Mathematics - 2020-09-17 (page 1 of 2)

4:27 AM

no

Sophie

4:59 AM

is anyone familiar with this notation? Looks like multiplication...

It's from Serre, a course in arithmetic

which part?

SL$_2(\Bbb R)$ is the special linear group

Sophie

@geocalc33 the $gz$ thing. Can you just multiply a matrix by a complex number?

5:17 AM

This is how $GL[2,\Bbb C)$ acts on $\Bbb CP^1 = \tilde {\Bbb C}$. Act on $(z,1)$.

Oh I'm dealing with that right now

Btw Ted hey! Turns out I've been roped into differential geometry :P

Specifically trying to read this: www-math.mit.edu/~helgason/lie_groups_24-1.pdf

Roped? Ha!

Haha, yeah, I'm doing a reading group set up by my tentative advisor

This paper looks like it'll be nifty but he's pulling out some stuff real fast and my natural verify everything line by line instinct is causing some fun times lol

5:34 AM

Symmetric spaces are very cool.

Yeah for sure! What've you been up to?

Nothing ... Finding out how f***ed up my back is.

Ah darn, I hope you're doing alright

Just deteriorating :)

You still in TX ?

Yeah, I'll probably be home for the year. Save money on rent which is nice

5:38 AM

Make sure you vote!

Yeah for sure! I was originally skeptical but there's apparently a chance that Texas is a swing state so voting is actually kinda relevant here

Count on Putin :D

Sophie

@TedShifrin I find it a little weird because you're talking about a group of functions so I was expecting the notation $g(z)$ instead

Haha, likely, but you gotta go for it and see what happens

With linear maps, you usually write $Ax$, right?

But I don't disagree with you.

5:54 AM

Ted one thing I'd like to run by you

On my iPad now, so is it short?

So we have the hyperbolic disk with metric $ds^2 = \frac{dx^2 + dy^2}{(1-|z|^2)^2}$. We have that its isometry group is $(SU(1,1)/\pm I) \cup c(SU(1,1)/\pm I)$, and it acts transitively. Now we know that straight lines through $0$ are geodesics

And these guys are claiming that because the isometry group (I'll call it $I(D)$) acts transitively, that we in fact know all the geodesics: circular arcs perpendicular to the boundary

Conformally equivalent to the upper half-plane, and LFTs map lines to circles, etc.

Somehow this feels a bit weird to me, I guess the point here is that isometries should probably send geodesics to geodesics, and I'm guessing that if you push a line across one of these guys you get a circular arc

But would you get all geodesics from this? Naively I'd expect you need more than transitivity, like almost a sort of halfway "double transitivity"

You won't like it, but see my notes :)

6:00 AM

Maybe if $d(z,z') = d(w,w')$, then there's an isometry sending $z\mapsto w$ and $z'\mapsto w'$?

Hahaha, alright

Yes, isometries are transitive on the unit tangent bundle.

Anyhow, section 2 of chao 3 is hyperbolic geometry (half-plane model).

Gotcha, I'll make sure to take a look at that. Also more general stuff since in a few pages this book is gonna pull out Levi Cita lol

We can do more tomorrow when I'm at my computer if you need.

That might be a good idea yeah, thanks!

Later

6:33 AM

As many people are interested in online collaboration, the room Find your math partner has been created to connect people having common math interests with each other. Please first read the guideline of the room. If you have any question, request, suggestion, criticism, ... about this room, please send your messages in Later's Room.

Q: What is the transported metric?

7:25 AM

if anyone has an idea of how to answer this let me know

2

In this previous question: Transported Metric, I wanted to transport the Euclidean metric $ds^2=dx^2+dy^2$ into the first quadrant of the $(u,v)$-plane, such that for all curves $\gamma$ in the $(x,y)$-plane we have $L_{uv}\bigl(f(\gamma)\bigr)=L_{xy}(\gamma)$. I used a map $f:\Bbb R^2\to\Bbb R^{...

8:10 AM

What is a character of a representation

@LeakyNun

@BalarkaSen the trace of the group elements?

How do you say something is a character of a rep without reference to the rep

what do you mean

A map $\chi : G \to \Bbb C^\times$ satisfying (1) conjugacy invariance (2) $\chi(1) = 1$ (3) ... is a character of some rep

How do you say that

Hmm, irreducible characters of a group are a basis for the space of class functions

8:12 AM

orthogonal even

I could see there being a condition on the norm which maybe guarantees that your character corresponds to a rep?

How do I classify irreducible representations of $S_3$ using character theory

I know 3 irreps of $S_3$

Why aren't there more

why $\Bbb C^\times$

@BalarkaSen sum of squares

Orthogonality

please explain

I know 0 character theory

8:14 AM

1^2 + 1^2 + 2^2 = |S3|

I didn't even know what a character is until you told me

the dimensions are 1, 1, 2

Aha

Can I prove this

maybe?

