May 2009 - mathfun@mailman.xmission.com

Re: [math-fun] good introductory real-analysis text?
by Dan Asimov 29 May '09

29 May '09

That would seem most familiar and comfortable to students. The only reason I didn't mention the decimal construction is that it favors base 10, and for mathematical aesthetics it's nice to have a "neutral" construction, like Cauchy sequences of rationals. Of course, after constructing the reals as Cauchy sequences of rationals, one might want to show that Cauchy sequences of reals don't yield anything new (aka, the reals are complete). That leads students to feel with satisfaction that the reals is a natural object. --Dan << My favorite construction for the reals is to match the implicit "infinte decimals" that we grew up with. It's the ground state, hence "intuitive". There's no harm in mentioning the alternatives, nor in emphasizing there are a bunch of equivalents. >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] good introductory real-analysis text?
by Dan Asimov 29 May '09

29 May '09

I think constructing the reals in some natural way (probably Cauchy sequences is most intuitive) is pedagogically superior to presenting the reals as a set satisfying certain axioms. First of all, when something is defined by axioms, its mere existence isn't obvious. When it's constructed (essentially assuming no more than set theory, whose axioms are too abstruse to go into in a course like this), then it must exist unless set theory is self-contradictory. Second, when something is defined by axioms, it's easy to feel like you're flying blind. Someone has set you down in front of an instrument panel (the axioms), and your only options for staying aloft is to fiddle with the buttons and dials. It's as though you can't see out the windows. But when something is constructed in a simple and natural construction (which should of course come sometime after a demonstration that there are irrational numbers that "ought to" exist), then you can in a sense see and feel what you have wrought. Finally, the whole process of rigorously constructing a mathematical object is a core concept in math, and I can't think of a better place to have it introduced than with the reals. --Dan Jim wrote: << . . . I should also say that I'm NOT looking for a book that constructs the reals (via Dedekind cuts or Cantor sequences or Conway games). My view is that a student's first exposure to real analysis should be based on an axiomatic DESCRIPTION of the reals, not a formal CONSTRUCTION of them. My point of view might be expressed succinctly, with only a small amount of distortion, in the slogan "Why construct the reals when they already exist?" . . . On the other hand, I love a good (friendly) argument, so if any of you think I'm going in the wrong direction, please offer your counter-arguments! >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] The Axiom of Choice is Wrong
by Mike Speciner 29 May '09

29 May '09

Intuitively, if I'm prisoner j, the probability that the chosen equivalence class representative agrees with reality starting at or before my position (j) is 0, and so my chance of being correct is still 1/2. One could ask what the expected value is for how far along the sequence I have to look to match with any of the chosen representatives, and I'd guess that if that expected value exists it would have to be infinity.

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[math-fun] Sightsing this Sunday?
by Andy Latto 29 May '09

29 May '09

Trau sent an email a few weeks ago saying that there would be a sightsing this Sunday at Lizzie's ice cream, but he hasn't sent anything recently, and he usually sends something the week of the sing. Did he send something that I didn't get? Is the sing on, as far as you know? Andy -- Andy.Latto(a)pobox.com

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Re: [math-fun] good introductory real-analysis text?
by James Propp 29 May '09

29 May '09

Michael Kleber suggested Mattuck's book. Mattuck is a good teacher, but I suspect that he wrote the book for MIT students, so I'm not sure how suitable it'd be for UMass Lowell students. UML does have some fine students, but the average UML math major is not as well- prepared as the average MIT math major. I should also say that I'm NOT looking for a book that constructs the reals (via Dedekind cuts or Cantor sequences or Conway games). My view is that a student's first exposure to real analysis should be based on an axiomatic DESCRIPTION of the reals, not a formal CONSTRUCTION of them. My point of view might be expressed succinctly, with only a small amount of distortion, in the slogan "Why construct the reals when they already exist?" Don't get me wrong; one of the formative experiences of my young mathematical career was getting a copy of Kirshner and Wilcox's "The Anatomy of Mathematics" as a gift when I was thirteen and devouring it from cover to cover. I loved the thrilling progression from Peano Postulates to the construction of the real numbers, with each successive number system incorporating the one before. But most of my students won't become mathematicians, let alone logicians or model theorists (constructions are good for establishing relative consistency but it's a rare student who would worry about this). I want my students to become principled and savvy practitioners of real analysis who know what the main tools are and who know when caution is required in wielding them. So I'm looking for a book by someone who shares that spirit. On the other hand, I love a good (friendly) argument, so if any of you think I'm going in the wrong direction, please offer your counter-arguments! Jim

