arXiv subjects

Xinjie Jiang

Publications and source records attributed to Xinjie Jiang.

5 recordsLinked to original sources

arXiv2026

Evolution of hypersurfaces in $(n+1)$-dimensional light-cone

In this paper, we investigate the evolutionary processes of hypersurfaces within half of the $(n+1)$-dimensional light-cone. Depending on the evolutionary processes, our focus extends to exploring variational problems associated with a smooth function $f(S_1,\cdots,S_n)$, where each $S_r$ denotes the $r$-th elementary symmetric polynomial, defined as the sum of all possible products of $r$ distinct principal curvatures. We present several fundamental properties related to these variational problems. Furthermore, we examine a curvature-type flow defined locally within the light-cone, establishing its perpetual existence and smooth convergence to a circle whose length is preserved and equal to that of the initial curve.

math.DG↗

arXiv2026

A fourth-order area-preserving curve flow in centro-equiaffine geometry

In this paper, inspired by the work of Guan and Li (2015), we introduce a fourth-order centro-equiaffine invariant curve flow via the affine Minkowski formula. Without any smallness assumptions on the initial curve, we establish the long-time existence of the flow and prove that, as $t \to +\infty$, the evolving curve preserves its enclosed area and converges smoothly to a round circle up to the action of $\mathrm{SL}(2)$.

math.DG↗

arXiv2024

VrdONE: One-stage Video Visual Relation Detection

Video Visual Relation Detection (VidVRD) focuses on understanding how entities interact over time and space in videos, a key step for gaining deeper insights into video scenes beyond basic visual tasks. Traditional methods for VidVRD, challenged by its complexity, typically split the task into two parts: one for identifying what relation categories are present and another for determining their temporal boundaries. This split overlooks the inherent connection between these elements. Addressing the need to recognize entity pairs' spatiotemporal interactions across a range of durations, we propose VrdONE, a streamlined yet efficacious one-stage model. VrdONE combines the features of subjects and objects, turning predicate detection into 1D instance segmentation on their combined representations. This setup allows for both relation category identification and binary mask generation in one go, eliminating the need for extra steps like proposal generation or post-processing. VrdONE facilitates the interaction of features across various frames, adeptly capturing both short-lived and enduring relations. Additionally, we introduce the Subject-Object Synergy (SOS) module, enhancing how subjects and objects perceive each other before combining. VrdONE achieves state-of-the-art performances on the VidOR benchmark and ImageNet-VidVRD, showcasing its superior capability in discerning relations across different temporal scales. The code is available at https://github.com/lucaspk512/vrdone.

cs.CV↗

arXiv2024

An eternal hypersurface flow arising in centro-affine geometry

In this paper, the existence and uniqueness for a specific centro-affine invariant hypersurface flow in $R^{n+1}$ are studied, and the corresponding evolutionary processes in both centro-affine and Euclidean settings are explored. It turns out that the flow exhibits similar properties as the standard heat flow. In addition, the long time existence of the flow is investigated, which asserts that the hypersurface governed by the flow converges asymptotically toward an ellipsoid via systematically investigating evolutions of the centro-affine invariants. Furthermore, the classification of the eternal solutions for the flow is provided.

math.DG↗

arXiv2022

A nonlocal curve flow in centro-affine geometry

In this paper, the isoperimetric inequality in centro-affine plane geometry is obtained. We also investigate the long-term behavior of an invariant plane curve flow, whose evolution process can be expressed as a second-order nonlinear parabolic equation with respect to centro-affine curvature. The forward and backward limits in time are discussed, which shows that a closed convex embedded curve may converge to an ellipse when evolving according to this flow.

math.DG↗