Mathematics - 2020-06-03 (page 2 of 2)

Mathphile

6:00 PM

$$\sum_{x=1}^{\infty}\sum_{n=1}^x \frac{1}{f(n)}$$
For what $f(n)$ does ithe above sum converge if it converges at all?

Knight

I will surely follow that sir.

Ted Shifrin

You haven't changed the problem, @Knight, other than rephrasing, as I immediately did, mod $n$.

Balarka Sen

Yeah it's not a reduction, you're just looking at the problem modulo $n$.

Knight

@TedShifrin Actually, I have problem with modular arithmetic symbols. Can you please write it (if possible) without that mod? (story if I'm too demanding)

Balarka Sen

If you want to rephrase that's alright but you should rephrase it correctly

Ted Shifrin

6:01 PM

Anyhow, you can easily write a proof for your example if you consider $2$ cases. Is $n$ divisible by $3$? If not, then $3$ and $n$ are relatively prime. Then you use the Euclidean algorithm to say ....

"I have a problem with modular arithmetic symbols"? What the ... ?

Knight

HAHAHAHAHA you were about to curse

Tobias Kildetoft

@Knight Well you started it. You asked him if he rogered something.

Knight

We didn't have Modular Arithmetic ever in our course. We had half a page in 9th Grade on "Fundamental Theorem of Arithmetic"

Balarka Sen

Just learn it

Knight

@TedShifrin Yes, after that ...

@TobiasKildetoft Didn't get you

Ted Shifrin

6:07 PM

@Knight: Just sit down and write it out. Some multiple of $3$ will be congruent to $2$ mod $n$, so adding that many $3$'s to $n-2$ will get you a multiple of $n$.

Knight

Yes.

Ted Shifrin

That is a proof in total generality.

You only have to check that you need no more than $n-1$ copies of $3$.

Knight

And if $n$ is divisible by 3? Let's say $n=153$

Ted Shifrin

Then you add up $n/3$ copies of $3$.

Knight

then we have 152 threes

Ted Shifrin

6:11 PM

Huh?

Yeah, and $n/3$ is way less than $n-1$.

Knight

Okay, we need just 51 threes, ha?

Ted Shifrin

The "interesting" part of the argument is what I said above — that you never need to use "too many" copies of $3$.

Knight

So, sir is this a right way of proving? That is I'm taking particular collections and then proving "if I can take appropriate members such that their sum is divisible by $n$" ?

Ted Shifrin

You're never going to prove the general result by considering examples. Think about how many separate arguments you'd have to try to give.

Knight

Yes

And I cannot even talk about a general collection because it is absurd to say that general collection of $n$ entries from $$0, 1 , \cdots, n-1$$ would be this ...

(I mean anything)

Ted Shifrin

6:19 PM

That's why mathematics is interesting. There are general methods of proving challenging things. In this case, I did not think of the trick that Balarka used.

Knight

;-)

Thorgott

another reason it's interesting is that there's no too general method for proving challenging things

Knight

So, I'm and was going in the wrong direction, ha?

Balarka Sen

@Thogott Jacob Lurie enters the chat

Ted Shifrin

Yes, although trying particular examples helps convince you that the result is true. But it gives no insight into a general approach.

Edward Evans

6:20 PM

@TobiasKildetoft oh nice, am I right in saying that the "usual" definition takes a $K[H]$-module $M$ and tensors with $K[G]$ over $K[H]$?

Thorgott

oof

Edward Evans

where $K$ and $G$ and $H$ are the obvious notations

Balarka Sen

Random: If any of you like science fiction, read Roadside Picnic. It's a great book.

Have not read anything quite like it

Thorgott

I feel bad for having wasted my afternoon thinking about coproducts instead of studying what I need to study

Edward Evans

I got 12.5% in the modular forms exam

Knight

6:22 PM

Thank you so much @TedShifrin. I would like to be sorry I was disrespectful to you or to any other member, I would like to be sorry to others too.

Edward Evans

so

be happy

Ted Shifrin

No need to be sorry, @Knight. I don't know how old you are, but it seems like you should learn some basic modular arithmetic if you're going to play around with things like that.

