Mathematics - 2017-07-29 (page 1 of 3)

Ted Shifrin

00:00

Talk to you soon, @heather :)

Daminark

Basically, you have something a bit weaker than two spaces being homeomorphic, and you can compute certain nifty algebraic things which make your life easier that involve this relaxation. That's the garbage explanation, at least

Later it'll get more clear

See you!

Paul Plummer

I feel bad for homotopy theorists, why do spheres have to be so difficult

Daminark

Gotta be on guard against arguments which are spherical

Ted Shifrin

Demonark: You should declare a one-month moratorium on so-called humor.

Paul Plummer

Totally, aspherical is where it is at

Daminark

00:12

@TedShifrin A whole month? My consciousness would probably just dissipate into a cloud or something at that point

Ted Shifrin

Seems an attractive thought ... :)

Daminark

Like 45 seconds is reasonable, one month tho...

Paul Plummer

How is that hyperbolic geometry book treating you? @Daminark

Daminark

I got rekt pretty fast, I'll first do Ted's stuff, then Riemannian geo

I am doing hyperbolic dynamics though

Actually my lecture today didn't end up getting too far into the dynamics side because I started by giving what I thought would be a 10 minute overview on stuff like, okay what's a manifold, what's a Riemannian metric, etc

10 minutes became 50 :P

PVAL-inactive

It was more on the hyperbole side then?

There's lots of hyperbole in all my lectures.

Daminark

00:16

snaps eyy

Ted Shifrin

I warned you about time dilation, Demonark.

Daminark

This is true, although the thing is that our professor came today

So part of the reason it took as long as it did was that our professor often wanted to expand quite a bit on my points

So, at one point I mentioned tangent spaces and was gonna say

Ted Shifrin

Hand him the chalk? :P

Daminark

"Alright, so you can think about curves on a manifold and define tangent vectors by derivatives composition with functions from the manifold to $\mathbb{R}$. This lets us talk about tangent spaces and things just work as they do how we defined it in $\mathbb{R}^n$. Moving on"

Ted Shifrin

That's way too glib.

Unless you're just doing everything locally anyhow.

PVAL-inactive

00:20

Yeah or just think about everything embedded.

Daminark

But Schlag wanted to treat that point in more detail. Similar for stuff like vector fields, Riemannian metrics, and the like. In particular, Schlag wanted to emphasize how you the fact that chart compatibility is what made these quantities coordinate-free

PVAL-inactive

is this a basic manifolds course?

Ted Shifrin

Too much technical stuff already in that dynamics stuff.

No, PVAL.

Daminark

We're doing dynamical systems, and my lecture is kicking off hyperbolic dynamics, where things are taking place in a Riemannian manifold, so I was gonna give a rehash of definitions

Paul Plummer

Is this a class?

Or is this some independent stuff

Eric Silva

00:24

@Daminark did everyone get confused

PVAL-inactive

I don't know how to give talks on something non-trivial that requires this kind of basic material without assuming people know it.

Daminark

Not an official one, it's this bootcamp that our analysis professor organized where we're going through various things. We give lectures from the books and do psets

Eric Silva

I was kind of zoned out when he went off

PVAL-inactive

Like I am now imagining myself trying to explain what a diff form is in the beginning of all my talks.

Daminark

@EricSilva I think many were, Schlag did his calculations rather quickly so I think many were looking at it like "It's very plausible to me that this makes sense" and not much better than that

Eric Silva

00:26

Ah, I mean when you're doing things mostly in proper format like he was the notation can be cumbersome

And honestly I think a lot of that basic stuff should be done on your own anyway

Daminark

Though I think they got the general point of, this is a diffeomorphism so we can change coordinates and not screw up. Then again it's tricky, many people were iffy on the manifolds stuff in Schlag's class

Ted Shifrin

I have mixed feelings about such endeavors, especially with hyper-technical things. But you can't be pedantic and do a graduate course on each thing.

Eric Silva

Wrestling with the indices takes practice

Also unclear if he actually wants me to tell you guys what connections are and some basic Riemannian stuff or if he was just giving you a "sneak peek"

Daminark

@PVAL in general this is true, and like you kinda want to go into the book we're using after taking a course on analysis, topology, and manifolds, but in reality most just had analysis

Paul Plummer

Yah the moment I see indices in a lecture there is a 95% I zone out

At least if it is any more complicated than let blah be indexed by $I$

Daminark

00:31

Same @Paul

PVAL-inactive

I think the goal in a talk where the audience is varied is to explain why something is cool, and actually describe a few of the more interesting/important arguments in the work.

