Hey everyone!

@TedShifrin I'm not sure if this matters, but I don't know that $\sup \{a_k \mid k \ge n \}$ converges to the same limit as $a_n$. In fact, the result I am trying to prove now I am going to use in my proof that $\sup \{a_k \mid k \ge n \}$ converges to the same limit as $a_n$.

Hi Demonark and Alessandro.

@Daminark Harro eburybuan

@user193319: I surmised you were trying to show that limsup = lim when the sequence converges. So you can't use it, no. But you need to use the fact that the sequence converges in your proof. How can you do that?

Okay. I just wanted to make sure that the result I am currently trying to prove doesn't rely on the other. Let me think about it a little longer.

Do you know something equivalent to convergence for sequences of real numbers?

That they are Cauchy? Bounded?

Sorry...not bounded.

But convergence is equivalent to being cauchy

How's it going Ted and Balarka?

Maybe that's helpful, @user193319.

@Balarka Peter said to polish up what I wrote but to leave it at that

Balarka is on another never-sleeping binge, I assume. I'm doing fine. :)

I'm trying to get some physics done

@Daminark Ah ok

So there's not much work after this?

Yeah, just a bit of reorganization and editing. I did find a paper when searching around about symmetric products of algebraic curves, but right now AG is just Sanskrit to me so I'm gonna flag that and return to it once I'm more ready. Between that and maybe reading up to be sure I complement understand D-T, I'm basically done now, so time to start a new thing

Cool

Hi @Ted

Also I found a paper on something called "Equivariant Dold-Thom" but I'm not even sure what that means

prolly in the equivariant homological setting

> time to start a new thing

Gotta go Forster

FORSTER FORSTER FORSTER

The point is I'm not sure what equivariant even means, just in general

And lol Forster wouldn't be a bad idea actually

:thinking:

It means you have a G-action

I see

And you want whatever you are doing to respect that G-action

eg a map f : X --> Y of G-spaces is G-equivariant if f(gx) = gf(x)

Ah, that sounds pretty cool

a CW structure on X is G-equivariant if G acts cellularly

i.e., G permutes cells

Oh so that's why you were talking about the $S_n$-equivariant CW stuff yesterday

ya

equivariant topology is a big thing

Mike does that

Nifty

So you should understand how to compute cohomology using invariant forms :)

Mhm, I should learn that someday

hey @robjohn good to see you again!

Invariant forms?

If you have a $G$ action on $M$, is every closed form cohomologous to an invariant form?

oh wow, @robjohn exists?

yes, since a very while.

Cool question. I guess I need to do an averaging constriction?

@Demonark, if $G$ acts on $M$ (say by left multiplication), then a $k$-form $\omega$ on $M$ is $G$-invariant if $L_g^*\omega = \omega$.

and then ask if the average is cohomologous to the form

or something

So you might want $G$ to be compact, for starters.

True

If it's not, averaging might be difficult. But it's worth exploring.

is there an analogue to the usual averaging trick for noncompact groups

ive never seen that

Can you have a bi-invariant volume form on a non-compact group? Related question.

Tanegntial, but I remember you can't have a bi-invariant metric on GL_n(R)

Bi-invariant volume form is weaker than bi-invariant metric.

Right, I'm thinking about that

Well you can have a bi-invariant metric on a noncompact group so

@EricSilva: Is it true that with respect to an invariant metric (or volume form?) that $G$ has finite volume iff $G$ is compact?

Hmm, what's an example of a bi-invariant metric on a noncompact group?

$\mathbb{R}^{n}$

anything like K x R^n

where K is compact

Oh, duh.

in fact those are the only ones IIRC

yeah that's true

Ugh I get distracted by math when I do physics and physics when I do math

I have to get work done

Change your potential energy and work shall be done.

i think it's enough to have semisimplicity

math.stackexchange.com/questions/2562232/… someone?

@TedShifrin At the cost of decreasing my kinetic energy I am afraid

I shall think about your invariant forms question

But you're at rest, Balarka.

@TedShifrin hi :)

and report back

Hi @Liad

for now, off I go

Bye.