@Edward, was 15% a top mark? :D

Edward Evans

yeah 15% was an A+ :( Just missed it

lol

Ted Shifrin

I think one time I taught differential topology (granted, I had cancer during the semester and wasn't even there to teach the last few weeks), the high score on the final was something like 60%.

Edward Evans

oh damn

Balarka Sen

6:23 PM

@EdwardEvans Hot damn

Thats what I will never do number theory

Thorgott

wew

Ted Shifrin

LOL, maybe the modular forms exam was a bit off. Hard to tell what people know and what they don't if the high score is 15%.

Thorgott

what on earth happened there

Edward Evans

I had a super bad mental health semester last semester so that kind of destroyed modular forms for me

Ted Shifrin

Oh, too much hanging around this chat room. We're sorry.

Edward Evans

6:25 PM

algebraic number theory went well, but that's because I wrote my bachelor's dissertation on that, so it wasn't really a strong effort lol

Hopefully the course on modular forms will be offered again next year though, I'd like to actually understand what's going on

@TedShifrin my life is RuInEd

Ted Shifrin

Oh, I was wrong. The high score was 68%.

Balarka Sen

yeah a major problem for me is that i tend to stress out and fight for grades and that makes me have a total breakdown some point of time every semester

its not worth it

Ted Shifrin

Not worth stressing out.

Just pretend you're teaching some dummy in here.

Like me.

Balarka Sen

Haha

Yeah actually this chat has been so helpful; often I get sick of the semester and sick of math and then I come back here and suddenly I'm productive and happy

Edward Evans

@Balarka I just didn't engage with the material in the first semester I guess, I found it interesting but couldn't get stuck into it lol

Balarka Sen

6:28 PM

I learn a lot from talking to everyone here

i know what you mean, that happened to me with a couple courses. i had some linear optimization course last semester and was like meh for the first half of the semester

Ted Shifrin

Weird anti-social progression, @Balarka :D

Balarka Sen

did not do well in midterms, but came back home for a break because i was feeling really shit, and found some lecture notes on geometric topology and optimization -- that was an immediate motivation and regained my interest in the course in the second half

Fargle

wait wait wait hold on wait

what's this "math" thing everyone's going on about?

Edward Evans

haha

I jokingly asked the prof what a modular form was just before the exam

and then had to quickly check if I actually did know the definition

Ted Shifrin

It's just a section of a line bundle.

Fargle

6:30 PM

one of my favorite things to do at the tail end of review sessions in math classes is to ask philosophy of math questions

Done.

@Ted :3

@TedShifrin by jove

That's probably not the greatest use of time/effort at that frazzled time, @Fargle.

Fargle

"Any more questions, before we go?" "Yeah---just a quick one---I know that three rocks exist, but does 'three' exist absent an object?"

no, it's not

but I never claimed to be a good person :)

Ted Shifrin

6:31 PM

Have you stayed in touch with our mutual friend, @Fargle?

Balarka Sen

Lmao

Fargle

not in some time, but that's a good idea, I should shoot them an email

Ted Shifrin

@Edward: Have you seen our chatroom denizen who belongs at your school? I haven't seen him here in ages.

Balarka Sen

the Boulomenos

whats his real name

Edward Evans

@TedShifrin a message I sent to him ages and ages ago changed from "received" status to "read" status about 2 weeks ago

Ted Shifrin

6:32 PM

Matthein?

Lukas is his real name.

Balarka Sen

Mathein is not his real name haha

Yeah

Edward Evans

but otherwise I haven't heard a peep, he missed his seminar talk as well

Ted Shifrin

Oh oh.

I hope he's OK.

Edward Evans

Indeed, I keep forgetting to speak to his dissertation supervisor

I was even in his office ysterday -.-

Thorgott

is your uni back in business already? or were you just there for one of the exams?

