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In this work, we show that complete non-compact manifolds with non-negative Ricci curvature, Euclidean volume growth and sufficiently small curvature concentration are necessarily flat Euclidean space. [ABSTRACT FROM AUTHOR]

This study addresses the challenges of maritime traffic management in the western waters of Taiwan, a region characterized by substantial commercial shipping activity and ongoing environmental development. Using 2023 Automatic Identification System (AIS) data, this study develops a robust feature extraction framework involving data cleaning, anomaly trajectory point detection, trajectory compression, and advanced processing techniques. Dynamic Time Warping (DTW) and the Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN) algorithms are applied to cluster the trajectory data, revealing 16 distinct maritime traffic patterns, key navigation routes, and intersections. The findings provide fresh perspectives on analyzing maritime traffic, identifying high-risk areas, and informing safety and spatial planning. In practical applications, the results help navigators optimize route planning, improve resource allocation for maritime authorities, and inform the development of infrastructure and navigational aids. Furthermore, these outcomes are essential for detecting abnormal ship behavior, and they highlight the potential of route extraction in maritime surveillance. [ABSTRACT FROM AUTHOR]

On a complete |$p$| -nonparabolic |$3$| -dimensional manifold with non-negative scalar curvature and vanishing second homology, we establish a sharp monotonicity formula for the proper |$p$| -Green function along its level sets for |$1

In this work, we extend the existence theory of non-collapsed Ricci flows from point-wise curvature lower bound to Kato-type curvature lower bound. As an application, we prove that any compact three-dimensional non-collapsed strong Kato limit space is homeomorphic to a smooth manifold. Moreover, similar result also holds in higher dimension under stronger curvature condition. We also use the Ricci flow smoothing to study stability problem in scalar curvature geometry. [ABSTRACT FROM AUTHOR]

AIS ship-type code categorizes ships into broad classes, such as fishing, passenger, and cargo, yet struggles with finer distinctions among cargo ships, such as bulk carriers and containers. Different ship types significantly impact acceleration, steering performance, and stopping distance, thus making precise identification of unfamiliar ship types crucial for maritime monitoring. This study introduces an original classification study based on AIS data for cargo ships, presenting a classifier tailored for bulk carriers, containers, general cargo, and vehicle carriers. The model's efficacy was tested within the Changhua Wind Farm Channel using eight classification algorithms across tree-structure-based, proximity-based, and regression-based categories and employing standard metrics (Accuracy, Precision, Recall, F1-score) to assess the performance. The results show that tree-structure-based algorithms, particularly XGBoost and Random Forest, demonstrated superior performance. This study also implemented a feature selection strategy with five methods, revealing that a model trained with only four features (three ship-geometric features and one trajectory behavior feature) can achieve high accuracy. Conclusively, the classifier effectively overcame the challenges of limited AIS data labels, achieving a classification accuracy of 97% for ships in the Changhua Wind Farm Channel. These results are pivotal in identifying abnormal ship behavior, highlighting the classifier's potential for maritime monitoring applications. [ABSTRACT FROM AUTHOR]

Motivated by the torus stability problem, in this work, we study Kähler metrics with almost non-negative scalar curvature on complex torus. We prove that after passing to a subsequence, non-collapsing sequence of Kähler metrics with almost non-negative scalar curvature will converge to flat torus weakly. [ABSTRACT FROM AUTHOR]

In this paper, we study the hypercritical deformed Hermitian-Yang–Mills equation on compact Kähler manifolds and resolve two conjectures of Collins–Yau [Moment maps, nonlinear PDE, and stability in mirror symmetry, preprint (2018), https://arxiv.org/abs/1811.04824]. [ABSTRACT FROM AUTHOR]

It is conjectured by Song–Tian that the normalized Kähler–Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle is uniformly smooth away from the singular fibers. In this work, we adapt the method in the work of Hein–Tosatti on collapsing Calabi–Yau metrics and obtain a uniform spatial Hölder estimate along the normalized Kähler–Ricci flow for all time away from the singular fiber. [ABSTRACT FROM AUTHOR]

In this work, we consider sequences of C 2 metrics which converges to a C 2 metric in C 0 sense. We show that if the scalar curvature of the sequence is almost non-negative in the integral sense, then the limiting metric has scalar curvature lower bound in point-wise sense. [ABSTRACT FROM AUTHOR]

In this work we consider compact Kähler manifolds with non-positive mixed curvature which is a "convex combination" of Ricci curvature and holomorphic sectional curvature. We show that in this case, the canonical line bundle is nef. Moreover, if the curvature is negative at some point, then the manifold is projective with canonical line bundle being big and nef. If in addition the curvature is negative, then the canonical line bundle is ample. As an application, we answer a question of Ni [Comm. Pure Appl. Math. 74 (2021), pp. 1100–1126] concerning manifolds with negative k-Ricci curvature and generalize a result of Wu-Yau [Comm. Anal. Geom. 24 (2016), pp. 901–912] and Diverio-Trapani [J. Differential Geom. 111 (2019), pp. 303–314] to the conformally Kähler case. We also show that the compact Kähler manifold is projective and simply connected if the mixed curvature is positive. [ABSTRACT FROM AUTHOR]