Eric Silva

I tend not to mind them so much but I prefer that calculations be left as exercises cause I will basically always sit down and do them later

PVAL-inactive

rather than go through everything formally.

Eric Silva

@Daminark it was actually kind of weird to see schlag derail the way he did cause he's generally not fond of going on super abstract tangents

Like last year he didn't mention any of the stuff he did today until we were almost through Ted's book

Daminark

Well, to be fair I was doing things somewhat formally

The context made sense but I think Schlag didn't want to just have a bunch of words and Greek symbols plus maybe 2 pictures or something

Eric Silva

00:36

Also for the future @Daminark: give more examples/non-examples, and draw pictures, I remember you defined a foliation and it was super technical, that could've done with a quick picture

Ted Shifrin

whispers to Eric: he's an algebraic formalist

Eric Silva

Lol

PVAL-inactive

A foliation is a fiber bundle where you can glue things on top and bottom instead of just on the sides.

Eric Silva

Also next time ask me for chalk, I always carry fancy Japanese chalk and I couldn't read your scrawl for most of the lecture

Daminark

Actually I was originally gonna present the juggernaut theorem as "It's basically saying that if you have a diagram which commutes up to delta, you can epsilon-jiggle this map and get something which actually commutes"

Eric Silva

00:38

Schlag would've been bemused lol

And then probably said "I'm very confused"

Ted Shifrin

I've been known to say "I'm confused."

Paul Plummer

I don't know what the juggernaut theorem says, but that makes enough sense to me

Daminark

That was the theorem I showed you yesterday with 5-6 quantifiers floating around all at the same time or something

Eric Silva

Yeah im aware

I ended up reading this chapter cause it looked interesting

Daminark

I first saw that theorem and was like "Yeah here we've got some words and symbols"

How'd you like it?

Eric Silva

00:42

Anosov diffeomorphisms are pretty interesting

PVAL-inactive

Anosov diffeomorphisms are pseudo-interesting.

Ted Shifrin

I thought I was joking when I said to let me know when you prove geodesic flow on negatively curved manifolds is Anosov.

Paul Plummer

what about pseudo-Anosovs @PVAL-inactive

although they are not diffeo

Ted Shifrin

We did that theorem in the third quarter of my graduate dynamical systems course my first year of grad school.

PVAL-inactive

All the ones i know are.

@Paul

Daminark

00:44

I don't think we're doing that theorem exactly

Ted Shifrin

No, it's too hard for this.

Paul Plummer

You are right @PVAL-inactive For some reason I was thinking they where not

Ted Shifrin

You need a bunch of Riemannian geometry, stable/unstable manifolds, etc.

Daminark

Like we don't have curvature at our disposal yet, all we're doing is invariant cones, stability of hyperbolic sets, and stable/unstable manifolds

Eric Silva

I was actually talking to someone abt that after the lecture, that surprised me

Ted Shifrin

00:45

Too bad I threw out all those notes :(

Daminark

I pity whoever is doing next Friday

Actually holy crap there's more

Ted Shifrin

I don't know why Schlag picks such recondite, technical stuff.

Actually, that theorem probably isn't so bad.

Daminark

When I defined a hyperbolic set in class today he was actually saying "Okay so unless you already know this definition this won't make sense, let's have an example, a simple one, not the hyperbolic toral automorphism"

Ted Shifrin

I reiterate what Eric said (and I said it during your first lecture, too) — try to use examples and non-examples as much as you can.

Daminark

Maybe it's just that there isn't much more literature to do dynamics from? At least this type

PVAL-inactive

00:49

Well at some point you'll end up in the same place.

Eric Silva

I think examples are more important than all the theory building tbh

Daminark

Yeah that's pretty true. Also what would a picture of a foliation be anyway? Like I don't actually have a picture in mind

Ted Shifrin

Sheets of paper in a stack ...

Eric Silva

Integral curves of a flow?

PVAL-inactive

translations of irrational slope on a torus.

Paul Plummer

00:51

What has that book done to you :P

Mike Miller

A decomposition of manifolds into submanifolds in a reasonable way.

