Mathematics - 2016-12-22 (page 6 of 6)

Ted Shifrin

7:00 PM

right ... so on such a region every vector field with curl = 0 is a gradient.

GFauxPas

right now

Balarka Sen

@Kari Exterior derivative is $0$.

Ted Shifrin

@Kari: It means the associated vector field has curl = 0.

@Balarka: He doesn't know forms.

Kari

Yep, I'm not at forms yet

Balarka Sen

Ah, ok, sorry.

Ted's got a better explanation there then

Kari

7:01 PM

No worries! I should really know them by now

Ted Shifrin

Ah, @GFauxPas: so you need some serious real analysis and a year of algebra would be good. But it also depends where you're hoping to do you Ph.D. in pure math.

@Kari: If you like YouTube lectures, you might find mine helpful for rigorous multivariable calculus/analysis.

GFauxPas

yeah so im looking as to which schools offer algebra, for example

Kari

Math 3500 and 3510, @Ted?

Ted Shifrin

Yes, @Kari.

Kari

I've studied Fréchet derivatives on Banach spaces so I'm gonna skip a fair bit of the directional stuff :-b

Adeek

7:04 PM

hey @TedShifrin

Kari

My differentiation course was overkill in all the wrong ways. Instead of just studying the implicit and inverse function theorems on $\Bbb{R}^n$, we did them between Banach spaces. The lecturer was a bit too advanced.

Adeek

hey ted do you want to come for 1 sec in facebook ?

want to tell you something

Ted Shifrin

Right, @Kari. Well, the proof of the Inverse Fn. Thm. I like and did in those lectures is the one that works in Banach spaces, but perhaps you didn't do concrete applications. So pick and choose what's interesting/important for you. The differential forms come many lectures later.

Balarka Sen

Not great for intuition, IMO. Once you know the proof for R^n it doesn't take much to generalize in Banach spaces

Ted Shifrin

Hang on, Karim. I'm actually trying to do something else.

Kari

7:06 PM

Will do!

Yep, I had to go back to examples in $\Bbb{R}^n$ to get some idea of what was going on, @Balarka

Ted Shifrin

Well, @Balarka, the "standard" proof in $\Bbb R^n$ (e.g., in Spivak and Munkres) does not generalize.

Adeek

okay @TedShifrin

Balarka Sen

@TedShifrin Hmm, do they not do it by using the Banach fixed point theorem?

Ted Shifrin

not remotely, @Balarka. They prove openness by using compactness and finding a critical point.

Balarka Sen

Interesting, never seen that one

Mary Star

7:08 PM

Could someone of you take a look at my question about cardinality?

-1

Q: Cardinality: Is the last sum correct?

We have the set $A=\{a_1, a_2, \ldots \}$, the $a_i$'s might be finitely or infinitely many. We have that $\mathbb{Q}(A)=\left \{\frac{f(a_1, \ldots , a_n)}{g(a_1, \ldots , a_n)} : f,g\in \mathbb{Q}[x_1, \ldots , x_n], g\neq 0, a_1, \ldots , a_n\in A, n\in \mathbb{N}\right \}$. We have that $...

tillmo

user254665 is right. Where is the problem?

Mike Miller

I don't really see why $\Bbb R^n$ is more concrete than $\ell^2$. You never use that there are finitely many real coordinates in any important way, you just use various important properties.

Ted Shifrin

For proofs, perhaps. For exercises, I totally disagree.

Adeek

I am so happy @TedShifrin the prof told me my paper looks great after I put a lot of hours into it

now I am officially done with my semester

Ted Shifrin

Congrats, Karim. That's nice!

Adeek

7:20 PM

yeah

I mean to be honest her class we did a lot of things however I learned so many things. It was too much but I should been more disciplined with respect to time.

Next semester I will be very discplined

Kari

I can relate

I'm going to lectures next term

... hopefully :-b

Ted Shifrin

most of my students missed no lectures ... the ones that skipped tended not to do very well (I wonder if there's correlation).

