Mathematics - 2020-11-13 (page 1 of 2)

zacts

00:35

that 3b1b link is really cool.

I'd love to learn more about knot theory too.

I've got to dive into this Calculus text at the moment however. I'm sure to have some questions.

Myprofileissaved

01:05

If F:M->N is a smooth map between manifolds and A subset of M, is the restriction of f to A (f|_A) is necessarily a smooth map from A to N?

Thorgott

what does it mean for a map from A to N to be smooth

Myprofileissaved

that I can find a chart (U,i) U open in A (subspace topology) and i:U->R^k (some k) smooth homeomorphism

Thorgott

what does that have to do with the given map?

Myprofileissaved

sorry. You are right I had a mind glitch. For every a-in-A I can find (U,i) smooth chart in A containing a, and (V, j) smooth chart in N containing f(a), such that j∘f∘i^1 has continuous partial derivatives of all order

*each of it's components has partial derivatives of all order..

continuous partial derivatives

Thorgott

01:26

then the answer is no, for the rather contrived reason that there may be points in A for which you won't be able to find any charts at all

if you make the natural assumption that A is a submanifold, everything works out, of course

Ted Shifrin

Usually the notion is that there is a local smooth extension to a neighborhood of any point of $A$ (when $A$ is not a submanifold).

Myprofileissaved

Is isolated points is a case sample for what you have in mind?

Ted Shifrin

What is a smooth chart in $A$ when $A$ is a random subset?

With my definition, the answer is that it's tautologically true. Without my definition, it makes sense only when $A$ is a submanifold.

Thorgott

you can always find a chart around an isolated point, cause the point is homeomorphic to R^0 and an open neighborhood of itself

Myprofileissaved

Is there a simple example where I cannot find such charts?

Thorgott

01:38

take the cross X in R^2

no chart around the origin

Myprofileissaved

Ok I get it. Thank you

user193319

02:01

How do I prove that the derivative of a meromorphic function is meromorphic? Let $f(z)$ be meromorphic, and let $z_0$ be a singularity of $f'(z)$. I want to argue that $z_0$ is a pole of $f'(z)$. My thought was to consider the Laurent expansion of $f'(z)$ at $z_0$, and then argue by contradiction. Then either $z_0$ is removable or it is essential.

I thought that if $f'(z)$ had a Laurent expansion at $z_0$, then I could modify it (change the coefficients) to get a Laurent expansion of $f(z)$ at z_0$...But the $(-1)$-th coefficient seems to be giving me trouble...

Thorgott

why not just differentiate the Laurent series of $f$?

epic_math

08:43

scream boys, scream.

user2103480

09:20

@AlessandroCodenotti
HoTT lecturer: "you don't need any algebraic topology for this"

HoTT proofs: literal translations of algebraic topology into simplicial sets

Alessandro Codenotti

09:57

lol

Are you taking another HoTT course?

user 6232128

one can never be too HoTT

:p

geocalc33

10:29

ehllo

epic_math

'lo

user 6232128

o

geocalc33

you can do a geometric mean of two functions?

epic_math

ya

geocalc33

nice I never thought of that

because it's usually done with numbers

epic_math

10:33

functions also give numbers

it's a sort of generalizing the geometric mean

user 6232128

under suitable restrictions

epic_math

ya

seems weird but the argument of the zeta function can be a function

under suitable restrictions

geocalc33

Li(x)=(h(x)f(x))^1/2

anyone familiar with logarithmic integral?

epic_math

yes I always work with it

geocalc33

would it be good to express Li(x) like I wrote above?

i.e. as a geometric mean of h(x) and f(x)

epic_math

10:38

which logarithmic integral do you mean? the logarithmic integral or the offset logarithmic integral?

geocalc33

offset

epic_math

@geocalc33 I am not sure but I don't think such functions are possible

geocalc33

10:51

I'll ask on main

I doubt it's useful but now I'm curious

0

Q: Express logarithmic integral as geometric mean of two functions

The offset logarithmic integral is defined as: $$ Li(x)=\int_2^x \frac{1}{\log(t)} ~dt$$ I want to express $Li(x)$ as the geometric mean of two functions $f(x)$ and $h(x)$ s.t. $h(x)\ne f(x).$ That is, $$ Li(x)=\sqrt{f(x)h(x)}.$$ Can it be done?

user2103480

11:14

@AlessandroCodenotti nono, just the topology one, but I already see some analogies

Alessandro Codenotti

Ahh makes sense

epic_math

11:44

Hey guys listen

northerner

What is the point of functional arrow notation if $f: x \rightarrow x^2$ is same as $f(x) = x^2$? How's using the arrow any better, are there situations where it's more clear?

