Mathematics - 2018-08-05 (page 2 of 2)

Gabriel Romon

16:11

re-discovered this in my bookmarks: lkozma.net/inequalities_cheat_sheet/ineq.pdf

Secret

16:32

my impression of rationals if we have uncountable magnifying glasses

G. Ünther

another pretty picture

Secret

If we have a magnifying class of magnifying power at least $\omega_1$ it will outpace the density requirement of dense sets and allow us to see the holes made up by the irrationals

since the rationals, being countable, can only be dense countably steps deep

geocalc33

I was reading about fixed points the other day. In a loose sense a fixed point is a point on a function that stays in the same place after a transformation

Secret

yup

G. Ünther

That's pretty much it

Secret

16:40

and in dynamical systems, they control a lot of the dynamics

geocalc33

haha lol

so wait, does a function that has the same inverse as itself, have multiple fixed points along the line y=x?

let's say you have a sliding parameter on this function so that it flows along y=x

Secret

You mean a fuction f such that for all x $f(f(x))=x$?

geocalc33

yeah

Secret

$f^2(x)=x$ and $y=x$ means:

geocalc33

so like $ y=s/x $ where $s$ is a sliding parameter

Secret

16:53

$f(y)=f(x)$

G. Ünther

well take for example $f(x) = 1/x$, then it just has one fixed point for positive values, namely $x = 1$.
The function $f(f(x))$ then has fixed points everywhere

Secret

so for those (x,y) where y=x is true, if f(x) is a fixed point, then so is f(y), otherwise since f is involutive, all points make 2 cycles under the map f

G. Ünther

for f(x) = 1/x, yeah

Secret

likewise, if f(x) is not a fixed point, then if y=x is true, then f(y) is also not a fixed point, for any involution f

G. Ünther

All points of the function that intersect with y = x are fixed points

geocalc33

16:57

oh thats pretty neat

G. Ünther

because there the output of the function is equal to the input, so badabing, a fixed point

Secret

ah right, I was mistakenly thinking about f(x,y)

geocalc33

so if let's say you traced out a nodes position along the line y=x in finite steps they would all be fixed points?. I'm trying to try to model a real world network with certain physical constraints and try to view each node along y=x as a fixed point

not sure if that's an accurate way to view it

/an application of it

mind you the nodes are equally spaced along the line $y=x$ and there are finitely many of them

you can view them all as "points" i guess

but in my network the nodes are all people and the links are connections. it's a social network sorta application

G. Ünther

hmmm, I can't really see a function there that could have fixed points, but networks, graphs and such stuff are things I know absolutly nothing about, so yeah idk

geocalc33

yeah ill probably get a book about network science. well thank you for explaining that concept to me

G. Ünther

17:12

no problem, I hope my explanation helped you kind of

ÍgjøgnumMeg

18:54

The series of videos on alg. geo. that I linked earlier follow these notes exactly btw, if anyone is looking for some nice algebraic geometry notes

Also, hello chat

Leaky Nun

wtf, $\log$ in desmos is base 10

G. Ünther

I really don't like it when people write just log, if they mean base 10, why don't they write lg, and when they mean base e, why don't they write ln?

Leaky Nun

$\operatorname{lng}$

G. Ünther

d'oh

Krijn

19:11

'supsupsup

Leaky Nun

'infinfinf

Krijn

Oh god, the jokes.

Secret

Still trying to find a system which there is topological mixing but no chaos

I wonder if irrational rotations on a circle counts...

Alessandro Codenotti

Hi @Krijn long time no see! What are you up to those days?

Krijn

I'm working in consultancy.

ÍgjøgnumMeg

19:14

I have just seen Peter Scholze write "rigid-anal. geometry" on a blackboard

Krijn

But also looking into Crypto quite a lot

Cryptography, not the blockchain industry that is

The knowledge-gap is closing so that's nice

What about you, @AlessandroCodenotti

G. Ünther

Does anyone know how to calculate lyapunov exponents for a given 2D map with equations for x and y? Like actually calculate. I don't get this on wiki:
$$\displaystyle \Lambda =\lim _{t\rightarrow \infty }{\frac {1}{2t}}\log(Y(t)Y^{T}(t))$$
https://en.wikipedia.org/wiki/Lyapunov_exponent
No idea what that's even supposed to mean or how I calculate it for a certain map...

Alessandro Codenotti

@Krijn Just finished my Bachelor, I'll start my Master in Bonn in October

Krijn

You were doing your Bachelor all that time? You seem much more advanced

Kudos to you

Alessandro Codenotti

Thanks, that might be because I like delving deeper into the topics I encounter in lectures and find interesting

Krijn

19:34

What will your master be on?

