Mathematics - 2019-11-18

ÍgjøgnumMeg

12:09 AM

@Leaky what's wrong with having different norms? :o

Leaky Nun

@ÍgjøgnumMeg nothing it's just confusing

ÍgjøgnumMeg

Fair :) We're actually looking at $K \otimes_{\Bbb Q} \Bbb R$ in ANT now

ffs

Leaky Nun

deja vu

ÍgjøgnumMeg

it's very cool lol

Mason

3:12 AM

@ShineOnYouCrazyDiamond. Of course, I can apply the formula but my question is if the software has a tool already. Doesn't make sense to rewrite code when you can just use the tools that have already been built out.

CowperKettle

4:34 AM

eurocoder

6:37 AM

Is lim_{x->c} f(x)^{g(x)} = lim_{x->c} f(x)^{lim_{x->c} g(x)} ?

BaiduryaMathaddict

8:56 AM

@eurocoder yeah that is so

The basic thing about limits in general, is that if the argument (i.e. x) tends to a certain value, it does so in every part of the function

Even if there is a composition of functions whose limit is required, you could break it into sub-functions whose arguments all tend to the same value and find their individual limits and arrive at a conclusion

Slereah

10:47 AM

Hello the math

I have questions on cut locus

Paper I'm reading (homepage.univie.ac.at/james.grant/papers/NullInj/…) claims that there's only two ways a geodesic can be non-minimizing

1) another geodesic of the same length exists
2) the two points are conjugate points

But isn't it possible to have such a thing due to topology, ie on the circles $[0,1]$, the curve $p = 0$ to $q = 3/4$ isn't minimizing

Or are such points always conjugate points?

Wrongapedia claims that on $S^2$ only antipodal points are conjugate so it wouldn't seem so

Astyx

The circle is $S^1$

$S^2$ is the sphere

Slereah

11:02 AM

I know, but the same argument applies there as well

Astyx

What does conjugate mean ?

Slereah

Any curve going 'round more than $\pi$ in the azimuthal angle isn't gonna be minimizing

@Astyx There's a non-zero Jacobi field $J$ such that $J(\gamma(0)) = J(\gamma(1)) = 0$

Astyx

The curve from $p = 0$ to $q = 3/4$ given by $\gamma(t) = 3/4t$ goes through $\gamma(t_0) = 1/2$ when $t_0 = {2\over 3}$

And 0 and $1/2$ are antipodal

In $S^1$

Slereah

Yes.

Astyx

I think you misread the paper

they mention $\gamma(0)$ and $\gamma(t_0)$, not $\gamma(1)$

Slereah

11:16 AM

Hm

Let's see

Slereah

12:08 PM

ah yes you're right

The antipodal point is the limit beyond which minimizing is lost

Now to find the actual proof of this

Apparently it's from Whitehead

Written in 1933

Yikes

Gonna have to translate it a bit in modern notation

It's old enough that he uses coördinates

ÍgjøgnumMeg

1:00 PM

afternoon y'all

Slereah

hello

ÍgjøgnumMeg

Wassup :)

Slereah

not much

looking at some cut locus business

ÍgjøgnumMeg

bleh

Slereah

alas

Can't be $\mathbb{R}^n$ all the time

ÍgjøgnumMeg

1:04 PM

sad

Slereah

If the universe was zero dimensional things would be a lot easier

God I hate the universe

ÍgjøgnumMeg

yeah it's pretty lame

Ultradark

1:27 PM

the universe is idiot

it's very low quality

extremely stupid

and just stupid

0

Q: What is the growth rate of $h(n)?$

$$h(n) = \#\{ \pi(x)\pi(n-x),x\le n\}$$ What is the growth rate of $h(n)?$ Prime counting function is in light green for reference Updated plot: It seems to grow about the same as $\pi(x)$ for this sample set of points, which makes sense because $f(x)$ is defined using the multiplication ...

One of the best questions ever

@tigre you really need to work on your mathematical precision

ÍgjøgnumMeg

@ultradark https://math.stackexchange.com/questions/74347/construct-a-function-which-is-continuous-in-1-5-but-not-differentiable-at-2/74383#74383

is the best answer on the site

Ultradark

okay that answer is pretty good

Secret

1:50 PM

Yet another attempt at defining finite numbers starting from the absolute infinite:

(because the Zohar does that, so why don't set theory)

ÍgjøgnumMeg

Don't mess with the Zohar

Secret

aw come on, Ayin is really really cool stuff

Slereah

it's not peer reviewed

Secret

http://www.esotericonline.net/group/kabbalah/forum/topics/ayin-the-concept-of-nothingness-in-jewish-mysticism-by-daniel-c-m
This linguistic professor translates the whole thing into english

ÍgjøgnumMeg

way too heavy for my ailing mind

hey I got 72% of the points on my last pset

cheering

Secret

1:54 PM

alright, will do that elsewhere

cannot think of any earth level maths atm

Slereah

Addition

is one

Secret

link.springer.com/chapter/10.1007/978-3-319-97148-3_32

ebooks are paywalled though, but it seems addition is still a major topic in mathematics education

Slereah

One thing I have never actually done is number theory on decimal numbers

I don't know why the various tricks for multiplication and division work in decimal representation

Why are multiples of 3 such that $\sum_i d_i$ is a multiple of 3???

