arXiv subjects

Lu Xu

Publications and source records attributed to Lu Xu.

94 records · Page 6Linked to original sources

arXiv2016

A semi-automatic computer-aided method for surgical template design

This paper presents a generalized integrated framework of semi-automatic surgical template design. Several algorithms were implemented including the mesh segmentation, offset surface generation, collision detection, ruled surface generation, etc., and a special software named TemDesigner was developed. With a simple user interface, a customized template can be semi- automatically designed according to the preoperative plan. Firstly, mesh segmentation with signed scalar of vertex is utilized to partition the inner surface from the input surface mesh based on the indicated point loop. Then, the offset surface of the inner surface is obtained through contouring the distance field of the inner surface, and segmented to generate the outer surface. Ruled surface is employed to connect inner and outer surfaces. Finally, drilling tubes are generated according to the preoperative plan through collision detection and merging. It has been applied to the template design for various kinds of surgeries, including oral implantology, cervical pedicle screw insertion, iliosacral screw insertion and osteotomy, demonstrating the efficiency, functionality and generality of our method.

cs.GR↗

arXiv2015

On the uniqueness of $L_p$-Minkowski problems: the constant $p$-curvature case in $\mathbb{R}^3$

We study the $C^4$ smooth convex bodies $\mathbb{K}\subset\mathbb{R}^{n+1}$ satisfying $K(x)=u(x)^{1-p}$, where $x\in\mathbb{S}^n$, $K$ is the Gauss curvature of $\partial\mathbb{K}$, $u$ is the support function of $\mathbb{K}$, and $p$ is a constant. In the case of $n=2$, either when $p\in[-1,0]$ or when $p\in(0,1)$ in addition to a pinching condition, we show that $\mathbb{K}$ must be the unit ball. This partially answers a conjecture of Lutwak, Yang, and Zhang about the uniqueness of the $L_p$-Minkowski problem in $\mathbb{R}^3$. Moreover, we give an explicit pinching constant depending only on $p$ when $p\in(0,1)$.

math.AP↗

arXiv2013

KAM for Hamiltonian partial differential equations with weaker Spectral Asymptotics

In this paper, we establish an abstract infinite dimensional KAM theorem dealing with normal frequencies in weaker spectral asymptotics Ω_{i}(ξ)=i^d+o(i^{d})+o(i^δ), where $d>0, δ<0$, which can be applied to a large class of Hamiltonian partial differential equations in high dimensions. As a consequence, it is proved that there exist many invariant tori and thus quasi-periodic solutions for Schrödinger equations, the Klein-Gordon equations with exponential nonlinearity and other equations of any spatial dimension.

math.DS↗

arXiv2010

Convexity estimates for level sets of quasiconcave solutions to fully nonlinear elliptic equations

We establish a geometric lower bound for the principal curvature of the level surfaces of solutions to $F(D^2u, Du, u, x)=0$ in convex ring domains, under a refined structural condition introduced by Bianchini-Longinetti-Salani in \cite{BLS}. We also prove a constant rank theorem for the second fundamental form of the convex level surfaces of these solutions.

math.AP↗