Kari

There definitely is

I went to most of my ones last term but there's one that was pretty boring and I skipped sometimes

I'm not familiar with the method of writing integrals in the form below and manipulating. Usually I'd just parameterise and move on from there. Does the way below give any major advantages, @Ted? $$ \oint_{C} \dfrac{\text{d}z}{\text{z}} = \oint_{C} \dfrac{\text{d}x + i \text{d}y}{x+iy} $$

Ted Shifrin

You want to avoid real and imaginary as much as possible and actually use complex functions.

Kari

Yea, this is the first time I've seen it done like that

The font looks really cool in the LaTeX on here as opposed to the default Computer Modern Roman

Looks a bit more stylish in the chat

Ted Shifrin

7:34 PM

There are, of course, multitudinous fonts you can use in your own typesetting if you wish.

Kari

Do you happen to know the one used on here, @Ted?

Ted Shifrin

Nope. It looks like Times Roman.

Pissed off layman

what is logically wrong with the phrase "for up to"?

Kari

It can't be!

The $z$ looks different, not to mention some integral sign subtleties :-)

Can you use it in an example, @Pissedofflayman?

Pissed off layman

P is true for up to x.

Kari

7:43 PM

That just seems strangely phrased

What's the specification on $x$?

Ted Shifrin

$P(k)$ holds for $1\le k\le x$?

Pissed off layman

0

Q: Irritated by upper bounds pretending in adverts that they mean something "guaranteed to keep your baby dry for up to 12 hours"

To me the meaning is clearly that the baby will remain dry for up to 12 hours. I don't see why it is irritating?

phrase-meaning

Ted Shifrin

LOL

Kari

Hahahahaha!

What is this? Hahaha!

Pissed off layman

:D

Ted Shifrin

7:47 PM

I guess the point is that they should say "for no longer than 12 hours," and it's likely to be for only 3 or 4. :P

Alessandro Codenotti

I do my homeworks up to 24 hours per day :P

Ted Shifrin

You could even say up to 48 hours per day :P

It's just as true.

Alessandro Codenotti

Nah, I'm not such a good student

JeSuis

Bonsoir @TedShifrin and Hi chat

Ted Shifrin

Bonsoir, JeSuis.

Sophie

7:50 PM

it's weird to think that there exist functions that grow faster than any elementary function

Kari

Why's it weird?

JeSuis

@TedShifrin If I have $\Bbb{Q}(j,\sqrt{2})$ is Galois because slitting field of $(X^3-2)(X^2-2)$ right ?

Ted Shifrin

Nope.

You would need $\root3\of 2$.

JeSuis

arf

$(X^3-3)(X^2-2)$

Ted Shifrin

Easy to fix.

Agh.

You're getting worse.

JeSuis

7:53 PM

why ? We have $\Bbb{Q}(j,\sqrt{2})=\Bbb{Q}(\sqrt{-3},\sqrt{2})$

Ted Shifrin

Wait. What is $j$?

Some people use it for $i$. What do you mean?

JeSuis

a non trivial cubic roots of unity

Ted Shifrin

Horrible notation unless you announce it clearly.

So you're still wrong.

JeSuis

$j=\frac{-1+i\sqrt{3}}{2}$, sorry @TedShifrin it's common notation "here"

Ted Shifrin

So what polynomial gives you $\Bbb Q(j)$?

Tobias Kildetoft

7:55 PM

Wow, I have never seen a non-Greek letter used for that

JeSuis

$1+X+X^2$

Ted Shifrin

Now finish your problem.

Me either, @Tobias.

JeSuis

Ok silly me, $(1+X+X^2)(X^2-2)$

@TobiasKildetoft you mean $\zeta_3$ something like that

Tobias Kildetoft

@JeSuis Or probably at least 5 other Greek letters

JeSuis

yep, in "france" we often use $j$

Sophie

7:58 PM

unreasonable french strike again

Ted Shifrin

The problem is that engineers (and physicists) often use $j$ for $i$.