So you take the space of class functions on $G$, you put an inner product $\langle \varphi,\psi\rangle = \frac{1}{|G|}\sum_{g\in G} \varphi(g)\overline{\psi(g)}$

The irreducible characters of a group form an orthonormal basis of the space of class functions

Wrt that inner product

8:16 AM

eigenbasis?

So in particular, how many irreps of a group are there? The number of conjugacy classes. Also playing with this should give the dimension formula Leaky's talking about

what's the endomorphism

Who said anything about an eigenbasis?

But yeah I misspoke for some reason

nobody

@AminIdelhaj Why is this true

I have no intuition

Class functions is center of $\Bbb CG$

$\Bbb CG$ decomposes into matrix algebras by Wedderburn

Oh

Got it

8:18 AM

@BalarkaSen because of Schur

No but $\Bbb CG$ is direct sum of matrix algebras

Take dimension

$|G| = \sum n_i^2$

you should compute the character of dual, tensor, hom first

$n_i$ is dimension of each irrep

No I don't want to

alright

I just need to classify irreps of $S_3$

I can do it now

Thanks for being of no help, representation theorists

8:20 AM

anytime

lol jk

:P

I don't understand characters

Too hard man

if you do my exercise then you'll understand them better

Lmao

8:21 AM

Ok I'll try

and I guess computing the characters of your irreps

also like Amin said, there are 3 conjugacy classes in S3, so there are 3 irreps

that's another way if you know the number of conjugacy classes

yeah it's not immediately clear to me why that is true either

actually $a^2 + b^2 + c^2 = 6$ gives (1, 1, 2) as the only solution

it should be maybe

that's the number of dimensions of class functions

8:23 AM

Ah ok yes of course

dimension of $Z(\Bbb C G)$

why can't you talk in terms of ring theory

I don't understand this sudden concreteness from algebraists

trace of a matrix? wtf?

thats engineering

trace of endomorphism then, if that's better

trace is a mysterious concept

$g \in G$ acts on any $\Bbb CG$-module $V$

there is no good basis free definition of trace

its by construction engineering

$\operatorname{Hom}(V,V) = V^\ast \otimes V \xrightarrow{f \otimes v \mapsto f(v)} k$

the composite is the trace

8:26 AM

Hom(V, V) = V^* o V is noncanonical

Very mysterious I'm telling you

trace is just the derivative of the determinant

Lmfao

@TobiasKildetoft Uh oh

you have caught me shitting on rep theory

@BalarkaSen Heh

I didn't actually read the previous messages

Just came off a meeting where I was meant to do a demo for the customer. Somehow the demo had stopped working without anything being changed, so I spent most of the meeting getting it to work

11:43 AM

Hej!

what does it mean by complex numbers have a unique multiplication, whereas $\Bbb{R^2}$ may or may not be endowed with one of several different multiplicative operations?

@Stupidquestioninc Who claims that?

@TobiasKildetoft My book on metric space in isometries section

I am trying to make sense out of isometries too

And $Bbb{R^2}$ has multiplication? Never seen it?

There is a unique multiplication on the complex numbers (up to isomorphism) which makes it both a field and a $2$-dimensional algebras over the reals

There are three different 2-dimensional algebras over the reals

How much background do you have in algebra?

Umm... Do you know any link so that I can learn about the stuff you mentioned?

Hey guys, I did my undergrad in Mechanical Engineering really long time back (15 years). Since then I moved to programmming & have been programming for 15 years. In Mechanical Engineering, I didn't do any pure math at all. It was totally Applied Math and I used to be pretty good at it.

Due to some personal interests, I am trying to learn basic number theory - mainly congruences. I have been at it for more than a month & I am finding it really difficult - this is so different from Applied Math. If you look at my questions on math.SE, you would realize how difficult I am finding it.

11:54 AM

@TobiasKildetoft I just started learning group theory if you mean by abstract algebra.

@Stupidquestioninc Ok, so I assume you are not familiar with ring theory

@TobiasKildetoft True.

@user93353 As far as I can tell, what you need is more practice with the basic concepts. So you need to do a ton of exercises using these before moving onto any of the actual results like Fermat

@TobiasKildetoft - I have been at the basic concepts for a month now.

Also, as a programmer, you will need to get used to the fact that "mod" is not considered a function from integers to integers in math, like it is in most programming contexts

A month is not that long to digest completely new concepts

11:58 AM

@TobiasKildetoft - I understand the mod % in programming is remainder in math

Well, not quite. But close

@TobiasKildetoft So you mean In order to know about multiplication in $R^2$ you need to learn ring theory?