I don't do measure theory, Liad.

:(

Alessandro likes it. Eric knows it.

i got something in complex analysis bothering me too.

How egalitarian of you.

@Ted I think that to get a bi-invariant volume form you just need semisimplicity, cause you can get a left-invariant form, and then get right-invariance by looking at the adjoint representation

@EricSilva: Does that argument work for metrics too?

maybe you want connected

@TedShifrin i want to show that this term in the picture goes to zero (in absolute value)

something about it isnt gonna work for metrics

i havent crystallized the argument in my head yet

$P$ is poassion kernal formula

cute typo, Liad

hm where?

well, misspellings everywhere, but poisson turned into passion .... sorta.

:P well remember this is not my language :)

@Abcd: Are you using $\log(x+y) = \log x + \log y$??

@TedShifrin No, I just took log of both sides of the equation.

lol

doesn't it needs to be $P(r e \ ^ {i\theta} , e\ ^ {it})$ @TedShifrin ?

How did you take log of the left side?

@TedShifrin I took log of first term then log of second term.

i wonder what the world would look like if log were additive

we'd probably all be dead

:P

I think Abcd is dead right now.

@EricSilva How?

@Abcd But thats $\log(x + y) = \log(x) + \log(y)$

I was making a joke

@TedShifrin Hmm , I have misused log.

Uh huh.

you think ?

@TedShifrin Any other method to solve the problem?

Be careful when misusing logs. A tree might fall on you.

Try using $\cos^2x+\sin^2x = 1$?

@Liad: You want to show it goes to $0$ as what happens?

@ted the sin and cosine are exponents

I understand that, @Abra.

@TedShifrin I used that and got: $2^ {\sin^2 x - 1/2}+ 2^ {\cos^2x -1/2 }= 2^ {\sin^2 x + \cos^2 x}$

@TedShifrin $r \to 1$

Incorrectly, @Abcd?

but i think there is a mistake with the $t$ in $P(..,t)$

do you agree?

@TedShifrin No, just divide the entire equation by $\sqrt 2$

$2^{\sin^2 x}+ 2^ {\cos^2x}= 2\sqrt 2$, divide by $\sqrt2$.

Oh, go back to the original, Abcd. Stop it with that.

try using euler form

What if you let $u=2^{\sin^2x}$?

@Liad: I haven't thought about Poisson kernel in years. Ask Eric.

I don't remember the notation, in particular.

@TedShifrin $\dfrac{2}{u}+ u = 2\sqrt 2$

Precisely. Now you can finish.

Yes, thanks.

Anyone familiar with quadratic forms over the p-adics?

@TedShifrin so using the adjoint rep just gives you nothing useful when you try to apply

Why is that, Eric?

Back later.

Whether $x_n$ has an infinite limit or finite limit, is it true that $\displaystyle \lim_{n \to \infty} x_n \le \limsup_{n \to \infty} x_n$?

@TedShifrin barely

@user193319 surely

someone can take a look at my question here :math.stackexchange.com/questions/2562232/…?

Has anyone here gone through the proof of noether's theorem?

Which one?

I didn't realize there were multiple. The one regarding the connection between symmetry and conservation laws

are there multiple versions of that one specifically?

hi everyone

@David Noether has done some stuff in algebra (Noetherian rings), so it's plausible that there's a Noether's theorem in that context

Ah yes

Suppose we have a matrix $A$ of coordinates of points in space, where each row represents the coordinates of a point. Does anyone here know if the first right-eigeinvector of $V$ of the SVM $USV=A$ of this matrix $A$ is particular, in some sense?

I'm interested in pursuing this just because of its landmark status historically and its aesthetics. However it appears I would have to go through the langrangian formulation of mechanics. I'm curious as to how far through this I would have to go.

anyone aware of calling solutions to PDE's formal?

So I don't know the proof but the formal statement is quite tricky

@DavidReed You have to know what a lagrangian is, what an action is, how we use calculus of cariations

I think also it requires a bit of symplectic geometry?

