ℝolliℵg M∀th Thr∑a∂

altho "would u fly in an airplane that depended on subtle distinction in math" isn't the best way to think about math

also i think there is a physical significance, no? don't you need lebesgue integrals to do, like, integration on manifolds or whatever? (haven't studied this stuff yet)

I have a copy somewhere of this eccentric but interesting book on numerical methods from Dover in which the guys gives an interesting example about why airplane windows are shaped the way the air. At some point way back they were rectangular and, even though the equations describing the stress and strain on the window were smooth, the discontinuity/lack of derivatives at the corners would end up causing cracks. Things that make you go...

The author put the book on line, so here it is, page 38: http://www-personal.umich.edu/~jpboyd/aaabook_9500may00.pdf

Easiest way to think of R vs. L integration is in the former you are thinking of how x->y and using the continuity of the given function to bound the y's in the sums, but in L integration you instead for each y consider the set of x that map to that given y. The latter requires creating a certain apparatus to keep track of how the sets work and ignoring certain sets as "too wild."

Simplest example is a function which is zero almost everywhere (a.e) on the interval 0,1 but takes on some other arbitrary values on some countable subset like, say, the rational numbers. Riemann integration won't work. but Lebesgue integration gives the expected answer zero.

Hamming's actual argument is pretty interesting and subtle i think -- the guy was no slouch. Ultimately the math with regards to an airplane corresponds to real physical things, and these real physical things are what matter, not our models of them, and since R and L agree where they're both defined, and since they're both defined in these types of physical systems, by construction, then the genuine differences between them cease to matter up to a certain point.

Its an argument that we shouldn't think of math in isolation from the reasons we invent it.

I get what he is saying and it's not wrong, I guess, but he seems to be implying that Lebesgue integration is useless, which is a bit of a stretch.

Its an argument that we shouldn't think of math in isolation from the reasons we invent it.

― lollercoaster of rove (s.clover), Monday, December 2, 2013 2:35 PM (2 hours ago) Bookmark Flag Post Permalink

booooooooring :-P

jr -- if you knew his work you'd recognize that he didn't mean it in that way, though yeah, in isolation it can take on that context.

he's actually the author of one of my favorite essays on the relationship of math and physics: http://web.njit.edu/~akansu/PAPERS/The%20Unreasonable%20Effectiveness%20of%20Mathematics%20(RW%20Hamming).pdf

and also "you and your research" is an ur-classic http://www.cs.virginia.edu/~robins/YouAndYourResearch.html

that's great, thanks

OK, looking forward to reading those.

Dudes, you guys are still in academia and have jstor access.

u want me to send you a pdf?

I might, rabbit, I might.

Yes, please.

send me yo email, can't attach a pdf to ilxmail

nm here u go http://www.scribd.com/doc/188781630/2321982

Thanks. Just found it here too: http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.html
Along with :http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

Top level is here: http://www.dartmouth.edu/~matc/MathDrama/reading.html The Geometry in Art and Architecture link looks really nice.

OK, that was well worth reading. Thanks, guys.

we should do reading group some time. my greatest regret is not having taken a course in complex analysis.

Took me much longer than I would have liked to grok what was going on in that subject. btw, did you see the thing in Vanity Fair about the guy who reverse engineered device Jan Vermeer used to paint his masterpieces?

like "girl with a pearl earring"?

Yes, only the one that he reproduced was The Music Lesson.

I linked it on some art thread where I don't think anyone looked at it but I'll put it here too: http://www.vanityfair.com/culture/2013/11/vermeer-secret-tool-mirrors-lenses

Anyway most important thing about complex variable

this is the book i wanna read

http://usf.usfca.edu/vca//vca.jpg

Things:
Assumption that the derivative of a function does not depend on the direction you are coming from is a pretty strong one which is why analytic functions have so many things you can say about them.

Function log z can't be defined without branch cut and related fact that integral 1/z on a loop is non-zero

You spend a lot of time switching back and forth from complex variables to rectangular coordinates to polar coordinates.

Have a copy somewhere. It's pretty nice, spends more time trying to generate intuition than most.

Just came across notes on branch cuts by my advisor's advisor: http://math.mit.edu/classes/18.305/Notes/n00Branch_Points_B_Cuts.pdf

heh, a prof in my dept wrote a 1200 page, 2 volume book on the complex logarithm

Really? Who was that?

http://ebooks.cambridge.org/content/978/05/1156/619/6/9780511566196i.jpg

they're enormous

Visual Complex Analysis is terrific (especially if you haven’t been exposed to Michael Spivak’s Calculus or “Baby Rudin”)!

Book club?

http://www.madore.org/~david/math/hyperbolic-maze.html

super down for a book club. would have to start after finals tho

I’d participate!

I started reading the recently published Computability: Turing, Gödel, Church and Beyond edited by Copeland, Posy, and Shagrir. It’s a servant of all. But so far, so good. Especially enjoyed Martin Davis’ essay, “Computability and Arithmetic.” It explores Hilary Putnam and Yuri Matiyasevich’s work on Hilbert’s tenth in a comprehensible way.

super down for a book club. would have to start after finals tho

Good luck!

by servant of all you mean written for a too-general audience?

and thanks!

not sure how much brain i have to tackle another math topic at the moment, but i'm for a reading group as a general notion and i'd try to follow along a bit at least.

i started skimming along the complex analysis stuff and not-incorrectly thought "fibration" so i'm glad i'm building some intuitions.

re. that computability volume, a, uh, friend of mine has to write a review of it pretty soon, so any tips on what worked/didn't from your point of view would be appreciated (my friend hasn't started reading the book yet but the review is overdue, story of his life)

http://i.imgur.com/gxC8u1S.png

why do they schedule exams at 9am? who can even think that early?

http://math.berkeley.edu/~wu/AMS_COE_2011.pdf

Professional development (PD) for in-service math teachers is
generally taken to be \feel-good sessions". Some believe that its
main goal is to give teachers encouragement and sharpen their
pedagogical skills.
Others believe that teachers should be exposed to fun mathematics
(such as the Konigsberg bridge problem or taxicab geometry),
even in the face of their inability to deal with bread-and-butter issues
such as how to teach fractions, why negative times negative
is positive, what similarity means, or why the parallel postulate
is important.

anyone want to take a stab at 'why negative times negative is positive'? seems like a good one.

because negative divided by positive is negative.