Eric Silva

True that

Mike Miller

Look at pictures of foliations that actually exist. The foliated atlas definition isn't any good when you're trying to understand what a foliations is, it's good for being careful about smoothness issues.

PVAL-inactive

I still like my theyre fiber bundles where you can glue things on the top and bottom as well as the usual sides.

Daminark

@Paul Lol I legit used to think that I was like, pah I don't want technicality, I want pictures. Now look

Mike Miller

00:52

I don't understand what that means at all.

Daminark

Oh okay those pictures make sense, I think

PVAL-inactive

like you can glue open sets with the grain of the fiber.

instead of just attaching end to end pieces against the grain of the fiber.

Daminark

Hmm

Eric Silva

Literally half the chapter was lists though @Daminark

Idk if I like brin and stucks book

Daminark

Yeah, I dunno, it's a book I'd definitely prefer for use in a class

Mike Miller

01:04

I guess. I don't like thinking of it as patched together though. I generally like to think of the whole object and use the patches as a tool.

Daminark

Since I think there's definitely a sense in which you're just in an ocean of stuff, it's nice to see... I dunno, the "insight", if it's fair to call it that? (I'm bad at wording so like all things I say do not read into it too much)

Somehow like Rudin, great book for a class, but if you're just reading it for a book club or something I dunno if I'd roll with that necessarily

Paul Plummer

Yah, it can be hard to figure that out the insight or main idea, especially if a book is just a lists and symbol pushing (maybe it isn't that but that is the impression I have gotten from here)

Eric Silva

I read rudin by myself way back when and think that's honestly the better way to read it

Daminark

My issue with just grabbing Rudin and reading it is a problem I have with many analysis books. It sorta puts stuff out in a certain way which is conducive to some idea of presentation but not how you should think about it

Take, say that if $a_n\to L$, then $\frac{1}{a_n} \to \frac{1}{L}$

You'd want to do that through the typical, okay calculate the difference, you get $|\frac{a_n - L}{a_nL}|$, so the top tends to $0$ and you just need to find the right bound on the bottom. Why should such a bound exist? Because $L\ne 0$ and $a_n\to 0$, perfect! Now figure out exactly how this bound should work

I remember looking back on how Spivak did something vaguely similar and was just straight up horrified, like he just wrote some stuff which didn't really mean anything, just pull a rabbit out of a hat and gg, you know?

Rudin is rather guilty of that, which is why I think one should read it under some form of guidance

Secret

01:50

[Random]

Poncaire recurrence inspired:

Let a list be $s_0=(a,b,c,d)$. Clearly this has $|S_4|$ number of permutation. Now consider a string $S=s_1,s_2,s_3,...$ where each $s_i$ is understood to be permutation of $s_0$ . Find the maximum length of $S$ such that once this is exceeded, then

at least 3 elements in terms of abcd must repeat itself

as if they are beginning to form a portion of a periodic pattern

e.g. a..a..a.........

It is easy to see that once all 24 permutations of abcd are writtened down, we have a string of the form $s_0s_1...s_{23}$

Now, to check for patterns at this scale, it is sufficient to consider out of all 24, how many of these have a in the 1st,2nd,3rd,4th position

BAYMAX

03:24

If $A$ is a $3 x 3 $ matrix with integer entries such that $det(A) = 1$,then number of such matrices $A$ are ?

how do i approach this!

Is that infinite

secondly if $A$ is a $3x3$ matrix such that $AA^{t} = I$ , then number of such matricces are ?

Topologicalife

Hi there.

Husnain Raza

Hello

Topologicalife

is $$f(x,y)=\begin{cases} x\cdot \sin\dfrac{1}{xy} & \text{if}& xy\neq 0\\0 & \text{if}& xy=0\end{cases}$$ differentiable at the origin? I don't think so cause the limit $$f'((0,0);(a,b))=\displaystyle\lim_{h \to{0}}{\dfrac{1}{h}}ha\cdot \sin\dfrac{1}{h^2ab}= \displaystyle\lim_{h \to{0}}a\cdot \sin\dfrac{1}{h^2ab}$$ doesn't exist.

Husnain Raza

03:42

Is there any way to explicitly find a formula or algorithm that calculates this sequence? oeis.org/A000066

Secret

2

Q: Group of integer orthogonal matrices

Let $O_n(\mathbb Z)$ be a group of orthogonal matrices B st. $B*B^T=I$ with entries $b_{ij} \in \mathbb Z$. How do I show that $O_n(\mathbb Z)$ is a finite group and find its order. I need to show also that symmetric group $S_n$ is a subgroup of $O_n(\mathbb Z)$. So it needs to satisfy associati...

