Mathematics - 2019-12-28 (page 3 of 3)

RPO

17:05

Hi

is it possible to write a tuple with conditions ?

e.g. $G = (f(x) | x = D, |G| = 4)$

Thorgott

17:22

What are you trying to do

Shootforthemoon

Hi, here ([1]: https://i.sstatic.net/g40dH.png
[2]: https://i.sstatic.net/SJRsl.png) in a generalization of the implicit function theorem I don't understand the fact that the domain is an open subset of $R^{n+m}$

Can anyone help?

It's in Italian, but if necessary I'll translate it

Alessandro Codenotti

Che cosa ti confonde esattamente?

Shootforthemoon

Non mi è ben chiaro perchè il dominio è in $R$ elevato a m+n

la potenza

Alessandro Codenotti

Se questa è una generalizzazione la versione "base" è con funzioni $\Bbb R^2\to\Bbb R$?

Shootforthemoon

Sì

o a partire da funzioni di tre variabili

Alessandro Codenotti

17:37

Ok allora nella versione base hai funzioni $\Bbb R^{1+1}\to\Bbb R$ e ti serve un minore $1\times 1$ dello Jacobiano non nullo (quindi di fatto basta chiedere $\partial _yF(x_0,y_0)\neq 0$)

Shootforthemoon

Mh, ma non abbiamo due come dimensioni n?

Alessandro Codenotti

no in questo caso hai $n=m=1$

Shootforthemoon

Ah, cioè ogni vettore ha una sola componente?

Alessandro Codenotti

No nel dominio ne hanno due, nel codominio una

Shootforthemoon

perciò per n bisogna guardare al codominio?

Alessandro Codenotti

17:44

$m$ è la dimensione del codominio, $n$ la differenza fra quella del dominio e quella del codominio (la prima deve essere più grande)

Shootforthemoon

Ah, capito, grazie mille

Alessandro Codenotti

Dove studi per curiosità?

Shootforthemoon

Grazie anche per l'Italiano :P

Politecnico di Milano

é un corso misto tra ingegneria biomedica e Medicina e chirurgia

Alessandro Codenotti

Ah capito

Shootforthemoon

purtroppo in matematica saltiamo molti argomenti

Maximilian Janisch

17:54

Perché tutti parlano italiano? 🙃

Alessandro Codenotti

Warum nicht?

Maximilian Janisch

Pourquoi non?

Alessandro Codenotti

Why not?

Shootforthemoon

Perché no?

Maximilian Janisch

Mi piace l'italiano ma il mio è male

@AlessandroCodenotti Warte, du warst ja in Bonn 👍 Ok dann schlage ich vor, dass wir auf Deutsch umstellen 😉

Alessandro Codenotti

17:58

Da die Vorlesungen in Bonn auf Englisch sind, ist mein Deutsch ziemlich schlecht :S

Maximilian Janisch

@AlessandroCodenotti umso besser, wenn wir uns nur noch auf Deutsch unterhalten hehe

Alessandro Codenotti

Vielleicht...

There's many people here in the chat that speak German

Astyx

18:15

Oui

Thorgott

18:39

Ita est.

Ted Shifrin

19:07

Niemand darf auf deutsch reden.

@Shoot: First of all, that text committed the sin of writing $I\in\Bbb R^n$ instead of $\subset$. That upsets me for an actual text, and not a student. What is the problem with having the domain be an open subset of $\Bbb R^{m+n}$? This generalizes $2+1$ that you had before.

Thorgott

This reminds me of my analysis prof who used to write $(a_n)_n\subset\mathbb{R}$ for sequences. Not as upsetting as $I\in\mathbb{R}^n$, but it still makes me shudder.

Ted Shifrin

Well, if you think of a sequence as an indexed subset, it's fine.

Thorgott

19:24

The problem is that the indexing doesn't happen in $\mathbb{R}$

Shootforthemoon

@TedShifrin True, that generalizes. Forgive me but I'm quite struggling with these concepts. For example I don't get how these n dimensions relate to the vector X we choose, while Y has m components

Ted Shifrin

You're just writing the variables in $\Bbb R^n$ as one vector variable and the variables in $\Bbb R^m$ as the other vector variable. It's the best notation to use. You can see this stuff explicitly in my YouTube lectures.

But I am very upset by the error in the text.

