Loewner's "forgotten" theorem
Let $f(t)$ be a smooth and periodic function of one real variable. Then the planar curves $t\mapsto \big(f'(t),f(t)\big)$ and $t\mapsto \big(f''(t)-f(t),f'(t)\big)$ both have non-negative rotation number around every point not on the curve. These are the two simplest cases of a b...
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| Main Authors: | , |
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| Format: | Article (Journal) Chapter/Article |
| Language: | English |
| Published: |
7 Sep 2021
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| In: |
Arxiv
Year: 2021, Pages: 1-10 |
| DOI: | 10.48550/arXiv.2109.03051 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.48550/arXiv.2109.03051 Verlag, lizenzpflichtig, Volltext: http://arxiv.org/abs/2109.03051 
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| Author Notes: | Peter Albers, Serge Tabachnikov |
| Summary: | Let $f(t)$ be a smooth and periodic function of one real variable. Then the planar curves $t\mapsto \big(f'(t),f(t)\big)$ and $t\mapsto \big(f''(t)-f(t),f'(t)\big)$ both have non-negative rotation number around every point not on the curve. These are the two simplest cases of a beautiful Theorem by C. Loewner. This article is expository, we prove the two statements by elementary means following work by Bol [3]. After that, we present Loewner's Theorem and his proof from [7]. |
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| Item Description: | Gesehen am 10.08.2022 |
| Physical Description: | Online Resource |
| DOI: | 10.48550/arXiv.2109.03051 |