Balarka Sen

6:37 PM

Anyone got good literature recommendations

Edward Evans

@Thorgott just back for exams

Baden-Württemberg was I think one of the worst hit Bundesländer, so they're being quite cautious here

Tobias Kildetoft

@EdwardEvans There is no "usual" one. There is the one you mentioned, and there is the one using maps from the group. The result is only the same in "nice" cases

Edward Evans

I see, the definition I had was using maps from the group, but I didn't find it as nice as the tensor product definition

Tobias Kildetoft

Each has its advantages

Edward Evans

for instance the proof of Frobenius-Reciprocity was pretty much immediate for the tensor product definition but some horrible calculations for the other definition

Tobias Kildetoft

6:41 PM

The one with maps has the advantage that you can restrict to some subset of maps (like continuous ones if you have a topological group)

Edward Evans

I see

Tobias Kildetoft

and this allows you to still get an induced module in more restrictive settings, such as continuous reps

Edward Evans

(e.g. $\operatorname{Gal}(\overline{\Bbb Q}/\Bbb Q)$)

Tobias Kildetoft

whereas the tensor product construction will in general not behave as nicely

Edward Evans

lol

I see

Ted Shifrin

6:42 PM

@Fargle: I dunno if you're FB friends with our mutual friend. He and I had a silly punny exchanged on there a few months ago.

Edward Evans

interesting, I'll probs see more of that within the seminar

Tobias Kildetoft

For a specific example, the tensor product does not yield an algebraic representation of an algebraic group, even if we induce the trivial rep of a closed subgroup

Fargle

I don't think I am. I don't think I have any educators from my time out east added on FB.

Tobias Kildetoft

But the other one does

Ted Shifrin

I don't think you added me, either, @Fargle.

Of course, I should be boycotting FB entirely. :(

Fargle

6:46 PM

same, but there are too many people in my life I only have contact with through FB

Ted Shifrin

Yup.

Alexander Gruber

7:07 PM

@BalarkaSen ever tried dostoevsky?

Alessandro Codenotti

@BalarkaSen What kind?

@EdwardEvans Two out of three with them being so close sounds very good to me

Balarka Sen

@AlexanderGruber Yeah kinda

@Alessandro I dunno, maybe some mind blowing science fiction

That would not hurt

Alessandro Codenotti

Have you read something from Dick already?

Balarka Sen

Ah yeah no I haven't

I am more of a J G Ballard guy

Alessandro Codenotti

I suggest the Minority Report to begin with then, it's very short so it's not too bad even if you don't like it after all :P

Balarka Sen

7:14 PM

Nice, I will try that

Alessandro Codenotti

(the movie adaptation is alright, not great, not terrible, you can safely skip it)

Balarka Sen

I would rather not watch a Tom Cruise movie

Alessandro Codenotti

There's also Colin Farrell in the movie who is a good actor in my opinion

But I'm not too fond of Tom Cruise either

Balarka Sen

Oh The Lobster guy

Yeah he's good mane

Alessandro Codenotti

Right, that's him. He was also in The Killing of a Sacred Deer (the movie Lanthimos made after The Lobster)

Balarka Sen

7:18 PM

The Lobster was a little hard to digest for me lol

Undoubtedly a good movie but very weird

I didn't know if I ought to laugh or take it seriously

Leaky Nun

'sup y'all cultured people

Alessandro Codenotti

It's not a light movie for sure, but definitely a really good one

Leaky Nun

I watch Pulp Fiction

Balarka Sen

What do you watch/read @LeakyNun

Pulp Fiction is good

@Alessandro I mean the gallows and the humor in "gallows humor" was a bit too contrasting each other

Speaking as a connoisseur of black comedy

Semiclassical

hi chat

Balarka Sen

7:23 PM

Hi SemiC

Did you I tell you T. S. Eliot is an anagram of "Toilets"

Semiclassical

yes

i'd seen that in a simpsons comic, lol

Balarka Sen

lol whoops

Semiclassical

looking at the book: there's a lisa simpsons adventure which is sorta like alice in wonderland / gulliver's travels / etc

one of the footnotes notes that the line "eat a peach" is a reference to Eliot's Prufrock, noting after: "It's most unfortunate that Eliot's parents named him Thomas Stearns rather than Stearns Thomas. That way, you see, he would have been S.T. Eliot...and Bob and Otto could tell you what that spells backwards!"