Shootforthemoon

ahahah yes that was really awful, even to me

@TedShifrin ok, this evening @TedS' lectures marathon for me :p

Ted Shifrin

It's generalizing write $z=\phi(x,y)$ as $\vec Y = \phi(\vec X)$, where in this case $\vec Y = z$ and $\vec X = (x,y)$.

No, for the implicit and inverse function theorem you only need the one lecture with the statement(s) and example(s). You don't need the proofs.

Shootforthemoon

@TedShifrin ahh, I see, thanks

Ted Shifrin

19:29

The introductory, less fancy, version is 3500 days 39-40; the general version is 3510 day 20.

Shootforthemoon

Great! Thanks very much!

Thorgott

A way to think about it is as sort of similar to a system of linear equations (after all, differentiable just means locally well-approximable by linear functions). You have n+m variables and m equations, so once after fixing n of the variables, you have m variables and m equations left, which - with a regularity assumption, the invertibility of the Jacobian - yields you a unique solution. The remaining n variables can be thought of as giving you n degrees of freedom.

Ted Shifrin

Indeed, in the first lectures I listed, I talk about how one should understand this as the generalization of solving linear equations (pivot variables, free variables).

Thorgott

In concept, a lot of multivariable analysis is trying to do linear algebra with things that aren't really linear :p

Ted Shifrin

Absolutely. That's why I created my course/text putting the two subjects together.

anakhro

19:59

@Thorgott that's "calculus" in a nutshell.

Shootforthemoon

@Thorgott Thanks!

Thorgott

20:26

np

btw @Balarka, my aforementioned friend just returned from his vacation, but he told me he hasn't seen the Janson-Suen inequality before. However, in the meanwhile, I have checked Random Graphs from Luczak, Rucinski and Janson himself; didn't have the time to ponder all the details, but it has a proof of the inequality, which is self-contained and accessible, so you might consider checking that out.

user76284

20:37

Let $F_X$ be the CDF of a random variable $X$. Then $F_{\max \{A, B\}} = F_A F_B$. Is there a similar expression for the quantile function (inverse CDF) $F^{-1}_{\max \{A, B\}}$ in terms of $F_A$, $F_B$, $F^{-1}_A$, $F^{-1}_B$, and similar terms?

Ted E

20:53

Most areas of mathematics are developed to solve some already known problems - this is essentially necessary for enough mathematicians to gain interest to develop them. Can someone tell me what types of problems intersection theory in algebraic geometry is developed for?

Oh @loch is here, you were the person I was going to @ lol

Or does it go the other way? Intersection theory was already interesting in classical AG, where it seemed actually intuitive, and now we're just putting it in a more general framework to deal with more general schemes?

Hi @MatheinBoulomenos

MatheinBoulomenos

Hi @TedE

Ted E

21:09

You have some good answers on main

MatheinBoulomenos

thanks

Ted E

What do you study?

MatheinBoulomenos

number theory, mostly

Ted E

algebraic or analytic? (or both) and from an AG perspective or no?

MatheinBoulomenos

more algebraic than analytic and with lots of AG

Ted E

21:10

Oh nice

Are you interested in intersection theory in AG by any chance?

MatheinBoulomenos

regarding your question on intersection theory: the theory of general schemes allows to capture more even for intersections of classical varieties

Ted E

How so?

I get that intersecting lines with curves for example just 'gives' the correct intersection numbers, which is quite beautiful (when dealing with them as schemes)

MatheinBoulomenos

there's a simple example: consider in the plane the line y=0 and the parabola y=x^2. These intersect non-transversally which is captured by the fact that the scheme-theoretic intersection is non-reduced

Ted E

Ah yeah, that's one of the ones I had in mind

I also liked intersecting $x=0$ with $y^3=x^2+ax$ (I think it was, as $a$ varies)

MatheinBoulomenos

but the modern approach seems to intersection theory seems to use derived algebraic geometry to capture even more information

there's a nice talk by Lurie on that on youtube

Ted E

21:13

Actually, that's why I'm reading intersection theory

Since it seems to be the entry point to motivating DAG

MatheinBoulomenos

maybe you already know this talk, then: youtube.com/watch?v=htTL0VvfsvM

Ted E

I didn't, I'll watch it

MatheinBoulomenos

as far as motivations for DAG go, this is the best I've seen so far

this derived approach to intersection theory is really hot in NT right now as well

Ted E

Does transverse intersection only make sense for smooth subvarieties of a smooth variety?

