ℝolliℵg M∀th Thr∑a∂

thx for the recommend ... just grabbed the .djvu of "groups and symmetry" and also his (?) "basic topology" too

http://web.math.princeton.edu/generals/examiner.html

princeton oral math exams

some nice questions in there

If f_n is a seqeunce of integrable functions, when is int(f_n)
convergent? Can you give an example where this fails?

Set a(n) = 1/n + ... + 1/2n. Compute lim a(n) as n --> infinity.

Let En be a sequence of measurable sets in [0,1] with m(En) --> 1. Does
there exist a subsequence whose intersections all have measure > 1/2?

sorry for dropping the ball on those combo coin flipping problems. i wanted to write up a nice proof for 2a that everyone could understand but i'm too busy writing a dumbass masters thesis right now. anyways 2a uses reflection principle and for 2b you can find a bijection between unique maximums and multiple maximums by reflecting about the peak /\ in a clever way. the proof of 2b my friend showed me was actually an original one that he's getting published so i prob shouldn't write it here. but i'll link to it later

bonus (easy) analysis question with multiple answers:

find a sequence X_n in R such that lim_{n->infty} X_n - X_{n-1} = 0 but lim_{n->infty} X_n doesn't exist

Let En be a sequence of measurable sets in [0,1] with m(En) --> 1. Does
there exist a subsequence whose intersections all have measure > 1/2?

i think this is 'yes' but cannot prove it

after posting and thinking about those for 5 mins i realized none of those questions make sense, i think there's some missing info

like, En = [0,1] for all n is an answer to that one you mention Lamp. m(En) -> 1 and m(Ei int Ej) = 1 > 1/2 for all i, j (taking the sequence itself as subsequence)

either that or princeton oral exams are a scam

i need to revisit rudin someday, never got a good handle on measure they/real analysis and ended up doing actuary stuff so having that basis for probability would be valuable if only for exams

ilx poster eteaoe convinced me to take measure theory in this very thread. it's ok, a little too dry for me but i didn't pursue stochastic calculus or any of the fancy applications of it, so it mostly just looked like an overly fussy way of doing integration to me at the same. this is my favourite result from measure theory https://en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma which states that

if the sum of probabilities of a random sequence of events is finite, the probability that infinitely many of them occur is zero

my advisor was super psyched to teach a real analysis class (he was an algebraic topology/knot theory guy) bc he said he felt like he really "got" the subject, but the course was still, as you say, pretty dry. your synopsis is about where i landed, but having gotten into stats since i feel like i shd take another stab.

like, En = [0,1] for all n is an answer to that one you mention Lamp.

I took it to mean "is it the case that, for EVERY sequence E_n....."

Once you reach M such that, for m > M, m(Em) > .75 then the maximum disjunction between Em1 and Em2 for m1,m2 > M is strictly less than .25 + .25 = .5 so their intersection has to have measure > 1/2

So, any subsequence starting from that point

starting from that point on

ohhh got it

hmm ok and since it's cauchy you can always find such a point. might there not be some weird ass measurable sets that would contradict though?

Don't think you need to bring Cauchy into it, limit is already defined. But I think what is showed is only pairwise intersection, it is possible that out on the tail the intersection of all the sets might have measure < 1/2.

we've been working on the questions flopson posted at work and found theres lots of solutions for that question. will post our solution when im back at my desk. kept getting caught up on the wording at first (also read the last > as -->)

xp oh right i got limit definitions mixed up

For countable intersection, calling the subsequence Fi, just go out far enough so that 1- m(F1) < 1/4, 1-m(F2) < 1/8, 1-m(F3) < 1/16, etc.

Note also that sets are given to be measurable, so their countable intersection, as it is here, will also be measurable and there will be no weird or wild behavior

http://www.lel.ed.ac.uk/~heycock/proof.html

How to prove it

Proof by example
The author gives only the case n=2 and suggests that it contains most of the ideas of the general proof.

Proof by intimidation
"Trivial"

Proof by vigorous handwaving
Works well in a classroom or seminar setting.

Proof by cumbersome notation
Best done with access to at least four alphabets and special symbols.

Proof by exhaustion
An issue or two of a journal devoted to your proof is useful.

Proof by omission
"The reader may easily supply the details."
"The other 253 cases are analogous."
"..."

Proof by obfuscation
A long plotless sequence of true and/or meaningless syntactically related statements.

Proof by wishful citation
The author cites the negation, converse, or generalization of a theorem from the literature to support his claim.

Proof by funding
How could three different government agencies be wrong?

Proof by eminent authority
"I saw Karp in the elevator and he said it was probably NP-complete."

Proof by personal communication
"Eight-dimensional coloured cycle stripping is NP-complete (Karp, personal communication)."

Proof by reduction to the wrong problem
"To see that infinite-dimensional coloured cycle stripping is decidable, we reduce it to the halting problem."

Proof by reference to inaccessible literature
The author cites a simple corollary of a theorem to be found in a privately circulated memoir of the Slovenian Philological Society, 1883.

Proof by importance
A large body of useful consequences all follow from the proposition in question.

Proof by accumulation of evidence
Long and diligent search has not revealed a counterexample.

Proof by cosmology
The negation of the proposition is unimaginable or meaningless. Popular for proofs of the existence of God.

Proof by mutual reference
In reference A, Theorem 5 is said to follow from Theorem 3 in reference B, which is shown to follow from Corollary 6.2 in reference C, which is an easy consequence of Theorem 5 in reference A.

Proof by metaproof
A method is given to construct the desired proof. The correctness of the method is proved by any of these techniques.

Proof by picture
A more convincing form of proof by example. Combines well with proof by omission.

Proof by vehement assertion
It is useful to have some kind of authority relation to the audience.

Proof by ghost reference
Nothing even remotely resembling the cited theorem appears in the reference given.

Proof by forward reference
Reference is usually to a forthcoming paper by the author.

Proof by semantic shift
Some of the standard but inconvenient definitions are changed for the statement of the result.

Proof by appeal to intuition
Cloud-shaped drawings frequently help here.

http://varianceexplained.org/r/empirical_bayes_baseball/

great stats blog post featuring this striking graph

http://varianceexplained.org/figs/2015-10-01-empirical_bayes_baseball/unnamed-chunk-11-1.png

woaahhh duuude
https://84c67cd8f568acc648fb74bc321df20db70c2600.googledrive.com/host/0B3p9nx7jwyf9MjFtY3d1aXVBMjA/fourier.gif

can someone explain that to me? i was bad at trig

The Fourier in the URL gives it away, I think. It's an animation on the first 4 orders of fast Fourier approximations of a square wave. You can see it getting closer to pure square wave the further it goes down the page.

New biography of John Horton Conway is out. Same author as the Coxeter bio, rave reviews from all the right people.

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