Astyx

You'd write $f: x\mapsto x^2$

The notation $f:A\to B$ lets you define abstract functions from A to B without being specific

epic_math

The first proof of the abel Ruffini theorem was 500 pages but a proof was made later which was only 6 pages long

What are other examples when a very large proof was condensed very much?

geocalc33

proof of the PNT

epic_math

Should I ask this on main as a soft question?

northerner

11:49

oops mistake on my part

Astyx

Pedantically, in order to write $f(x) = x^2$ you have to define $x$ first, whereas in $f:x\mapsto x^2$ x is a dummy variable

epic_math

Please tell if there is a duplicate of this on main before I ask this.

northerner

@Astyx I'm not sure I get it. Why can't one just as easily assume x is a dummy variable with f(x)?

Astyx

I mean the correct expression for $x$ to be a dummy variable is $\forall x, f(x) = x^2$

epic_math

@geocalc33 can you tell how long was the original and the condensed proof?

northerner

11:53

what exactly do you mean by dummy variable, that it doesn't matter what set it's coming from? @Astyx

Astyx

But you could have a function such that $f(x) = x^2$ only for specific $x$

No, I mean that $x$ can take any value in the domain and the statement holds

But it's not a big deal anyway, it's just that you have to be careful what things mean

geocalc33

@epic_math not sure

epic_math

Is this question on main?

Should I ask this?

Astyx

I could define $f: \Bbb R\to \Bbb R$, $f: x\mapsto x$ and write f(x) = x^2 for $x\in \{0,1\}$

So a reader that is not too careful could miss the "for $x\in \{0,1\}$" and be confused

The notation $f:x\mapsto x^2$ prevents that

Some people also use $f(x) := x^2$

epic_math

Okay I am going to ask this. I will link the question.

Astyx

11:58

Then again, it's not a big deal

epic_math

0

Q: Examples of very long proofs which were condensed to a very small size

While reading this article, I learned that the original proof of the Abel Ruffini theorem was about 500 pages, which was later condensed to just 6 pages. Immediately after reading this, a question came to my mind: What are some examples where a very large proof was condensed to a very small size?

soft-question big-list

Merosity

I think with that whole thing on prime gaps that Zhang found a couple years back, an independent and better proof was found not too long after

epic_math

I got an upvote

Lol who did that

It was not a remarkable question but yeah

Hey guys thanks for your support(the upvotes) may it get many answers

Merosity

I think this is probably already a thing that was asked actually

dunno, I just vaguely remember seeing it, maybe it was on math overflow though

Mike Miller

12:28

137

Q: Extremely messy proofs

Currently in my undergraduate courses I am being taught how to set up various machinery using slick, short proofs and then how to apply that machinery. What I am not being taught, largely, is what came before these slick, short proofs. What did mathematicians do before so-and-so proved such-and...

big-list ho.history-overview

Secret

12:49

big scary subscript

from some DFT review paper

Merosity

wewwwww

DFT meaning Density Functional Theory I'm guessing from the bra and ket?

epic_math

13:08

Hey I got a weird question in my mind

What is the total area of the black region of the mandelbrot set?

Not an approximation, I want an exact form.

The answer is approximately 1.506484

See this:arxiv.org/pdf/1410.1212

geocalc33

Does anyone know what this is? ⭕️

user2103480

@epic_math a

where a is defined to be the total area

geocalc33

yep

it's A=1 unit

user2103480

exactly

problem solved

epic_math

What the hell

This answer is trivial.

