| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
The real line is really neat! (powers of it as well)
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From: [email protected] (Paul R. Chernoff)
Newsgroups: sci.math
Subject: Re: Reference for plane topology theorem of H. Hopf ?
Message-ID: <[email protected]>
Date: 10 Jan 91 02:41:55 GMT
References: <[email protected]>
Sender: [email protected] (USENET Administrator)
Organization: University of California, Berkeley
Lines: 27
In article <[email protected]> [email protected] (John Franks) writes:
>I am looking for a reference to the following theorem. I have been
>told it is due to H. Hopf.
>Suppose X is a compact connected subset of the plane. A "chord"
>for X is a horizontal line segment both of whose endpoints are in X.
>Theorem: If there is a chord for X of length 1 then there is a chord
> for X of length 1/n for every positive integer n.
>I think sometimes a weaker version of this result is given where
>X is the graph of a continuous function defined on some closed interval.
------------------------------------
See R. Boas, "A Primer of Real Functions", 3rd Edition, p. 92
where it is shown that if f is a continuous real function with
f(0) = f(1) then the graph of f has horizontal chords of length
1/n for every positive integer n. See also Note 16, p. 188,
which discusses the history, and includes a reference to H. Hopf,
"Uber die Sehnen ebener Kontinuen und die Schliefen geschlossener
Wege", Comment. Math. Helv. 9 (1937), 303-319.
----
# Paul R. Chernoff [email protected] #
# Department of Mathematics ucbvax!math!chernoff #
# University of California chernoff%[email protected] #
# Berkeley, CA 94720 #
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 1369.1 | GUESS::DERAMO | Dan D'Eramo | Fri Jan 11 1991 18:57 | 31 | |
Path: ryn.mro4.dec.com!shlump.nac.dec.com!news.crl.dec.com!deccrl!bloom-beacon!snorkelwacker.mit.edu!apple!portal!cup.portal.com!ts From: [email protected] (Tim W Smith) Newsgroups: sci.math Subject: Re: Reference for plane topology theorem of H. Hopf ? Message-ID: <[email protected]> Date: 11 Jan 91 11:01:09 GMT References: <[email protected]> <[email protected]> Organization: The Portal System (TM) Lines: 20 "Challenging Mathematical Problems With Elementary Solutions Volume II" by Yaglom and Yaglom has the simpler form of this problem (problem #119): Let a=1/n be the reciprocal of a positive integer n. Let A and B be two points of the plane such that the segment AB has length 1. Prove the every continuous curve joining A to B has a chord parallel to AB and of length a. Show that if a is not the reciprocal of an integer, then there is a continuous curve joining A to B which has no such chord of length a. For a complete characterization of the possible sets of chord lengths parallel to AB, see H. Hopf[26]. And in the bibliograpghy, there is 26 Hopf, H., Comm. Math. Helv. 9, 303 (1937) Tim Smith | |||||