitations follow: (i) the rich, unstructured Management Discussion and Analysis (MD&A) narrative of the 10-K filing is discarded, and (ii) the resulting scores are difficult for auditors to trust because they carry no transparent, standards-aligned rationale. Recent large language model (LLM) systems have shown that multi-agent collaboration is more robust than a single LLM for anomaly detection, but no study has systematically transferred this paradigm to listed-company statement fraud. We propose FraudDebate-Agent, a four-role multi-agent system in which a Quantitative Analyst agent scores 28 raw accounting items and 14 ratios with gradient-boosted and tabular attention models, a Narrative Auditor agent quantifies tone, linguistic uncertainty, and year-over-year textual novelty of the MD&A with FinBERT, and an Industry Peer agent uses retrieval-augmented generation to measure industry-relative anomaly. A Critic–Debate agent then orchestrates a pair-wise Evidence-based Multi-Agent Debate (EMAD) that reconciles disagreement across modalities and arbitrates a reconciled fraud-risk assessment, which is aggregated over a tri-modal evidence graph. Our contributions are as follows: (1) the first use of an evidence-grounded debate mechanism for accounting fraud, which materially reduces LLM hallucination; (2) a numerical–textual–peer evidence graph that fuses heterogeneous signals; and (3) an explainable report aligned with the PCAOB AS 2401 fraud-risk taxonomy. On AAER-labelled firm-years linked across a SEC financial dataset and EDGAR-CORPUS, FraudDebate-Agent improves the area under the ROC curve and the rare-event ranking metric NDCG@k over the strongest single-modality and single-LLM baselines while producing substantially more faithful explanations. We frame the system as a fraud-risk screening and risk-ranking tool for AAER-labelled misstatement risk rather than a determination of fraudulent intent. We report results over multiple seeds to reflect real-world stochasticity and discuss limitations and cross-domain applications. Full article

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