Mathematics

4:00 PM

Do people here have a preferred notation for derivatives?

@AkivaWeinberger hmm, I wonder if this will work: $\sin x = 0$ has solutions $x = n \pi$ for all $n \in \Bbb{Z}$ and then $n\pi$ is nonzero since $3< \pi < 4$ can be proved and $3n < n \pi < 4n$ for all nonzero $n$

s.harp

there are so many things to chose from and when I'm starting to write something it frustrates me

Akiva Weinberger

@Secret You don't want $n\pi$ to be zero, you want it to be an integer

and the range $(3n,4n)$ contains many integers

anakhro

@s.harp what class?

Akiva Weinberger

@s.harp For LaTeX $y'$ is the easiest

anakhro

4:01 PM

Usually it is personal preference, but often there is a nicest looking way.

But it's kind of dependent on what you are doing.

s.harp

differential geometry

Érico Melo Silva

@s.harp why does it matter

s.harp

I used to like stuff like $\varphi_* $ but if you try to do that with a group action and add the point it acts on it looks like $R_{g,*,x}$ which is trash

Érico Melo Silva

just do what works best for whatever situation/what u like most and causes least confusion

Ted Shifrin

Heya @Eric, DogAteMy, @anahkhro, @s.harp ...

Érico Melo Silva

4:03 PM

hello Ted

s.harp

@ÉricoMeloSilva but thats the thing isnt it, when I look at the choices they all look messy

Ted Shifrin

Sometimes, overly pedantic notation just isn't worth it.

Érico Melo Silva

eh it's not that big a deal i think

Ted Shifrin

<--- has no idea what's being discussed

You back in Chi-land, Eric?

Érico Melo Silva

nah im in berkeley still

anakhro

4:04 PM

Maybe use $T_p$ in that case, @s.harp

Ted Shifrin

Oh.

Érico Melo Silva

I'm going back in a week bc this visit overlapped with my spring break

anakhro

So it is $T_p(R_g)$.

Érico Melo Silva

im done with my visits though

Ted Shifrin

I'm sure you're disappointed :)

Érico Melo Silva

4:05 PM

im very tired of asking the same 5 questions to 5000 people

the free hotels and food were of course very nice

Ted Shifrin

Perhaps you exaggerate just a little bit.

Érico Melo Silva

yes but only a little

anakhro

@TedShifrin I think I figured out the tangent bundle thing shortly after you left but then I woke up this morning thinking I did it wrong. If we have the coordinates $x_i$ on the base and $y_i$ on the fibre of $TM$, the tangent space $T_pZ$ of the zero section is the span of the partials of the $y_i$ or the $x_i$?

I thought it was $y_i$ initially

But then that doesn't make sense for the tangent space of the fibre

Or does it

Ted Shifrin

No, $x_i$, of course. The $y_i$ variables go vertically. As I said, thinking about a general vector bundle avoids a lot of confusions.

Érico Melo Silva

do any of y'all know if there's a thing for physics departments akin to the AMS group rankings?

Akiva Weinberger

4:16 PM

@Secret If you wanna approach Niven's proof

let $f(x)=x(x-\pi)$

and look at $\int_0^\pi(f(x))^n\sin x~dx$

Secret

Ah I read and discussed that months ago with the number theorists here already.

Akiva Weinberger

That's definitely positive

and you can show it's an integer multiple of $1/b^n$ or something like that

where $\pi=a/b$

Semiclassical

Hi chat

Akiva Weinberger

I forget the details

Somehow you get a contradiction

Secret

The random is I thought I found a very intuitive proof of irrationalilty of pi, only to forget that I need to show sin (p) is not an integer

and yup, Niven's proof is one of the examples of auxillary function proofs in transcendence theory

The function f is specially chosen so that it dies in just the right way to trigger the contradiction

it is somewhat related to the diophataine approximation proof strategies, but not quite the same

whereas diophatine approximation proofs rely on chopping out the tail of the difference between the number y and any rational, and show that difference dies much faster than if y is rational, triggering the contradiction

Semiclassical

4:22 PM

It’s not really the same thing , but this reminds me of nothing so much as exp(pi sqrt(163)) being very close to an integer value

Secret

exp(pi sqrt(163)) is transcendental, right?

anakhro

@s.harp on second though, I can't think of a really bad spot for $T_pf$ notation for the derivative of $f$, other than the fact it is in general more clunky than something which only uses a subscript, or only uses a letter out front.

