We consider embedded hypersurfaces evolving by fully nonlinear flows in which the normal speed of motion is a homogeneous degree one, concave or convex function of the principal curvatures, and prove a noncollapsing estimate: Precisely, the function which gives the curvature of the largest interior sphere touching the hypersurface at each point is a subsolution of the linearized flow equation i...

This paper presents a four-step training method for increasing the efficiency of support vector machine (SVM) by simplifying the shape of separation hypersurface. First, a SVM is initially trained by all the training samples, thereby producing a number of support vectors. Second, the support vectors, which make the hypersurface highly convoluted, are excluded from the training set. Third, the S...

The Ricci curvature at an isolated singularity of an immersed hypersurface exhibits a local behaviour echoing a global property of the Ricci curvature on a complete hypersurface in euclidean space (i.e., the Efimov theorem [12], that sup Ric ≥ 0). This local behaviour takes the form of a near universal bound on the decay of the Ricci curvature at a simple singularity (eg. a cone singularity) in...

Let R be a hypersurface in an equicharacteristic or unramified regular local ring. For a pair of modules (M, N) over R we study applications of rigidity of Tor(M, N), based on ideas by Huneke, Wiegand and Jorgensen. We then focus on the hypersurfaces with isolated singularity and even dimension, and show that modules over such rings behave very much like those over regular local rings. Connecti...

Abstract. We calculate the fibre integrals of the affine hypersurface in a torus in the form of their Mellin transforms. Especially, our method works efficiently for an affine hypersurface defined by a so called “simpliciable” polynomial. The relations between poles of Mellin transforms of fibre integrals, the mixed Hodge structure of the cohomology of the hypersruface, the hypergeometric diffe...

In this paper we develop the theory of finite-type invariants for homologically nontrivial 3-manifolds . We construct an infinite-dimensional affine space with a hypersurface in it corresponding to manifolds with Morse singularities. Connected components of the complement of this hypersurface correspond to homeomorphism type of spin 3-manifolds. This suggests the natural axiomatics of Vassiliev...

Abstract In this paper we consider the prescribed mean curvature flow of a non-compact space-like Cauchy hypersurface bounded geometry in generalized Robertson–Walker space-time. We prove that preserves space-likeness condition and exists for infinite time. also convergence setting manifolds with boundary. Our discussion generalizes previous work by Ecker, Huisken, Gerhardt others respect to cr...

We give an elementary introduction to the first author’s theory of intersection spaces associated to complex projective varieties with only isolated singularities. We also survey recent results on the deformation invariance of intersection space homology in the context of projective hypersurfaces with an isolated singularity.