In this work, we consider sequence of metrics with almost non-negative scalar curvature on torus. We show that if the sequence is uniformly conformal to another sequence of metrics with bounded Ricci geometry, then it converges to a flat metric in the volume preserving intrinsic flat sense, the measured Gromov–Hausdorff sense and L p sense. Moreover, similar stability also holds on manifolds with non-positive Yamabe invariant. [ABSTRACT FROM AUTHOR]

Motivated by the work of Li and Mantoulidis [C. Li, C. Mantoulidis, Positive scalar curvature with skeleton singularities, Math. Ann. 374(1–2) (2019) 99–131], we study singular metrics which are uniformly Euclidean (L ∞) on a compact manifold M n (n ≥ 3) with negative Yamabe invariant σ (M). It is well known that if g is a smooth metric on M with unit volume and with scalar curvature (g) ≥ σ (M) , then g is Einstein. We show, in all dimensions, the same is true for metrics with edge singularities with cone angles ≤ 2 π along codimension-2 submanifolds. We also show in three dimensions, if the Yamabe invariant of connected sum of two copies of M attains its minimum, then the same is true for L ∞ metrics with isolated point singularities. [ABSTRACT FROM AUTHOR]

We show the existence of complete negative Kähler–Einstein metric on Stein manifolds with holomorphic sectional curvature bounded from above by a negative constant. We prove that any Kähler metrics on such manifolds can be deformed to the complete negative Kähler–Einstein metric using the normalized Kähler–Ricci flow. [ABSTRACT FROM AUTHOR]

We prove that compact complex manifolds admitting metrics with negative Chern curvature operator either admit a |$d d^c$| -exact positive |$(1,1)$| current or are Kähler with ample canonical bundle. In the case of complex surfaces we obtain a complete classification. The proofs rely on a global existence and convergence result for the pluriclosed flow. [ABSTRACT FROM AUTHOR]

In this article we study the local estimates of Ricci flow when the curvature of the initial metric is of almost PIC1 and use it to study complete manifolds with weakly PIC1 and maximal volume growth. [ABSTRACT FROM AUTHOR]

In this work, we show that along a particular choice of Hermitian curvature flow, the non-positivity of the first Ricci curvature will be preserved if the initial metric has Griffiths non-positive Chern curvature. If in addition, the first Ricci curvature is negative at a point, then the canonical line bundle is ample. [ABSTRACT FROM AUTHOR]

We prove some uniqueness result for solutions to the heat equation on Riemannian manifolds. In particular, we prove the uniqueness of L p solutions with 0 < p < 1 and improves the L 1 uniqueness result of Li (J Differ Geom 20:447–457, 1984) by weakening the curvature assumption. [ABSTRACT FROM AUTHOR]

This study proposes a multi-ship collision avoidance and route generating algorithm based on the general requirements of the International Regulations for Preventing Collisions at Sea (COLREGs) and the artificial potential field (APF) method. The velocity potential field is used as the field function of APF method. The algorithm consists of two modes, the course-changing and the track-keeping modes based on the velocity potential of vortex and dipole, respectively. The course-changing mode guides the ship to turn away from the obstacles according to the vector field of vortex potential. The track-keeping mode steers the ship back to and along on a pre-designed track in accordance with the vector field of dipole potential. The data of distance to the closest point of approach (DCPA), time to the closest point of approach (TCPA) and bearing angle evaluated from a maneuvering simulation are the acquired parameters of the proposed collision avoidance algorithm. The algorithm is straightforward and very simple to implement, and is suitable for the real time and distributed intelligent collision avoidance system. Simulation results indicate that the anti-collision formulation can avoid collision safely with the desired distance and indicate that the algorithm proposed can work effectively. [ABSTRACT FROM AUTHOR]

We produce complete bounded curvature solutions to the Kähler-Ricci flow with existence time estimates, assuming only that the initial data is a smooth Kähler metric uniformly equivalent to another complete bounded curvature Kähler metric. We obtain related flow results for nonsmooth as well as degenerate initial conditions. We also obtain a stability result for complex space forms under the flow. [ABSTRACT FROM AUTHOR]

In this note, we study the problem of the uniqueness of solutions to the Ricci flow on complete noncompact manifolds. We consider the class of solutions with curvature bounded above by C / t when t > 0 and prove a uniqueness result when initial curvature is of polynomial growth and Ricci curvature of the flow is relatively small. [ABSTRACT FROM AUTHOR]

In this work, we obtain some existence results of Chern–Ricci Flows and the corresponding Potential Flows on complex manifolds with possibly incomplete initial data. We discuss the behaviour of the solution as t → 0 . These results can be viewed as a generalization of an existence result of Ricci flow by Giesen and Topping for surfaces of hyperbolic type to higher dimensions in certain sense. On the other hand, we also discuss the long time behaviour of the solution and obtain some sufficient conditions for the existence of Kähler-Einstein metric on complete non-compact Hermitian manifolds, which generalizes the work of Lott–Zhang and Tosatti–Weinkove to complete non-compact Hermitian manifolds with possibly unbounded curvature. [ABSTRACT FROM AUTHOR]