:'(

geocalc33

13:28

wait 1.506-484-4223

epic_math

Gimme the exact answer

Yeahh

geocalc33

expanding via binary and exptrapolating Czech cocycle data I think I get...

yep, A=1 unit

epic_math

And what is the area of the burning ship fractal?

geocalc33

A=1 unit I think

epic_math

Exactly 1?

geocalc33

13:32

no it depends on the unit you're using

epic_math

Tell me the unit, man

Okay the exact area of mandelbrot is almost impossible to calculate

So an easier question:

Area of Koch snowflake?

geocalc33

that should be easy to google @epic_math

epic_math

With the length of the side of the initial equilateral triangle being 1️

The calculators calculate with finite iterations, I want the answer with infinite iterations.

I can myself calculate with finite iterations

geocalc33

Koch snowflake

The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch. The Koch snowflake can be built up iteratively, in a sequence of stages. The first stage is an equilateral triangle, and each successive stage is formed from adding outward bends to each side of the previous stage, making smaller equilateral triangles...

epic_math

Oh the area is $\frac{4\sqrt{3}}{5}$

Easy to calculate

Very easy

Using limits

Fractals are fun

geocalc33

13:41

exactly

epic_math

I don't think are techniques are advanced enough to calculate area of mandelbrot

See a mandelbrot zoom and then you will realize

Mandelbrot Zoom 10^227 [1080x1920]

►

geocalc33

I don't like the Mandelbrot zoom

epic_math

It's aesthetic

Rather hypnotizing

geocalc33

yeah it's just not my cup of tea

epic_math

It's my donut of tea

Tohpohlozy

geocalc33

13:44

it's my sphere of tea

my cup handle is broken

epic_math

Means you can easily decrease the genus

Which field of math you love?

Analysis/geometry/algebra

Considering number theory in analysis

And topology in geometry

geocalc33

exactly

epic_math

Your profile says you know all of those

Oh shit

I again fell on a mandelbrot set

Just IN: Riemann hypothesis declared solved by clay mathematics institute

Oh man

geocalc33

exactly

epic_math

Lol

geocalc33

13:52

I doubt the RH has been solved

it's a pretty thorny problem

epic_math

'twas but a joke

RH is insane

geocalc33

exactly

epic_math

exactly

geocalc33

speaking of RH

epic_math

arxiv.org/pdf/math/0307136

Enjoy

Task: figure out the mistake in the proof.

geocalc33

13:58

how do the critical zeros differ from $\zeta(s)$ to $2\zeta(s)$

@epic_

epic_math

And he handles infinity and zero without even thinking about it

Zero and infinity can never be friends.

Edward Evans

afternoon

geocalc33

hello ed

Astyx

hey

epic_math

Heya @EdwardEvans

Sayan Chattopadhyay

14:03

Hello @EdwardEvans

epic_math

ed is a vip everyone greets him

Mike Miller

@epic_math Boring, if you want the good stuff you go to vixra

Astyx

The formatting of this article is horrific

2 line space is unreadable

epic_math

terrible

@MikeMiller vixra is even less reliable.

feynhat

How was it accepted though?

Mike Miller

14:10

It's the general math section which has very little review

Astyx

I think Arxiv articles are not necessarily accepted

Mike Miller

@epic_math I mean you're looking at junk, I would think you would want reliably amusing junk

feynhat

@MikeMiller oic

epic_math

oh thanks, you found what I like

lol

feynhat

@Astyx I meant, how was it accepted on Arxiv. I mean articles on Arxiv also go through some review, right?

Apparently not.

epic_math

14:13

arxiv is where you go when you can't go to researchgate.

Mike Miller

No, authors go through some review to get access (and papers are glanced at to make sure they're not gibberish, when they're not in general math)

Astyx

There might be some light reviewing to check it's not a complete troll, but there isn't any peer review AFAIK

epic_math

there are more RH proofs on arxiv than sensible articles.

