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.
― art, Tuesday, 11 August 2015 02:34 (eight years ago) link
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....."
― Guayaquil (eephus!), Tuesday, 11 August 2015 06:06 (eight years ago) link
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
― Eternal Return To Earth (James Redd and the Blecchs), Tuesday, 11 August 2015 10:29 (eight years ago) link
So, any subsequence starting from that point
― Eternal Return To Earth (James Redd and the Blecchs), Tuesday, 11 August 2015 10:30 (eight years ago) link
starting from that point on
― Eternal Return To Earth (James Redd and the Blecchs), Tuesday, 11 August 2015 10:43 (eight years ago) link
ohhh got it
― flopson, Tuesday, 11 August 2015 14:53 (eight years ago) link
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?
― flopson, Tuesday, 11 August 2015 15:09 (eight years ago) link
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.
― Eternal Return To Earth (James Redd and the Blecchs), Tuesday, 11 August 2015 16:11 (eight years ago) link
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 -->)
― dead (Lamp), Tuesday, 11 August 2015 16:16 (eight years ago) link
xp oh right i got limit definitions mixed up
― flopson, Tuesday, 11 August 2015 16:43 (eight years ago) link
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.
― Eternal Return To Earth (James Redd and the Blecchs), Tuesday, 11 August 2015 17:24 (eight years ago) link
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
― Eternal Return To Earth (James Redd and the Blecchs), Tuesday, 11 August 2015 17:42 (eight years ago) link
http://www.lel.ed.ac.uk/~heycock/proof.html
How to prove it
Proof by exampleThe 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 handwavingWorks well in a classroom or seminar setting.
Proof by cumbersome notationBest done with access to at least four alphabets and special symbols.
Proof by exhaustionAn 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 obfuscationA long plotless sequence of true and/or meaningless syntactically related statements.
Proof by wishful citationThe author cites the negation, converse, or generalization of a theorem from the literature to support his claim.
Proof by fundingHow 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 literatureThe author cites a simple corollary of a theorem to be found in a privately circulated memoir of the Slovenian Philological Society, 1883.
Proof by importanceA large body of useful consequences all follow from the proposition in question.
Proof by accumulation of evidenceLong and diligent search has not revealed a counterexample.
Proof by cosmologyThe negation of the proposition is unimaginable or meaningless. Popular for proofs of the existence of God.
Proof by mutual referenceIn 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 metaproofA method is given to construct the desired proof. The correctness of the method is proved by any of these techniques.
Proof by pictureA more convincing form of proof by example. Combines well with proof by omission.
Proof by vehement assertionIt is useful to have some kind of authority relation to the audience.
Proof by ghost referenceNothing even remotely resembling the cited theorem appears in the reference given.
Proof by forward referenceReference is usually to a forthcoming paper by the author.
Proof by semantic shiftSome of the standard but inconvenient definitions are changed for the statement of the result.
Proof by appeal to intuitionCloud-shaped drawings frequently help here.
― flopson, Friday, 11 September 2015 17:15 (eight years ago) link
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
― flopson, Thursday, 8 October 2015 02:31 (eight years ago) link
woaahhh duuudehttps://84c67cd8f568acc648fb74bc321df20db70c2600.googledrive.com/host/0B3p9nx7jwyf9MjFtY3d1aXVBMjA/fourier.gif
― flopson, Saturday, 17 October 2015 20:47 (eight years ago) link
can someone explain that to me? i was bad at trig
― flopson, Saturday, 17 October 2015 23:29 (eight years ago) link
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.
― koogs, Sunday, 18 October 2015 02:23 (eight years ago) link
New biography of John Horton Conway is out. Same author as the Coxeter bio, rave reviews from all the right people.
― Dover Blecch (James Redd and the Blecchs), Wednesday, 28 October 2015 14:00 (eight years ago) link
https://upload.wikimedia.org/wikipedia/commons/0/0f/SquareWaveFourierArrows%2Crotated.gif
― + +, Wednesday, 28 October 2015 16:48 (eight years ago) link
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 CHESSBOARDMost 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 SUBDIVISIONA 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.
