ℝolliℵg M∀th Thr∑a∂

also this https://npmjs.org/package/bayes

new riddle

60 immortal impotent chameleons
15 red, 20 green, 25 blue
When two chams of different colors meet, they both change to the third color
When two like-coloreds meet, nothing happens
If all chameleons changed to one color, that configuration would be stable
Question: why does such a configuration never arise?

ok here's a sketch of an answer, which i think is correct.

first recognize that from any 'stable state' we can only get there from any state that include two colors with exactly the same quantity. they can then meet eachother to go all to the remaining color. so now the problem is to equalize any two colors.

if we start equalizing colors, each step we take to move them closer together involves shifting one up by two and the other down by one. this means that they can only approach each other (or run away from each other) by steps of three. since the colors given differ, pairwise, by five, five, and ten, none of which three divides evenly, we can never get to a state where two are equal, and hence never to a state where all are the same color.

ok now for a bit of random further speculation. writing this down actually leads me to an invariant: the difference between any two colors, mod 3, should never vary. furthermore, given the distance between any two colors (mod 3), we know the distance from the first to the third. so if we forget about needing to keep a positive number of chameleons, and if we forget about what they sum to (we can add those conditions in later, somehow?) then we get there are "just" 4 basic quotients going on here:

0 1 1, 0 2 2, 1 1 2, 1 2 0, where the third distance is the sum of the first two, mod 3.

the situation in the above problem is # 3, and the only "unsolvable" one.

i'm sure there's some classic result in group theory or something that this is a very tiny special case of. do you have any reference like that flopson?

wait i fucked up the last a bit. the invariant is weaker

the full list is, unsurprisingly 9 (3 x 3) of which some are symmetric.

we have
0 0 0, 0 1 1, 0 2 2
1 0 1, 1 1 2, 1 2 0
2 0 2, 2 1 0, 2 2 1

of these, the six unique situations (under color permutation) are 0 0 0, 0 1 1, 0 2 2, 1 1 2, 1 2 0, and 2 2 1

we're actually in 2 2 1, which is one of _two_ unsolvable situations (i.e. ones which do not contain a zero).

(or rather the invariant is ok i think its that it just gives more situations than i first listed)

Think I got it. Didn't look at s. clover's solution but maybe I saw it out of the corner of my eye and it inspired me.

1) The next to last step of making them all the same is to have two of the them equal so n of color 1, n of color 2, m of color 3. The n of color 1 meet the n of color 2 and they all become color 3.
2) The simplest operation you can do is have two chamos of different colors meet. What does that do? Decreases the number of each of their colors by 1 and increases the number of the 3rd color by 2.
3) If we think of the differences of the numbers mod 3 the above operation does not change that, since it is just subtracting 1 mod 3 from each population. Obviously if numbers are to be equal they should be equal mod 3. The original numbers 15, 20,25. are 0,2,1 mod 3. Hence all operations will essential permute these moduli, we can't make them equal and we can't get there from here.

Which seems to be exactly what Sterling was going for now that I look.

^_^ nice job fellas

Got thrown off for a while by the fact that they were all divisible by 5 which had nothing to do with anything.

homology is very confusing

feel like it's gonna pay off in a huge way though

Afraid I can't keep up with you guys on that stuff. Did hear that Sylvain Cappell just proved a big theorem though.

OK, that is not quite the case.

usually find these kind of things pretty cheezy but this is a p good lil vid http://vimeo.com/77330591

is there a simple piece of software / web app out there that does nothing more than support computation w/ bayes' rule when repeatedly updating prior probabilities in light of new evidence?

Google Docs, maybe.

also the thing is in 'day-to-day' math its less whether or not you 'believe' in large ordinals or double-negation-elimination or the axiom of choice but ppl just pick a background setting sufficient for the work they're doing (often not explicitly). and then if you can show that you get different results with or without choice, that just makes it more interesting.

Well said.

Separately, when I was in college I tried to envision contemporary algebra or analysis rebuilt with both weak (e.g. Peano) and strong (e.g. New Foundations) programs. My view of modern mathematics was idyllic, but naïve. Later, when I discovered music and poetry, I appreciated the universal adoption of, for example, the axiom of choice, for the same humanitarian reasons I appreciated iambic pentameter and the diatonic scale: universality, expressiveness, simplicity, etc. Nonetheless, the best advice I’ve received on the subject: match the program to the problem and move on.

I should confess that I hated the “match the program to the problem and move on” strategy when I first encountered it, but like the apocryphal Feynman algorithm:

• Write down the problem.
• Think real hard.
• Write down the solution.

from a comp sci perspective these questions become much more worthwhile since "does it compute" is a pretty important question, with "how does it compute" coming a close second.

of course its also worth remembering the famous hamming quip:

"Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane."

ha, that's brutal

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.