Mathematics - 2024-04-18 (page 1 of 2)

12:58 AM

I'm thinking about offering a math prize like the able prize but significantly less money

2:01 AM

https://mathoverflow.net/questions/469414/automorphisms-and-mellin-transforms

Is this a good question? 🤔

AMDG

2:30 AM

Huh, pretty neat. $\left(\ln w+iw\right)e^{\ln w+iw}=z\ln z$ shows why $W_n(z \ln z) = \ln z$ over the reals yet again.

I will never get bored of the complexes

Quite satisfyingly, because of getting into complex arithmetic, I also solved $r=\theta$ as $-iW_n(ix)$.

~~There. The complexes are finally well-ordered.~~

is this intersection an open interval? I feel like it should be $(-1, 0]$, but visualizing the intersection makes me think it should be $(-1, 0)$

but this set is supposed to be closed, but maybe there was a typo in the notes or something

AMDG

Sorry I misspelled "canonical ordering" as "well-ordered".

Also clearly all $r=b\theta$ for $b\neq 1$ are fanfics. True fans recognize the canon lore.

robjohn

2:47 AM

@SillyGoose Ask yourself, "is $0$ in the intersection?" Is there any of the sets being intersected which do NOT contain $0$?

3:11 AM

hm i see so i guess they all should contain $0$

robjohn

3:22 AM

So the intersection contains $0$.

why is the euler mascheroni constant so important?

it is the limiting difference between the harmonic series and the natural log

Xander Henderson

@JohnZimmerman what do you mean by "so important"?

It naturally pops up in a few places---particularly in analytic number theory---but I am not sure that makes it important.

@XanderHenderson fair enough

EE18

3:39 AM

Hoping for someone to give me a hint as to how to proceed with the following:

The latter is what I have as yet. I've not been able to figure out how to use the hypotheses in order to show that the $k_i$ must be the same, which would be enough to prove my claim

But maybe my whole strategy is ugly and there's a much quicker and more elegant way to do this

Does the Paley-Weiner theorem apply to closed surfaces?

paley wiener applies to gaussian (rapidly decreasing)

gaussian is not closed.

closed=compact w/o boundary

sphere can be ruled out for obvious reasons

2d gaussian btw**

7:06 AM

@EE18 why k_i

leslie townes

@EE18 lots of ways to organize it. maybe prove by induction on N that if i and j are naturals with min(i,j) = N and h(i) = h(j), then i = j. for the induction step, if min(i,j) = N+1 and h(i) = h(j), then writing i' = i-1 and j' = j-1 we have min(i',j') = N and from h(i) = h(j) one can deduce h(i') = h(j') [why?] so that i' = j' by an inductive hypothesis and hence i = j.

The Empty String Photographer

By definition $h(n) = f^n(c)$. There is nothing to prove

Nothing to prove that such $k_i$ exist

8:56 AM

Quora, when seeing MSE: internal screaming

9:12 AM

let $M$ be a smooth manifold. let $x \in M$. i want to define the tangent space $T_xM$ as the space of all derivations over smooth functions at $x$ (with appropriate addition and scalar multiplication defined on the derivations).

I am a bit confused because wikipedia states that a derivation as $D: C^\infty(M) \to \mathbb{R}$, so it sends a smooth function $f: M \to \mathbb{R}$ to a real number.

Is that right? I would have thought that it should send the smooth function evaluated at the point $x$ to a real number, so $f(x = x') \mapsto Df(x) \lvert_{x = x'}$

Expressing the zeta labyrinth in terms of its automorphic representations and canonical geometric evolution dynamics

I think i dropped the ball with that title

oh well - at the end of the day it's just a question

Swan

@YourLordJoyBoy That was the idea behind the name. I had just finished that book when I created this account and that book was my whole personality for almost a month.

9:38 AM

@SillyGoose what does what you just said mean?