so same basic joke but different approach

Balarka Sen

Lmao nice

It just tickles me to no end that the originator of this is Nabokov

Semiclassical

i suspect it's one of those jokes that is independently rediscovered with some frequency :)

Alessandro Codenotti

7:36 PM

A thing which always amuses me in bookstores is that Lolita, by Nabokov and Reading Lolita in Tehran, by Nafisi, are often next to each other on the shelves when the books are listed by alphabetic order on the author's last name

Semiclassical

i mean, it would perhaps explain his insistence on going as T.S. Eliot and not T. Eliot.

Balarka Sen

Haha

re Lolita, I tried reading Pale Fire once but gave up quickly

Semiclassical

(i stole that observation from here, to give credit: simonleyland.net/t-s-eliot-toilet-and-turds)

Alessandro Codenotti

I only read Lolita by Nabokov many years ago. Maybe I'll try something else at some point

Semiclassical

theisleofthanetnews.com/2019/04/08/…

now I find myself wondering if that's an actual quote or just made up

i see that as a quote online, but i've yet to see an actual source for it

linda Oiladali

7:46 PM

please I f is aperiodic function what is the ideate prove that $$\forall a\in\mathbb{R},\forall n\in \mathbb{Z}, \int_a^{a+nT} f(x) dx=n\int_a^{a+T} f(x) dx$$

Thorgott

8:03 PM

try splitting the integral up

Semiclassical

(presumably it's "a periodic function with period T", not an aperiodic function)

(but that's something that should be said rather than requiring us to guess)

Alejandro Bergasa Alonso

Is this chat to answer quick questions? This is my first time here

Semiclassical

the policy is "just ask, don't ask to ask". that is: if you've got a question, then ask it, and if someone is interested they'll answer

AfronPie

8:37 PM

I was able to solve and understand this question. After multiplying 1*2*3*...*20, and doing a prime factorization, I found that 11*13*17*19 didn't pair up (b/c they did not have even powers) and thus there is no way to split the set.

However, my teacher asked a follow up question: If it is possible to split a big set S into two other sets (using the same conditions as the original question) except this time instead of the set being S = {1,2,...20}, the restriction on S is that it may contain any 20 positive consecutive integers. i have no idea how to do this, any help/hints would be greatly appreciated.

Mathphile

8:51 PM

3 hours ago, by Mathphile

$$\sum_{x=1}^{\infty}\sum_{n=1}^x \frac{1}{f(n)}$$
For what $f(n)$ does ithe above sum converge if it converges at all?

any clues?

Mike Miller

@BalarkaSen I'm surprised, it's pleasant reading to me and not particularly difficult I don't think

I think it's often suggested as an artistic masterpiece when it's just a really well done gimmick

But I like the gimmick a lot, nobody has done it as well even though it's been tried since

Leaky Nun

leanprover-community.github.io/sphere-eversion @MikeMiller

Ted Shifrin

9:15 PM

@AfronPie Here's a warm-up question. Can you list me two consecutive positive integers with no primes in the list?

AfronPie

8,9

Ted Shifrin

LOL. Oops, I typed twenty and it came out two :D

Hawk

In real analysis, (a) to define (formally?) the double improper integral $\int\int_\mathbb{R^2} f(x,y) dxdy.$, do we just define it as a double limit $\lim_{x,y \to \infty} \int_{-x}^{x} \int_{-y}^{y} f(x,y,)$? (b) To show that $\int\int_\mathbb{R^2} (1 + x^2 + y^2)^(-s) dxdy.$ exists iff $s > 1$, is it a sufficient proof to just directly compute it and show the eventual one-dimensional integral has a limit iff s > 1?

Ted Shifrin

In general, can you give $n$ consecutive composite numbers?

AfronPie

@TedShifrin haha I was confused for a sec

Ted Shifrin

9:16 PM

My apologies.

