We give a criterion for checking the Cohen-Macaulayness of the tangent cone of a monomial curve by using the Gröbner basis. For a family of monomial curves, we give the full description of the defining ideal of the curve and its tangent cone at the origin. By using this family of curves and their extended versions to higher dimensions, we prove that the minimal number of generators of a Cohen-M...

Finding the position of a radiative source based on time-difference-of-arrival (TDOA) measurements from spatially separated receivers has been widely applied in sonar, radar, mobile communications and sensor networks. For the nonlinear model in the process of positioning, Taylor series and other novel methods are proposed. The idea of cone constraint provides a new way of solving this problem. ...

We consider several examples of mechanical systems the configuration spaces which have form smooth manifolds with a unique singular point: two intersecting (or tangent) curves on two-dimensional torus, four four-dimensional torus common point, and cone (cusp) in $${{\mathbb{R}}^{6}}$$ . The main problem presented this paper is calculation (co)tangent space above point by means various theoretic...

We show that for any Ricci-flat manifold with Euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross-section. Similarly, for any noncollapsing limit of Einstein manifolds with uniformly bounded Einstein constants, we show that local tangent cones are unique if one tangent cone has a smooth cross-section. 0. Introduction By Gromov’s compactness theo...

We study the generalization of the idea of a local quiver of a representation of a formally smooth algebra, to broader classes of finitely generated algebras. In this new setting we can construct for every semisimple representation M a local model and a noncommutative tangent cone. The representation schemes of these new algebras model the local structure and the tangent cone of the representat...

We study Hironaka's idealistic exponents over Spec(Z). give an interpretation of the tangent cone, directrix, and ridge. The main purpose is to introduce notion characteristic polyhedra deduce from them intrinsic data on exponent.

We study the Zariski tangent cone TX π −→ X to an affine variety X and the closure TX of π−1(Reg(X)) in TX . We focus on the comparison between TX and TX , giving sufficient conditions on X in order that TX = TX . One aspect of the results is to understand when this equality takes place in the presence of the reducedness of the Zariski tangent cone. Our other interest is to consider conditions ...