 1:19 AM
Can someone give me a proof that if a and b are integers. If a divides b^2, then a divides b? What happens if $a=b^2$?

2 hours later… 3:45 AM
@Mathguru No, because it's false. Eg, $4|36$ but $4\nmid6$ 4:08 AM
@Mathguru like 4 divides 36? 4:30 AM
@Mathguru consider the case of 36, which is a multiple of 4 5:08 AM
How to get the proper partial fractions for integration, logic behind them? 5:30 AM
@Thorgott: thanks for suggesting the reference. Ted’s book also has a chapter on compactness and defines sup ||Tx||, which I think I must know.
Given a continuous map $f: \mathbb R\to \mathbb R$, suppose that B is a borel $\sigma$ algebra, how do I show that $f^{-1}(B)$ is also a borel $\sigma$ algebra on R?
I showed that $f^{-1}(B) \subset B$.
But I don’t know how to show the reverse containment. :(
If I could show that $f^{-1}(B)$ contains all open subsets of R, then I’ll be done. "a" borel sigma algebra on R?
is 'borel sigma algebra' here = a sigma algebra generated by some topology?
this might just be the fact that the inverse image of a topology is a topology. the algebras can be different e.g. if you give both domain and codomain the usual topology on R and let B denote the usual borel sigma algebra, but let f be a constant map, f^{-1}(B) is not B but the trivial sigma-algebra
and in particular it does not include all open subsets of R in the sense of the usual topology on R 6:19 AM
A compact invariant set $\Gamma$ of diffeomorphism $f$ is called a basic set if it is topologically transitive and locally maximal ($\Gamma = \cap_{n \in \Bbb{Z}} f^{-n}(U)$) for some neighborhood $U$ of $\Gamma$
I am trying to understnad the locally maximal part
any intuitive argument

1 hour later… 7:31 AM
For anyone with a reasonably strong math background who is interested: I am trying to organize and advanced (intro graduate-level) set theory study group. We will probably use Kunen and follow the syllabus from here: people.math.wisc.edu/~miller/old/m771-11/index.html
2
If you are interested, please check out our chatroom! chat.stackexchange.com/rooms/138052/…
2
I would like to start as soon as we have 4 or 5 people who are seriously interested.
The study group will probably run for about 4 months, with weekly meetings to discuss the textbook and problems.
A main goal of this group will be to understand Godel's and Cohen's proofs of the independence of the continuum hypothesis from ZFC. Will someone please tell me how does the idea of partial fractions work
If I have (x+3)^3 in the denominator should I write it has A/x+3 + B/(x+3)^2 + C/(X+3)^3 ?
If a quadratic is not factorable should I take the numerator as Ax+B?
What is I had a cubic that is not factorable should I take the numerator has Ax^2+Bx+C? 8:30 AM
How is partial fractions written for dx/(1+x)((1+x^2)(1+x^3)? 8:52 AM
@leslietownes yes.
@leslietownes I think it means that the smallest sigma algebra that contains all open sets in R.

3 hours later… 11:48 AM
@Koro If $f$ is a constant map then $f^{-1}(B) = \{\Bbb R,\emptyset\}$.

2 hours later… 1:43 PM
How is dt/(t(2t-t^2)^1/2) integrated?

1 hour later… 2:46 PM
in this article of Quanta Magazine, it says in isometric embedding of $C^k$ of flow, if k>1/3, it's a energy-conserving smooth flow, but if k<1/3, it becomes energy-dissipating turbulence. It also says in the isometric embedding of $C^r$ of any manifold with a fixed boundary, if r>3/2, the manifold can be crumpled without buckling, but if r<3/2, the manifold can only be crumpled with buckling.
but I only know how to differentiate a function a positive integer number of times k to define $C^k$-diffeomorphism. In the above article, it says k or r can be a fraction number. I wonder how to define the kth derivative when k is a fraction number.

2 hours later… 4:20 PM
@onepotatotwopotato thanks :). I understand that I misunderstood the question. i said that too. where's my thanks?
koro, when does the term start up? is this prepping for class or are things already going? 4:41 PM
Hi everyone.
Is anyone here familiar with Stirling's Approximation (i.e., $n!\approx n\ln(n)-n$)? there are 20,000 things called 'stirling's approximation,' but that's definitely one of them (for ln(n!), not n!) Ah, I see, right
I'm trying to formulate the binomial coefficient a choose b by using Stirling, but hang on, let me give it another shot before I ask here. there are some SE posts on this. i won't paste and spoil if you're working on it, but they're out there 5:30 PM
leslie, thanks to you too. As it so happened, one of my classmates was also giving me that example (this was after my comments to you). Then, I came here and saw 1potato2potato saying the same thing. I understood. :)
@leslietownes the term started already :).
I'm the only one in class who did undergrad in engineering. Well, there's one more.
But he was in academia after UG/PG, I was not. I worked in industry related to the engineering. 😬 sounds like you're the cool kid in a class full of nerds 6:22 PM
I found an error in the book I'm reading. They wrote that if we equip weak/Mackey topology on E, with a closed subspace G, then G gas weak/Mackey topology
But in reality they meant that E/G has weak/Mackey topology
I have a question: does the subspace G always have Mackey topology?
This is true for weak topology
@leslietownes do you know perhaps? If not, I'll just post a question. I don't suppose anyone else would. i do not know off the top of my head. it wouldn't surprise me if the answer were no if G is somehow 'ugly' despite being closed
it's a good question 7:04 PM
I see. Yeah, I suppose it must be either false, really hard to prove, or an open problem for it to not be included in the book I'm reading.

2 hours later… 9:22 PM
What's the proof that relative countable compactness of A is equivalent to every sequence in A having a convergent subnet?
2  I find these things easier to think about in the language of filters. Using the usual correspondence between nets and filters, (2) is equivalent to saying that every filter on $X$ containing $A$ has an accumulation point in $X$. So, suppose $\overline{A}$ is not compact, and we will find a filte...

I suppose it's false in general so lets assume we are in a topological vector space
@AlessandroCodenotti hi. Do you happen to know/have a reference for this? 10:08 PM
0  Are the following definitions of relative countable compactness equivalent? $\overline{A}$ is countably compact Every sequence in $A$ has a convergent subnet Note that for regular spaces, the analoguous definitions are equivalent for relative compactness. Also, the equivalence always holds if...  11:13 PM
Method of stationary phase rules :) 11:24 PM
It is quite useful to NERDS @leslietownes not enough hugs as a child? 11:44 PM
Perhaps, too many...