What topics, I mean

Alessandro Codenotti

Logic and set theory probably

Hi @Mathei @Eric

Krijn

Ever did a course on intuitionistic logic?

I had a prof at my old uni who was pretty good at that stuff, and it's a fun subject in logic

Alessandro Codenotti

Nope, only classical first order logic so far

Krijn

19:57

Ah well, logic is fun!

I'm off for the night

If @BalarkaSen is around sometime he should drop me a comment

MatheinBoulomenos

20:10

Hi @Alessandro

Leaky Nun

Abend @MatheinBoulomenos

seram

Alessandro Codenotti

I pinged you yesterday and earlier but I guess you weren't actually online

MatheinBoulomenos

@AlessandroCodenotti yeah yesterday I was online very shortly and wrote something in the garbology room and then I went off again

I read what you wrote though

Alessandro Codenotti

Did you read the measure theory thing I tagged you in? You might find it interesting

MatheinBoulomenos

sounds interesting about infinite products of measures

didn't look it up in Tao's book though

Alessandro Codenotti

20:14

After reading that I read some more analysis stuff written by Tao, I like his style

Leaky Nun

does the maximum modulus principle imply that if $f:D \to \Bbb C$ is analytic and non-zero then $\nabla |f| \ne 0$?

Mike Miller

20:28

@LeakyNun And non-constant, but yes.

Leaky Nun

intersting

Mike Miller

MMP is good stuff.

Leaky Nun

@MatheinBoulomenos did our swedish friend respond?

@mercio hoe ben je

ÍgjøgnumMeg

vilken svensk väääänn

MatheinBoulomenos

@LeakyNun yes

he's doing doing well and he was just busy with personal stuff

Leaky Nun

20:31

I see

@MatheinBoulomenos ANT is dank

MatheinBoulomenos

he said he'll come back with questions for Ted and me next week

ÍgjøgnumMeg

Hey @Mathein :)

and @Leaky

MatheinBoulomenos

hi @ÍgjøgnumMeg @LeakyNun @MikeMiller

@LeakyNun yeah I really like it

Leaky Nun

schee dog @ÍgjøgnumMeg

@MatheinBoulomenos A for analytic :P

MatheinBoulomenos

...

no comment

Leaky Nun

20:32

got you

r e k t

ÍgjøgnumMeg

as isch scho z'spöt zum an schönen tag wünscha

lol

Leaky Nun

what did you say?

Mike Miller

i do not get excited about analytic number theory

Leaky Nun

it is too soon to wish a good day?

ÍgjøgnumMeg

too late

Alessandro Codenotti

20:33

@LeakyNun Top ten anime betrayals

Leaky Nun

i see

MatheinBoulomenos

ANT is really popular in Afghanistan

Afghanistan National Television

Afghanistan National Television (Pashto: ملی تلویزیون‎ Da Afganistan Milli Telvizoon, Persian: تلویزیون ملی‎ Telvizoon-e Milli Afganistan) is the state-owned television channel in Afghanistan, launched in 1977 and part of the Radio Television Afghanistan (RTA) public broadcaster. == Exclusive 2008 speech == RTA became famous worldwide when Afghan President Hamid Karzai made a live speech to the world minutes after dozens of insurgents attempted to assassinate him at an Afghan military parade. The assassination attempt was thwarted by the Afghan National Army. The scene of the attempt was also...

ÍgjøgnumMeg

lol

Leaky Nun

@MikeMiller but do you get excited about prime number theorem?

mercio

@LeakyNun ik ben moe

Mike Miller

20:49

A little bit

Daminark

22:16

@Leaky eh the statement of the prime number theorem is fun but its proof is whatever. Learn about the Tate module of an elliptic curve instead

Iza_lazet

22:55

is there a proof of the prime number theorem which is conceptually elegant, or is that impossible?

user31415

do you consider the usual proof not conceptually elegant?

Iza_lazet

if you mean magical elegant formulas that fall from the sky, sure

user31415

not sure which part is "magical"

Iza_lazet

depending on which proof you are alluding to, they all reference some analytic formulas that can be checked when you know them, but not really possible to discover by yourself.

user31415

23:19

As far as I know there are mostly two proofs (the analytic proof, and the elementary proof). At least the analytic proof, it's magical that you can link primes to L functions via Euler product, but I guess that's also an insight somewhat passed down since Euler. But beyond that, modulo analytic details the proof strategy seems very clear (that you are trying to find a zero free region)

rschwieb

23:35

@MatheinBoulomenos Is "pointwise integral" the same as "localizations are domains"?

TRiG

23:46

Too cool for regular vectors?

I'm trying to prove that mathematics forms a meta-abelian group, which would finally confirm my suspicions that algebreic geometry and geometric algebra are the same thing.

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$Mathematics$ Mathematics