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

100%

Pawłucki W.

2002

tom 79

nr 2

171-184

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.

Michael's theorem for Lipschitz cells in o-minimal structures

63%

Czapla M. , Pawłucki W.

tom 117

nr 2

101-107

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.

On the implicit function theorem in o-minimal structures

63%

Ambroży Z. , Pawłucki W.

nr 1

19-21

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.

Extension of $C^{∞}$ functions from sets with polynomial cusps

44%

Pawłucki W. , Pleśniak W.

tom 88

nr 3

279-287