Mathematics - 2018-01-05 (page 3 of 3)

Mr. Xcoder

17:18

What I do when I am bored :)

Secret

$$\prod_{y \in \Bbb{R}}\left(\int (\cdot) dx\right) \sin x$$

Now do this >:D

(Integrating sin x uncountably many times)

Mr. Xcoder

What does $(\cdot)$ mean?

Secret

it means put whatever that is on the right hand side into it. For example:

$$\prod_{n=1}^3 \left( \frac{d (\cdot)}{dx} \right) \sin x = \frac{d^3}{dx^3}\sin x = -\sin x$$

Mr. Xcoder

Oh like $\left ((\cdot) + 1\right)5=6$? :P

Secret

yup

Mr. Xcoder

17:23

Cute

Secret

I use that notation because nobody bother to apply an operator uncountably many times (for reasons), thus there is no literature symbol for such operations

Mr. Xcoder

I see

@Secret Is that really possible?

Secret

That I am not really sure, as I said, almost no one is interested in doing uncountable number of operations

(for unknown reasons)

But I suppose on the reals, one can define a net in order to compute uncountable limits

and then having the operators to be indexed by an uncountable set

Mr. Xcoder

I think my formula doesn't apply here (:P:P:P), otherwise I'd joke that (:P:P:P):$$
\prod_{y \in \Bbb{R}}\left(\int (\cdot) dx\right) \sin x = 0^{\infty\%2}\cdot \sin(x)\cdot(-1)^{(\infty / 2) \% 2}+0^{(\infty+1)\%2}\cdot \cos(x)\cdot(-1)^{((\infty + 1) / 2) \% 2} + \sum_{k=0}^{\infty-1}\frac{Cx^k}{k!}$$

@Mr.Xcoder Don't even think I am serious, just a joke :P

Secret

Well, on the other hand, there does exists a notion of integration that is basically like a countable version of what I wrote: Those are the path integrals that appear in QFT and functional analysis

Mr. Xcoder

17:29

QFT? Quantum Field Theory?

Secret

yup

Mr. Xcoder

Uhm huh, ok

(and in the meantime, we haven't even studied integrals in school...)

Secret

The precise maths is not very well defined and the mathematical physics community are still scratching heads trying to ground that from category theoric appeoaches to defining some weird objects

Mr. Xcoder

You know what, nevermind

Secret

41

Q: Mathematics of path integral: state of the art

I was told that one of the most efficient tools (e.g. in terms of computations relevant to physics, but also in terms of guessing heuristically mathematical facts) that physicists use is the so called "Feynmann path integral", which, as far as I understand, means "integrating" a functional (actio...

So, measure theory, category theory, and it seems the notion of nets were indeed used

The major issue is to construct examples

O, and semiclassical might be able to tell slightly more

user193319

18:21

Quick question: If $\{f_n\}$ is a sequence of complex-valued functions that are bounded and that uniformly converge to $f$, is $f$ necessarily bounded too?

MatheinBoulomenos

18:40

@user193319 yes. uniform convergence is the same as convergence in the norm $\|f\|=\sup |f(x)|$. By the reverse triangle inequality, every norm is continuous, so convergence in the norm implies convergence of the norms $\|f_n\| \to \|f\|$ (using the supremum norm)

Daminark

Yo

MatheinBoulomenos

you can also do this with explicit epsilonics, but I'd prefer not to do that

Hi @Daminark

Leaky Nun

> explicit epsilonics

-- der Sylowmeister, 2018

Semiclassical

Epsilontology

Daminark

18:58

Petition to retitle analysis to "Greek literature"

Faust

Morning

Ted Shifrin

Howdy, @Faust

Faust

So happy i got a hole bunch of bursarie money today =)

Ted Shifrin

money down a hole?

Daminark

Hey Ted and Faust!

Faust

19:01

lol

a whole buncha

Ted Shifrin

hi Demonark

Faust

also got a meeting monday to do discuss doing a research project this summer =)

should i know what i want to do when i go into the meeting?

Ted Shifrin

well, gotta learn stuff to research, then

I have no idea what sort of meeting this is, with whom ... but, generally, yeah, probably.

Faust

well i asked a prof whos more ot what i kinda of like

Ted Shifrin

So, yeah, it shouldn't just be "all the cool kids are doing research, so I suppose I have to also."

