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1
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A decomposition of a set definable in an o-minimal structure into perfectly situated sets

100%
Pawłucki W.
Annales Polonici Mathematici
|
2002
|
tom 79
|
nr 2
171-184
EN
A definable subset of a Euclidean space X is called perfectly situated if it can be represented in some linear system of coordinates as a finite union of (graphs of) definable 𝓒¹-maps with bounded derivatives. Two subsets of X are called simply separated if they satisfy the Łojasiewicz inequality with exponent 1. We show that every closed definable subset of X of dimension k can be decomposed into a finite family of closed definable subsets each of which is perfectly situated and such that any two different sets of the decomposition are simply separated and their intersection is of dimension < k.
2
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Examples of functions $𝐶^𝑘$-extendable for each 𝑘 finite, but not $𝐶^∞$-extendable

100%
Pawłucki W.
Banach Center Publications
|
1998
|
tom 44
|
nr 1
183-187
EN
In Example 1, we describe a subset X of the plane and a function on X which has a $𝐶^k$-extension to the whole $ℝ^2$ for each 𝑘 finite, but has no $𝐶^∞$-extension to $ℝ^2$. In Example 2, we construct a similar example of a subanalytic subset of $ℝ^5$; much more sophisticated than the first one. The dimensions given here are smallest possible.
3
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Subanalytic version of Whitney's extension theorem

64%
Kurdyka K., Pawłucki W.
Studia Mathematica
|
1997
|
tom 124
|
nr 3
269-280
EN
For any subanalytic $C^k$-Whitney field (k finite), we construct its subanalytic $C^k$-extension to $ℝ^n$. Our method also applies to other o-minimal structures; e.g., to semialgebraic Whitney fields.
4
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Michael's theorem for Lipschitz cells in o-minimal structures

64%
Czapla M., Pawłucki W.
Annales Polonici Mathematici
|
2016
|
tom 117
|
nr 2
101-107
EN
A version of Michael's theorem for multivalued mappings definable in o-minimal structures with M-Lipschitz cell values (M a common constant) is proven. Uniform equi-LCⁿ property for such families of cells is checked. An example is given showing that the assumption about the common Lipschitz constant cannot be omitted.
5
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O-minimal version of Whitney's extension theorem

64%
Kurdyka K., Pawłucki W.
Studia Mathematica
|
2014
|
tom 224
|
nr 1
81-96
EN
This is a generalized and improved version of our earlier article [Studia Math. 124 (1997)] on the Whitney extension theorem for subanalytic $𝓒^{p}$-Whitney fields (with p finite). In this new version we consider Whitney fields definable in an arbitrary o-minimal structure on any real closed field R and obtain an extension which is a $𝓒^{p}$-function definable in the same o-minimal structure. The Whitney fields that we consider are defined on any locally closed definable subset of Rⁿ. In such a way, a local version of the theorem is included.
6
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On the implicit function theorem in o-minimal structures

64%
Ambroży Z., Pawłucki W.
Banach Center Publications
|
2015
|
tom 107
|
nr 1
19-21
EN
A local-global version of the implicit function theorem in o-minimal structures and a generalization of the theorem of Wilkie on covering open sets by open cells are proven.
7
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A remark on the Lion-Rolin Preparation Theorem for LA-functions

64%
Pawłucki W., Piękosz A.
Annales Polonici Mathematici
|
1999
|
tom 71
|
nr 2
195-197
EN
A correct formulation of the Lion-Rolin Preparation Theorem for logarithmic-subanalytic functions (LA-functions) is given.
8
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Extension of $C^{∞}$ functions from sets with polynomial cusps

45%
Pawłucki W., Pleśniak W.
Studia Mathematica
|
1988
|
tom 88
|
nr 3
279-287
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