Mathematics - 2020-09-13

5:05 AM

o/

why is this room not appearing in chat.so search result? I've landed here from Google search...

Is division considered a usual way to "scale" numbers in Mathematics? Because I usually scaled numbers in computer using multiplication. I am no Mathematician but I'd like to know why 3B1B used division to scale polynomial values instead of multiplication.

shelton Benjamin

5:22 AM

Where i can read the proof that in disc method frustum best estimate surface area and i. Volume disc approximate better also how to check if the infinitesimal I took integrates to correct value

geocalc33

6:22 AM

Is there a simple equation for y=1/x revolved about y=x?

Calvin Khor

trying to read a paper in french..."fonctions convenablement décroissantes à l'infini"...does this mean functions decreasing to infinity? A little odd...?

Leaky Nun

@CalvinKhor link?

Calvin Khor

@geocalc33 in R^3? its a quadric surface (hyperboloid of two sheets)

its an old one but I can screenshot @LeakyNun gimme a sec

geocalc33

@CalvinKhor yes but is the cartesian equation of it simple?

Calvin Khor

oh, this isn't one of the old ones that I had to try very hard to find. Its here, page 228 (page 8 of the pdf)

geocalc33

6:34 AM

I know that it would be $z^2-y^2-x^2=1$ rotated by a certain amount

Calvin Khor

@geocalc33 OK, that's simple to me, so I would say yes

ack no page 7 sorry. I edited the comment but it was correct

geocalc33

@CalvinKhor okay, I'm trying to derive the cartesian equation

so I have to rotate the re-arranged hyperboloid $z=(1+x^2+y^2)^{1/2}$

but the only way to do this is with parametric equations, like there's no explicit cartesian equation for it right?

in $\Bbb R^2,$ $y^2-x^2=1$ rotated by 45 degrees has a nice rep. as $y=1/x$

Calvin Khor

ok, what about $y^2 - x^2 = c$?

geocalc33

I mean to say that in $\Bbb R^2$ both the standard form and rectangular form are simple

Calvin Khor

2 mins ago, by Calvin Khor

ok, what about $y^2 - x^2 = c$?

shelton Benjamin

6:46 AM

At least tell me a room where i can ask my question

geocalc33

what about it?

Calvin Khor

I'm asking you?

mr5

@sheltonBenjamin is it related with 3d stuffs?

geocalc33

that's in $\Bbb R^2$

shelton Benjamin

Yes integrals

geocalc33

6:48 AM

I'm working with $z^2-y^2-x^2=1$

Calvin Khor

And you think $y^2 - x^2 = c$ has no relation to this?

geocalc33

it does, I just can't figure out this problem

Calvin Khor

the slices are precisely of this form. So if you want to see if it has a nice form after the correct rotation, then it suffices to rotate this and check the form, right?

Also @geocalc33 do you speak french... :)

mr5

do you Mathematicians code your equation also to confirm if it's working?

geocalc33

@CalvinKhor I know a few phrases

Calvin Khor

7:03 AM

@geocalc33 ok, not enough to help me translate my sentence I guess. Thanks anyway

user21820

7:16 AM

6

Q: Leapfrogs puzzle -- Least number of moves needed to interchange the pegs

This is a question from the book "Thinking Mathematically" by Burton and Mason. Question: Ten pegs of two colors are laid out in a line of 11 holes as shown below. I want to interchange the black and white pegs, but I am only allowed to move pegs into an adjacent empty hole or to jump over one p...

^ I would like to draw attention to this question (not mine), because although I believe it can be solved by some simple invariant, I am unable to find it.

So I was hoping someone here might have some good ideas.

Ping me if anyone is interested to discuss ideas! =)

Leaky Nun

8:22 AM

> Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive – even shocking – that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré[3] and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Cantor, a devout Lutheran,[4] believed the theory had been communicated to him by God.[5] Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God[6] – on one occasion equ…

if this isn't the benchmark of success I don't know what is

user21820

@LeakyNun Wait a minute. What do you mean by success and what do you mean by benchmark?

