Mathematics

Could someone tell me what is the 'obvious' homeomorphism between $Y_F$ and $\prod X_i$ in Henno's answer of this question math.stackexchange.com/questions/911028/… ?

Ted Shifrin

It's just the projection onto those factors, @bella.

bella

@TedShifrin $\pi:\prod X_i\to Y_F$?

Ted Shifrin

the other way

$Y_F$ sits inside the giant cartesian product

But except for the integers in the finite set $F$, you have $x_i=p_i$.

bella

12:27 AM

@TedShifrin so the projection will be $\pi: Y_F\to prod X_i$?

Ted Shifrin

Yes.

bella

@TedShifrin and in this case how is given $\pi$?

Ted Shifrin

The product of the projections $\pi_i$ for $i\in F$.

0celo7

" But it is still reasonably straightforward, and the
application of Moser’s lemma suggested in both references is not really necessary."

ayy

apparently I'm just bad at math

these people apparently saw the original article though

0celo7

12:58 AM

I think I have ze preuve

bella

@TedShifrin I don't understand how $\pi: Y_F\to prod X_i$ works. Doing an analogy with the typical projection $\pi_j:\prod X_i\to X_j$ ($\pi_j(x_i)=x_j$) I have, for this particular case $\prod_i(x_i)=\prod_i(p_i)=x_{which index goes here?}$

Faust

1:21 AM

should i delete a question if i have answered it?

2

Q: decomposing and the summing a sum.

Prove: $$\sum_{k\leq n} k \left\{ \frac {n}{k} \right\} = n^2\left(1 - \frac {\pi^2}{12}\right) + O (n \log n) \quad \text{ where } \{x\} = x-\lfloor x \rfloor $$ We have that $$\sum_{k\leq n} k ( \frac {n}{k} - \lfloor \frac {n}{k} \rfloor ) = \sum_{k\leq n} n -\sum_{k\leq n} k \lfloo...

number-theory

Faust

1:52 AM

A resounding maybe

Adeek

yo @0celo7 want to verify argument with me that I was solving ?

It is in complex analysis

Semiclassical

3:39 AM

@Faust you could post your solution so that people can appraise it

Faust

i posted it as an edit or should i post it as an answer @Semiclassical

Semiclassical

post as an answer

Faust

kk thx

0celo7

I managed to draw something in inkscape

it took absolutely forever

Semiclassical

I'm typing up a boundary value problem solution and ugh I had forgotten how tedious it is

$$V_{III}(r,\phi)=\sum_{l=0}^\infty \frac{C^l}{r^{l+1}}P_l(\cos\theta) \\ V_{II}(r,\phi)=\sum_{l=0}^\infty \left(A_l r^l+\frac{B^l}{r^{l+1}}\right)P_l(\cos\theta)$$

$V_{III}(b,\phi)=V_{II}(b,\phi)$ (okay, that's not bad)

0celo7

3:52 AM

@Semiclassical cover page i.gyazo.com/d535d8079b4d9bd3efe06f33afba55f6.png

Semiclassical

$V_{II}(a,\phi)=V_0$ (still not bad)

$\left(-\dfrac{\partial V_{III}}{\partial r}+\dfrac{\partial V_{II}}{\partial r}\right)_{r=b}=\dfrac{k}{\epsilon_0}\cos\theta$ (ah g******)

@0celo7 I approve

Tyger

4:16 AM

I need help with this question P(C | A, B) and A and B are independent events.P(C) = 0.7 P(A) = 0.2 P(B) = 0.5 P(A | C) = 2/7 P(B|C) = 3/7. Im in a discord group with my friends and we are getting different answers

any help is appreciated thank you

0celo7

4:32 AM

@BalarkaSen @MatheinBoulomenos "the space $C^\infty(E)$ is the inverse limit $\varprojlim L^2_s(E)$" lmao

Faust

4:56 AM

MOrning @TedShifrin

Morning, @Faust

Hey there!

how ya doing?

Hi Demonark

hi

5:08 AM

@TedShifrin hi

Ted Shifrin

@Tyger: I don't understand your notation. What does $P(C|A,B)$ mean? Do you mean $P(C|(A\cap B)$, i.e., the probability that $C$ occurs given that both $A$ and $B$ occur?

hi Karim

Adeek

@TedShifrin I aced my complex analysis midterm

Ted Shifrin

Cool. Congrats.

Adeek

I only have 1 more year of classes, then I don't have to take anymore exams finally

been doing exams now for what 6 years

Tyger

@TedShifrin Yes i meant the probability that $C$ occurs given that both $A$ and $B$ occur. Sorry about that

Adeek

5:11 AM

btw I am keeping a moustache btw. My wife tells me I look like her dad lol @TedShifrin

I shaved my beard, but I will keep my moustache to stroke it while thinking

Ted Shifrin

Ha.

Adeek

hahaha

0celo7

Is it me or does Besse have a shitload of typos

Ted Shifrin

@Tyger: That's what I figured.

Adeek

hey @0celo7 do you want a cool complex analysis problem ?

0celo7

5:13 AM

no

Adeek

okay no complex analysis problem for you then :P

Ted Shifrin

I guess you want to do something like $P(A\cap B\cap C) = P(A\cap C|B\cap C)P(B\cap C)$ or something like that, @Tyger.