Apply similar argument here except noting you are in the special orthogonal group of dimension 3, which is a subgroup of the orthogonal group of dimension 3

---
A plot of

$$f(x,y)=\begin{cases} x\cdot \sin\dfrac{1}{xy} & \text{if}& xy\neq 0\\0 & \text{if}& xy=0\end{cases}$$

literally go wild at the origin

Secret

04:08

both the x and y partial derivatives blow up to $-\infty$ near $(0,0)$

BAYMAX

Yes nice @Secret

but when $A$ is a $3 \times 3$ matrix

with integer entries such that $AA^{t} = I$

then number of such entries are?

I found that

the columns of $A$ are unit vectors

here $[e_{1},e_{2},e_{3}]$

so I think there are 18 of them

but I think there are more

Secret

$AA^T=I$ means the matrix $A$ is an element of the orthogonal group, which includes all A s with $\det = 1 $ and $\det = -1$

BAYMAX

oh I left the case of $det(A) = -1$

Secret

yeah, it consists of all rotations and improper rotations

BAYMAX

so 36

but still i think I am missing more

Secret

04:13

the identity is the obvious one. I suspect -I is also one of them

BAYMAX

entries with combination of 1 and -1

found 48 of them yes

Secret

nvm ,those are considered under unit vctors

BAYMAX

yes

Martin Sleziak

Isn't this the same question as what you are asking @BAYMAX? Group of integer orthogonal matrices

BTW that post would probably deserve a better title.

Oh, I see that Secret already linked to this question.

BAYMAX

yes@MartinSleziak

Martin Sleziak

04:25

Oh, I see that Secret already linked to this question. Sorry for the unnecesary ping then.

BAYMAX

I did not get that

ping

np

I figured 48

of them

is that correct?

Martin Sleziak

So $2^n n!$ in general?

BAYMAX

oh

need to see that carefully

Martin Sleziak

BTW another post on main: How many $3 \times 3$ integer matrices are orthogonal?

BAYMAX

nice,thanks :)

Secret

04:39

Meanwhile, I wonder how does one differentiate $f(x,y)$ by first principles... The trouble is that in higher dimensions, the possible paths are no longer just +h or -h, but all possible $C^0$ functions $f$ and thus first principle might look something like...:

Let $\vec{f}(h)$ be the directed version of $f$ parametrised by $h$. Then

$$f'(x,y)=\lim_{h\to 0} \frac{f(x+\hat{\frac{d\vec{f}}{dh}}\cdot \hat{x},y+\hat{\frac{d\vec{f}}{dh}}\cdot \hat{y})-f(x,y)}{h}$$

actually, that does not really took care of those $f$ that are not differentiable, need to think harder...

(NB That conflict of notation will be fixed in the newer version...)

AppleMyEye

A question that's too soft/trivial for main site: Say I have a magic fair coin that is head/tails, except at the first toss resulting in a tail it becomes unfair and will only land tails. What is the expectation for number of tosses to a head? I think it's infinite since there is a non-zero probability that I'll never see a head. Am I correct?

Secret

actually, we can rewrite $f(x,y)$ as $f(\vec{r})$ and then we can compute $f'(\vec{r})$ which give a jacobian (and the known formula for that is $\lim_{\vec{h}\to \vec{0}}\frac{||f(\vec{r}+\vec{h})-f(\vec{r})||}{||\vec{h}||}$). So if a point is not differentiable somewhere, then the jacobian at that point has to fail to match up with the limit of the jacobian near that point. To be investigated...

Meanwhile, another weird derivative notion in wikipedia:

Arithmetic derivative

In number theory, the Lagarias arithmetic derivative, or number derivative, is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis. Remark: There are many versions of "arithmetic derivatives", including the one as in this article (Lagarias Arithmetic Derivative), Ihara's Arithmetic Derivative, and Buium's Arithmetic Derivatives. == Definition == For natural numbers the arithmetic derivative is defined as follows: p ′ ...

Secret

04:56

@AppleMyEye Is the first toss always land a tail or only have a probability of 1/2 to be a tail?

Topologicalife

Hi. Is true that $D_f(a)(tu) = t D_f(a)(u)$?