I understand the concept of mod, I understand the concept of congruences, but it's so difficult to retain it. As you said, I need a lot of exercises - is there a book which comes with a hell of a lot of solved exercises I can practice with? @TobiasKildetoft

@Stupidquestioninc Not necessarily for that specific case, but the general ideas are exactly what ring theory is about

@TobiasKildetoft so where do I start from?

12:01 PM

@user93353 I don't really have a good recommendation unfortunately. The book I used to teach from has a decent number of exercises on these things, but it is too expensive for this purpose

@Stupidquestioninc I think the remark about a unique multiplication on the complex numbers can probably just be ignored for now, unless it was made as part of something important

@TobiasKildetoft - can you give me the name of the book? Do the exercises come with solutions? Or alteast answers?

@TobiasKildetoft ok thanks

If the latter is the case, we can probably help you understand it if you supply the greater context

@user93353 I don't recall if there were answers to the exercises (I think not). The book is Concrete Abstract Algebra by Lauritzen

But that book is aimed towards algebra rather than number theory, so even though there are a decent number of exercises on congruences, there are not as many as you probably would want

I have a physics question, but I'm not a physics major so it'll effectively be a layman question, what place in stackexchange would be the best to ask it?

Anyone else has any recommendations on Congruence/Number theory books which are written for someone who is dumb at pure math & the books drills down everything to a lot of detail assuming the reader doesn't know much. And also has a lot of exercises & solutions to them?

12:07 PM

I basically want to ask what would happen if we designed a valve that allows atoms of a gas to pass through it in only one direction? It would mean when placed inside a cylinder of gas, due to atoms always being in motion, one side, to which the motion is permitted by the valve will have more density than the other side to which motion is not allowed

If you can read Danish, I can point to two introductory books that should have some exercises

@TobiasKildetoft No i don't read Danish :)

(I did not expect you to)

But I think learning Danish would most likely be easier than learning Pure Math

user400188

12:23 PM

You could learn pure math using Polish notation if you wanted to make it harder. The notation is as different from other pure math notations as pure math notation is from most of math.

@user93353 Jones and Jones Elementary Number Theory

@EdwardEvans - thank you. I'll look it up

Look up André Weil's Basic Number Theory for a laugh

@EdwardEvans - what do you mean? Is it not a good book? Or is it too difficult?

It's called "Basic Number Theory" but it's far from what one would consider "basic" if one were just starting out

Luyw

1:18 PM

Hello!
by "What is the probability that a week (7 days) passes till the event that exactly 3 computers fail the same day occurs?", do you understand that the event happens on the 8th or 7th day?

1:41 PM

the more I read up on things like Alcubierre drive, string theory, Casimir effect, etc the more I feel like I should have chosen physics major and specialized in cosmology rather than choosing math major and specializing in stochastic calculus

:(

it's so so so exciting

especially Alcubierre drive

I should know more stochastic calculus

Clarinetist

1:57 PM

@BalarkaSen Stochastic calculus is so exciting. It's pushed me to go for a PhD program; I hope to at least study it or something related to it

José Carlos Santos

2:51 PM

@EdwardEvans A number theorist once told me that it took him twenty five years to understand what André Weil was trying to achieve in that textbook.

3:53 PM

@Clarinetist let's create a separate group chat just for probabilists and especially for stochastic calculus experts

Thorgott

Is there a nice description of $\operatorname{End}(G\ast H)$ ($G,H$ groups)?

Sayan Chattopadhyay

@TheTestosteroneFanatic Don't know about the physics side of things, but from my very limited mathematical knowledge, I find the mathematical ideas presented in modern physics very interesting.

@JoséCarlosSantos that’s unsettling

4:20 PM

@Thorgott Hmm, there ought to be I think, at least to some extend

Some part of it should come from the universal property of the free product

Thorgott

yeah, that gives $\operatorname{End}(G\ast H)\cong\operatorname{Hom}(G,G\ast H)\times\operatorname{Hom}(H,G\ast H)$ (but only as groups, not as monoids)

4:46 PM

@LeakyNun something something MB-complete problems

lol

does anyone mind helping in forming a question for a possible post?

Post it and I'll see if I can help

chat.stackexchange.com/transcript/message/55568172#55568172 @Astyx

The post is easier for me to explain in details what I'm trying to do/the goal, just unsure on how to form the actual question :/

I don't get what you're trying to say in that message

Mind showing me an example ?

4:55 PM

sure

let's say i have a long integer: 1238654648800985656688878...'
for more practice example, let's also say it's a fixed string (30 digit/char long).

Now the goal here is to find the *shortest* calculation that will give back said number/integer. By shortest, i mean that it have to be shorter than the actual length (in character) of the integer above.

divide the number by 2?