Astyx

Imagine inventing something so dope you're convinced God has to have something to do with it

Sayan Chattopadhyay

If $V(I) = V(J)$ for two ideals $I,J$ what can we say about $I,J$?

Leaky Nun

@SayanChattopadhyay $\sqrt{I} = \sqrt{J}$

Astyx

What's V ?

Leaky Nun

8:27 AM

@user21820 I was just referring to the part about him being rejected by contemporaries

Sayan Chattopadhyay

Lol @LeakyNun I was coming from there itself, I was looking to say something more than that

Leaky Nun

@Astyx the vanishing set -- $V(I) := \{ \mathbf{x} \mid \forall f \in I, f(\mathbf{x}) = 0 \}$

Calvin Khor

@LeakyNun sorry to ask again but can you help with my french :) A "No" is fine

Astyx

cheers

Leaky Nun

or alternatively, $V(I) := \{ \mathfrak p \mid I \subseteq \mathfrak p \}$

user21820

8:28 AM

@LeakyNun Ah. But that isn't success, no?

Or do you mean it's a truly useful invention that is rejected by contemporaries?

Leaky Nun

@CalvinKhor I don't know what they mean

Calvin Khor

oki, thanks

geocalc33

@CalvinKhor I think I figured out the hyoperbal problem

Sayan Chattopadhyay

@LeakyNun Can you say anything more than this?

Astyx

@CalvinKhor "Convenablement décroissante à l'infini" means the function decreases at infinity, not to infinity

Calvin Khor

8:32 AM

@Astyx that makes a lot more sense! thank you!

Astyx

And convenablement just means that it's nice enough for things to work

Unless you have a more specific definition in the textbook

Leaky Nun

@SayanChattopadhyay what do you mean

Calvin Khor

no, its not been defined prior (just 7 pages into a paper). Thank you :) @Astyx

Astyx

link to the paper ?

Calvin Khor

@Astyx here

Astyx

8:35 AM

Thank you

Calvin Khor

you're welcome :)

geocalc33

@CalvinKhor I got $xy-1=z^2$ for the hyerboloid of two sheets

so it looks like it is pretty simple

Calvin Khor

yup

Koyadovic

8:52 AM

Hello all

geocalc33

9:46 AM

What is the explicit cartesian equation for y=1/x revolved about y=x?

is it actually $xy-1=z^2?$

I know that it's a two-sheeted hyperbola^

user330477

9:59 AM

What is the largest open set where $e^{\sqrt{z}}$ is analytic?

Balarka Sen

1:44 PM

@user330477 There is no such object as "the largest open set". It is analytic on $\Bbb C \setminus \Bbb R_{\leq 0}$, if you like.

Alessandro Codenotti

Hi @Balarka

Balarka Sen

It is also analytic on $\Bbb C$ with a "topologists' sine curved negative real axis" thrown out.

Hi @Alessandro

Edward Evans

waddup @Balarka @Alessandro

Balarka Sen

Damn I saw a flag on a massive message

Lmao it was about mask abolition

user330477

2:07 PM

@BalarkaSen Then shouldn't $e^{\sqrt{z}}+e^{-\sqrt{z}}$ be analytic on $\mathbb{C} \setminus (-\infty,0]$? Correct me if I am wrong?

Edward Evans

@Balarka on main or what?

Mike Miller

2:39 PM

@BalarkaSen lmao

sevdaicmis

3:26 PM

hi all how are you doing

long time i've been away from this chat

Alessandro Codenotti

4:01 PM

Sanity check, if $(X_i)_{i\in\Bbb N}$ is a family of subspaces of a space $X$ with $X_i\subseteq X_j$ whenever $j\leq i$, then $\varprojlim X_i=\bigcap X_i$, right?

Ted Shifrin

6:35 PM

@user330477 That's an interesting question. Why is it not analytic everywhere?

Belated hello to a @Balarka, @Edward, @MikeM, demonic @Alessandro.

Mike Miller

Hi

Ted Shifrin

How's your topology course going, Mike?

