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Valeurs critiques asymptotiques d'une fonction définissable dans une structure o-minimale

100%

D'Acunto D.

Annales Polonici Mathematici

2000

tom 75

nr 1

35-45

We prove that the set of asymptotic critical values of a $C^1$ function definable in an o-minimal structure is finite, even if the structure is not polynomially bounded. As a consequence, the function is a locally trivial fibration over the complement of this set.

A Note on a Theorem of Lion

100%

Ambroży Z.

Bulletin of the Polish Academy of Sciences. Mathematics

2013

tom Vol. 61, no 1

1--7

In this note we bind together Wilkie's complement theorem with Lion's theorem on geometric, regular and 0-regular families of functions.

Intersection of Generic Rotations in Some Classical Spaces

88%

Nowak K. , Tomkowicz G.

Bulletin of the Polish Academy of Sciences. Mathematics

2016

tom Vol. 64, no. 2/3

105--107

Consider an o-minimal structure on the real field R and two definable subsets A, B of the Euclidean space Rn, of the unit sphere Sn or of the hyperbolic space Hn, n ≥ 2, which are of dimensions k, l ≤ n−1, respectively. We prove that the dimension of the intersection σ(A) ∩ B is less than min{k, l} for a generic rotation σ of the ambient space; here we set dim ∅ = −1.

The Whitney property of a fiber of a definable mapping

75%

Kocel-Cynk B.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

tom Vol. 48, no 2

141--147

We prove the following theorem Let S be a polynomially bounded o-minimal structure on (R,+,.) and let f : A --> [R^n] be a continuous, definable function on a compact definable set [A is subset of R^m]. Then there exist a positive real number [alpha belongs to R+] and a definable function C : f(A) --> R+ such that for any x [belongs to] f(A) and any two points p and q in the same connected component of [f^-1](x) there exists a piece-wise analytic curve gamma joinning p and q in [f^-1](x) with length [gamma is less than or equal to C(x)][...]. As a consequence we obtain the regular separation with parameter for definable sets.

Hausdorff limits of one parameter families of definable sets in o-minimal structures

75%

Kocel-Cynk B.

Czasopismo Techniczne. Nauki Podstawowe

2016

tom Y. 113, iss. 1-NP

73--80

We give an elementary proof of the following theorem on definability of Hausdorff limits of one parameter families of definable sets: let A [subset of] R×Rn be a bounded definable subset in o-minimal structure on (R,+,∙) such that for any y∈(0,c), c>0, the fibre Ay:={x∈Rn:(y,x)∈A} is a Lipschitz cell with constant L independent of y. Then the Hausdorff limit lim[y→0] Āy exists and is definable.

W prezentowanej pracy przedstawiamy elementarny dowód następującego twierdzenia o definiowalności granicy Hausdorffa jednoparametrowej rodziny zbiorów definiowalnych: niech A [pozdbiór] R×Rn będzie ograniczonym zbiorem definiowalnym w strukturze o-minimalnej typu (R,+,∙) takim, że dla dowolnego y∈(0,c), c>0, wókno Ay:={x∈Rn:(y,x)∈A} jest komórką Lipschitza ze staą L niezależną od y. Wtedy granica Hausdorffa lim[y→0] Āy istnieje i jest definiowalna.

Interior distance of two points in a fiber of a subanalytic map

75%

Kocel-Cynk B.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

tom Vol. 48, no 2

149--155

The aim of the paper is to prove the following theorem: Let N and M be two analytic manifolds and let K [is subset of] N be a compact subanalytic subset. For any continuous subanalytic map f : K --> M there exists a constant C > 0 such that for any y [belongs to] f(K) and any two points p, q in the same connected component of the fiber [f^-1](y) there exists a subanaiytic curve joining p and q in [f^-1](y) of length less than C.