Let X be a Stein manifold of dimension at least 3. Given a compact codimension 2 real analytic submanifold M of X, that is the boundary of a compact Levi-flat hypersurface H, we study the regularity of H. Suppose that the CR singularities of M are an O(X)-convex set. For example, suppose M has only finitely many CR singularities, which is a generic condition. Then H must in fact be a real analy...

Let f be a polynomial in Q[X1, . . . , Xn] of degree D. We focus on testing the emptiness and computing at least one point in each connected component of the semi-algebraic set defined by f > 0 (or f < 0 or f 6= 0). To this end, the problem is reduced to computing at least one point in each connected component of a hypersurface defined by f −e = 0 for e ∈ Q positive and small enough. We provide...

A smooth, strongly $\mathbb{C}$-convex, real hypersurface $S$ in $\mathbb{CP}^n$ admits a projective dual CR structure addition to the standard structure. Given smooth function $u$ on $S$, we provide characterizations for when can be decomposed as sum of and function. Following work Lee pluriharmonic boundary values, characterization using differential forms. We further tangential vector fields...

Introduction. In my note on A 5 curve theorem generalizing the theorem of Carnot, I introduced the notion of curve cross ratio. This extension of the ordinary cross ratio is the simplest situationally invariant (3.4) case of the generalized or hypersurface cross ratio of n + l pairs of hypersurfaces in w-space which is the subject of the following lines. The generalized cross ratio is a t the s...

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...

A new method for analytic inversion of Radon type integral transforms is proposed. Key words: Regular hypersurface family, Funk-Radon transform, Principal value integral, Reconstruction, Hyperbolic algebraic curve MSC 53C65 44A12 65R10 1 Introduction We present a uniform reconstruction method for a class of geometric integral transforms for submanifolds of codimension 1. The reconstruction does...

In this short communication, we announce results from our research on the structure of complete noncompact oriented weakly stable minimal hypersurfaces in a manifold of nonnegative sectional curvature. In particular, a complete oriented weakly stable minimal hypersurface in Rm, m > or = 4, must have only one end; any complete noncompact oriented weakly stable minimal hypersurface has only one e...

We derive a general integral formula on an embedded hypersurface for general relativistic space-times. Suppose the hypersurface is foliated by two-dimensional compact “sections” Ss. Then the formula relates the rate of change of the divergence of outgoing light rays integrated over Ss under change of section to geometric (convexity and curvature) properties of Ss and the energy-momentum content...

We deposit a prescribed amount of liquid on an umbilical hypersurface of the hyperbolic space Hn+1. Under the presence of a uniform gravity vector field directed towards , we seek the shape of such a liquid drop in a state of equilibrium of the mechanical system. The liquid-air interface is then modeled by a hypersurface under the condition that its mean curvature is a function of the distance ...