évidemment @Sophie

(and now I suppose you'll insult my French again ...)

Sophie

your french is okay or better

Ted Shifrin

LOL

Sophie

your french $\geq$ okay

Ted Shifrin

JeSuis ne me s'en plaint pas. :)

JeSuis

8:01 PM

@TedShifrin if your french is only okay... my english is zero

I suppose that the Galois group is the Klein group

Ted Shifrin

You suppose correctly.

JeSuis

thanks

Sophie

let $\alpha=[a_0;a_1,a_2\dots] $ be an algebraic number of degree at least 3, and let $f$ be a function. Is it possible that $a_n<f(n)$ for all $n$? And can we find such a number and such a function?

Mike Miller

Somehow without the "four-" I take Klein group to mean the fundamental group of the Klein bottle.

Tobias Kildetoft

@Sophie Unless you put more restrictions on $f$ then yes, just define $f(n) = a_n+1$

Ted Shifrin

8:06 PM

@MikeMiller Only a topologist could even think that ...

SteamyRoot

I actually have the same problem

Sophie

bah f is an elementary function, that is a composition of $\ln$ $\exp$ and multiplication and addition, etc

Mike Miller

@Ted Seems pretty natural.

SteamyRoot

But, then again, I study the fundamental groups of (infra-)nilmanifolds.

Mike Miller

Not that I dislike the Klein 4-group. It's one of my favorite subgroups of SO(3).

Semiclassical

8:09 PM

Reminds me of another math 'joke' I like: If $f(x,y)=x^2+y^2$, then what is $f(r,\theta)$?

Sophie

$r^2+\theta^2$

GFauxPas

that caught me off guard semiclassical

Mike Miller

@SemiC I love that one

Ted Shifrin

This is precisely why I hate the usual sloppy notation in calculus courses (with Leibniz notation, of course).

GFauxPas

What sloppy notation are you referring to?

Semiclassical

8:11 PM

It's a good one.

Ted Shifrin

$\dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx}$

$y$ representing $f(x)$ and $f(u)$, of course.

Mike Miller

I really want to get up and work but there's an adorable dog in my lap and it's hard to bear loving him

GFauxPas

it's true if you define $u$ the right way

Ted Shifrin

You have a bear in your lap, @MikeM? Must be heavy!

@GFauxPas, no.

Alessandro Codenotti

I have the same problem, just without dog or any other excuse...

Ted Shifrin

8:12 PM

The $y$ on the left is $f(g(x))$ and the $y$ on the right is $f(u)$.

GFauxPas

I'm missing something. $u = g(x)$

Ted Shifrin

Anyhow, I'm just saying we lead ourselves into utter confusion with this notation, but we do it anyhow.

Semiclassical

I should probably go see if my advisor is around, but I'm in my afternoon ground state right now

Ted Shifrin

@Semiclassic: Soon it'll be Xmas holidays (if it isn't already) and your adviser won't be there.

GFauxPas

oh

oh I see what you mean

Kari

8:13 PM

Afternoon ground state, @Semiclassical?

JeSuis

@TedShifrin Comment le groupe symétrique $S_n$ peut-il agir sur l'ensemble des parties de cardinal 3 de {1..n}?

GFauxPas

well the $y$ is built into the differential so it doens't HAVE to be the same on both sides

Ted Shifrin

Comment pas? @JeSuis

GFauxPas

but I see why its confusing

Ted Shifrin

$\{i,j,k\} \rightsquigarrow \{\sigma(i),\sigma(j),\sigma(k)\}$.

Semiclassical

8:14 PM

A physics joke. The ground state of a quantum system is its lowest energy state. @Kari

GFauxPas

because ideally $\dfrac {\mathrm dy}{\mathrm dx} = \dfrac {\mathrm d}{\mathrm dx} y$

Ted Shifrin

I got it, @Semiclassic :P

Mike Miller

bear *moving him

GFauxPas

that is a problem

what do I do :(

Ted Shifrin

No, that's not the point, @GFauxPas. There are truly two different functions being represented by the letter $y$.

mercio

8:15 PM

@sophie would a function growing like the Ackermann function be okay ?