Faust

19:03

i wanna do something realting to dif geometry or topology or algerbra

or some conmbination

Ted Shifrin

That's super broad and vague.

You aren't very far in any of those :(

Faust

nope

Daminark

Do algebra

Ted Shifrin

So basically research will just mean working some really hard problems or learning some stuff from a book.

Faust

but its not until the summer

and ill be better at all of those by then

well not dif geo

Daminark

19:05

If you're trying to pick up background in the meantime so that your summer stuff is more substantial I'd probably recommend picking something (not necessarily algebra, I'll understand :'( ) and focusing on it

Faust

can i like google something to see what options i have?

the prof i asked speciliazes in Operator algebras and
Noncommutative geometry whatever those are

but hes one of only two profs that does geometry at my uni

Given a vector space V if U is a subspace how do we know that U is non empty?

Daminark

It contains 0

Faust

thats what im tring to prove

Daminark

That's an axiom of a vector space

Faust

facepalm

Liad

19:24

@TedShifrin hi

Leaky Nun

@Faust ...

Ted Shifrin

19:40

hi @Liad

Liad

@TedShifrin thanks for the help the other day with Laurent series , it was really helpful. and btw , you help like 4-5 other students too with it :P

Ted Shifrin

Oh, how did I do that?!

You mean, you went and taught them? That's the best way for you to learn.

Liad

yea they had no clue either how to get the series because we did not see any examples, so i gave them yours :P

Ted Shifrin

I hate teachers who do no examples.

4

Mats Granvik

$$\log (x)=\lim_{n\to \infty } \, \frac{n \sum _{k=1}^{n-1} \frac{x^{k-1} \left(\sum _{j=0}^k (-1)^j \binom{n+1-1}{j} (k-j)^{n-1}\right)}{(x-1)^{n-1}}}{\sum _{k=1}^n \frac{x^{k-1} \left(\sum _{j=0}^k (-1)^j \binom{n+1}{j} (k-j)^n\right)}{(x-1)^n}} \; \text{ if } x>0$$

Alessandro Codenotti

19:42

Hi chat

Hi @Ted

Ted Shifrin

hi demonic @Alessandro :)

HNY

ignores Mats

Alessandro Codenotti

HNY?

Abcd

I am unable to understand this question asked on Math.SE

@AlessandroCodenotti Happy new year

Ted Shifrin

Happy New Year, Alessandro :)

Abcd

Could someone explain what the first line of the question means?

Alessandro Codenotti

19:44

Oh, thanks, to you too!

Ted Shifrin

@Abcd: It means you're allowed to arrange the numbers in any order when you assign them to A,B,...

Abcd

But what is A ?

Is it the sum?

Ted Shifrin

No, it's one of the numbers.

Abcd

Oh, I was misinterpreting the question all this while.

I thought it's a set, a permutation of digits from 11 digits.

Ted Shifrin

Nope.

Abcd

19:46

Fine.

user9143057

Are there any good function analysis notes online?

*functional analysis

Ted Shifrin

(You can edit for a minute or so and fix typos.)

Alessandro Codenotti

@user9143057 I like those people.math.ethz.ch/~salamon/PREPRINTS/funcana.pdf

user9143057

@AlessandroCodenotti Thanks.

Vrouvrou

20:02

hello @AkivaWeinberger

are you here

Semiclassical

hi chat

Alessandro Codenotti

Hi @Semi

Akiva Weinberger

@Vrouvrou Momentarily

Daminark

21:07

@AlessandroCodenotti that's the book my class is using actually!

Alessandro Codenotti

It's even possible you suggested it to me, I don't remember how I stumbled on those notes

Wait weren't you using Kolmogorov-Fomin?

Daminark

21:23

Last year, this year it's the ones linked up there. And lol that'd make sense

Ted Shifrin

@Alessandro: How's the running over of innocent bystanders proceeding?

Alessandro Codenotti

Still no casualties

I consider that quite the achievement

Daminark

Italy's got some ninjas

Ted Shifrin

I don't think I'll be returning to Italy any time soon.

Gabriel Romon

@Ted Salut et meilleurs voeux !

Daminark

21:26

@Ted it'd be good practice for your reaction times

Ted Shifrin

Bonne année, @Gabriel.

Demonark: Let's start with yours!

Salut, @Astyx! Bonne année.