Semiclassical

I’m not sure it’s proven to be transcendental—so few things are—but there’s no good reason to expect it to be algebraic

Secret

I see

s.harp

@anakhro I decided to just take $d F$ for the derivative, this sometimes conflicts but it is the simplest right now

Semiclassical

4:26 PM

Actually, the Wikipedia page on the Heegner numbers (to which that exp is related) says that it is transcendental

Akiva Weinberger

It's $i^{i\sqrt{163}}$, and I'm pretty sure it's been proven that an algebraic raised to an algebraic power is transcendental

under certain conditions which I forget

Exponent should be irrational

anakhro

Yeah when you do differential forms, you might change your mind, @s.harp

DF might be a little safer.

Semiclassical

@AkivaWeinberger better have some conditions, lest one runs into the usual proof of there being a rational number of the form irrational ^ irrational

s.harp

Indeed, but $DF$ has a meaning for me that I am loathe to overload

anakhro

on the otherhand, you can invent new notation

Semiclassical

4:34 PM

But the Lindemann-Weierstrass theorem does suffice here

Akiva Weinberger

Oh that's what it's called?

Oh

No

Gelfond–Schneider theorem

In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. == History == It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider. == Statement == If a and b are algebraic numbers with a ≠ 0, a ≠ 1, and b irrational, then any value of ab is a transcendental number. === Comments === The values of a and b are not restricted to real numbers; complex numbers are allowed (they are never rational when they have an imaginary part not equal to 0, even if both the real and imaginary parts are rational). In general...

I want this^

Semiclassical

L-W implies e^a is transcendental for nonzero algebraic a

Ah, so sqrt(2)^sqrt(2) isn’t algebraic

Hadn’t accounted for that

Akiva Weinberger

OK so guess what

No ideas for Secret Santa yet again

so I bought what I bought last year

which is a dry-erase board and some markers

and maybe I'll convince him to tape it to the wall in his dorm room or something

or on his door

so his roommates/suitemates can mess with it also

vzn

4:53 PM

in The h Bar, 2 hours ago, by vzn

Abel prize winner Uhlenbeck 1st female winner + noted for contributions to mathematical physics eg gauge theory http://www.abelprize.no/c73996/seksjon/vis.html?tid=74011&strukt_tid=73996

anakhro

@TedShifrin why don't you like calling the zeros of vector fields singularities? Just wondering, as I recalled you sighing about it a little while back.

Ted Shifrin

5:08 PM

They're a singularity in the sense that the one-dimensional line bundle given by the vector field has no fiber there. But to me a zero of a function is not a singularity.

anakhro

5:25 PM

So you would prefer to call them "zeros" rather than singularities?

Ted Shifrin

Yup indeed.

The "singularity" terminology is a remnant from thinking of slope fields in classical differential equations.

anakhro

Do you know of any good references on 2D geometry of vector fields?

Like the whole elliptic/hyperbolic zero stuff?

I thought Robinson had it but he doesn't.

Ted Shifrin

My favorite sophisticated ODE book is Hirsch-Smale.

anakhro

I will have a look there to see if I can find it.

One thing that bugs me is that hyperbolic tends to mean two different things.

Either that the singularity is generically stable (eigenvalues have non-zero real parts), or that its eigenvalues have different signs for the real parts

Wow I didn't realize my question about linearisation got a downvote. :(

Oh it was just yesterday, too.

@TedShifrin would you be fine with me answering it with the explanation yielded by your comment?

I wouldn't accept it, as I was going to wait to see if anyone else had anything to say.

user123

5:47 PM

Hi , im studying representation theory and i got this question - does $Q_{2^n}$ has a faithful one-dim. rep. ?

Ted Shifrin

I haven't even seen the question.

user123

over $\Bbb C$

anakhro

@TedShifrin it was the one that spawned the discussion about $T_{(p,0)}TM \cong T_pM\oplus T_pM$.

Ted Shifrin

You need to ask our rep theory experts, @user123, who aren't here right now. Wait for @MatheinBoulomenos or @TobiasKildetoft.

LOL, oh, @anakhro. :)

vicious spawn

user123

alright :P thanks!

anakhro

5:49 PM

Indeed.

Transcript for

$Mathematics$ Mathematics