Sophie

do we know anything about the asymptotic behavior of Ramsey numbers? Say what is the order of magnitude of $R(6.02\times 10^{23},6.02\times 10^{23})$

mercio

don't Ramsey numbers grow really fast ?

GFauxPas

Is it resolved by letting $y$ equal the function rather than the function's image?

Sophie

@mercio I think so, I haven't been able to prove even absurdly weak bounds

Semiclassical

A similar question: What notation would one use to write a number of similar size as a Ramsey number?

JeSuis

8:17 PM

@TedShifrin Car je ne vois pas comment dénombrer le nombre de partitions de $S_6$ en partitions de 3-3 i.e. of the form $\{A,B\}$ where $\vert A\vert=3$ and $B$ as well. I can count it by "hand" but with group action ?

GFauxPas

$\Large{X}$

Alessandro Codenotti

the usual proof of existence of Ramsey numbers gives (huge) upper bounds in terms of smaller Ramsey numbers

Kari

Something to do with upward facing arrows maybe, @Semiclassical?

Semiclassical

Knuth up-arrow notation?

Ted Shifrin

@JeSuis: The action I defined is clearly transitive. You need to figure out the stabilizer of a particular element.

Kari

8:18 PM

Oh I'm thinking of Graham's number.

mercio

there is a point where the hugeness of numbers can only be compared via the strength of the logical axioms needed to define them

4

JeSuis

@TedShifrin Ok thanks

Semiclassical

I'm not sure even up-arrow notation would help

GFauxPas

I can't even imagine how big the universe is in meters

mercio

the observable universe*

GFauxPas

8:27 PM

the non-observable one is even bigger, probably

whatever that means

Balarka Sen

8:50 PM

@MikeMiller Well, on a brighter note, you've got an excuse now :)

I used to hang out with a bunch of stray cats and gave them better food than stuff from the street dustbins. But then my family moved to an urban wilderness far away from that place...

Mike Miller

I had to move him eventually. I'm writing (!!)

Not well. But I'm writing.

Balarka Sen

Certainly an improvement. Keep at it!

I've seen a person writing a paper (I used to be in his office on a regular basis). He's extremely slow and took hours to write/format a paragraph, but he sticked to it and hardly moved from his chair. I just didn't get how it was possible.

Mike Miller

it's not

Balarka Sen

maybe he played tetris when i wasn't looking, who knows

GFauxPas

you being there made him nervous, clearly

Balarka Sen

9:00 PM

nah

a'right, gonna push off to bed. night

GFauxPas

don't go

Akiva Weinberger

@TedShifrin So there's a diffeomorphism from this to $\{x:1\le\|x\|\le2\}$ with the boundaries identified, whose universal cover is $\Bbb R^3\setminus\{0\}$.

Mike Miller

$S^2 \times S^1$, $S^2 \times \Bbb R$ are better names

Akiva Weinberger

So I think I can take three linearly independent vector fields on $\Bbb R^3\setminus\{0\}$ — always up, always to the right, always forward — and then use the cover map to get from this to $S^1\times S^2$. Would that work?

@MikeMiller Right, I meant specifically the subset of $\Bbb R^3$ there

Mike Miller

@AkivaWeinberger This does not work as stated. You have an action of $\Bbb Z$ on $\Bbb R^3 \setminus 0$. Your vector fields had better be preserved by this action.

the action being $v \mapsto 2v$, or whatever

Akiva Weinberger

9:13 PM

@MikeMiller Right, yeah — that action is just scaling, right? So the directions of the vectors are consistent but the magnitudes aren't

Mike Miller

the action on the tangent space at $v$ is multiplication by $2$

Akiva Weinberger

I can make them unit vectors in the tangent space or something I think

Mike Miller

How do you intend to do that?

Akiva Weinberger

or make the magnitude $2^r$ or something

Pissed off layman

That must have been a direct flight?