MatheinBoulomenos

from the abstract of a recent paper by Galatius/Venkatesh that seems to be very much discussed among some number theorists right now "speaking informally, the locus of crystalline Galois representations of $\mathrm{Gal}(\overline{\Bbb Q}/\Bbb Q)$ is obtained by intersecting the space of Galois representations of $\mathrm{Gal}(\overline{\Bbb Q}/\Bbb Q)$ with the space of local geometric representations of $\mathrm{Gal}(\overline{\Bbb Q}/\Bbb Q)$ and the intersection is
not, in general, transverse."

Ted E

21:19

I see

MatheinBoulomenos

@TedE hmm, good question. Maybe, one might say that a non-smooth variety doesn't intersect itself transversally

Ted E

What is meant by locus in that context?

Does it mean something like taking the direct sum of all irreps, and taking the kernel?

Nah that wouldn't make sense I guess, since it wouldn't give a space

MatheinBoulomenos

no, you have some "space" (probably a stack in general, in any case a geometric object) such that every point corresponds to a galois representation

Ted E

Is $\text{Gal}(\overline{\Bbb Q}/\Bbb Q)$ an algebraic group here?

Oh, I see

We can deal with intersections of stacks (that aren't represented by schemes) then?

MatheinBoulomenos

I'm no expert, but I think so, yes

Ted E

21:23

Hmm, I wonder how

MatheinBoulomenos

What do you study? Are you a student?

Ted E

I'd be interested in seeing how intersections of stacks classifying G-torsors and H-torsors behaves

I'm not a student atm, taking a gap year, but I did do my undergrad in math

MatheinBoulomenos

ah I see

I'm in the process of finishing my undergrad myself

Ted E

Oh nice, and then phd?

MatheinBoulomenos

masters first, I'm European

MatheinBoulomenos

21:37

@TedShifrin are you around? long time no see

Ted E

21:53

@MatheinBoulomenos I find the part at 12:30 slightly confusing

Maximilian Janisch

Hi there

Ted E

It seems he's moved onto $C,C'$ subvarieties of $\Bbb CP^n$, and he's taking $C$ to be given by some homogeneous polynomial, and then he says $C'$ has ring of functions $R$. Is $C'$ by assumption affine for some reason?

Hi there @MaximilianJanisch

Maximilian Janisch

@TedShifrin Damn not bad

MatheinBoulomenos

Hi there @MaximilianJanisch

Ted E

Oh wait, I just realised he said "Lets consider a special case, which is not really applicable to our situation"

Maximilian Janisch

21:56

Ok so I found out that at least two people in this chat play chess

Ted E

Well, I only just started playing really a few days ago haha

Do you play on lichess?

Maximilian Janisch

Yes

You too?

Ted E

Yes, Balarka invited me there

Or was it Leaky

One of those two lol

Maximilian Janisch

I'm actually pretty active there

What's your username

Peter

Has anyone a software which can factor with the quadratic sieve the 89 digit composite number :

(5^164+3^164+1)/268850372931522596788077287 ?

Ted E

21:58

TedE112412413 (I tried TedE and a bunch of combinations, and it kept failing, so I mashed some numbers)

Maximilian Janisch

Oh hi @Peter. Somehow I could've guessed who asked this question without even seeing the username ;)

Ok I'll follow you on lichess @TedE

Ted E

Would you like to play a 3+2 game?

Maximilian Janisch

sure lets go

Ted Shifrin

22:25

heya @Mathein!

Ted E

@MaximilianJanisch I wonder if he just had a script

Maximilian Janisch

@TedE There were some videos of him playing 1/4+0 against Stockfish level 1

which is super crazy because that thing has almost 0 delay between moves

Also there is one guy who beat Stockfish level 1

Ted E

You mean he beat stockfish level 1 in actual chess (in 1/4+0), or he won in the 'fastest player challenge' by flagging it?

Maximilian Janisch

He (not the fast guy, somebody else) won against it in 1/4+0 chess

Ted E

Oh gotcha

Maximilian Janisch

22:30

lichess.org/jFTjYAbQ

there is a live playback feature

Ted E

Doesn't stockfish play the same way in response to your moves, so you could choose a move sequence?

