Bi-Lipschitz A-triviality of map germs and Newton filtrations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Topological K-equivalence of analytic function-germs

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Finiteness theorem for topological contact equivalence of map germs

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

On pairs of regular foliations in R(3) and singularities of map-germs

We study germs of pairs of codimension one regular foliations in R(3) . We show that the discriminant of the pair determines the topological type of the pair. We also consider various classifications of the singularities of the discriminant.

On the number of singularities in generic deformations of map germs

Let f:C-n, 0 --> C-p, 0 be a K-finite map germ, and let i = (i(1),..., i(k)) be a Boardman symbol such that Sigma(i) has codimension n in the corresponding jet space J(k)(n, p). When its iterated successors have codimension larger than n, the paper gives a list of situations in which the number of Sigma(i) points that appear in a generic deformation of f can be computed algebraically by means of Jacobian ideals of f. This list can be summarised in the following way: f must have rank n - i(1) and, in addition, in the case p = 6, f must be a singularity of type Sigma(i2.i2).

Polar Multiplicities and Euler obstruction of the stable types in weighted homogeneous map germs from C-n to C-3 n >= 3

In this article we show that for corank 1, quasi-homogeneous and finitely determined map germs f : (C-n, 0)-> (C-3, 0), n >= 3 one can obtain formulae for the polar multiplicities defined on the following stable types of f, f(Delta(f) and f(Sigma(n-2,1)(f), in terms of the weights and degrees of f. As a consequence we show how to compute the Euler obstruction of such stable types, also in terms of the weights and degrees of f.

Light microscopy and computer three-dimensional reconstruction of the blood capillaries of the enamel organ of rat molar tooth germs

We performed a light microscope and a computer three-dimensional reconstruction study of serial sections of the molar enamel organ of 3- and 5-day-old rats perfused with Indian ink through the arterial system. The tooth germs were fixed in Bouin's solution, embedded in paraffin, sectioned and stained with haematoxylin and eosin. For the three-dimensional reconstruction, light micrographs of the serial sections were digitized, and aligned using the serial EM Align software downloaded from http://synapses.bu.edu/tools/. After alignment, the boundaries of the India-ink-filled blood vessels were manually traced with a mouse using the software IGL trace (version 1.26b), also downloaded from the above website. After tracing, a three-dimensional representation of the blood vessel contours was generated in a VRML format and visualized with the help of the software Cortona Web3D viewer (version 4.0) downloaded from http://www.parallelgraphics.com/products/cortona. Our results showed that in regions where ameloblasts are polarized the capillaries are arranged in three distinct levels: (1) penetrating and leaving capillaries in relation to the outer enamel epithelium; (2) capillaries crossing and branching inside the stellate reticulum; and (3) capillaries branching and anastomosing profusely within the stratum intermedium, thereby forming an extensive capillary plexus intimately associated with the cells of the stratum intermedium. The existence of a conspicuous capillary plexus intermingled with cells of the stratum intermedium, as shown in our results, suggests that some molecules produced by cells of the stratum intermedium could be released into the capillary plexus and thereafter carried to the dental follicle.

K-bi-Lipschitz equivalence of real function-germs

In this paper we prove that the set of equivalence classes of germs of real polynomials of degree less than or equal to k, with respect to K-bi-Lipschitz equivalence, is finite.

Pre-weighted homogeneous map germs finite determinacy and topological triviality

In this paper we introduce the notion of G-pre-weighted homogeneous map germ, (G is one of Mather's groups A or K.) and show that any G-pre-weighted homogeneous map germ is G-finitely determined. We also give an explicit order, based on the Newton polyhedron of a pre-weighted homogeneous germ of function, such that the topological structure is preserved after perturbations by terms of higher order.

Multiplicity of Boardman strata and deformations of map germs

We define algebraically for each map germ f: Kn, 0→ Kp, 0 and for each Boardman symbol i = (il, . . ., ik) a number ci(f) which is script A sign-invariant. If f is finitely determined, this number is the generalization of the Milnor number of f when p = 1, the number of cusps of f when n = p = 2, or the number of cross caps when n = 2, p = 3. We study some properties of this number and prove that, in some particular cases, this number can be interpreted geometrically as the number of Σi points that appear in a generic deformation of f. In the last part, we compute this number in the case that the map germ is a projection and give some applications to catastrophe map germs.

Stable singularities of co-rank one quasi homogeneous map germs from (ℂn+1, 0) to (ℂn, 0), n = 2, 3

In this article, we investigate the geometry of quasi homogeneous corank one finitely determined map germs from (ℂn+1, 0) to (ℂn, 0) with n = 2, 3. We give a complete description, in terms of the weights and degrees, of the invariants that are associated to all stable singularities which appear in the discriminant of such map germs. The first class of invariants which we study are the isolated singularities, called 0-stable singularities because they are the 0-dimensional singularities. First, we give a formula to compute the number of An points which appear in any stable deformation of a quasi homogeneous co-rank one map germ from (ℂn+1, 0) to (ℂn, 0) with n = 2, 3. To get such a formula, we apply the Hilbert's syzygy theorem to determine the graded free resolution given by the syzygy modules of the associated iterated Jacobian ideal. Then we show how to obtain the other 0-stable singularities, these isolated singularities are formed by multiple points and here we use the relation among them and the Fitting ideals of the discriminant. For n = 2, there exists only the germ of double points set and for n = 3 there are the triple points, named points A1,1,1 and the normal crossing between a germ of a cuspidal edge and a germ of a plane, named A2,1. For n = 3, there appear also the one-dimensional singularities, which are of two types: germs of cuspidal edges or germs of double points curves. For these singularities, we show how to compute the polar multiplicities and also the local Euler obstruction at the origin in terms of the weights and degrees. © 2013 Pushpa Publishing House.

Topological K-classification of finitely determined map germs

We consider smooth finitely C 0-K-determined map germs f: (ℝn, 0) → (ℝp, 0) and we look at the classification under C 0-K-equivalence. The main tool is the homotopy type of the link, which is obtained by intersecting the image of f with a small enough sphere centered at the origin. When f -1(0) = {0}, the link is a smooth map between spheres and f is C 0-K-equivalent to the cone of its link. When f -1(0) ≠ {0}, we consider a link diagram, which contains some extra information, but again f is C 0-K-equivalent to the generalized cone. As a consequence, we deduce some known results due to Nishimura (for n = p) or the first named author (for n < p). We also prove some new results of the same nature. © 2012 Springer Science+Business Media Dordrecht.

Bi-Lipschitz A-equivalence of K-equivalent map-germs

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)