AfronPie

Yes I looked this up, there is a prime gap of 20 numbers starting at 887 until 907 with just composite numbers in between

Ted Shifrin

Do you see a way to write it for general $n$ without looking up anything?

That's the first step to tackling your question, anyhow. But I still suspect there's no way to do it.

@Hawk: No, you can't do it symmetrically. You have to go to $-\infty$ and $\infty$ in independent ways.

In principle, you have to justify comparing polar coordinate improper integrals to cartesian (according to (a)).

AfronPie

@TedShifrin I am not sure how to write it for general n without looking up anything.

Ted Shifrin

You have to think about a way to choose $N$ so that you know that $N+2$, $N+3$, \dots will all be composite.

Thorgott

that double limit seems to be a non-standard notion of a principal value

it probably isn't well-behaved at all

Hawk

9:26 PM

You mean I have to do it like $\lim_{x,y \to infty} \int_{-x}^a \int_{-y}^b + lim_{x \to infty} \int_{a}^x \int_{b}&y$? Is that the formal definition? Or there such thing as "upper/lower" sum for these type of integrals (b) I have to justify change of variables?

Mary Star

Hello!! Is someone of you familiar with Huffman code?

I have applied the Huffman code for a given sequence of letters and the respective frequencies. Now it is asked for the total weight of the code. How do we caclculate that?

Ted Shifrin

@Hawk: Well, it's like $\lim_{m,n\to\infty}\int_{-m}^n$, but for the double integral.

I'm saying you need to justify why the improper integral $\int_0^{\infty} r\,dr$ exists tells you that the improper double integral $dx\,dy$ exists (or not).

Hawk

(a) Thats what I wrote right? Any other alternative would go back to the symmetric one (b) Can't I just cite the change of variable property? I am not seeing why we have to justify this.

Ted Shifrin

@Hawk: You don't have to split it up, but your way with two separate integrals saves letters.

I'm not worried about the change of variables, although you're going to be comparing integrals on big disks (radius $R$) with integrals over big rectangles. So there's some work to do.

You can compare disks and squares, but then that still doesn't justify the independence of $m$ and $n$ in my definition.

Hawk

9:45 PM

(a) You mean to say your suggested method (which I am not seeing) is more economical than mine right? (b) So you are saying I have to justify a circle with growing radius R will cover R^2? Isn't this stuff covered by the change of variables?

Thorgott

taking the limit of integrals over increasing squares and over increasing balls is not equivalent in general

Ted Shifrin

If you knew that squares sufficed (instead of general big rectangles), comparing squares and disks is quite straightforward, but you need to do it. You need to sandwich each in the other.

No one is disputing that disks of radius $R$ will cover the plane as $R\to\infty$. That is not the question; nor is the question about the change of variables theorem.

Hawk

I might not be understanding you but you are saying I have to show that int_disks \leq int_squares \leq int_disks?

Ted Shifrin

Yes, if you want to show that the polar coordinate improper integral shows that you get the limit for squares. But you still have to discuss that squares are good enough to give the general improper integral in this case.

So it's worth thinking, even in one variable, about the kinds of functions where $\lim_{m\to\infty}\int_{-m}^m f(x)\,dx$ exists but the improper integral does not. What is it about your function that says you won't have such problems? Proof?

Hawk

i mean its not discontinuous, but you might be thinking of something deeper

Ted Shifrin

9:54 PM

Let's stick to continuous functions, yes.

Hawk

i m just listing properties at this point, but I think definitely the nonegativity is important.

Ted Shifrin

Aha.

So can you prove what I said works for a nonnegative continuous function ?

Hawk

because for otherwise the "half limit" in my definition (that's the correct one right? Because idk what your economical one is) may not exist

and hence the sum can't be properly defined

You mean to prove that a nonnegative continuous function over square = circle?

*disk

Ted Shifrin

No. Why does the improper integral exist for a nonnegative functions if you know just that $\lim_{m\to\infty} \int_{-m}^m$ exists?

Hawk

The point is that we don't right?

Ted Shifrin

10:04 PM

Huh?

