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[math-fun] Terminology
by dasimov＠earthlink.net 19 Mar '06

19 Mar '06

When I first learned the rigorous definition of a function f: X -> Y, the terminology was unequivocal: X is the domain of f, and Y is the range of f. The latter is unconditionally the case, irrespective of whether f is onto. Nowadays, so many high school course and ill-considered calculus texts have redefined "range" to mean f(X) -- what I was taught is called the image of f -- that in teaching I've leaned away from using the word range to mean either thing. Now I call the classical range by the term "codomain" (and I still call image "image"). It's a shame when the cognoscenti feel obliged to follow the ignorami. --Dan

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Re: [math-fun] etymology
by Phil Carmody 18 Mar '06

18 Mar '06

From: "Fred lunnon" <fred.lunnon(a)gmail.com> > Subject: Re: [math-fun] etymology > > One way and another, this terminology is rather overworked. > I'm currently trying to recast elementary geometry in terms of > Clifford algebra, so frequently trying to avoid referring simultaneously > to normal subgroups, vectors normal to a surface, and (the only > remotely justifiable use of the word) normalised coordinates. What issue do you have with vectors being normal to a surface? That term is not just entirely justifiable, but I'd contend that it's the single best justified use of the term out of the three you give. The OED dates that usage back to 1650. No other use of the word as an adjective has a longer history. Phil () ASCII ribbon campaign () Hopeless ribbon campaign /\ against HTML mail /\ against gratuitous bloodshed [stolen with permission from Daniel B. Cristofani] __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com

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Re: [math-fun] Re: proofs
by Schroeppel, Richard 18 Mar '06

18 Mar '06

Dylan Thurston <dthurston(a)barnard.edu> sent this. -- Rich ---------------- On Sun, Mar 05, 2006 at 02:35:19PM -0500, dasimov(a)earthlink.net wrote: > In Fall, 2002, the Russian mathematician Grigori Perelman posted to > the math ArXiv website some papers claiming to prove W. Thurston's > geometrization conjecture (TGC) > <http://en.wikipedia.org/wiki/Geometrization_conjecture>. > ... > (I believe there is at least one piece of the TGC proof that Perelman has > promised to post on the ArXiv, but has not yet done so.) That's right; furthermore, it seems unlikely he will post it at this point. > ... That October, John Morgan of Columbia published an article > written in August, 2004 that discussed Perelman's work, but he dd > not affirm that the proof was correct. John Morgan is now more optimistic, and is currently co-writing an article that will have the proof; this article will be peer reviewed. > Assuming the absence of reliable automated theorem-checkers for this > kind of proof, how do we decide when Perelman's work is correct? As someone else said: it's a sociological thing. > (Incidentally, he has apparently never submitted it to a > peer-reviewed journal, ... He now apparently says that he's not interested in getting it published. > but some sources say it's already gotten much more careful scrutiny > than most such submissions would.) Yes, but it's also harder than many other papers. Peace, Dylan Thurston

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Re: [math-fun] Cardinality puzzle - answer
by dasimov＠earthlink.net 18 Mar '06

18 Mar '06

Frankliin wrote: << Here's a related problem. Can you describe a *maximal* uncountable antichain? More difficult: describe an (uncountable) maximal antichain where every set is infinite. (I don't know the answer to this second question. It might be: no, no such description is possible.) >> This lasts kind of antichain can be shown to exist using Zorn's lemma, but here's an construction of an uncountable maximal antichain using only infinite subsets of Z: It consists of a) all subsets intersecting every set {2n, 2n+1} in exactly one element and b) all subsets of the form S_n = Z - {2n, 2n+1}. This is clearly an antichain. If any other subset X could be added, X would contain at least one element from each {2n, 2n+1} to avoid being a subset of one of the S_n's. But this means X has a subset of type a), contradiction. --Dan P.S. Here's an old cardinality puzzle in the same vein: What's the largest size of a collection of subsets of Z such that any two of them intersect in a finite set?

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[math-fun] G ö del meets Schoenberg meets Dylan
by Paul Reiners 17 Mar '06

17 Mar '06

Gödel Leaves Vienna: http://www.leipzig48.com/compositions/kurt/ A dodecaphonic, musical impression I wrote of Kurt Gödel leaving Austria for the U.S.

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Re: [math-fun] Stirling Numbers?
by dasimov＠earthlink.net 14 Mar '06

14 Mar '06

Hah! I hadn't even noticed my post hadn't gone through. After posting I found a recursion formula & calculated the terms up through 8 by hand, which were enough numbers to find the sequence in OEIS. (So I wrote a follow-up post saying Never mind, I learned that they're Bell numbers. Apparently that post is stuck, too.) And Jim, thanks for recommending the Rota book, but if you happen to know an asymptotic formula I'd love if you'd post it. --Dan I wrote about 10 days ago: << Suppose you want to count all possible partitions of the set X_n = {1,2,...,n}. I.e., you want to know how many [collections of mutually disjoint nonempty subsets of X_n] have union = X_n. . . . >> Michael Kleber wrote: << These are the Bell numbers, B(n). A000110 in the EIS, where you can find formulas and references to your heart's content. >>

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[math-fun] etymology
by Cordwell, William R 14 Mar '06

14 Mar '06

Does anyone know why "normal subgroups" are "normal"? Is it because they are the kernel of a homomorphism, and that's the way it's supposed to be? Also, does the use of "normal" for a field come from the relationship to a normal subgroup, in Galois Theory? Thanks, Bill Cordwell

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[math-fun] 3D geometry program
by Bernie Cosell 13 Mar '06

13 Mar '06

My wife has taken a fancy to playing around with 3d objects. Things like making a cube with circles inscribed in each face and rotating it around to see what the various projections look like, or fiddling with a menger sponge [and again, rotating it around to see what it looks like], or intersecting tetrahedra... Anyhow, I've been looking around some and I haven't had much/any luck finding a (windows) app that'll do it -- it seems that there are a lot of 2D drawing programs, and 3D modeling/animation programs (that are generally way too complicated (and generally expensive) for the sort of hacking my wife wants to do). Any of you know of an app that might do the job? Thanks!! /Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--

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[math-fun] BBC's "Negative Numbers"
by Henry Baker 10 Mar '06

10 Mar '06

Fascinating audio program on negative & complex (root of negative) numbers -- highly recommended: http://www.bbc.co.uk/radio4/history/inourtime/inourtime.shtml http://downloads.bbc.co.uk/rmhttp/downloadtrial/radio4/inourtime/inourtime_…

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Re: [math-fun] Cardinality puzzle - answer
by dasimov＠earthlink.net 10 Mar '06

10 Mar '06

Franklin wrote: << OK. I'm still refining this problem. The problem now is to describe a maximal uncountable antichain of subsets of Z, where no component set is either finite or counter-finite (i.e., Z-S must not be finite). And yes, I chose the word describe deliberately; the existence of such a set is easy to establish >> Hey, you could've said "nice example" when I solved your last question (:-)>. Anyhow, I've found as cute a little antichain as you could ask for that consists solely of subsets of Z that are each infinite and co-infinite. (I'm not sure how hard this is, but it seemed hard to me.) Without Zorn's Lemma, however, I don't yet see how to extend it to a maximal antichain. --Dan

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