arXiv subjects

Karim Bekka

Publications and source records attributed to Karim Bekka.

7 recordsLinked to original sources

arXiv2024

Unstability problem of real analytic maps

As well-known, the $C^\infty$ stability of proper $C^\infty$ maps is characterized by the infinitesimal $C^\infty$ stability. In the present paper we study the counterpart in real analytic context. In particular, we show that the infinitesimal $C^ω$ stability does not imply $C^ω$ stability; for instance, a Whitney umbrella $\mathbb{R}^2 \to \mathbb{R}^3$ is not $C^ω$ stable. A main tool for the proof is a relative version of Whitney's Analytic Approximation Theorem which is shown by using H. Cartan's Theorems A and B.

math.AG↗

arXiv2021

Correction to : The Kuo condition, an inequality of Thom's type and (c)-regularity

We correct a statement of a theorem on characterisation of the (c)-regularity we gave in Topology 37 (1998), 45--62. This theorem was used in the paper in the proof of two theorems on the (c)-regular stratification. In this note we give a weaker version of the theorem as an alternative lemma which ensures the (c)-regularity condition and turn to be sufficient for the proof of the theorems.

math.AG↗

arXiv2020

Equivalence of Kuo and Thom quantities for analytic functions

Sufficiency of jets is a very important notion introduced by Rene Thom in order to establish the structural stability theory. The criteria for some sufficiency of jets are known as the Kuo condition and Thom type inequality, which are defined using the Kuo quantity and Thom quantity. Therefore these quantities are meaningful. In this paper we show the equivalence of Kuo and Thom quantities. Then we apply this result to the relative conditions to a given closed set.

math.AG↗

arXiv2020

Characterisations of V-sufficiency and C^0-sufficiency of relative jets

We consider the problems of sufficiency of jets relative to a given closed set. In the non-relative case, criteria for r-jets to be V-sufficient and C^0-sufficient in C^r mappings or C^{r+1} mappings have been obtained. In particular, it is shown that V-sufficiency and C^0-sufficiency in C^r functions or C^{r+1} functions are equivalent. In this paper we discuss characterisations of V-sufficiency and C^0-sufficiency in the relative case, corresponding to the above non-relative results. Applying the results obtained in the relative case, we construct examples of polynomial functions whose relative r-jets are V-sufficient in C^r functions and C^{r+1} functions but not C^0-sufficient in C^r functions and C^{r+1} functions, respectively. In addition, we give characterisations of relative finite V-determinacy and also relative finite C^r contact determinacy.

math.AG↗

arXiv2020

On the relative Kuo condition and the second relative Kuo condition

The Kuo condition and the the second Kuo condition are known as criteria for an r-jet to be V-sufficient in C^r mappings and C^{r+1} mappings, respectively. In a previous paper we considered the notions of V-sufficiency of jets and these Kuo conditions in the relative case to a given closed set, and showed that the relative Kuo condition is a criterion for a relative r-jet to be V-sufficient in C^r mappings and the second relative Kuo condition is a sufficient condition for a relative r-jet to be V-sufficient in C^{r+1} mappings. In this paper we discuss several conditions equivalent to the relative Kuo condition or the second relative Kuo condition.

math.AG↗

arXiv2012

Briançon-Speder examples and the failure of weak Whitney regularity

It is easy to find real algebraic varieties with weakly Whitney regular stratifications which are not Whitney regular, and we give such an example in section 3 below. No examples are known among complex analytic varieties however, so that the natural question arises : do Whitney regularity and weak Whitney regularity coincide in the complex case ? As a test, in this paper we study the well-known Briançon-Speder examples. We investigate systematically all of the (infinitely many) Briançon-Speder examples, and establish in particular that none of these examples are weakly Whitney regular. We determine all the complex curves along which Whitney regularity fails and all the complex curves along which weak Whitney regularity fails. It turns out that for each example there are a finite number of curves $γ_i$ such that weak Whitney regularity fails precisely along those curves tangent to one of the $γ_i$ at the origin.

math.AG↗

arXiv2010

Uniform radius and regular stratifications

In this paper we investigate how germs of real functions can change under deformation. In particular we look at deformations of germs of isolated singularities from R_n to R_k (n >= k) and the relation with there natural stratification in some tame categorie (algebraic, analytic, semi-algebraic, subanalytic, o-minimal straucture polynomially bounded). The word tame in this paper will refer to one of these categories.

math.AG↗