Akiva Weinberger

9:21 PM

Yeah

Pissed off layman

When do you come back?

Akiva Weinberger

In ten days or so

Pissed off layman

cool

GFauxPas

are you there on your own? family?

Akiva Weinberger

9:36 PM

Family

And visiting sister who was there on her own

Kari

@robjohn Do you happen to know which packages are used to obtain the LaTeX font here? It seems to be a times variant so I tried \usepackage{mathptmx} but the integral signs still don't look as aesthetic.

(I ask because I recall your name in the hyperlink to LaTeX in chat)

Ted Shifrin

9:57 PM

Glad you made it safely, DogAteMy. :)

Sophie

how many knights can you put in a 8*8*8 chessboard such that none of them threaten each other and the knights move along a plane in the usual way?

Mike Miller

thirteen

Ted Shifrin

Stop being so easily distracted, @MikeM. Put the dog back.

Vrouvrou

What i can do to this $$(2^{n\gamma}+2^{n(\gamma+\gamma^*)}+2^{n\gamma})^{\frac{\gamma^*}{\gamma}}$$

Mike Miller

I can't write, man.

Ted Shifrin

10:03 PM

It doesn't have to be correct syntax or even complete sentences. Just get the math down in some form. It'll get reorganized and rewritten later.

Perturbative

Kari

Everything ok, @MikeMiller?

Perturbative

Hey everyone, quick question, in the definition of a basis, does the third basis element need to be unique or can it just be $B_1 \cap B_2$?

Kari

Topological basis? I thought there'd need to be a $B_3 \subset B_1 \cap B_2$.

Perturbative

i.e. $B_3 = B_1 \cap B_2$ would be valid? (In reference to the picture I linked above)

Kari

10:07 PM

Oh, yea I don't see why not.

Perturbative

@Kari, Yep, basis for a topology

Kari

Uniqueness isn't an issue.

Vrouvrou

any help for this

What i can do to this $$(2^{n\gamma}+2^{n(\gamma+\gamma^*)}+2^{n\gamma})^{\frac{\gamma^*}{\gamma}}$$

Ted Shifrin

@Perturbative: If $B_1\cap B_2$ happens to be in your basis, you're golden.

Perturbative

@Kari Okay well, then if that's the case then wouldn't every topology $\mathcal{T}$ be a basis for itself? I was trying to prove this and that was the only issue I was having

Kari

10:09 PM

I can't say I've thought about it before right now, but wouldn't that be ok?

Ted Shifrin

Sure, @Perturbative, but the idea is to have an optimally chosen smaller collection of sets. The topology itself is unwieldy in general.

Perturbative

@TedShifrin, Okay great. Thanks for your help! @Kari, you too!

Kari

Any time :-)

Perturbative

@TedShifrin, Also in the definition of a topology, is there any particular reason why the union of any subcollection of $\mathcal{T}$ must be in $\mathcal{T}$, but only the intersection of finite subcollections of $\mathcal{T}$ must be in $\mathcal{T}$? (I can post a picture if this is a nonstandard definition)

Mike Miller

@Kari I am just a very slow writer.

One sentence every month or so.

Kari

10:16 PM

Oh wow!

Ted Shifrin

@Perturbative: Consider what happens if you intersect all the open intervals $(-1/n,1/n)$ in $\Bbb R$.

Kari

(It's a pretty standard definition as far as I know, @Perturbative)

Perturbative

@TedShifrin You'd get the empty set $\emptyset$

Kari

Not quite, @Perturbative

Ted Shifrin

Try again, @Perturbative.

Kari

10:20 PM

What's $\lim_{n \to \infty} \pm 1/n$?

Perturbative

Whoops my bad, you'd get $0$

Ted Shifrin

Is $\{0\}\subset\Bbb R$ an open set?

Perturbative

Definitely not. I see where you're going with this now

Ted Shifrin

I was just leading you to answer your own question.