I'm trying to understand the step between the 4th and the 5th limit here: drexel28.wordpress.com/2011/06/02/…

I think he did $$ \lim_{t\to 0} \dfrac{D_f(a)(tu)}{tu} = \lim_{t\to 0} \dfrac{tD_f(a)(u)}{tu} = D_f(a)(u)$$

err

$$\lim_{t\to 0} \dfrac{D_f(a)(tu)}{t} = \lim_{t\to 0} \dfrac{tD_f(a)(u)}{t} = D_f(a)(u)$$

right?

Secret

If $D_f(a)(tu)=\nabla f \cdot tu|_{a}$ then by linearity of dot product, I can see how the t will move out. However, I am not confident to say whether it still holds when $D_f(a)(tu)$ is defined abstractly like in the notes

Topologicalife

05:11

Oh I see, thanks.

AppleMyEye

@Secret First toss is fair (1/2 probability of head or tail)

Secret

05:33

Let$ \{heads,tails\}=\{1,0\}$
Then expected value for heads for the (n+1)th throw should be:

$Pr(tail)[Pr(tail|tail)]^n+Pr(head)Pr(tail)[Pr(tail|tail)]^{n-1}+Pr(head)Pr(head)Pr(tail)[P(tail|tail)]^{n-2}+\cdots+Pr(all heads)$

begin{Roughwork}
1/211111(0)+1/2(1/211111(0))+1/2(1/211111(0)+1/2(...)))
0+1/2(0+1/2(0+1/2(...)))
end{Roughwork}

$=\sum_{k=0}^{\infty}\left(\frac{1}{2}\right)^{k+1}*0+\lim_{k\to \infty}\left(\frac{1}{2}\right)^k=0$

ANDY MURAY

Hi Everyone ,Could anyone help me to solve this math.stackexchange.com/questions/2375456/… problem where i got stuck any help will be really appreciated

Secret

Sorry typo:

$E(heads)=\sum_{k=0}^{\infty}\left(\frac{1}{2}\right)^{k+1}*0+\lim_{k\to \infty}\left(\frac{1}{2}\right)^k*1=0$

AppleMyEye

@Secret If I parse that correctly, expectation for number of heads is 0, hence my "...expectation for number of tosses to a head..." is infinite, yes?

Secret

I think it means, If you allow unrestricted number of flows, then it should take on average infinite number of throws to get a head. If you, however restrict the number of throws to some finite number n, then on average you need a huge but finite number of throws (to be exact: $2^n$) to get a head

I think this and possibly an extension to this question may be an interesting question to the main, On an unrelated note, I am not sure how a bayanesian will handle this question though

AppleMyEye

05:54

@Secret Thanks for taking the time! Tempting to post it on main, but I think it would be laughed at/down-voted to oblivion. Or perhaps my feeling that it's completely trivial is wrong... but thanks again in any case!

ANDY MURAY

@Secret could you please help me

Secret

Unfortunately, I am terrible at number theory. You need to wait for the number theory guys to get on and ask them

---
hmm... it seems for a bayanesian (if I understood correctly,) the answer becomes a bit more interesting: Suppose you flip a tail, then since the coin become locked in tails once the tail is shown (not sure how this can be incoporate as a notion of evidence), as the number of throws increases, the bayanesian will update its probability distirbution such that eventually $Pr(tails)\to 1$

this means, $Pr(head)\to 0$ as number of throws increases to infinity

in such case, it kinda agrees with the frequentist that it gets an increasingly small probability of heads as n increases

However, for the case where there are many consecutive heads, as the number of heads accumuate, $Pr(heads) \to 1$ unless a tail occur, which then $Pr(head)$ will start to decrease towards zero because tails will start to accumulate

This means, the probability distribution will become highly sensitive to the previous throws under a bayanesian view, while the frequentist will give a small number if the throws are finite, and 0 if the throws are infinite

I am, however not sure how to calculate expectation values under the bayanesian model, though

Typhon

06:45

I've been setting up various ways to determine the largest number of mutually coprime elements in some range of integers and I have to say... this is very interesting. As a function of length and and element there is quite a TON relating to modular arithmetic.

I'd sit down and explain it to all of you but in the past day of analyzing these numbers and stuff... I'm not even sure what's going on. If everything working right and someone can turn it into a lower bounded function one day... Legendre's Conjecture would be trivially true.

