In this paper we give a simple proof for the interior gradient estimate for curvature and Hessian equations. We also derive a Liouville type result for these equations. §0. Introduction The interior gradient estimate for the prescribed mean curvature equation has been extensively studied, see [9] and the references therein. For high order mean curvature equations it has also been obtained in [1...

In this paper, a new ear anatomy feature edge extraction method based on Hessian matrix is proposed. Stable edge is obtained from principal curvature image across scale space. Firstly, the side face image that includes an ear is filtered and forms Gaussian pyramid. Secondly, the 2D gray image in the pyramid was regarded as a surface, maximum and minimum principal curvature and their direction w...

In this work, we study the spectra and eigenmodes of the Hessian of various discrete surface energies and discuss applications to shape analysis. In particular, we consider a physical model that describes the vibration modes and frequencies of a surface through the eigenfunctions and eigenvalues of the Hessian of a deformation energy, and we derive a closed form representation for the Hessian (...

Limited-memory BFGS quasi-Newton methods approximate the Hessian matrix of second derivatives by the sum of a diagonal matrix and a fixed number of rank-one matrices. These methods are particularly effective for large problems in which the approximate Hessian cannot be stored explicitly. It can be shown that the conventional BFGS method accumulates approximate curvature in a sequence of expandi...

In this paper, we study gradient Ricci expanding solitons (X, g) satisfying Rc = cg +Df, where Rc is the Ricci curvature, c < 0 is a constant, and Df is the Hessian of the potential function f on X . We show that for a gradient expanding soliton (X, g) with non-negative Ricci curvature, the scalar curvature R has at least one maximum point on X , which is the only minimum point of the potential...

Given a smooth surface S in the Heisenberg group, we compute the Hessian of the function measuring the Carnot-Charathéodory distance from S in terms of the Mean Curvature of S and of an “imaginary curvature” which was introduced in [2] in order to find the geodesics which are metrically normal to S. Explicit formulae are given when S is a plane or the metric sphere.

We show (a) that any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in C with the Euclidean metric is flat; (b) that any space-like entire graphic self-shrinking solution to the Lagrangian mean curvature flow in C with the pseudo-Euclidean metric is flat if the Hessian of the potential is bounded below quadratically; and (c) the Hermitian counterpart of (b) for the...

In the neural-network parameter space, an attractive field is likely to be induced by singularities. In such a singularity region, first-order gradient learning typically causes a long plateau with very little change in the objective function value E (hence, a flat region). Therefore, it may be confused with “attractive” local minima. Our analysis shows that the Hessian matrix of E tends to be ...