User dave l renfro - MathOverflow

Answer by Dave L Renfro for What are Normal Sets (Fréchet)?

2013-01-28T14:50:29Z

The most complete study in English of Fréchet's work that I know of is a series of three long papers (total of 217 pages) by Angus Ellis Taylor that were published in the 1980s:

A study of Maurice Fréchet: I. His early work on point set theory and the theory of functionals, Archive for History of Exact Sciences 27 #3 (1982), 233-295.

A study of Maurice Fréchet: II. Mainly about his work on general topology, 1909–1928, Archive for History of Exact Sciences 34 #4 (1985), 279-380.

A study of Maurice Fréchet: III. Fréchet as analyst, 1909–1930, Archive for History of Exact Sciences 37 #1 (1987), 25-76.

Near the top of p. 256 of the first paper Taylor writes:

In a number of theorems Fréchet deals with $V$-classes that are complete and separable. He calls them normal. This terminology has not survived; in later developments of abstract topology the word normal is given an entirely different meaning.

Answer by Dave L Renfro for Is there a maximum to the amount of disjoint non-measurable subsets of the unit interval with full outer measure?

2013-01-23T21:32:57Z

In 1917 Luzin and Sierpinski proved there exists continuum many pairwise disjoint subsets of the interval $[0,1]$ such that each of these subsets has outer Lebesgue measure $1.$

Nikolai N. Luzin and Waclaw Sierpinski, Sur une décomposition d'un intervalle en une infnité non dénombrable d'ensembles non mesurables [On a decomposition of an interval into nondenumerably many nonmeasurable sets], Comptes Rendus Académie des Sciences (Paris) 165 (1917), 422-424.

See Gallica site for C. R. Paris volumes or Internet Archive copy of volume 165

Although I've mentioned this paper in various internet groups several times over the past 10+ years, I don't think I've ever mentioned why I find it so fascinating. First, that such an amazingly strong result was published so early -- little more than a decade after non-measurable sets were known to exist. Second, that such a result is so rarely mentioned in texts that deal with measure theory, despite that fact that very little background is needed to state the result, which can also easily be done in a one-sentence footnote.

Answer by Dave L Renfro for On the set of divergence to infinity for sequences of positive continuous functions

2012-05-23T13:23:17Z

What follows are some additional comments about this topic.

Sierpinski's 1921 paper was written without knowledge of Hahn's 1919 paper, this being a time (end of WWI) when the flow of information and journals was intermittent and/or temporarily suspended.

p. 348 in Volume 2 of Hans Hahn's Collected Works (1996) includes these remarks about Hahn's 1919 paper:

"... Hahn sets himself in Über die Menge der Konvergenzpunkte einer Funktionenfolge the task of finding out if this property gives a complete characterization of such sets. He not only proves it, but also enters the study of Baire functions by finding a characterization of the sets of convergence of functions of any given Baire class.

Jolanta Wesolowska has published versions of Hahn/Sierpinski's result for sequences of functions belonging to various other classes of functions, for example On sets of convergence points of sequences of some real functions (MR 2001d:26003; Zbl 1035.26006) and On sets determined by sequences of quasi-continuous functions (MR 2002i:26002; Zbl 1002.26004) and On sets of discrete convergence points of sequences of real functions (MR 2005f:26010; Zbl 1070.26005). Incidentally, I was in attendance when she first presented her work (represented by the first paper above) to people outside her immediate research group [1] (this work was her 2000 Ph.D. Dissertation at Uniwersytet Gdański), and it created a minor buzz among those attending, who found it simply amazing that the kinds of questions she was working on had not been thoroughly worked over before. (I should point out that the literature of real functions and point set theory is pretty much everywhere dense with minutia on most anything you can imagine, and more.)

[1] She gave this talk on 26 May 2000 at Summer Symposium in Real Analysis XXIV, held at The University of North Texas (Denton, Texas).

(Next Day) Yesterday I forgot to post a couple of references in English to a proof of the Hahn/Sierpinski result. A proof can be found on pp. 307-308 of the 1978 3rd English edition (and presumably, on the same pages for any of the other English editions) of the 1935 3rd edition of Hausdorff’s Set Theory [1957 (MR 19,111a; Zbl 81.04601); 1962 (MR 25 #4999); 1978 (Zbl 488.04001); 1991 (Zbl 896.04001); 2005] and on pp. 185-186 of Kechris’ Classical Descriptive Set Theory [MR 96e:03057; Zbl 819.04002].

