On an affine variety X defined by homogeneous polynomials, every line in the tangent cone of X is a subvariety of X. However there are many other germs of analytic varieties which are not of cone type but contain “lines” passing through the origin. In this paper, we give a method to determine the existence and the “number” of such lines on non-degenerate surface singualrities.

We study the multiplicity modulo 2 of real analytic hypersurfaces. We prove that, under some assumptions on the singularity, the multiplicity modulo 2 is preserved by subanalytic bi-Lipschitz homeomorphisms of R. In the first part of the paper, we find a subset of the tangent cone which determines the multiplicity mod 2 and prove that this subset of S is preserved by the antipodal map. The stud...

In this paper we present some new properties of the metric dimension defined by Bouligand in 1928 and prove the following new projection theorem: Let dimb(A − A) denote the Bouligand dimension of the set A − A of differences between elements of A. Given any compact set A ⊆ R such that dimb(A−A) < m, then almost every orthogonal projection P of A of rank m is injective on A and P |A has Lipschit...

As part of an investigation of the radar backscatter properties of axisymmetric conducting shapes, a study has been made of backscattering by a cone-sphere. This body is comprised of a circular cone with a 25° included cone angle, terminated by a tangent spherical segment. Measurements of backscatter cross sections over all aspect angles are presented for wavelengths in the range from one-tenth...

Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semi-algebraic subsets of the parameter space. We consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisfy standard probabilistic regularity conditions. We show that the assumptions of Chernoff's theore...

A Richardson variety X α in the Orthogonal Grassmannian is defined to be the intersection of a Schubert variety X in the Orthogonal Grassmannian and a opposite Schubert variety Xα therein. We give an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any T -fixed point of X γ α, thus generalizing a result of Ra...

We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, Vk and Rk, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of Vk and Rk are analytically isomorphic, if the group is 1-formal; in particular, the tangent cone to Vk at 1 equ...

Many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semi-algebraic subsets of the parameter space. We consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisfy standard probabilistic regularity conditions. We show that the assumptions of Chernoff's theore...

In this paper, we study the Hilbert series of tangent cone Gorenstein monomial curves in 4-dimensional affine space. We give an explicit formula for reduced a noncomplete intersection curve whose is Cohen-Macaulay.

Fat conic section and fat conic spline are defined. With well established properties of fat conic splines, the problem of approximating a ruled surface by a tangent smooth cone spline can then be changed as the problem of fitting a plane fat curve by a fat conic spline. Moreover, the fitting error between the ruled surface and the cone spline can be estimated explicitly via fat conic spline fit...