@Felix.C Usually you call a solution to an ODE or PDE formal if you can write down an expression for it, but the expression makes sense only under certain unstated conditions. Or we might call a solution formal if it involves some other unknown function which cannot itself be easily found

@Kevin If you had to quantify the workload prereq wise in terms of how far into lagrangian mechanics I would have to go prior to pursuing this theorem in terms of a typical "semester's worth of material", how many semesters would it be?

In my book I have that if a map $x$ is continuous and regular and one-to-one, then $x^{-1}$ is continuous. My question: is the $one-to-one$ part a requirement or not. It's not stated clearly in my book. I can read it as either a requirement or a result of the previous two conditions.

@KevinDriscoll Ah ok..many thanks..

math.uchicago.edu/~may/REU2017/REUPapers/Hudgins.pdf @David

I saw a brief talk on Noether's theorem, and the person who gave the talk wrote this paper

@Daminark Thank you!

From that talk, I remember the formal statement of the theorem basically had to do with a Hamiltonian vector field (satisfies some differential equation involving its energy and the symplectic form), and you had a Lie group acting on the manifold

Yes it's a big deal. Einstein referred to her as the most important woman in the history of mathematical physics for her proof of it

Very appreciate of your help

But yeah I think if your Hamiltonian vector field is invariant under a Lie group action with some momentum mapping, then that mapping is an integral of the vector field or something. This goes through some of the math and physics involved in that, though I'm not sure how accessible it is

I'm not worried so much about the math prereqs--more the physics side of it

@DavidReed nope, I was thinking about Noether's normalization theorem in algebra

No worries, mathematicans generally think of her contributions in terms of ACCP and the like.

How do you comprehend this? Is the one-to-one a requirement or a result of conditions 1 and 3?

Can you send the conditions?

@Felix.C What's the context? "Formal solutions" is a terminology you use in the context of partial differential relations and h-principles, as far as I know.

@Daminark These REU papers are a great place to pick stuff up.

I should try and look for some which I would want to read

Yeah it's good shit

@TedShifrin ...but will it be done by you, or be done by the environment? :P

@DavidReed From the physics perspective.... I'm really not sure. You odn't really need a whole semester of E&M to get to Hamiltonian and Lagrangian dynamics. But you probably do need Physics 1 concepts like momentum, energy, etc. Otherwise you had no clue where the Lagrangian/Hamiltonains are coming from

I have a year of undergrad physics

So really just the formulation of lagrangian mechanics would be it

@DavidReed Ok yea, then that's at most a few weeks of thinking. You just need enough calculus of variations to understand the derivation of the Euler-Lagrange equations, plus doing some examples where you can work out particular Lagrangians

@BalarkaSen PDE's = partial differential equations...Kevin gave a already a nice explanation

I know. I am just saying formal solutions mean a lot of different things.

context!

sends Balarka back to his physics room

Yea, get outta here you nonsense spouting wrench toting physics person!

Done with physics for now

don't get torqued up now

@KevinDriscoll Physics is rubbish. I believe in a new kind of science.

grabs popcorn

Figure out what quadratic differentials have to do with semiclassical methods.

@Semiclassical I do all my work without torsion

that's my challenge :P

though semiclassical in this case really just means "WKB"

Went ot the wiki page. Saw the name 'Teichmuller'. Noped the FUCK out.

lol

Teichmuller is great

not as a person, perhaps

yeah

@Balarka don't leave me in suspense, what's this new kind of science?

It's a new kind

OH WOW! That dude was a Nazi too? I had no idea. I only know of Teichmuller due to its association to Mochizuki

And I dont wanna be within 10 lightyears of what that guys doing

yup he was a dedicated nazi

0/10

but Teichmuller theory is like a solid 9/10

just says there's no correlation between you as a person and the math you do

Helped organize a boycott of Landau’s lectures because he was Jewish

@BalarkaSen i wonder if there is a correlation tho

Do we get anything in physics directly from Nazi scientists? I'm not aware of anything named after a Nazi, but Im also now sure who was a Nazi and who wasnt