Granted... that's not a trivial act to do but I'm sure you all get my point.

oh and btw I've been thinking heavily about the Collatz Conjecture as well and I'm thinking that maybe some other stuff might help with that. Technically my inductive proof of doing blocks of 12 at a time is just stuck by 12n + 3 and 12n + 7. So that must mean the heart of the Conjecture lies in those two forms. Proving for those would resolve the conjecture.

Mostly... just posting stuff so I don't forget tomorrow.

plus I wanted to tell you all

perhaps I'll get lucky and have a breakthrough before the semester starts in a month.

probably not

but at the very least I'll have groundwork done in a possible part to proving it.

Mats Granvik

08:25

@Typhon
https://oeis.org/A275345

Kirill

Have a nice daytime and a pretty season!

Secret

lol ,the first sequence of OEIS has something to do with groups

oeis.org/A000001

Alessandro Codenotti

@TedShifrin which Frobenius theorem problem?

Secret

<s> oeis.org/A199999 last known sequence in the A series </s> actually not

Kirill

we set $p_0>0$ as a starting value, $t_0=0$ starting time. Let $h=t_1-t_0$. Then we set $p_1 = p_0 + hxp_0$. How to put $hxp_0$ in words? That is an initial population, with a factor $x$, multiplied by time-step? What does the term mean?

Or, if the population is already "scaled" by the factor $x$, why should it be multiplied by $\Delta t$?

Secret

08:38

~~strikeout~~

Kirill

08:53

So, $t_1 - t_0$ is not a time unit, but e.g. n time units? So, $p_1 = (nx)p_0$, so that we scale the initial population $p_0$ with a factor $x$, then we scale the resulted population $xp_0$ with factor $x$ again, and so $n$ times? That is the model?

Dattier

10:22

Hi, $$\text{Is it true that : }\forall a,b\in \mathbb R_+, \exp(a)+\exp(b)-1 \leq \exp(a+b) $$

SteamyRoot

Rewrite it as $(e^a - 1)(e^b-1) \geq 0$

Mats Granvik

Can someone tell me if this is the correct recurrence for the characteristic sequence of twin primes together with the number 2:

$t(1)=1$

$t(2)=0$

$n>2:$
$$t(n) = \left(1-\left({\prod_{d|n}}_\limits{d<n} t(d)\right)\left({\prod_{d|(n-2)}}_\limits{d<n} t(d)\right)\right)\left(1-\left({\prod_{d|n}}_\limits{d<n} t(d)\right)\left({\prod_{d|(n+2)}}_\limits{d<n} t(d)\right)\right)$$

?

Kirill

11:37

how to prove that $\sum_{k=1}^{\infty}\frac{1}{2^{k-1}}=2$?

SteamyRoot

11:50

Working with epsilon definition of the limit, I guess?

Prove that $$\lim_{n \to \infty} \left(2 - \sum_{k=1}^n\frac{1}{2^{k-1}}\right) = 0$$

Kirill

12:06

@SteamyRoot have proved that one. Just transform to the common geometric series and take the limit

so, $\sum_{k=1}^{\infty}\frac{1}{2^{k-1}}=\sum_{k=0}^{\infty}(\frac{1}{2})^k = \frac{1}{1-\frac{1}{2}}=2$

Kirill

12:18

I will have another question: if we want to show that $\lim_{z \to a}e^{z}=e^a$ in the complex plane, we say $\mid e^z - e^a \mid = \mid e^a \mid \cdot \mid e^{z-a}-1 \mid \to 0$ for $z \to a$. Do we use the absolute value to ensure that the things are $>0$, or are we obligated to use them to be able to compare complex numbers? I M not sure about the point.

user462339

How does one reach a chat user who doesn't appear to be online? Also, is doing so rude, if one is just going to ask them a mathematical question?

By reach a chat user, I mean, through the sites mechanics, and not by emailing them etc.

skullpatrol

There's no way, through the site mechanics, to reach them if they aren't online.

user462339

12:33

Interesting.

I would expect you would be able to atleast send them a notification on the main site.

skullpatrol

Just leave them a @name in chat and it will show up in their inbox when they come to the site.

user462339

Oh, I see.

Thank you.

skullpatrol

Also you can do the same on one of their answers/questions.

But don't be too persistent, that would be rude.

After all, people are not getting paid here to tutor.

user462339

Sure. I was only considering my options. I am trying to understand a few things, and hopefully they will come online at some point.