@geocalc33 that wouldn't truly make it shorter... it depend on the integer

I tried with certain string, where dividing work, but the remainder is missing

if i have to include the remainder, then the calculation isn't shorter than the whole integer

Factorizing the number would be good probably

@Astyx do you have an example? (didn't yet tried factorizing)

Can you embed a projective module in Hilbert space?

5:09 PM

@geocalc33 How does that make any sense?

Thorgott

lol

$12876420756023476058721634760127634076123 = 7^1 \times 53^1 \times 1597^1 \times 4289^1 \times 781789970699^1 \times 6481420415270233639^1$

robjohn

@TedShifrin once or twice?

Now arguably for this one it doesn't work

@Astyx Yeah :/

I wish i knew some algorithm or theorem that could help with this

but i don't know of any

5:12 PM

@TobiasKildetoft my bad 😁

the only method that worked so far was bruteforcing and generating every possible calculation of N length (shorter than the whole integer + also give back the original integer as result)

this work but take a tremendous amount of time, especially since i have to recalculate every time i do a new query

(since i can't store every result of every calculation)

There's a reason we use decimal expansion

@Astyx what do you mean by that?

To store numbers

If you look at machines they use binary expansion

And there is no more "efficient" way of storing numbers than binary expansion, because you use every possible combination of symbols

@Astyx so you mean what I'm trying to do can't be done easily? or at all?

5:15 PM

I mean you're asking something weird

As everyone on SE/SO is :D

we all have/are entitled to our reasons for asking things, saying it's weird is like saying the obvious

What operations do you allow yourself to use ?

Are you restraining yourself to basic symbols ?

What do you mean by calculation ?

@Astyx hmm, for now yeah, only using basic symbols (+, x, -, /) but i wish to use more, just not sure what to add

You can come up with an infinite amount number of symbols and assign each one of them to a specific integer

And every integer would be represented by a single symbol

But I'm guessing you don't want to do this

calculation = it's basically the equivalent translation of "calcul" in French (on google translate).

I was always unsure of the right word for this, as it work in French but probably not so much in English, when talking about computation/anything that is computed (which give back a result)

5:20 PM

Are you french ?

@Astyx yeah

tant que tu ne me donnes pas une définition formelle de ce que tu entends par calcul je peux toujours te donner une réponse farfelue

French too i see :D

@Astyx I think it would probably be better if we talked English here? Unless you don't mind if we talk in a separate room in French

don't know if it's fine doing it here too

I doubt anyone cares

I can speak english

fine by me :)

5:22 PM

Anyway, if you allow more symbols, I could use decomposition in base 16 (hexadesimal representation)

Or even larger base decomposition

@Astyx yeah, i actually do want more symbols, I'm just unsure on what to add

I could take base $10^{31}$ and have a unique symbol for every number up to $10^{31}$

parlez en francais si vous voulez

@loch long time no see

sounds good yeah. But what logic/way would you use to make this clear that Symbol = Specific number?
A database lookup?

allons-y tout les personnes dans le chat. allons-y parle le français maintenant

5:25 PM

you could divide a square into two halves, and divide on of its half into two again and so on

And color either the largest square, or the second largest, or the third largest ...

And have an infinite number of symbols I can correspond to integers

It depends what you mean by symbol

@Astyx yeah, i guess

In the real world you only have a limited number of symbols you can differentiate

Computers only have 2 (up to some extent)

I said i did try this method, but the way you described, i have to say i didn't :) (doing halves and assigning them to symbol as you said above).

It's a promising idea yeah

well yes and no

c'est fondamental

5:28 PM

At some point it's not going to be possible to differentiate symbols

But that's another discussion

@Astyx yeah...

that remind me of a certain project that tried to do the same thing, and somehow succeeded: github.com/librarianofbabel/libraryofbabel.info-algo

they basically found a way to make a fixed length book, into a seed (not really an hash) which can be found back and vise versa (at least that's how i understood it, probably not 100% like this)

the only problem is that the seed is still very long, but smaller to the fixed length book

it seems he doesn't store every possible seed -> data but just progressively generate it on demand, which is why it's interesting

Silent

Above pic is from the lecture note I have.

Then, it goes to find radius of covergence of $\sum_{n=1}^{\infty}z^{k_n}$ (where $(k_n)$ is strictly increasing sequence in $\Bbb N\cup\{0\}$) like this:

$\limsup_{n\to\infty} |a_n|^{1/n}=1$ and hence $R\ge1$. By considering $z=1$, we see $R=1$.

I am wondering why we needed to look at z=1 to deduce R=1? Doesn't Cauchy-Hadamard itself say R=1?