Mike Miller

Who cares

This is dork shit

epic_math

tell about that website to mochizuki. He can publish his work there

robjohn

14:14

@epic_math how about the MandelBuddha?

feynhat

Oh okay. I didn't know how Arxiv worked.

epic_math

@robjohn that one photo isn't real

Astyx

Benoît B. Mandelbrot

epic_math

@Astyx =benoit benoit B. mandelbrot mandelbrot

robjohn

@epic_math pardon? what do you mean it isn't real?

epic_math

14:17

I mean that isn't the original fractal.

feynhat

open.spotify.com/track/1KOV7p4LBEFkTitQ0vcmgy

robjohn

@epic_math the alpha channel is the Manelbrot set

epic_math

@feynhat does that singer even know what a fractal is?

feynhat

@feynhat btw this is the same guy who composed many of Portal 2's soundtracks.

epic_math

songs are shit and we are here to talk mathematik

Balarka Sen

14:19

I quickly scrolled back to 5 hours and there's still no math

robjohn

@epic_math that is the Buddhabrot

Balarka Sen

Why do I even come back here

feynhat

lmfao.

robjohn

@BalarkaSen so?

epic_math

@BalarkaSen "What is the area of the mandelbrot set"-epic_math, ~1 hour ago

balarka left immediately

geocalc33

14:20

It's friday

epic_math

and tomorrow is diwali

geocalc33

oh I have math (notation)

I came up with a new math letter

epic_math

can new mathematics notation be even made? Will anyone even accept it?

geocalc33

hey Leibniz notation stuck around

epic_math

tomorrow is the death date of Leibniz

Eduardo C.

14:25

Hello everyone. Why is it that for the converse we had to define $\delta_n$ as such? Why is it that this delta in specific makes (5) false?

epic_math

what are X, Y, E, f and p?

Eduardo C.

X and Y are metric spaces, E is a subset of X, f is a function from E to Y and p is a limit point of E.

Astyx

We just needed a positive sequence converging to 0

Eduardo C.

Oh, so there's nothing special with this delta? I could've chosen another delta_n going to 0 and this would work?

Astyx

yes

Eduardo C.

14:33

@Astyx It got me confused because this (delta_n) was somewhat specific, and it runs away from main purpose of the proof, I think. Thanks

epic_math

there was a question that I couldn't solve I wanna ask it now

how can we prove this:

help please

geocalc33

made an ugly graph

epic_math

too ugly to be in existence.

lol

what is the equation?

geocalc33

epic_math

I asked for the equation

btw the graph is not ugly

I can see its beauty

geocalc33

14:50

what do you think of this one? @epic_math

epic_math

beauuutiful

geocalc33

ty

Myprofileissaved

16:16

Let Mn be the square matrices on R, let Gln be the non-singular matrices, a let O(n)\subset Gln be the orthogonal matrices. I am using the map F:Mn->Mn F(A)=AA^T to build smooth atlas, using the constant rank theorem, on O(n). So far I know that F has constant rank on Gln and proved that O(n) is a regular surface using the regular value theorem (restricting F to Gln and using the identity matrix as the regular value). I am stuck with building a smooth chart on O(n) with the constant rank.

user21820

← 1 message moved from Basic Mathematics

Myprofileissaved

If someone has an idea, it will be super helpful

anakhro

@MikeMiller what's the equation called that relates the Riemann curvature tensor to the shape operator?

I think it was something like $R(X,Y)Z = g(S(Y),Z)S(X) - g(S(X),Z)S(Y)$?

Mike Miller

16:34

I dunno this stuff, sorry.

anakhro

That's fine. Maybe Ted knows.

@TedShifrin is there a name for the equation that relates the Riemann curvature tensor to the shape operator? I think it was something like $R(X,Y)Z = g(S(Y),Z)S(X) - g(S(X),Z)S(Y)$?

Thorgott

17:33

how does one motivate hyperbolic geometry?

geocalc33

good question lol.

@anakhro Codazzi equations?

schn

18:02

This is from a book in mathematical statistics, the first chapter on inference theory where estimators are explained. One would like to estimate the unknown parameter $\theta$ of a known distribution $F$. In the example, why is it that when $F$ is unknown, $\theta=F$? $F$ can be seen as a function of $\theta$ (and vice versa), but if $F$ is unknown, shouldn’t this “rule” (function) not exist, i.e. $F=F$ and $\theta=\theta$?

geocalc33

well your distribution has to be $F=\theta$ if you have no other knowledge of the distribution

Mike Miller

@Thorgott Failure of the parallel postulate

Clearly there are two ways it can fail

schn

@geocalc33 Intuitively it makes sense, but it seems that with $F=\theta$, $F$ is actually known, namely $F(x)=x$, or?

