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...

Here we prove a global existence theorem for the solutions of semi-linear wave equation with critical non-linearity admitting positive definite Hamiltonian. Formulating parametrix in globally hyperbolic curved spacetime, derive an apriori pointwise bound solution nonlinear terms initial energy, from which follows straightforward way. This is accomplished two steps. First, based on Moncrief's li...

Let $(M^{2n},g)$ be a real hypersurface with recurrent shapeoperator and tangent to the structure vector field $xi$ of the Sasakian space form$widetilde{M}(c)$. We show that if the shape operator $A$ of $M$ isrecurrent then it is parallel. Moreover, we show that $M$is locally a product of two constant $phi-$sectional curvaturespaces.

Numerical relativity describes a discrete initial value problem for general relativ-ity. A choice of gauge involves slicing space-time into space-like hypersurfaces. This introduces past and future gauge relative to the hypersurface of present time. Here, we propose solving the discretized Einstein equations with a choice of gauge in the future and a dynamical gauge in the past. The method is i...

We show that the diagonal light-like solution with 16 supersymmetries in elevendimensional supergravity derived in our previous paper can be generalised to nondiagonal solutions preserving the same number of supersymmetries. Although the metric coefficients of this generalised light-like solution depend on only a single null coordinate, this class of solutions contains a subclass equivalent to ...

The 3-metric gij and extrinsic curvature (second fundamental form) Kij are the fundamental variables describing the geometry in any space+time decomposition of the Einstein equations. (g,K) describe the local geometry of a single space-like hypersurface M , and it is then natural to describe the evolution of the space-time geometry by a 1-parameter family (g(t), K(t)), describing the local geom...

We show that the diagonal light-like solution with 16 supersymmetries in elevendimensional supergravity derived in our previous paper can be generalised to nondiagonal solutions preserving the same number of supersymmetries. Although the metric coefficients of this generalised light-like solution depend only on a single null coordinate, this class of solutions contains a subclass equivalent to ...

A special case of a theorem due to Orlov states that for a hypersurface X ⊂ Pn−1 of degree n given by the equation W = 0, there exists an equivalence between the bounded derived category D(cohX) of coherent sheaves on X and the homotopy category HMF(W ) of graded matrix factorizations. We first give a description of this result, and present some methods for doing calculations with it. In the la...

We initiate a systematic local study of singular Levi-flat real analytic hypersurfaces, concentrating on the simplest nontrivial case of quadratic singularities. We classify the possible tangent cones to such hypersurfaces and prove the existence and convergence of a rigid normal form in the case of generic (Morse) singularities. We also characterize when such a hypersurface is defined by the v...