ℝolliℵg M∀th Thr∑a∂

https://upload.wikimedia.org/wikipedia/commons/0/0f/SquareWaveFourierArrows%2Crotated.gif

some nice riddles from this new ams math grad student blog

http://blogs.ams.org/mathgradblog/2015/12/06/math-puzzlesriddles-part-ii/#more-26560

LIFE ON A CHESSBOARD

Most of you are probably familiar with various versions of Conway’s famous “Game of Life”. This riddle pertains to a particularly simple version, played on an 8×8 grid of what are usually envisioned as light-up squares. The setup is as follows: initially, some subset of the squares are lit up (the “starting configuration”). At each stage, a square lights up if at least two of its immediate neighbors (horizontal or vertical) were “on” during the previous stage. Note that in this version of Life, squares do not ever turn from “on” to “off”.

It’s easy to see that for the starting configuration in which eight squares along a diagonal of the board are lit up, the entire board is eventually covered by “on” squares. Several other simple starting configurations with eight “on” squares also result in the entire board being covered. Is it possible for a starting configuration with fewer than eight squares to cover the entire board? (If yes, find it; if no, give a proof!)

THREE-WAY CAKE SUBDIVISION

A group of three (mutually distrustful) mathematicians are attempting to divide a cake between themselves. They have a knife, but no measuring utensils of any kind. The mathematicians need to agree on a procedure for subdividing the cake in which each mathematician has a role in the subdivision and assignment of cake pieces. This procedure must satisfy the following “fairness” condition: for each mathematician X, if X has “perfect play”, then X can guarantee him or herself at least one-third of the cake, regardless of the actions of the other two mathematicians.

In the two-person case, a solution is furnished by the following simple procedure: one person (either one) cuts the cake into two pieces. The other person then chooses a piece for him or herself, with the remaining piece going to the one who originally divided the cake. This procedure evidently satisfies the fairness condition (with one-third replaced by one-half); the question is then to devise a suitable procedure for three mathematicians (or any number of mathematicians, if you are feeling bold!).

Note that one is not allowed to assume anything about the other players, even rational self-interest or perfect play on their part. For example, one (flawed) procedure might be to have person A cut the cake into three pieces, and then have A, B, and C then choose their own pieces in some order with A going last (say B, C, and then A). Although one might argue that A has an incentive to divide the cake as equally as possible (since it seems likely that A would receive the smallest piece), we do not assume that A can or will do so. Thus A might (perhaps by accident) cut the cake lopsidedly into one large and two extremely small pieces, violating the fairness condition from the point of view of C.

The cake eating one is particularly badly posed, at first I thought it was Vickrey Clarke Groves but I'm not even sure what the question is

holy shit, springer made basically all its math books (at least) older than 10 years old available online free. the entire graduate texts in mathematics series (including like lang's algebra, whitehead on homotopy theory, thurston on singular homology), the entire lecture notes in mathematics series, the entire universitext series, and much else besides.

this is such a trove of amazing stuff.

(lots of philmath stuff too from the synthese series, including great books by beth, etc).

what?!?

You sure you don't need jstor or university library account or something to access?

If not, please provide link.

http://link.springer.com/search?facet-series=%22136%22&facet-content-type=%22Book%22&showAll=false

fantastic

now reading Lee - Introduction to Smooth Manifolds, the textbook for Geo Topo 2, the follow up to the point set and algebraic topology class i was in when i started this thread. seems like that class must have been way harder than GT1

you should liveblog it

one of my browser windows right now

http://i.imgur.com/u4t222I.png

lol

i would liveblog if we could embed LaTeX in ilx posts

stet needs to install mathjax

Aloha, suckers.

Finally finished David Foster Wallace's book on infinity--only took me three months. Sentences like "But if you can conceive, abstractly, of a progression like ω, ((ω + 1), (ω + 2),..., (ω + ω)), ω²,..., then you can get an idea--or at any rate an 'idea'--of the hierarchy and the unthinkable heights of ordinal numbers of infinite sets of infinite sets of the ordinals of infinite sets it involves."

It was the hierarchy and the unthinkable heights that slowed me down.

That looks like stuff from abstract algebra (which I got a d in), it's hella cool, though

There was something in that book that flummoxed me, I heard it had a few errors though so I don't know if it was a bona fide infinity paradox flummoxing or just rongness. It was something like "every number is adjacent to another number, but between any two numbers is another number". Seems like a contradiction but also self-evidently true, unless 'adjacent to' is not a well formed concept for infinite sets, or something. Anyone?

depends what you mean by adjacent, and depends if you're discussing the reals, the rationals, etc.

it's true that there is a number between any two real or rational number but there is so 'next' number in R or Q so no concept of adjacency. take a real number x, a number adjacent would be either the infimum of the set of numbers larger than it, inf (x,+infty), or the supremum of all numbers smaller than it sup (-infty,x). but since those sets lack the limit point x (they aren't complete iirc) those don't exist. i *think* that's correct. so no real paradox

but just intuitively, what is the "next" rational number after 1? 1.0000001? why not 1.0000000000000000001? and so forth

but the reals and rationale are well-orderable, so there's a next number; same for the complexes. def not necessarily the "intuitive " order i.e. not necessarily the one generating the natural density and the order may not respect the properties you expect with the usual arithmetic operations but you can have "nextness" with any set by the well-ordering principle (equivalent to the axiom of choice)

Thx

man, what is the axiom of choice not equivalent to

what's yellow and equivalent to the axiom of choice?

zorn's lemon!

