We describe algorithmic methods for the Gauss-Manin connection of an isolated hypersurface singularity based on the microlocal structure of the Brieskorn lattice. They lead to algorithms for computing invariants like the monodromy, the spectrum, the spectral pairs, and M. Saito’s matrices A0 and A1. These algorithms use a normal form algorithm for the Brieskorn lattice, standard basis methods f...

For an n-dimensional hypersurface in unit sphere, we introduce an abstract Willmore type called W (n,F)-Willmore functional, which generalizes the well-known classic Willmore functional. Its critical point is called the W (n,F)-Willmore hypersurface, for which the variational equation and Simons' type integral equalities are obtained. Moreover, we construct a few examples of W (n,F)-Willmore hy...

In this paper we introduce the concept of a bipolar variety of a real algebraic hypersurface. This notion is then used for the design and complexity estimations of a novel type of algorithms that finds algebraic sample point for the connected component of a singular real hypersurface. The complexity of these algorithms is polynomial in the maximal geometric degree of the bipolar varieties of th...

We give a full exceptional collection in the triangulated category of singularities in the sense of Orlov [16] for a hypersurface singularity of Fermat type, and discuss its relation with homological mirror symmetry for simple elliptic hypersurface singularities.

In this paper, the generalized helical hypersurfaces x=x(u,v,w) with a time-like axis in Minkowski spacetime E14 are considered. The first and second fundamental form matrices, Gauss map, shape operator matrix of x calculated. Moreover, curvatures hypersurface obtained by using Cayley–Hamilton theorem. umbilical conditions for given. Finally, Laplace–Beltrami is presented E14.

We construct solutions of u = e which blow-up precisely on a given space-like hypersurface of class H. For this purpose, we prove a general existence theorem for Fuchsian PDE in Sobolev spaces. The precise relation between the regularity of the data and that of the solution is shown to involve logarithmic symbols, in a model situation. A few further results on power nonlinearities are also incl...

Numerical relativity describes a discrete initial value problem for general relativity. A choice of gauge involves slicing spacetime 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 ill...

Abstract. We consider perturbations of Hamiltonians whose Fourier symbol attains its minimum along a hypersurface. Such operators arise in several domains, like spintronics, theory of supercondictivity, or theory of superfluidity. Variational estimates for the number of eigenvalues below the essential spectrum in terms of the perturbation potential are provided. In particular, we provide an ele...

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...

On a rationally connected variety we would like to use rational curves to obtain a similar result. Complete intersection curves are essentially never rational. (For instance, if X ⊂ P is a hypersurface then a general complete intersection with hyperplanes is rational iff X is a hyperplane or a quadric.) Therefore we have to proceed in a quite different way. Let X be a smooth, projective, ration...