Hawk

because existence of lim_{m \to \infty} Int_{-m}^m isn't sufficient to assert that the improper integral (\int_{-m}^a + \int_{a}^m) exists

Ted Shifrin

We were discussing when it would be sufficient, remember?

Hawk

when lim_{m \to \infty} Int_{-m}^m attains a finite value

Ted Shifrin

No.

What if that value is always $0$. Does that mean the improper integral exists?

Hawk

by what I previously justified, no because we would not be able to define the sum if the half limit DNE

Ted Shifrin

10:10 PM

Do you know a concrete example?

Hawk

i mean just \int x

if the two integrals are equal, that should be sufficient/

Ted Shifrin

I do not understand your second sentence.

Hawk

nvm that didn't make sense

Ted Shifrin

But, yes, $f(x)=x$ is the standard example I was looking for. You suggested that maybe for nonnegative functions it should be true. I'm suggesting you try to write a proof of that.

Hawk

hang on i missed the keyword nonnegative. but its because int_{-m}^m equals exactly the improper integral right? my example doesn't work because it is negative somewhere else

user462942

10:20 PM

Hi @TedShifrin :)

Ted Shifrin

It does not equal it, no, @Hawk. I have stated several times what you need to prove.

hi @Joanna

I think it really is useful, @Hawk, to write the single integral as I did. How do you relate $\int_{-m}^n f(x)\,dx$ to integrals of the form $I_m=\int_{-m}^m f(x)\,dx$?

Hawk

no i m saying if it exists

Ted Shifrin

If the limit exists, then the improper integral should exist and equal that limit. You need a proof.

Hawk

> How do you relate $\int_{-m}^n f(x)\,dx$ to integrals of the form $I_m=\int_{-m}^m f(x)\,dx$?

add the other half \int_{n}^m

Ted Shifrin

Not half!! ... You're assuming $n<m$? There are two cases, right? But you have to figure out how to handle them.

user462942

10:27 PM

@TedShifrin what's a subfield that's in the intersection of linear algebra, geometry, and analysis?

user462942

I'm looking for some direction to consider heading towards ...

Ted Shifrin

That's just way too vague. What does "geometry" mean there?

Hawk

i mean if it is n>m, u just have to flip it.

*flip the integral

Ted Shifrin

Representation theory is an answer I've given you numerous times. Most of geometric analysis involves PDE's, which is serious analysis, and linear algebra is everywhere.

So give me something specific, @Hawk. How are you going to prove something?

Hawk

so i was assuming -m < n then we just add \int_{-m}^n + \int_n^m. if for some reason we have m < n, then I guess m doesn't even matter. but i m not sure why we have to consider this case.

Ted Shifrin

10:30 PM

Fine, now prove something. How does this show the limit exists as $m,n\to\infty$?

Hawk

i mean u just pass the limit individual to each of the integrals

oh wait u want n -> infinity too

Ted Shifrin

That doesn't prove anything.

It's just throwing symbols around.

Hawk

i mean that sum equals int_{-m}^m. then the limit over them equals \lim \int_{-m}^m

user462942

@TedShifrin Yeah, my question is vague, tbh. A day or two ago, I looked over some course notes for representation theory of finite groups and symmetric groups, and the notes didn't seem all that interesting. Just one theorem / proof after another; I had thought that problems in representation theory would reduce to problems in linear algebra, but it didn't seem like that at all ...

Ted Shifrin

You are a little too fond of basic linear algebra. I'm thinking of Lie groups, not finite groups. That's where both some geometry and some more serious analysis comes in.

@Hawk: I don't see that that gives a proof yet, but you're close. How are you using nonnegativity of the function?

user462942

10:38 PM

@TedShifrin Yeah, exactly -- I don't know how to progress (and I would like to do so) when everyone around me is doing computational work with applications to bio and physics.

user462942

Lie groups -- I'll look into this, thanks :)

Ted Shifrin

Have you studied undergraduate differential geometry? Good idea to know some of that stuff before you learn Riemannian manifolds.