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.
― flopson, Friday, 18 December 2015 18:20 (eight years ago) link
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
― flopson, Saturday, 19 December 2015 01:27 (eight years ago) link
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.
― big WHOIS aka the nameserver (s.clover), Monday, 28 December 2015 08:33 (eight years ago) link
(lots of philmath stuff too from the synthese series, including great books by beth, etc).
― big WHOIS aka the nameserver (s.clover), Monday, 28 December 2015 08:34 (eight years ago) link
what?!?
― Die Angst des Elfmans beim Torschluss (James Redd and the Blecchs), Monday, 28 December 2015 12:23 (eight years ago) link
You sure you don't need jstor or university library account or something to access?
― Die Angst des Elfmans beim Torschluss (James Redd and the Blecchs), Monday, 28 December 2015 12:26 (eight years ago) link
If not, please provide link.
― Die Angst des Elfmans beim Torschluss (James Redd and the Blecchs), Monday, 28 December 2015 12:27 (eight years ago) link
http://link.springer.com/search?facet-series=%22136%22&facet-content-type=%22Book%22&showAll=false
― j., Monday, 28 December 2015 13:13 (eight years ago) link
fantastic
― flopson, Monday, 28 December 2015 18:39 (eight years ago) link
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
― flopson, Monday, 28 December 2015 19:46 (eight years ago) link
you should liveblog it
― j., Monday, 28 December 2015 19:49 (eight years ago) link
one of my browser windows right now
http://i.imgur.com/u4t222I.png
― big WHOIS aka the nameserver (s.clover), Monday, 28 December 2015 20:58 (eight years ago) link
lol
i would liveblog if we could embed LaTeX in ilx posts
― flopson, Monday, 28 December 2015 20:59 (eight years ago) link
stet needs to install mathjax
― pizza rolls are a food that exists (silby), Tuesday, 29 December 2015 01:50 (eight years ago) link
Aloha, suckers.
― Green Dolphin Street Hassle (James Redd and the Blecchs), Friday, 1 January 2016 21:43 (eight years ago) link
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.
― clemenza, Sunday, 3 January 2016 03:17 (eight years ago) link
That looks like stuff from abstract algebra (which I got a d in), it's hella cool, though
― lute bro (brimstead), Sunday, 3 January 2016 03:23 (eight years ago) link
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?
― ledge, Sunday, 3 January 2016 10:23 (eight years ago) link
depends what you mean by adjacent, and depends if you're discussing the reals, the rationals, etc.
― big WHOIS aka the nameserver (s.clover), Sunday, 3 January 2016 11:41 (eight years ago) link
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
― flopson, Sunday, 3 January 2016 18:35 (eight years ago) link
but just intuitively, what is the "next" rational number after 1? 1.0000001? why not 1.0000000000000000001? and so forth
― flopson, Sunday, 3 January 2016 18:37 (eight years ago) link
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)
― droit au butt (Euler), Sunday, 3 January 2016 19:35 (eight years ago) link
Thx
― Green Dolphin Street Hassle (James Redd and the Blecchs), Sunday, 3 January 2016 20:03 (eight years ago) link
man, what is the axiom of choice not equivalent to
― pizza rolls are a food that exists (silby), Sunday, 3 January 2016 20:15 (eight years ago) link
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.)
― big WHOIS aka the nameserver (s.clover), Sunday, 3 January 2016 20:57 (eight years ago) link
Yup
― Green Dolphin Street Hassle (James Redd and the Blecchs), Sunday, 3 January 2016 21:15 (eight years ago) link
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.
― j., Sunday, 3 January 2016 21:22 (eight years ago) link
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.
― droit au butt (Euler), Sunday, 3 January 2016 21:31 (eight years ago) link
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.
― droit au butt (Euler), Sunday, 3 January 2016 21:35 (eight years ago) link
well they're weird
― j., Sunday, 3 January 2016 21:40 (eight years ago) link
they're all weird if they're not capturing the flow of one moment into a next
― droit au butt (Euler), Sunday, 3 January 2016 21:44 (eight years ago) link