Maximilian Janisch

@TedE I think there is some randomness

on the lichess version

especially level 1

okay actually the game I linked you is really lame

because Stockfish just resigns on move 151

Here is a real win: lichess.org/FH6GH3Xq/black

Ted E

I think I'll give this a shot

Maximilian Janisch

hehe good luck

Ted E

I thought stockfish 1 was meant to be terrible lol

It just rekt me

Maximilian Janisch

22:36

Yeah it plays really bad but super fast

So I never managed to beat it for example in 1/4+0

Already in 1+0 it is easy to beat I think

My latest failed attempt: lichess.org/8AO2RVKh

also 1/2+0 is still hard: lichess.org/FS1q96TM

hi @Zacky

Zacky

Hello there!

Maximilian Janisch

@Shootforthemoon Do you have new insights on the implicit function Theorem? 😄

Shootforthemoon

Hey

yesm somehow XD

I've almost done with my research on it

Do u play chess @Max?

Maximilian Janisch

@Shootforthemoon Yes I do

:D

@TedE 1+0 is already (somewhat) easy: lichess.org/MYCSkWn5

Shootforthemoon

Nice, I play sometimes too

Maximilian Janisch

22:46

Also sometimes on lichess ?

Shootforthemoon

sure

Ted E

@MaximilianJanisch I'll have to try it out later on :). For now I'm just watching that video lecture posted above

Maximilian Janisch

have fun @TedE

@Shootforthemoon What's your username?

Shootforthemoon

luca51 but don't look at my score lol

Maximilian Janisch

Lets do an atomic game

Shootforthemoon

22:48

at least I've beaten @Leaky

Maximilian Janisch

@Shootforthemoon So did I 😄

Shootforthemoon

@MaximilianJanisch ok

Maximilian Janisch

what time control?

Shootforthemoon

7+2?

around that

or 5+5

Maximilian Janisch

okay lets go 7+2

Shootforthemoon

22:50

fine!

Trey

23:05

can someone explain me, how did Adriano replace 10 by k in this proof? math.stackexchange.com/questions/447923/…

Thorgott

Cause $10\le k$

Trey

so if have N and T such that N < T, I can replace N by T?

Victor

Can someone answer my question here? stats.stackexchange.com/questions/442555/…

Thorgott

i don't know what you mean by replacing

it's simply estimating

Maximilian Janisch

Is there anyone else here who wants to play chess ?

Trey

23:22

@Thorgott here's what I don't get it. He's trying to proove that $(k+1)^3 < 2^{k+1}$ he expands the polynomial to $k^3 + 3k^2 + 3k + 1$ and on the next line he changed it to $k^3 + 3k^2 + 9k + 10$. So to me, he doesn't seem to be proving the original problem. He's now prooving that $k^3 + 3k^2 + 9k + 10 < 2^{k+1}$ rather than $(k+1)^3 < 2^{k+1}$

Thorgott

One implies the other, because $k^3+3k^2+3k+1<k^3+9k^2+9k+10$

Trey

don't you have to prove that before?

Thorgott

well, Adriano probably considered this to be obvious

Maximilian Janisch

23:39

@Trey By the way a different "continuous" approach to this problem: We have for $k\geq e$ by taking the logarithm to basis $e$: $$k^3\le 2^k\iff 3 \ln(k)\le k\cdot\ln(2)\iff\frac{3}{\ln(2)}\le\frac{k}{\ln(k)}$$

But $\frac{k}{\ln(k)}$ is increasing for $k\geq e$ as can be proven by taking the derivative

So we need to solve $\frac{3}{\ln(2)}=\frac{k}{\ln(k)}$

It turns out that no symbolic solutions (without the Lambert W function) exist

but numerically we have $k\approx 9.94$

So in fact the above inequality holds for all $k\geq 9.95$

Admittedly this is overkill, one can also just check that $\frac{10}{\ln(10)}\geq\frac{3}{\ln(2)}$ to get the same result for all $k\geq 10$ (but in this result, non-natural numbers $k$ are also allowed)

Ok does anybody of you know something about Bernoulli walks

?

Because I am wondering

If I have a 1D-walk where each added Bernoulli r.v. has parameter $p\in[0,1]$, then what is the probability of ever crossing the zero line?

I think it should be $1-|p-(1-p)|=1-|1-2p|$

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$Mathematics$ Mathematics