A mystery

Secret

I recall a long time ago, Akiva shared a paper which basically gives the algebra behind carryovers and stuff under a given base b, which sort on touched upon that topic

ÍgjøgnumMeg

@Slereah because $10 \equiv 1 \bmod 3$

Slereah

2:06 PM

@ÍgjøgnumMeg Sounds reasonable enough!

I'm guessing they're not terribly hard to prove, but I just never looked into it

Secret

link.springer.com/article/10.1134/S0081543819030027

ÍgjøgnumMeg

I mean the point is that in base 10 you can write $k = k_0 + k_1\times 10 + \dots + k_n\times 10^n$

Secret

Random elliptic integral stuff found by google scholar "addition" + year 2019

ÍgjøgnumMeg

so modulo 3 you get $0 \equiv k_0 + k_1 + \dots + k_n \bmod 3$

Slereah

Indeed true

Secret

2:08 PM

You could do the same with 9, right, since $10 \equiv 1 \mod 9$?

ÍgjøgnumMeg

right

Secret

One thing I remember back in primary school is that the unusual property of 9 in base 10 allows you to make a lot of weird number results

like 1111^2 11111^2 etc. patterns

ÍgjøgnumMeg

yeah those are cool

there are lots of videos about such results on numberphile :P

Secret

yup

Rithaniel

4:32 PM

Actually, I'm pretty sure $1111^2=1234321$ in all bases 5 or higher, provided we're starting and ending in the same base

Ted Shifrin

Yes, @Rithaniel, of course. I think Secret is thinking of the binomial theorem for $(10+1)^n$ or something.

ÍgjøgnumMeg

Heya @Ted and @Rithaniel

Ted Shifrin

hi @ÍgjøgnumMeg

I thought by now you'd have a German name :P

ÍgjøgnumMeg

Not yet hehe

I like the faroese name

Ted Shifrin

4:48 PM

Well, yours is the only name I can't type (or abbreviate).

ÍgjøgnumMeg

I take pleasure in this minor inconvenience

Ted Shifrin

I figured you would.

See if I help you again!

ÍgjøgnumMeg

uh oh!

Alessandro Codenotti

5:29 PM

Typing @ alone has @ÍgjøgnumMeg as first suggestion though

(and Érico Silva as second, I guess the chat software doesn't understand accents)

ÍgjøgnumMeg

5:49 PM

Nise

Hi @Alessandro

Leaky Nun

@ÍgjøgnumMeg servus

ÍgjøgnumMeg

@Leaky meinst du servus? :D

okey hahaha

Leaky Nun

:P

@ÍgjøgnumMeg hast du Dirichletscher Einheitensatz gelernt?

ÍgjøgnumMeg

Werden wir in der nächsten Vorlesung sehen glaub ich :)

Leaky Nun

nice

ÍgjøgnumMeg

5:58 PM

Heute haben wir die Endlichkeit der Klassenzahl bewiesen

mit einer guten Schranke (wo ich das das erste Mal gesehen hab hab ich voll die schlechte Schranke benutzt :D)

#KeineBeistriche

Leaky Nun

Schranke?

ÍgjøgnumMeg

Bound

tigre

7:13 PM

@Ultradark

This is a very low quality post. Delete everything and just type "for each 𝑛, let ℎ(𝑛)=#{𝜋(𝑥)𝜋(𝑛−𝑥),𝑥≤𝑛}, what is ℎ(𝑛)'s growth rate ?".

I literally said this to you days ago

@Ultradark Why repeat this back to me

WeavingBird1917

Just a soft question, are conic sections (, complex numbers, quarternions considered "elementary" algebra? I'm looking for a book recommendation question

Sorry, clicked enter on iPad prematurely.

WeavingBird1917

7:35 PM

Sorry, I can't send a new message on the iPad and can't delete either. On computer now.

Are conics sections + loci, complex number and quaternions, techniques for solving higher order roots considered "elementary" algebra? I'm looking to polish my HS algebra + calculus before doing physics in uni. I've checked elementary algebra lists, but they are too basic. Is there a specific question/term I should be looking for which covers those topics?

Ted Shifrin

No, @WeavingBird1917, definitely not elementary algebra. Quaternions barely appear in the undergraduate college curriculum, and theory of equations went out of the curriculum over half a century ago. Some precalculus courses cover conics, but the best way to learn a lot of that material is with linear algebra.

Astyx

Hi chat

WeavingBird1917

Thank you. Conics, loci and complex numbers are scattered throughout my textbook (a general textbook for the maths course at my HS), but the problem is that it wasn't quite in depth, and I'm not sure what textbooks to use now for self study. There were higher order root solving, but these were quite "rigged" (we're usually given a solution, told to find the others). Is there an all encompassing area of maths to describe these topics?

