In this paper we prove the birational superrigidity and nonrationality of a hypersurface X ⊂ P of degree 6 such that the hypersurface X does not contain three-dimensional linear subspaces of P and the only singularities of X are isolated ordinary double points.

0. Introduction and Preliminary 2 0.1. Existence and uniqueness theorem 2 0.2. Equation of a chain 7 1. Nonsingular matrices 9 1.1. A family of nonsingular matrices 9 1.2. Sufficient condition for Nonsingularity 12 1.3. Estimates 16 2. Local automorphism group of a real hypersurface 23 2.1. Polynomial Identities 23 2.2. Injectivity of a Linear Mapping 33 2.3. Beloshapka-Loboda Theorem 42 3. Com...

The kinematic theory of Weingarten-Volterra line defects is revisited, both at small and finite deformations. Existing results are clarified and corrected as needed, and new results are obtained. The primary focus is to understand the relationship between the disclination strength and Burgers vector of deformations containing a Weingarten-Volterra defect corresponding to different cut-surfaces.

String Tension and String Susceptibility in two-dimensional generalized Weingarten model Masanori Hanada∗) and Fukuichiro Kubo∗∗) Department of Physics, Kyoto University, Kyoto 606-8502, Japan (Received December 27, 2006) We study the two-dimensional generalized Weingarten model reduced to a point, which interpolates between the reduced Weingarten model and the large-N gauge theory. We calculat...

For entire numbers r and s satisfying 0 ≤ n − 2, we showed that the index of (r, s)-stability a s)-linear Weingarten Clifford torus immersed into (n + 1)-dimensional unit Euclidean sphere, has linear combination their higher order mean curvatures Hr+1 Hs+1 being null, is exactly equal to 3 provided geometric condition involving Hr+2 Hs+2 satisfied.

A surface in homogenous space Sol is said to be an invariant surface if it is invariant under some of the two 1-parameter groups of isometries of the ambient space whose fix point sets are totally geodesic surfaces. In this work we study invariant surfaces that satisfy a certain condition on their curvatures. We classify invariant surfaces with constant mean curvature and constant Gaussian curv...

In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. the first part consider volume preserving by a family curvature functions including positive powers $k$-th mean curvatures with $k=1,\ldots,n$, $p$-th power sums $S\_p$ $p > 0$. We that if initial hypersurface $M\_0$ is smooth closed has sectional curvatures, then solution Mt flow...

The main idea of the support vector machine (SVM) classification approach is mapping the data into higher-dimensional linear space where the data can be separated by hyperplane. Based on the Jordan curve theory, a general nonlinear classification method by the use of hypersurface is proposed in this paper. The separating hypersurface is directly used to classify the data according to whether th...