Perturbative

I see, I prefer it that way :)

And the union of any (finite or non-finite) subcollection of $\mathcal{T}$ on a set $X$, would be at most $X$

Which is why we don't need the 'finite' condition for the union

Ted Shifrin

10:27 PM

no, not really ...

Kari

~~That seems reasonable though, @Ted~~

I just read it

Ted Shifrin

Uh huh

Kari

Try to prove it for two sets, @Perturbative

Then try and extend

Perturbative

$\mathcal{T}$ is just a collection of subsets of $X$. Let $K_i$ be an element of $\mathcal{T}$. $K_i \subset X$. Let $J_j$ be an arbitrary subcollection of $T$. Therefore $J \subset T$, and $K_i \in J_j$. Define $K_{j_i}$ to be a $K_i \in J_j$. Then $\bigcup_{j}\bigcup_{i}K_{j_i} = X$

mick

0

Q: Asymptotic to $f( 2 f^{[-1]}(x) ) $?

Let $f(x)$ be the half-iterate of $ 2 sinh(x).$ Im looking for an asymptotic to $f( 2 f^{[-1]}(x))$ for Large $x>0$. $^{[*]}$ means iteration here thus $^{[-1]}$ means functional inverse. For the $y$ th iteration of $ 2 sinh(x) $ we use the fixpoint at $0$ And use the so-called koenigs functi...

optimization asymptotics dynamical-systems tetration

Perturbative

10:38 PM

@TedShifrin, @Kari, I guess that was what I was trying to say earlier, in a more rigorous way. If it's nonsense then please let me know, no harm done

mick

Question was almost closed :/

Perturbative

Also it should be $J_j \subset \mathcal{T}$, not just $J \subset \mathcal{T}$

mick

Maybe you guys have ideas

Guess not ?

GFauxPas

How do you even know if a function has a fractional iterate

Before you talk about asymptotic growth

mick

Because of the fixpoint 0

GFauxPas

10:54 PM

There's some theorem you're invoking that I don't know about

mick

See the link in the OP : koenigs function. Also note that 2sinh has only 1 real fix and is strictly > x for x > 0

The derivative at 0 = 2 so koenigs applies

GFauxPas

And a Koenig function iterates to the original function?

Sophie

11:28 PM

in how many ways can I put the numbers 1 to n in an n by n matrix in a way that each number appears only once in each row and each column?

Perturbative

@Sophie $(n!)^n$

mercio

no

Sophie

I know that's an upper bound

do all those configurations work?

Perturbative

@Sophie, Sorry misread the question, that's the number of ways each number appears only once in each row

mercio

your best bet is to brute force the first few values and check what oeis says

Sophie

11:32 PM

combinatorics is the devil

GFauxPas

one of my professor's specialty is combinatorics

and I'm so bad at it

usukidoll

._. eww

GFauxPas

they don't need a course with combinatorics :( so he teaches other things

Perturbative

@Sophie, just guessing again, but I'd say $\Pi_{i = 0}^{n-1} (n -i)!$

GFauxPas

usukidoll I got excited and thought there was a bulbasaur hat

i bought pokemon moon but I told myself I wouldn't open it until my finals were over, they're over now and I'm waiting for an urge to play it

Kari

11:40 PM

I'm playing sun right now, @GFauxPas!

Only just started and ~~Hawaii~~ Alola is pretty cool!

Sophie

@mercio this is hard to brute force

usukidoll

pokemon sun and moon?

I heard it's based of Hawaii x.X

Lovsovs

Hey guys, a quick Q: Is the set of all diagonal matrices with exclusively real, positive diagonal entries connected? Is it even simply connected? Thanks.

Kari

What does it mean for a set like that to be connected, @Lovsovs?

Lovsovs

11:56 PM

Hmm, is the question not well-defined?

Kari

Nope, I just can't remember what it means to be connected

Lovsovs

It basically means that all the elements are "close" to each other, i.e., you can draw a continuous curve through all points.

Well, perhaps not all points for the same curve, but for any two points, and do so without leaving the set.

Sophie

how do you define distance between two matrices?