If a smooth function is evaluated, then this is just a composition with a function $\mathbb{R}\to\mathbb{R}$?

hm i see i guess that would not be what we want

is there a definition of pullbacks similar to the shown definition of a pushforward? The text I am using writes a definition like $(3)$, but it utilizes an inner product and that seems to be extra data than just the smooth manifold and its natural constructions

@SillyGoose what is the star supposed to mean

dual space?

the star $T^*_xM$ means cotangent space at $x \in M$.

yes

dual to $T_xM$

then $\alpha\in T_xM$

and $\beta\in T_{f(x)}^*M$

there's typos in the text

this is not an inner product but a dual product

$\langle T, x\rangle = T(x)$

its not called dual product, a second

oh oopsies that is my bad i made the typo

9:48 AM

Dual system

In mathematics, a dual system, dual pair, or a duality over a field K{\displaystyle \mathbb {K} } is a triple (X,Y,b){\displaystyle (X,Y,b)} consisting of two vector spaces X{\displaystyle X} and Y{\displaystyle Y} over K{\displaystyle \mathbb {K} } and a non-degenerate bilinear map b:X×Y→K{\displaystyle b:X\times Y\to \mathbb {K} }. Mathematical duality theory, the study of dual systems, has an important place in functional analysis and has extensive applications to quantum mechanics via the theory of Hilbert spaces. == Definition, notation, and conventions == === Pairings === A pairing or pair...

found it

anyway its a very canonical pairing between a vector space and its dual and it has nothing to do with Hilbert spaces or inner product spaces

ohh

you can consider it for Banach spaces no problem (for example)

is this literally just like if i start with a vector space $V$ then construct $V^*$ as the space of linear functionals over $V$, then by construction there is a product defined $v^*v$ where $v^* \in V^*$ and $v \in V$?

other than, indeed an inner product on a inner product space is a dual pairing in some sense, where the dual is isomorphic to the vector space

so its just a more specific context of this more general thing

@SillyGoose well, its literally evaluation of that functional at a vector

but the notation $\langle x, y\rangle$ is sometimes helpful because it does behave a lot like an inner product

good god a smooth manifold comes with so many natural constructions :P

9:52 AM

I don't know if it has anything to do with smooth manifolds, its just vector spaces really

oh i am just trying to learn a bit about smooth manifolds right now for a physics thing. and i am trying to construct all the necessary structures and see if each is a natural construction or if they require more data than just the manifold to specify

so far all are natural :P

well the list is not so long: (co)tangent bundle, pushforward and pullbacks of smooth maps, and exterior algebras of differential $k$-forms

@SillyGoose so here the mapping $f^*$ would be the transpose of the map $f_*$

there's a natural way to define, given $T:V\to W$, a map $T^*:W^*\to V^*$ given by $$\langle T^* \alpha, \beta\rangle = \langle \alpha, T\beta\rangle$$

this is known as a transpose in linear algebra

ah :D

or dual map maybe

i am trying to build up to learning about the holonomy of a connection...I am just about at starting to learn what connections are

9:57 AM

Then you can say, do it again and obtain $T^{**}$ and so on

and if $V, W$ are finite dimensional then $V\cong V^{**}$ and $W\cong W^{**}$ in a natural way, and we have $T^{**} = T$ for an appropriate interpretation of that equality

@SillyGoose I don't know any of that. I never had a course from differential.. well, anything

Thorgott

10:39 AM

@Jakobian It's not a typo, writing $\langle-,-\rangle$ for a pairing of vector spaces is something we do on occasion in topology or geometry.

Though agree it can be potentially confusing if it's not explained

11:29 AM

@Thorgott uhhh. I never said it was?

and pairing of vector spaces is literally something I've been explaining

Thorgott

ok, I didn't get what you mean then

two messages above me saying "there's typos in the text" is what I was referring to

Q: Understanding where my approach is incorrect

11:54 AM

-1

The gist of this approach is to encode the zeta function into a space, similar to how number theorists like to encode sequences into generating functions to leverage analytic properties. I will refer to this construction in general as the $\zeta$-labyrinth, and the main idea here is to link the a...

I will remember that downvote

weird how the previous question got +15 lol

And I will take no prisoners.

I think it's just so foreign looking that people don't know how to react

12:20 PM

@JohnZimmerman You wrote "There must be flaws in my reasoning perhaps fatal ones. Which part of this seems the least rigorous? I want to identify where to put my energy first."

The purpose of this site is not to peer review your proposed solutions