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[math-fun] The Axiom of Choice is Wrong
by Ray Tayek 28 May '09

28 May '09

http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/ When discussing the validity of the <http://en.wikipedia.org/wiki/Axiom_of_choice>Axiom of Choice, the most common argument for not taking it as gospel is the <http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox>Banach-Tarski paradox. Yet, this never particularly bothered me. The argument against the Axiom of Choice which really hit a chord I first heard at the Olivetti Club, our graduate colloquium. It's an extension of a basic logic puzzle, so let's review that one first. ... --- vice-chair http://ocjug.org/

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[math-fun] good introductory real-analysis text?
by James Propp 27 May '09

27 May '09

I'll be teaching real analysis in the Fall, and I'm looking for a good textbook to use. Do any of you have suggestions? I've looked at Kenneth Ross' book, which I heard was easier than Royden or "baby Rudin", but I think it's still going to be rough going for the average student in my class; the students at my current school are hard- working but typically not as prepared for proof-oriented mathematics as students at other schools I've taught at. Two texts that I'm tempted by (based solely on what I've read on the web) are "Yet Another Introduction to Analysis" by Victor Bryant and "A Friendly Introduction to Analysis" by Witold A.J. Kosmala. Do any of you know anything about either of them? To point out a (possibly forced) connection between my question and the charter of the math-fun email discussion group, let me add that one thing a good real analysis textbook should do, in my opinion, is make the material as fun as possible! Also, for those of you who might be interested in my ideas on how to teach completeness of the reals via an axiom about cutting R into two pieces, my current thoughts can be found at jamespropp.org/cut.pdf . Comments are welcome. Jim Propp

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[math-fun] Reuleaux bicycle
by Henry Baker 27 May '09

27 May '09

FYI -- http://www.china.org.cn/china/photos/2009-05/07/content_17738257.htm http://technology.timesonline.co.uk/tol/news/tech_and_web/article6366308.ece

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Re: [math-fun] Ford circles and intuition
by Phil Carmody 27 May '09

27 May '09

From: rwg(a)sdf.lonestar.org > Admittedly, pictures can deceive, but check out > the epsilon=1 case > on page 11 of http://www.tweedledum.com/rwg/rectarith12.pdf . Not reached page 11, got stuck on page 6: "Arbitrarily assign the value 1 to the sides of the little red square, and the value x to the adjoining blue square. ... you should get an equation that determines x. Answer: x = 7, " No, answer x=1, as a square, regardless of size, has value 1. (Ibid. p1) Phil

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[math-fun] Fwd: Distinguisher and Related-Key Attack on the Full AES-256
by rcs＠xmission.com 26 May '09

26 May '09

(A few funsters are interested in crypto.) This note is not about a security weakness, but it is a minor theoretical blemish on the long-key (256 bit) version of AES. An ideal cipher should behave as a (keyed) random permutation on the input space. It should not be statistically different from random, so related keys should still produce unrelated permutations. This note announces a minor defect. The security of the cipher is unaffected (in proper use), because your keys should always be chosen as unrelated random numbers anyway. Rich ----- Forwarded message from lloyd(a)randombit.net ----- Date: Fri, 22 May 2009 11:06:23 -0400 From: Jack Lloyd <lloyd(a)randombit.net> Reply-To: Jack Lloyd <lloyd(a)randombit.net> Subject: Distinguisher and Related-Key Attack on the Full AES-256 To: cryptography(a)metzdowd.com Alex Biryukov, Dmitry Khovratovich, and Ivica Nikolic gave a talk at the Eurocrypt rump session, 'Distinguisher and Related-Key Attack on the Full AES-256', with the full paper accepted to Crypto. Slides from Eurocrypt are here: http://eurocrypt2009rump.cr.yp.to/410b0c56029d2fa1d686823e3a059af8.pdf The q-multicollisions attack they describe may be a practical way of breaking a hash function based on AES. So this could have some interesting ramifications to SHA-3 candidates which use the AES round function; I'm not sufficiently familiar with those designs yet for it to be clear one way or another if they would in fact be vulnerable. (via zooko's blog) --------------------------------------------------------------------- The Cryptography Mailing List Unsubscribe by sending "unsubscribe cryptography" to majordomo(a)metzdowd.com ----- End forwarded message -----

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