(more seriously: so this is where constructive vs. nonconstructive proofs come into play, i think? if you give me two concrete reals that are not equal, i have a procedure that constructs a new real that is between them. but if you give me a single concrete real, i only have a non-constructive proof that there is a number that is the "next higher" and not a construction of it. since i can't get my hands on that "next" number, then i can't feed it into my first procedure and construct something in between.

i only know that it would be a contradiction for this next number not to exist, but to actually get my hands on it is hopeless.)

Yup

the nextness is always relative to some well ordering, though, i.e. to its existence, right? (i.e. sure there's guaranteed to be one, but it's still one relative to which anything would be 'next'.) maybe there being one is to be compared to there being one for the naturals or integers, if you want to test intuitions.

to s.clover: yes; on most ways of thinking about constructivity there's only countably many constructible reals but uncountably many reals so a well-ordering of the reals isn't gonna be constructive for every pair of reals.

to j: yes, but all claims about nextness are relative to an ambient ordering. generating one by choice gives you no intuitive info; there are other weird well-orderings of the natural numbers that won't match our intuitive ordering on the natural numbers.

well they're weird

they're all weird if they're not capturing the flow of one moment into a next

Glad I generated some discussion... DFW was wrong is my basic take-home message.

I mean the point is that by choosing a well-ordering you can give the rational numbers a notion of "adjacent to," but this notion has nothing to do with and need not be compatible with (indeed, CANNOT be compatible with) the natural notion of "between", and it's this lexical slippage that creates the apparent inconsistency.

By "compatible with" I might mean something like "if A is adjacent to B and B is adjacent to C and C is not A, then B is between A and C" which seems natural given the English words, but, y'know, slippage.

(more seriously: so this is where constructive vs. nonconstructive proofs come into play, i think? if you give me two concrete reals that are not equal, i have a procedure that constructs a new real that is between them. but if you give me a single concrete real, i only have a non-constructive proof that there is a number that is the "next higher" and not a construction of it. since i can't get my hands on that "next" number, then i can't feed it into my first procedure and construct something in between.

i only know that it would be a contradiction for this next number not to exist, but to actually get my hands on it is hopeless.)

― big WHOIS aka the nameserver (s.clover), Sunday, January 3, 2016 3:57 PM (5 hours ago) Bookmark Flag Post Permalink

...but it's also a contradiction for it to exist

ie, for any next number candidate you can say, give me the one epsilon closer

I mean the point is that by choosing a well-ordering you can give the rational numbers a notion of "adjacent to," but this notion has nothing to do with and need not be compatible with (indeed, CANNOT be compatible with) the natural notion of "between", and it's this lexical slippage that creates the apparent inconsistency.

― Guayaquil (eephus!), Sunday, January 3, 2016 7:09 PM (2 hours ago) Bookmark Flag Post Permalink

ok yeah this is otm

a non-ilx internet friend just posted this quote by terry tao to on twitter

There is a tradeoff between +∞ and negative numbers.

If one wants to keep many useful laws of algebra then one can use infinity, xor negative numbers, but it is difficult to have both at the same time.

Once one adopts the convention +∞ · 0 = 0 · +∞ = 0, then multiplication becomes upward continuous (i.e.: when both multiplicands increase, the product is what you would expect) but not downward continuous—so 1÷n → 0 works but 1÷n · +∞ ↛ 0 · +∞ fails.

This asymmetry ultimately forces us to define integration from below rather than from above, which leads to still other asymmetries, and finally to two versions of measure and integration theory.

Terence Tao, Intro to Measure Theory

ie, for any next number candidate you can say, give me the one epsilon closer

― flopson, Sunday, January 3, 2016 9:52 PM Bookmark Flag Post Permalink

right, but since the proof it exists is nonconstructive, you can't pick any particular candidate and actually execute that construction. that's why there's not a contradiction.

or i think i'm tangling myself here. that's why explicit infinitesimals are not a contradiction in a system like synthetic differential geometry without excluded middle. with regards to standard analysis, ignore all this :-)

hey so, math thread, as a programmer/computer science type and not a mathematician at all I stopped fucking with continuous domains after taking multivariable calculus in high school, so the diffeq and analysis and algebra sequence is pretty much unknown to me at any level of sophistication. Is there a good book/resource/PDF/set of lecture notes out there where I can learn, like, some "greatest hits" of analysis without tons of additional prerequisites? Like idk precisely what I'm asking for, I just have this sense that there's some Fun Facts About the Reals that I could get a sense of with appropriate scaffolding but without having to like consume three semesters' worth of course material.

i only took up to analysis 4 but you could always download a pdf of Rudin and read the definitions and theorems skipping the proofs

does anyone know if there's an R equivalent to STATA's .do files? i'm switching over to R from STATA cuz they don't own a license to the latter at my job and i feel like i've exhausted what i can get out of vba/excel, and i really liked those .do files when i was in school

If you are using R and you are only sort of a programmer, using R Studio will probably help you out a lot. It looks like .do files are just scripts, so yes, there is an equivalent, just save R commands to a text file and then run Rscript on it or load it into your interactive session with source().

sweet, thx silby

yep, also ask for a raise, you're a programmer now

Lol at the Rudin recommendation upthread.

i wanna make a pdf of that, like a Jefferson bible of Real Analysis

hey so, math thread, as a programmer/computer science type and not a mathematician at all I stopped fucking with continuous domains after taking multivariable calculus in high school, so the diffeq and analysis and algebra sequence is pretty much unknown to me at any level of sophistication. Is there a good book/resource/PDF/set of lecture notes out there where I can learn, like, some "greatest hits" of analysis without tons of additional prerequisites? Like idk precisely what I'm asking for, I just have this sense that there's some Fun Facts About the Reals that I could get a sense of with appropriate scaffolding but without having to like consume three semesters' worth of course material.

Michael Spivak’s Calculus and Needham’s Visual Complex Analysis