Mathematics

5:51 AM

$\lim_n\chi _{(n,n+1)}=0, \lim_n n\chi_{(0,\frac 1n)}$=0. As one sees by sketching the graphs, the trouble in these examples is that the area under the graph "escapes to infinity" as $n \to \infty$, so the area in the limit is less than one would expect.

I don't understand how/why the graphs tend to infinity as $n\to \infty$.

I think the area under the graph always remains 1 no matter what n you take.

So it is categorically wrong to say that the graph escapes to infinity.

$\chi$ is indicator function.

and we are considering Legesgue measure.

copper.hat

6:05 AM

What is $\chi$?

Koro

Indicator function.

$\chi_E(x) :=1$ if x is in E and 0 otherwise.

Overtherainbow

6:22 AM

If X is a real normed space and Y a closed subspace and we take $x\in X\setminus Y$ then we define $d(x,Y)=\inf\{||x-y||: y\in Y\}$. In the exercises they wrote that we need to show that for $t\in \Bbb{R}$ we have $d(tx,Y)=|t|d(x,Y)$. But now if I consider X to be the real line and $Y=[0,1]$ and x=2, t=3 then d(6,Y)=5 but and 3d(x,Y)=3? What am I doing wrong in my example

Ah I see Y needs to be a sub vectorspace, which it is not in my case?

copper.hat

6:50 AM

I presume by closed subspace they mean a closed linear subspace.

I don't know what you are talking about regarding the escaping bit.

good night! its after my bedtime :-)

Overtherainbow

Thanks I think I got it

Good night:)

Koro

7:45 AM

@copper.hat Good night :)

That escaping bit is from a paragraph following corollary 2.17 in Folland's book.

one potato two potato

Is it possible to calculate ${1\over\pi}\int\int_{\Bbb D}|\psi'_\alpha|^2dxdy$ without a calculation? $\psi_\alpha(z) = (\alpha-z)/(1-\bar{\alpha}z)$ for $|\alpha|<1$.

Jakobian

9:02 AM

@leslietownes I read it and it just confused me

@XanderHenderson Do you like complex made simple?

jay

9:17 AM

If I have two functions on $f,g:\mathbb{R}^d\to \mathbb{R}^d$ with $f(x)=g(x)$ and $g$ is invertible then $f$ is also inveritble $f^{-1}=g^{-1}$. What about if the two functions are such that $f(x)=g(x+T(x))$ and $g$ is invertible, then $g^{-1}\circ f (x)=x+T(x)$ but I cant say anything about $f^{-1}$ ?

Jakobian

9:28 AM

@jay did this come up in something linear?

jay

9:40 AM

no?

just got confused

when looking at notes

one potato two potato

10:09 AM

It seems the statement of Runge's theorem in RCA is stronger than in Stein

Koro

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ILikeMathematics

11:14 AM

I feel like long questions are usually less likely to get answered because people don't want to read it all, so if you include all your failed attempts and progress which is encouraged, you could actually increase the time it will take for you to get an answer, so it might be best to just include all of the context and quickly include what you've tried so it stays short
Any thoughts on this?

leslie townes

i think it's a good idea to include some attempt, and a bad idea to include all attempts.

ILikeMathematics

Yeah, there are lots of questions which include no attempt and just a statement and still are very well received (despite https://math.meta.stackexchange.com/questions/9201/proposal-discourage-questions-that-are-nothing-besides-a-problem-statement)
So trying to give all the context and one failed attempt is probably best

jay

If I have a differentiable function $f$ and some other function $g$ and I know that $\frac{d}{dx}\big((f+g)(x)\big)$ exists (for all $x$), then can I claim that $g$ is differentiable.

leslie townes

yes

jay

just translate it to a question about limits?

leslie townes

11:21 AM

how you phrase it is up to you, the key fact is that a difference of differentiable functions is differentiable

which does indeed boil down to a fact about limits

jay

coolio :) cheers

User1865345

11:46 AM

@copper.hat you might find it interesting: Chess Grandmaster ‘Likely Cheated’ in More Than 100 Matches, Report Finds.

si-LV-er_and_b-LA-ck

11:59 AM

$83 million for a chess playing app? 😳

leslie townes

i do wonder about that. someone is laundering money.

si-LV-er_and_b-LA-ck

yeah, wall street money

I guess they have to get rid of all that money buffet made breaking the NYSE reader board during the pandemic

ThunderGlove

(2cos²2x - 1) = 0 gives cos²2x = 1/2 which gives, x = npi/2 ± pi/6 but on simplifying it we get cos4x = 0 which x = (2n+1)pi/4, how are they equal?? (n belongs to the set of integers)

*in the second case, x = (2n+1)pi/8

jay

12:33 PM

very loosely speaking, what does it mean for a subset of $\mathbb{R}^d$, in particular surface of dimension $(d-1)$ to be $C^2$ ?

robjohn

12:59 PM

@ThunderGlove $\cos^2(2x)=\frac12$ means $\cos(2x)=\pm\frac1{\sqrt2}$, which means $2x=n\frac\pi2+\frac\pi4$ and $x=n\frac\pi4+\frac\pi8$, which is the solution to $\cos(4x)=0$

one potato two potato

1:46 PM

After a few hours of reading RCA, I spotted that Rudin denotes $U$ as an open disk.

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$Mathematics$ Mathematics