Description

• Title:
  Strict $C^1$-triangulations in o-minimal structures
• Creator:
  Czapla, Małgorzata;
  Pawłucki, Wiesław
• Description:
  Inspired by the recent articles of T. Ohmoto and M. Shiota \cite{OS1}, \cite{OS2} on $\mathcal C^1$-triangulations of semialgebraic sets, we prove here by using different methods the following theorem: {\it Let $R$ be a real closed field and let an expansion of $ R$ to an o-minimal structure be given. Then for any closed bounded definable subset $A$ of $ R^n$ and a finite family $B_1,\dots,B_r$ of definable subsets of $A$ there exists a definable triangulation \linebreak $h\colon |\mathcal K|\rightarrow A$ of $A$ compatible with $B_1,\dots,B_r$ such that $\mathcal K$ is a simplicial complex in $R^n$, $h$ is a $C^1$-embedding of each \rom{(}open\rom{)} simplex $\Delta\in \mathcal K$ and $h$ extends to a definable $C^1$-mapping defined on a definable open neighborhood of $|\mathcal K|$ in $R^n$.} This improves Ohmoto-Shiota's theorem in three ways; firstly, $h$ is a $\mathcal C^1$-embedding on each simplex; secondly, the simplicial complex $\mathcal K$ is in the same space as $A$ and thirdly, our proof is performed for any o-minimal structure. The possibility to have $h$ with the first of these properties was stated by Ohmoto and Shiota as an open problem (see \cite{OS1}).
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2018 Topological Methods in Nonlinear Analysis
• Date:
  2018.07.21
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 52, No 2 (December 2018); 739 - 747;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2018.033/15288 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2018.033/15289
• Subject:
  Triangulation;
  $C^1$-mapping;
  o-minimal structure;
  Primary;
  32B25;
  Secondary;
  14P10;
  03C64;
  32B20
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2018.033
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other