Mike Miller

Does there exist a bijective continuous map from $[0,1)^n$ to $[0,1]^n$? :)

I don't know how to prove not.

It's fine. Next week we cover closure, bases, subspaces, products, disjoint unions.

Getting past the preliminaries.

Ted Shifrin

Hmm, so the only way we know how to do $n=1$ is with disconnecting. I guess we'd need homology to generalize that.

Mike Miller

Yeah, I couldn't work out the details though.

Ted Shifrin

6:42 PM

Hmm ... well, an interior point must map to $(1,\dots,1)$. Pulling those out gets a contradiction. No, that's not necessarily the case. So we have to look at the whole boundary of $[0,1]^n$? Yuck.

Maybe there's a good reason we haven't thought about this question before :D

Mike Miller

Indeed

Alessandro Codenotti

How was the proof that space filling curves are not injective? I think something similar should work here

Ted Shifrin

That's usually invariance of domain.

I was thinking about that but didn't see it.

Mike Miller

Invariance of domain tells us only that $f(0,1)^n \subset (0,1)^n$

So the inverse image of the boundary is contained in the boundary of the domain; but part of the domain's boundary could fit inside (0,1)^n

Intuitively that seems preposterous but...

Ted Shifrin

I didn't understand that.

Mike Miller

6:51 PM

what part?

Ted Shifrin

So the question is why $[0,1)\times 0 \cup 0\times [0,1)$ can't wrap around the entire boundary. Surely that contradicts one-to-oneness nearby.

"But part of the domain's boundary could fit inside ..."

Mike Miller

We've only seen that $f^{-1}(\partial [0,1]^n) \subset \partial [0,1)^n$.

We haven't concluded that $f(\partial [0,1)^n) \subset \partial [0,1]^n$.

Ted Shifrin

Oh, now I see how to read your sentence. Thanks.

Mike Miller

Sure, I wasn't sure if I was presenting mangled English or mangled math. :)

Ted Shifrin

"fit" = "map"

Yeah, that doesn't seem to contradict continuity, but ugh.

How annoying.

Mike Miller

6:58 PM

Yeah. I gave up after a while.

Such maps might even exist. Who knows.

Ted Shifrin

I'm just trying to think if you can continuously map that boundary surjectively (and bijectively) to the boundary and that seems impossible. But your point stands.

Mike Miller

Well, there's definitely no continuous bijection $\Bbb R^{n-1} \to S^{n-1}$ by invariance of domain; it would be open, hence a homeomorphism.

Ted Shifrin

Right.

Mike Miller

And the boundary of the first guy is homeo to R^{n-1}, and the latter to S^{n-1}.

It just seems 'obvious' that the boundary can't snake inside and then back outside again without messing things up.

Now it's time to think about calculus.

Ted Shifrin

That's where I want to remove and use homology somehow.

I think I can handle calculus better.

Mike Miller

7:07 PM

Polar integrals and triple integrals this week. Logically, I'd prefer to do change of coordinates more generally after polar, but I understand why this is the standard order.

orientablesurface

Prove that a graph with 6 vertices contains either 3 mutually adjacent vertices or 3 mutually non-adjacent vertices

Ted Shifrin

Mostly you're only going to have them do general changes of variables in 2 dimensions, anyhow.

Mike Miller

Right. Triple integrals are already way too painful.

orientablesurface

this is just another way of formulating the friends and strangers problem, right?

Ted Shifrin

Well, I always enjoyed teaching this section, although I had to prepare some figures carefully ahead of time.

Sounds like it, @orientable, although I don't know it.

Mike Miller

7:51 PM

@TedShifrin When formalized, the question he's referring to is: "Can you draw a curve in [0,1]^2 from (0,0) to (1,1) which never touches the boundary except at time 0 and 1, and another curve from (1,0) to (0,1) satisfying the same hypotheses, do they intersect?"

AKA the Jordan curve theorem.

Oh no not at all

He's asking a graph theory problem

Alessandro Codenotti

That's a way of saying that the Ramsey number R(3,3) is <=6

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$Mathematics$ Mathematics