Adeek

I am really amazed by how amazing complex analysis. When I did it as undergrad I just half assed studied it, but now I really see its beauty

0celo7

@MikeMiller I thought I had an interesting approach to KW that used the solution of the Yamabe problem. Unless I can fix it, it's a no go. And I don't actually want to follow the method in the original paper :/

Daminark

@0celo7 this exchange amuses me

0celo7

5:16 AM

@Daminark why

Daminark

I dunno, I just kinda snorted air out of my nose somewhat harder than usual upon reading that

Joe Shmo

@TedShifrin as promised, email sent

Tyger

@TedShifrin: Thanks for the help

Daminark

5:34 AM

@Adeek let's hear this complex analysis problem anyway

Adeek

5:53 AM

@Daminark sureeeee

Suppose that f is holomorphic function on the disk prove that $\Sigma f(z^n)$ converges uniformly on any compact subset of the disk.

there is really easy way to solve this and hard way to do it

Daminark

Does it map from the disk to itself?

Adeek

anyway I will let you think about it. I am gonna go to sleep

no

that would be easy.

by Schwarz lemma

Daminark

Lmao rip, that's what I was hoping

Adeek

There hard way to do this and easy way to do this hehe

I will let you ponder about it. I am gonna go to sleep let us discuss tomorrow :D

Daminark

Okay

Adeek

5:55 AM

good nights @Daminark :)

Daminark

Wait this is barely complex analysis

I think this is just because the function has bounded derivative or something like that

0celo7

You can estimate the growth of $f$ using its derivative, which is bounded on a compact subset

Daminark

Let's say $f(0) = 0$ to make life easier, then $|f(z^n)| \le K|z|^n$, so $\sum |f(z^n)| \le K\sum r^n$ where $|z| = r < 1$

And then Weierstrass M-test does it

Rick_Roll

6:13 AM

How does one find the lower and upper estimates of an integral given a table of values?

Zee

6:23 AM

Is there anybody out there ?

Don’t be shy , I know your out there

Rick_Roll

I mean, I'm here

Zee

The classic man

Haven’t been rick rolled in years

Rick_Roll

I only wanted to know how to find the lower and upper estimates of an integral given a table of values, but that proved too much to ask. It would seem my name has gotten in the way of my question.

Zee

Why do you need that ?

And what do you mean by finding lower and upper estimate given a table of values

Rick_Roll

I'm not actually a 52 year old famous singer, but a young Calculus AB student who's being introduced to integrals.

Zee

6:30 AM

Ok , ask and you shall receive, AB calculus man

/boy

Rick_Roll

The problem reads as follows: A table of values of an increasing function f is shown. Use the table to find lower and upper estimates for integral sign(0->25) f(x)dx

x = 0,5,10,15,20,25. f(x) = -42,-37,-25,-6,15,36

The answer is (-475,-85)

Zee

What’s sign (0-...)

Notation wise

Rick_Roll

I don't have the math notation plugin installed so I was trying to make the integral symbol (Liebowitz notation). the 0 is a and the 25 is b. (upper limit and lower limit)

Zee

So your integrating f from 0 to 25?

Rick_Roll

Yes? But only trying to find the lower and upper estimates, bc not given the function.

Zee

6:35 AM

Ah ok

Alright so , am kinda too drunk for this right now , but that’s how I would go about it :

For the upper estimate take the strongest possible case

F is -42 at at 0 but then immediately becomes -37 from after 0 to 5

Then -25 from 5 to 10 and so on

MatheinBoulomenos

@0celo7 great stuff

Zee

And just add the area of the rectangles

Rick_Roll

First of all, it's a Monday night, how are you drunk? Second of all, okay... that makes sense... I don't know what this strongest case thingy is but the rest of what you said lines up with what I've been taught thus far.

Thanks so much for your help! :D

Zee

No problem, I gotta to sleep and I gotta sleep to do math

You get the method for the weakest integral too?

Rick_Roll

No...? Maybe?

Zee

6:42 AM

And I gotta drinks o sleep :D

Rick_Roll

Isn't it like, subtracting the lower limit from the rectangles?

I'm new at this so I don't know the proper way to say it. its also 22:45 where I'm at so I'm a little sleep-drunk as well. EDIT: ok, got it

Zee

Weakest case aka lowest estimate would be f is -42 from 0 to 5 and then -37 from 5 to 10 and sonon

The idea is you don’t know what the F is but you know it’s values at certain points so you assume F only takes those values and then see the worst case scenario and best case scenario

That was a bad explanation...

Maneesh Narayanan

in CSIR-TIFR-ISI-NBHM, 33 mins ago, by Maneesh Narayanan

please help us.

Zee

F is increasing so between -42 and -37 , for the lower estimate you assume it’s just -42 all the way to but not including -37

For the upper estimate you assume it’s -42 at a single point but then it jump to -37 from 0 to 5

Ya I made some typos

Rick_Roll

I mean, how does one get from there to the answer? Im mainly confused with the capital F and lowercase f. Are they the same? EDIT: okay doky, give me a few minutes and maybe ill get it

Zee

6:50 AM

Ya am just writing without thinking

Let’s say f is really really lame ?

F is totally depressed and lame , it really does not wanna do anything

Like me XD

Then what would f do between 0 and 5?

Rick_Roll

I got it! So like (-42*5) + (-37*5) + etc. etc. for lower limit and then (-37*5) + (-25*5) etc. etc. for upper limit? (height*width) for all rectangles. EDITED

Zee

Yes! Except it’s the other way around

Yes yes

Good job

Also note integrals don’t care about values at a single point only intervals

Rick_Roll

Wow, the 21st century is amazing. What a time to be alive. I'm amazed you could understand what I was trying to stay with my limited knowledge of mathematical language. Ok, I will take note of that in the future. I'll be back here one day once I get the hang of them some more for future questions.

Zee

I wish you luck and prosperity in your mathematical adventures

Well , I suppose my job here is done ?

Oh how beautiful it is to be young

I shall fly away now

Transcript for

$Mathematics$ Mathematics