Answer by Dave L Renfro for Articles with examples of Darboux functions without fixed points

2012-04-10T17:37:38Z

The standard reference for Darboux functions is

Andrew Michael Bruckner and Jack Gary Ceder, Darboux continuity, Jahresbericht der Deutschen Mathematiker-Vereinigung 67 (1965), 93-117. MR 32 #4217l; Zbl 144.30003

http://eudml.org/doc/146526;jsessionid=98522A06CD68A44763F32C1354F068AB

Given its publication date, I don't think you'll find much about fixed points of Darboux functions in this paper, but you should look over the paper anyway, plus it's an excellent survey that should be quite useful in general. However, and this is a useful search tip, you can use the paper to refine your search. Search google using the phrases "Darboux continuity" AND "Bruckner" AND "fixed point". The idea is that the hits you get will be those papers and other items that cite Bruckner/Ceder's paper, and thus you'll exclude the more superficially researched items.

http://www.google.com/search?q=%22Darboux+continuity%22+Bruckner+%22fixed+point%22

Answer by Dave L Renfro for Unique limits of sequences plus what implies Hausdorff?

2011-09-12T14:15:59Z

This past weekend, entirely by chance, I came across a published paper that used the term "US-space" for the class of topological spaces having the property that no sequence can converge to more than one point. The google search just below seems to bring up some things that might be of use to you:

http://www.google.com/search?q=%22US-space%22+convergence+sequence

Answer by Dave L Renfro for Newton and Newton polygon

2011-09-09T15:24:22Z

This was intended to be a comment on Bill Dubuque's answer, but I apparently don't yet have enough reputation points to comment, and in any event this is probably too long to appear as a comment.

Given Chrystal's intended audience, I'm surprised that he didn't mention Talbot's 1860 English translation and extensive commentary of Newton's Enumeration Linearum Tertii Ordinis. In Talbot's work, which is freely available on the internet, see the sections On the Analytical Parallelogram (pp. 88-104) and Examples (pp. 104-112). By the way, whoever scanned the book for google wasn't paying attention when the lengthy list of figures at the end of the book were being scanned, so I'm also giving the University of Michigan Historical Math Collection version, which has those figures correctly scanned.

Sir Isaac Newton's Enumeration of Lines of the Third Order, Generation of Curves by Shadows, Organic Description of Curves, and Construction of Equations by Curves, Translated from the Latin, with notes and examples, by C.R.M. Talbot, 1860.

http://books.google.com/books?id=6I97byFB3v0C

http://name.umdl.umich.edu/ABQ9451.0001.001

Answer by Dave L Renfro for Magnitude of Graham's Number?

2011-09-02T20:37:05Z

I realize this thread is from a year and a half ago (a possible spam post moved it into the current thread titles), but if anyone is still interested in these issues, about 9 years ago I posted in sci.math a lengthy essay on Graham's number and then followed it up with three more essays (and a promise for more, which I never got around to).

GRAHAM'S NUMBER AND RAPIDLY GROWING FUNCTIONS [2 March 2002] http://groups.google.com/group/sci.math/msg/0f3c8bab92145996

BIG NUMBERS #1 [8 April 2002] http://groups.google.com/group/sci.math/msg/403051f310ff3dfc

BIG NUMBERS #2 [8 April 2002] http://groups.google.com/group/sci.math/msg/d12962e3af2c74b7

BIG NUMBERS #3 [8 April 2002] http://groups.google.com/group/sci.math/msg/4f2ed8e0385b72f2

Answer by Dave L Renfro for What are some examples of colorful language in serious mathematics papers?

2011-06-14T20:43:11Z

Two that I like can be found on p. 756 of Edgar R. Lorch's Amer. Math. Monthly paper "Continuity and Baire functions" (Volume 78, 1971, pp. 748-762):

[...] the reader is reminded of the fact that sets which are of type F_sigma_delta_sigma or G_delta_sigma_delta and not of lower type--with respect to any of the classic topologies--are very thinly scattered through the literature. In fact, looking for them is almost like hunting for unicorns.

In order to penetrate further into this subject it is necessary to give an appropriate structure to T, the set of all coherent topologies. As mentioned earlier, this appropriate structure is itself a topology. This circumstance, that a collection of topologies is topologized, may seem a bit incestuous.

Comment by Dave L Renfro

2013-05-09T20:37:57Z

The best early 1800s and before survey I know of in English is pp. 152-159 (On the Singular or Remarkable Points of Curve Lines) in George Peacock's 1820 book "A Collection of Examples of the Applications of the Differential and Integral Calculus", which is freely available on the internet at catalog.hathitrust.org/Record/000578001 (I see that Benjamin Dickman's article by De Morgan mentions Peacock's book, but since I had already looked up the Peacock book . . .)