I have a linear algebra textbook I tried for a while, but it covers equation solving mostly, and other linear algebra "stuff" (maps, determinants, similarity, vector spaces). I'm mainly trying to learn calculus now (based on an SE textbook recommendation) and polish the areas above.

psa

7:51 PM

@TedShifrin might have it in his LA book?

it's called "a geometric approach"

so I'd assume so

WeavingBird1917

8:20 PM

I think I'll just plough through Spivak's Calculus and it might polish those areas along the way :P

psa

@WeavingBird1917 Calculus on Manifolds?

WeavingBird1917

Just "Calculus" @psa

psa

8:42 PM

@WeavingBird1917 I love that book.

tigre

@Ultradark Also, somehow it seems like trying to study the distribution of primes by multiplying prime counting functions is insane... An easier way to understand them would be the product $1\times \pi(n)$ right? It's quite circular

ROODAY

Quick/random question, if I wanted to reflect a set of 3D coordinates (as if they were seen through a mirror), and if the origin was the center of the image, I'd just flip the sign of the X value for each coordinate. If the origin is somewhere else, like say a corner of the bounding box these coordinates are in (for context I believe all my data is within a 1 meter cube), how would I go about doing the same reflection? (sorry if this is a trivial/silly question)

WeavingBird1917

@ROODAY It depends what axis you want to flip the coordinates over, doesn't it? But you would translate all the points, such that the corner becomes (0, 0), perform the reflection, then translate everything back.

Ultradark

@tigre I'm simply trying to understand the pattern of this sequence, it's really interesting

$$0,0,1,1,2,2,3,2,4,3,4,3,5,4,5,4,6,5,7,5,8,5,7,5,8,7,...$$

and it's one of the least understood sequences in all of mathematics

tigre

9:10 PM

@Ultradark I'm starting to think you're just trolling

Felix Crazzolara

Does someone here know by any chance, whats the dimensionality of the positive semidefinite cone of nxn matrices?

ROODAY

@WeavingBird1917 Ok, thanks!

ÍgjøgnumMeg

If anyone modular formy is about: let $M_k(\Gamma)$ be the space of meromorphic modular forms of weight $k$ wrt. $\Gamma \subseteq \operatorname{SL}_2(\Bbb Z)$. Does it seem reasonable for $f \in M_k(\Gamma)$ that $f \in M_k(A\Gamma A^{-1})$ for all $A \in \operatorname{GL}_2(\Bbb Q)^+$?

I'm hoping so.. hahaha

I guess I need to show $(f|_k A\gamma A^{-1})(z) = f(z)$ for all $A\gamma A^{-1} \in A\Gamma A^{-1}$ but it seems too unwieldy and I'm wondering if I need some other argument

Actually that's wrong: $M_k$ is the space of entire modular forms*

and I want all $A \in \operatorname{GL}_2(\Bbb Q)^+$ s.t. $A\Gamma A^{-1} \subseteq \operatorname{SL}_2(\Bbb Z)$

Ted Shifrin

10:03 PM

@FelixCrazzolara Assuming you mean symmetric when you say semidefinite, the vector space of symmetric $n\times n$ matrices has dimensions $n(n+1)/2$. The positive semidefinite cone is the closure of an open subset of this.

William Sun

I have a question: What is the significance of defining the class of Boolean rings?

I am reading through A&M but I haven’t been able to appreciate the significance in commutative algebra/algebraic geometry.

And I am able to prove the Stone’s representation thm for Boolean lattices as the lattice of Clopen subsets in a topological space.

But how can we utilize this representation? Does it have any nontrivial applications?

Leaky Nun

@WilliamSun

Boolean algebra

In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. Instead of elementary algebra where the values of the variables are numbers, and the prime operations are addition and multiplication, the main operations of Boolean algebra are the conjunction and denoted as ∧, the disjunction or denoted as ∨, and the negation not denoted as ¬. It is thus a formalism for describing logical operations in the same way that elementary algebra describes numerical operations....

William Sun

Yes and I have read through this wiki article.

But I don’t think it mentioned anything about the significance to study Boolean rings.

Actually I am looking for an “application” from studying the Boolean rings and its ring-theoretic properties, spectrum, localizations, etc.

Leaky Nun

10:35 PM

@WilliamSun did you look at the "application" section?

William Sun

Yes but mainly I am looking for the applications in math.

Alessandro Codenotti

They come up all the time in set theory and similar

William Sun

The only thing I know is that the spectrum of a Boolean ring is the example of a non-discrete spectrum that is “very separated”.

Recall that the spectrum is Hausdorff iff the ring is 0-dimensional, that is, every prime ideal is maximal. And such spectrum is also totally disconnected.

So I mean for me it is just an example of a “extreme” class of rings(very separated) and nothing else.

Leaky Nun

11:01 PM

then that's what it is

MadSpaceMemer

11:25 PM

Can you have two different planes in the $\mathbb{R}^2$ ?

For somereason I can't construct a simple example. It seems to my brain that if you have any kind of plane in the second dimension, then it just covers the whole place. Am I too tired or am I right?

wtf obviously I am right its called the PLANE LOL! ah jeeze!

Ted Shifrin

@MadSpaceMemer: $\Bbb R^2$ has precisely one $2$-dimensional subspace, i.e., the whole space.