@Eric i don't believe it

@Balarka Look at the paper by Reszutek

Ik that apparently there's some slight tendency for conservatives in academia in the US to gravitate towards certain fields

@Daminark That link doesn't work

more towards business and engineering, Eric

I just clicked on it and it did work

@Semiclassical Were any of those rocket guys legit Nazis?

straaange

can't remember where i read it exactly but i think geophysics was one of the things that had disproportionate amounts of conservatives compared to other part of academia

Not sure tbh

Try again. If not, look at the 2013 REU

hmm, I hadn't heard that

it might be a different field

it was something with a geo in it

“The V-2 assembly plant at the Mittelwerk, near the Mittelbau-Dora concentration camp, used slave labor, as did a number of other production sites. Von Braun was a member of the Nazi Party and an SS officer, yet was also arrested by the Gestapo in 1944 for careless remarks he made about the war and the rocket. His responsibility for the crimes connected to rocket production is controversial.”

@Daminark Is this the commutator paper?

So he’s a bit complicated

Yes. Look at the title

lmao

I think it’s less NASA employing former Nazis as them employing German scientists who collaborated with the Nazis

More here: smithsonianmag.com/smart-news/…

I am working on a problem from Rudin's RCA. I just showed that a certain collection of sets forms a $\sigma$-algebra and that certain function is a measure on this $\sigma$-algebra. I am now asked to describe the corresponding measurable functions and their integrals. My question is, what am I assuming is the codomain of these measurable functions>

?

Usually I'd guess $\mathbb{R}$

hi everyone, what does similar linear tranformation mean? I know similar matrices signifies same linear tranf. under different basis, but the notion of linear tranf. is independent of basis, so what would it mean?

@ManishKumarSingh two linear transformations f and g are similar if there is a bijective linear transformation h such that f = hgh^-1

yes i know the definition, i want to know does it signifies anything

like they do in case of matrices

@Daminark Okay. I'll give it a try. Thanks!

winterbash2017.stackexchange.com

27 minutes!

What's winterbash about?

@ManishKumarSingh hmm

@Daminark hats on avatars

Its all about hats. Everything is hats now. The economy runs on hats.

so hats like in Fourier transforms?

@Mathein no u

wheres my hat

where is it

Over there

@BalarkaSen Its in the Math Stack Excahnge loot box. It only costs 1,000 karma, or $2.

I can give you a paypal e-mail if you want to purchase some

hat neutrality

HAT NEUTRALITY

thats better

Caps lock makes everything better

you mean

CAPS LOCK MAKES EVERYTHING BETTER

@MatheinBoulomenos what do similar linear transformations mean?

can anyone help me with this problem I don't know how to start it Im given a function $1/2x^2+3x$ and interval [-7,0], I'm told to find the rram8 from 0 on top to -7 on bottom. I have a few questions how can I find the delta x usually its $b-a/n$ what is n in this case is it 8? and how can i find f(xk)?

@Kevin orthogonal?

@KevinDriscoll but what do they geometrically mean? btw the definition I see is invertible instead of orthogonal

similar linear transform make sense even if you don't have an inner product

yes, but what do they mean?

the meaning is the same as for matrices: they represent the same linear map after a base change

Oh ya sorry, changing context. Im used to only caring about arestricted class.

@MatheinBoulomenos but what if we don't want to consider basis?

@LeakyNun Let $A: V \to V$, $B: W \to W$, and $P: V \to W$ invertible, if $A = P^{-1} B P$ then you can find the action of $A$ on any element of $V$ by first using $P$ to go to the corresponding element in $W$, apply $B$, then go back to $V$. So in a certain sense $A$ and $B$ are the same. You can model the action of one by the action of the other.

yes, but what if we don't want to consider basis?

I never said anythign about a basis

Yay, a full quarter of my final was on ln(1-x)ln(x) in [0..1], which I did a whole thing on for an extra credit assignment in another course years ago

oh well

:) so I at least got a 25 lol

but then what is the geometric intuition?

Im just talking about their action on abstract elements in the vector space. No basis required.

like

linear transformations transform gridlines

if you're talking about gridlines, you're thinking about a basis