Comment by Dave L Renfro

2013-04-15T21:51:35Z

I don't know the answer to your question, but I posted about this result a few years ago in sci.math, and perhaps my comments and additional references could be of use: [Math Forum archive](mathforum.org/kb/message.jspa?messageID=5981611) of Riemann sums + integrability, sci.math, 7 November 2007. Also, this [prior post](mathforum.org/kb/message.jspa?messageID=5981311) in the same thread links back to a couple of 2002 posts that might also be of use.

Comment by Dave L Renfro

2013-01-30T15:53:29Z

A more appropriate place to ask this is at [Math StackExchange](math.stackexchange.com). Also, based on my experience there, you'll probably get some pretty good explanations.

Comment by Dave L Renfro

2012-09-26T18:15:19Z

I don't know the origin of such a construction (I haven't investigated it), but for those interested in its history, this construction can be found on [pp. 325-326](books.google.com/…) of the following paper: Ernest William Hobson, On inner limiting sets of points in a linear interval, Proceedings of the London Mathematical Society (2) 2 (1904), 316-326.

Comment by Dave L Renfro

2012-06-13T14:52:13Z

This [List of Elementary Mathematics Journals](mathpropress.com/elementaryJournals.html) at the MathPro Press webpages is at the low end of what you want. Not all their links are currently valid, however. For example, here's a correct link to [The Pentagon](pentagon.kappamuepsilon.org/Pentagon_Issues.php).

Comment by Dave L Renfro

2012-05-02T20:53:51Z

@AliReza Olfati: I believe Magill (reference at end) has proved that a locally compact Hasudorff topological space has a "countable Hausdorff compactification" if and only if it has an $n$-point Hausdorff compactification for each integer $n \geq 1,$ which implies that $\mathbb R$ does not have a Hausdorff compactification with a countably infinite remainder. Kenneth D. Magill, Countable compactifications, Canadian Journal of Mathematics 18 (1966), 616-620.

Comment by Dave L Renfro

2012-04-26T21:58:50Z

At groups.google.com/group/sci.math/msg/… there are some comments by Abhijit Dasgupta (made in 2006) in the sci.math thread "Every set is F_sigma-delta-sigma in the Feferman-Levy model?" (which I began in 2001) that might be of use.

Comment by Dave L Renfro

2012-04-18T15:47:51Z

The following paper does not address the specific question john asked (that I know of), but it may be of interest to some readers of this thread: Giovanni Mingari Scarpello and Daniele Ritelli, A historical outline of the theorem of implicit functions, Divulgaciones Matemáticas 10 (2002), 171-180. emis.de/journals/DM/vX2/art6.pdf I guess I should also mention the book The Implicit Function Theorem: History, Theory, and Applications by Steven G. Krantz and Harold R. Parks.

Comment by Dave L Renfro

2012-03-30T20:05:37Z

The following article may be of interest to you: Robert Gray, Georg Cantor and Transcendental Numbers, American Mathematical Monthly 101 #9 (November 1994), 819-832. tinyurl.com/br2e9jo

Comment by Dave L Renfro

2011-12-21T20:30:32Z

In case you decide to go with something that involves ruler and compass constructability and/or the theory of equations (by which I mean what was a standard undergraduate course from the late 1800s until its disappearance in the 1950s), you may find the following manuscript of use: pballew.net/Constructable_17gon.pdf I wrote this for situations such as you find yourself in, more as a secondary reference than as a primary reference.

Comment by Dave L Renfro

2011-11-22T19:39:04Z

I posted a fairly extensive survey on this topic in May 2002 at mathforum.org/kb/message.jspa?messageID=387148 and mathforum.org/kb/message.jspa?messageID=387149

Comment by Dave L Renfro

2011-11-02T16:43:54Z

@Steven Gubkin: I think you mean "high school", not "middle school", at least in the U.S.

Comment by Dave L Renfro

2011-10-20T14:15:21Z

@Thierry Zell: I'm pretty sure the main intended goal was to increase understanding. However, like many teaching innovations, the level of success (for either of the things you mentioned) varied greatly enough that those who thought it was a success and those who felt it was a failure could each find plenty of supporting evidence for their views.

Comment by Dave L Renfro

2011-09-27T20:16:20Z

Could you be thinking of "How to Write Mathematics" by Halmos? This was originally published in L'Enseignement Math. (1970) and then reprinted (along with some similar essays by others) by the AMS in 1973. math.uga.edu/~azoff/courses/halmos.pdf