Astyx

Bonne année !

Comment ça va ?

Ted Shifrin

Ça va bien, merci, et toi?

Astyx

Ça va ça va

Ted Shifrin

et Palaiseau?

Astyx

21:30

J'y suis pas encore

Mais je pense qu'ils vont bien

Dans la mesure du possible

Ted Shifrin

Oops, pas encore. Ma faute.

Astyx

Quoi de neuf ?

Ted Shifrin

Pas grand'chose chez moi.

Adeek

21:46

hi @TedShifrin

hi @Daminark

@Daminark have you learned about localization yet ? I would like to verify something with you.

@TedShifrin teach me french

:D

Daminark

@Adeek not exactly

Alessandro Codenotti

@Adeek If it's not hard stuff I might be able to help

Daminark

I do want to help you as much as I can but I think you overestimate me somewhat

Alessandro Codenotti

Balarka surely knows more but he's not here atm

Adeek

@AlessandroCodenotti It is okay I figured it out. But here is the deal. I am just working through Ravi Vakil note's

here is the issue

Show that $\phi : M \rightarrow S^{-1}M$ exist, by constructing something that satisfies the universal property.

@Daminark No I don't agree your great :)

@AlessandroCodenotti Define $S^{-1}M = (M \times S) / \sim$ such that $(m_1,s_1) \sim (m_2,s_2)$ iff there exists $s \in S$ : s(s_2m_1 - s_1m_2) = 0$.

Alessandro Codenotti

21:55

Ok, there should be an obvious candidate for a map $M\to S^{-1}M$

Adeek

This wasn't the issue however 1 moment.

Alessandro Codenotti

sure

Adeek

The universal property is that if $\phi : M \rightarrow N$ is $A$ module morphism such that $\forall s \in S$ : $\alpha_s : N \rightarrow N$ given by $n \mapsto sn$ is isomorphism. Then we need to constrcut $\psi : S^{-1}M \rightarrow N$ that fills the diagram.

such construction is given by $\frac{m}{s} \mapsto y$ where y is the unique element of N such that $sy = \phi(m)$.

The issue I was having is $S^{-1}A$ module morphism property, which actually hugely depend that the ring is commutative in order to get it. @AlessandroCodenotti

I forgot I was working with commutative ring, so that is why I was running into issues fixing that last step.

Alessandro Codenotti

Ok, so that's solved now?

Adeek

yeah

thanks for listening though

Alessandro Codenotti

22:00

you're welcome :P

Adeek

btw if Ravi Vakil notes is amazing.

if you want to ever learn algebraic geometry

Jacksoja

@Adeek give em to me adeek :)

Adeek

give what @Jacksoja ?:D

Jacksoja

Ravi Vakil notes Karim

Alessandro Codenotti

@Adeek I'll keep that in mind if I ever do

Adeek

22:08

they are online for free

Jacksoja

okay thanks Karim :)

@Adeek 764 pages?

Adeek

yh

Jacksoja

math.stanford.edu/~vakil/216blog/FOAGsept0611public.pdf

that is the one I found

CryinShame

22:43

@Daminark Btw, thanks for the help last night. I can see the path now from Constant rank Theorem. I'm ambivalent about taking the time to prove it myself though.

Leyla Alkan

23:19

Hi, Laczkovich proved that $\pi $ is irrational by first showing that ${\pi^2 \over 16}$ is irrational and then concluded that $\pi $ is irrational. I wanna ask you,does the fact that ${\pi^2 \over 16}$ is irrational imply that $\pi^2$ is irrational?

It might be odd to ask such a question but I need to show whether $\pi ^2$ is irrational or not

Balarka Sen

Suppose not, for a contradiction. Then $\pi^2/16$ is irrational, yet $\pi^2$ is rational.

What does it mean to say "$\pi^2$ is rational"?

Leyla Alkan

then ıt should be written as a/b where $a$ and $b\neq 0$are in $\mathbb Q$

Balarka Sen

$a$ and $b$ are in $\Bbb Z$ you mean. But yes.

Leyla Alkan

oh yes

Balarka Sen

If $\pi^2 = a/b$, $\pi^2/16 = ?$

Leyla Alkan

23:26

okay

ı got that

thanks

Adeek

23:45

@Jacksoja yeah that is correct

I am using the 2017 version

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$Mathematics$ Mathematics