Mike Miller

It is interesting to investigate both

love_sodam

What can we say if the Gaussian curvature is zero at some point?

Hmm seems locally plane

Mike Miller

18:19

No, that would be if the Gaussian curvature was zero everywhere in a little neighborhood around the point

A torus has many points of curvature 0

(It has maximal curvature along the circle encircling the whole donut, and minimal --- negative --- curvature on the circle encircling the donut hole)

Thorgott

ok, my question was too imprecise

Mike Miller

(So it must cross 0 at some point --- at the top and bottom of the circle)

Thorgott

how does one motivate the Poincaré distance?

Mike Miller

Radial lines through the origin should be isometric to R

Similarly circles meeting the boundary perpendicularly should also be geodesics and isometric to R

I think that determines the distance up to a constant maybe

anakhro

@love_sodam think of it in terms of the principal curvatures.

If you have a normal vector at the point with zero Gaussian curvature, then at least one of the principal curvatures is zero, and in the corresponding principal direction, the surface looks "flat" relative to this normal vector.

But the other principal curvature does not have to be zero.

geocalc33

18:24

@schn it says in the example, $F_{\theta}$ is completely unspecified. Say, $\theta$ is the true parameter of the distribution. How could you know anything about the distribution if you don't know anything about the parameters? I think it's basically saying that you need more data, more samples, to gain information about the parameter and hence about the distribution

maybe someone else can help better

schn

@geocalc33 Thanks, helped clarify it a bit.

love_sodam

So the dimension should be dropped

Ted Shifrin

@anakhro: I'm not sure what your context is. If you're talking about the shape operator, you have a submanifold and a normal-bundle valued second fundamental form. It's the Gauss equation that relates curvature tensor of submanifold, curvature tensor of ambient manifolds, and second fundamental form.

@love_sodam No, not planar points. Think of a cylinder. As anakhro said, you have only one principal curvature $0$; planar points they're both $0$.

love_sodam

18:40

Ok then what does it suggest?

Thorgott

hmm, interesting

Ted Shifrin

What does what suggest?

Thorgott

but is there, say, a "geometric" interpretation of the hyperbolic length of a curve?

Ted Shifrin

There's a cross-ratio interpretation, @Thor.

See Section 3.2 of my diff geo notes.

geocalc33

I have the dumbest question

how do you write f(x,y)=(2x,3y) as a function

Thorgott

18:45

oh, that sounds nice, I'll take a look

Ted Shifrin

Oh, sorry, @Thor: That's for the distance between two points (i.e., length of geodesic). For a random curve, I have no answer.

I mean, if you have a Riemannian metric, length of curve is length of curve.

Thorgott

that suffices, I'm just trying to get a minimal understanding of what Schwarz-Pick says geometrically

Ted Shifrin

Schwarz-Pick says that holomorphic maps are distance-decreasing.

Infinitesimally, the pullback of the metric is less than or equal to the metric.

You can reduce that to a statement about geodesic lengths, so my original comment probably does apply.

geocalc33

so it's fine as f(x,y) but I want it in an explicit form

Ted Shifrin

That's as explicit as it gets, @geocalc.

geocalc33

18:49

like f(x,y)=2x+3y

Ted Shifrin

You're talking nonsense.

geocalc33

I want to prove that a map is Lipschitz though

Ted Shifrin

So? Can't be any simpler than a linear transformation.

geocalc33

what about the nonlinear cases? for example $f(x,y)=(2^x,3^y)$ or something

Ted Shifrin

Any $C^1$ function is Lipschitz on a compact set. Otherwise, not true.

schn

18:59

Consider the log-likelihood function of a normal distribution, which is optimized to find maximum likelihood estimators. It is a function of $\mu$ and $\sigma$. As a function of $x$, it is of course positive definite and a maximum exists. However, as a function of the other two, it seems less obvious. What are some arguments for a maximum to exist?

wikimedia.org/api/rest_v1/media/math/render/svg/…

Thorgott

yeah, I just have to understand how the distance works in the first place for the statement "distance-decreasing" to become meaningful

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$Mathematics$ Mathematics