A novel training method has been proposed for increasing efficiency and generalization of support vector machine (SVM). The efficiency of SVM in classification is directly determined by the number of the support vectors used, which is often huge in the complicated classification problem in order to represent a highly convoluted separation hypersurface for better nonlinear classification. Howeve...

In previous work we designed an efficient procedure that finds an algebraic sample point for each connected component of a smooth real complete intersection variety. This procedure exploits geometric properties of generic polar varieties and its complexity is intrinsic with respect to the problem. In the present paper we introduce a natural construction that allows to tackle the case of a non–s...

The outward–pointing principal null direction of the Schwarzschild Rie-mann tensor is null hypersurface–forming. If the Schwarzschild mass spontaneously jumps across one such hypersurface then the hypersurface is the history 1 of an outgoing light–like shell. The outward– pointing principal null direction of the Kerr Riemann tensor is asymptotically (in the neighbourhood of future null infinity...

We introduce a new notion of Reeb invariant structure Jacobi operator and two kinds singular normal vector field N for real hypersurface M in the complex quadric Qm, m ? 3. If unit is U-isotropic, we give classification Hopf hypersurfaces with

It is proved that on a real hypersurface $M$ in the complex two-plane Grassmannian $G_2(\mathbb {C}^{m+2})$, structure Lie operator parallel if and only an open part of tube around totally geodesic {C}^{m+1})$ $G_2(\ma

We present algorithms to classify isolated hypersurface singularities over the real numbers according to the classification by V.I. Arnold (Arnold et al., 1985). This first part covers the splitting lemma and the simple singularities; a second and a third part will be devoted to the unimodal singularities up to corank 2. All algorithms are implemented in the Singular library realclassify.lib (M...

We consider the question of how well points in a quadric hypersurface ${M\\subseteq\\mathbb{R}^d}$ can be approximated by rational ${\\mathbb{Q}^d\\cap M}$. This contrasts with more common setup approximating manifold all ${\\mathbb{Q}^d}$. provide complete answers to major questions Diophantine approximation this context. Of particular interest are impact real and ranks defining quadratic form...

A criterion for a spacelike hypersurface in a conformally stationary spacetime to be a Cauchy hypersurface is given. The criterion is based on the forward and the backward completeness of a non-reversible Finsler metric associated to the hypersurface.