But there you see importance of certain key ideas from linear algebra (e.g., the spectral theorem, eigenvalues, etc.).

user462942

@TedShifrin No, for our curriculum, DG was a topic for a spring semester topics course. The DG material lasted only about 2 or 3 weeks. Some vector calculus different from Calc 3 and stuff about "charts". I'll revisit some of that material then :)

Ted Shifrin

Nah, I'm not caring about charts. I'm caring about curvature of curves and surfaces — actual mathematics. Download my differential geometry text (free) and play with it.

user462942

Oo I see

user462942

10:46 PM

Ok, will do ...

user462942

I also found Artin's Algebra book online ...

Hawk

i didn't use the nonnegativity of the function.

honestly I got a bit lost in this conversation

Ted Shifrin

@Hawk: Well, without nonnegativity, you have a counterexample. So you have to use it.

Hawk

i mean if it is negative somewhere the half integral might be unbounded

Thorgott

I'm not sure what you mean by "reduce". If everything just reduced to linear algebra, representation theory wouldn't be very interesting.

user462942

10:50 PM

@Thorgott Yeah, I feel like that's how people describe representation theory ...

Ted Shifrin

@Hawk: I would suggest thinking more pictorially (draw pictures) and not just so much with symbols. If $f$ is nonnegative, and $n<m$ as you wished, how do you relate $\int_{-m}^n$ to $I_m$ and $I_n$?

Thorgott

as not very interesting?

user462942

@Thorgott no, as a way to understand Algebra research questions better, I think ...

Hawk

\int_{-m}^{-n} + \int_{n}^m + \int_{-n}^n = \int_{-m}^m

user462942

@Thorgott for questions that might otherwise be intractable, but I'm not sure. In a seminar, a mathematician says that there are multiple ways that people describe the field of representation theory -- incorrectly. So, I'm a bit confused ...

Ted Shifrin

11:01 PM

@Hawk: Can you write down one mathematical sentence (say with inequality signs) relating $\int_{-m}^n$ to JUST $I_m$ and $I_n$?

Hawk

You mean $\int_{-m}^n \leq \int_{-m}^m$ for instance?

Ted Shifrin

YES! And ..., on the other side, ... ?

(And do you see where you're using nonnegativity of $f$?)

Hawk

well if its negative (sometimes) its not certain we can do this. \int_{-n}^n \leq \int_{-m}^n

Thorgott

well, one tries to understand groups by how they act on linear spaces

Ted Shifrin

OK, so now you should finally see that you have a proof of what you wanted. Please try to write an actual proof (with a positive justification of statements) rather than saying that without the hypothesis, "it's not certain we can do this."

Thorgott

11:07 PM

so linear algebra plays an indisposable role, of course, but things don't just "reduce" to linear algebra; groups can be much more complicated after all

Ted Shifrin

indispensable?

Hawk

hang on sorry I got so mixed up with the intermediate problem-solving what exactly does the proof show? That \int_{-m]^m is enough to justify the improper integral exists?

Ted Shifrin

Yes. Which, when you relate the integrals over squares to integrals over disks, will show you how the improper integral in polar coordinates tells you about the regular improper integral.

Thorgott

both work (though I probably meant to say indispensable)

Ted Shifrin

I'm indisposed to believe you.

user462942

11:15 PM

@Thorgott I see

user462942

@Thorgott so is representation theory a subfield of group theory?

user462942

can people do research in group theory without representation theory?

user462942

or is representation theory indispensable for doing group theory?

user462942

The curfew starts soon, I better head out for dinner and a walk. Be back later.

user462942

Bye, everyone ...

Thorgott

11:27 PM

I'm no group theorist, so I can't say for sure. I'd wager that there are some things in group theory you can do just fine without representation theory, but I'd also wager that every group theorist probably knows a lot of representation theory and has used it extensively at one or the other point in their career.
I don't think it's a subfield of group theory. Representation theory is very interdisciplinary. By nature, it involves both group theory and linear algebra, so only classifying it as either of that would already be a misnomer. Then, a lot of representation theory focuses on, say, t…

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