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A homotopy argument and its applications to the transformation rule for bi-Lipschitz mappings, the Brouwer fixed point theorem and the Brouwer degree

URN to cite this document: urn:nbn:de:bvb:703-opus-2448

Title data

Simader, Christian G.:
A homotopy argument and its applications to the transformation rule for bi-Lipschitz mappings, the Brouwer fixed point theorem and the Brouwer degree.
Bayreuth , 2005

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Abstract

The main purpose of the paper is to present an elementary self-contained proof of the change of variables formula for injective, locally bi-Lipschitz mappings. The proof is based on a homotopy argument. Various properties of bi-Lipschitz mappings are studied. As a by-product Lipschitz variants of the classical implicit function theorem and the local diffeomorphism theorem are proved. With the help of the homotopy argument a simple proof is given of Brouwer’s fixed point theorem and the main properties of Brouwer’s degree of mapping.

Abstract in another language

Hauptgegenstand der Arbeit ist der Beweis der Transformationsformel für n-fache Integrale unter injektiven, lokal bi-Lipschitzstetigen Abbildungen. Der Beweis beruht auf einem Homotopieargument. Es werden auch verschiedene Eigenschaften solcher Abbildungen studiert, z.B. dass sie offen sind. Als Nebenprodukt ergeben sich Lipschitzvarianten der klassischen Sätze über lokale Diffeomorphismen und implizite Funktionen. Aus dem Homotopielemma lässt sich auch ein sehr einfacher Beweis des Brouwerschen Fixpunktsatzes herleiten, sowie die Haupteigenschaften des Abbildungsgrades.

Further data

Item Type: Working paper, discussion paper
Keywords: Abbildungsgrad; Implizite Funktion; Brouwer-Fixpunktsatz; Homotopie; Koordinatentransformation; Transformationsformel für bi-Lipschitzabbildungen; bi-Lipschitz mappings; change of variables in multiple integrals for bi-Lipschitz mappings; degree of mapping; Brouwer's fixed point theorem
DDC Subjects: 500 Science > 510 Mathematics
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Language: English
Originates at UBT: Yes
URN: urn:nbn:de:bvb:703-opus-2448
Date Deposited: 25 Apr 2014 16:02
Last Modified: 04 Apr 2019 05:42
URI: https://epub.uni-bayreuth.de/id/eprint/813

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