Kirill

Is there any pages like Wolfram Problem Generator for multivariable calculus?

skullpatrol

12:43

I would suggest just posting your work here and if anyone wants to help they will.

Alessandro Codenotti

@skullpatrol Doing that will work only if the user was in chat recently (in the last few days) @user462339

skullpatrol

Yeah, 7 days.

Doing so on one of their answers/questions will always work :-)

Daminark

13:01

If you ever are talking to a non-math friend and want to give a video which has a really nice depiction of what math is all about, try numberwang

user84215

13:48

in Discussing Specific Topics, 4 hours ago, by aminliverpool

I do not know how to appreciate your warm participation in this event.

user84215

I think that no one except me has not even looked quickly at this event. I think it may be a good candidate to be recorded in the Guinness.

Paul Plummer

@aminliverpool I saw it in the sidebar here, but then saw it would be like 4am

where I am at

user84215

that is, you want to say that it is not a good candidate for the Guinness?

Kirill

14:03

how to get $x=r \cos\varphi$ from $z=r e^{i \varphi}$?

$z=r e^{i \varphi} = r (\cos \varphi + i \sin \varphi) = r \cos \varphi + ir \sin \varphi$ but further?

$x := Re(z), y := Im(z)$?

SteamyRoot

14:52

yup

Kirill

15:10

is that not contradictive? we get cartesian coordinates of the complex points?

Topologicalife

Hi.

Kirill

@Topologicalife hi

Topologicalife

Is the second answer of this question wrong? math.stackexchange.com/questions/908989/…

I don't see the claim "Then exists such a $δ$ for which $|a−x|<1/4$ when $x<δ$"

SteamyRoot

@Kirill Why would it be?

Balarka Sen

@Daminark I learnt some dynamics!

Well, that's a lie, but yeah.

Topologicalife

15:14

I think it should be "when $x > \delta$" and what he did there is to fix a $\epsilon = 1/4$.

and it should be "Then exists such a $\delta$ for which $|a - \sin{(1/x)}| < 1/4$.

Kirill

@SteamyRoot that doesn't reach my imagination: we get the coordinates in the complex plain. Can cartesian coordinates exists in the complex plain? I have always thought that the cartesian plain is built for real numbers, the complex one for complex ones. Is that true?

SteamyRoot

You can just see the complex plane as the real plane $\mathbb{R}^2$ where $(x_1,y_1) + (x_2,y_2) = (x_1+x_2,y_1+y_2)$ and $(x_1,y_1)\cdot (x_2,y_2) = (x_1x_2 - y_1y_2,x_1y_2+x_2y_1)$

Alessandro Codenotti

hi @Balarka

how was the workshop?

Kirill

@Topologicalife for $x \to 0 \quad \frac{1}{x} \to \infty$, so $\sin(\frac{1}{x})$ has infinitly many limit points.

Balarka Sen

@Alessandro I learnt a lot of stuff

Topologicalife

15:19

I know. What have to do that with my question...

Kirill

@Topologicalife you said you are stucked trying to understand why the limit does not exist, don't you?

Alessandro Codenotti

great

Topologicalife

No @Kirill, I asked if the second answer was wrong.

Alessandro Codenotti

So are you going to study more diff geo now?

Kirill

@Topologicalife I don't know and have no idea why do you need $\delta$-s and $\epsilon$-s there...

SteamyRoot

15:22

That second answer is already flawed from the first step

He writes $\lim_{x \to \infty} \sin\left(\frac1x\right)$ instead of the limit for $x \to 0$

Topologicalife

@Kirill I think he tried to use the definition of limit but he did it wrongly...

I think he tried to write: Suppose $\lim_{x \to \infty} \sin\frac{1}{x}$ exists and it's equal $a$. Then exists such a $\delta$ for which $|a-\sin\frac{1}{x}|<\frac{1}{4} = \epsilon $ when $x < \delta$. But there exists $k$ for which $\frac{1}{2k\pi}<\delta$ and $\frac{1}{2k\pi+\frac{\pi}{2}}<\delta$

And so on.

Kirill

@Topologicalife $\epsilon$ is not the one to choose, but $\delta$

SteamyRoot

Well, first the limit is wrong, as it goes to infinite rather than $0$

Kirill

you cannot say $\frac{1}{4}=\varepsilon$.

SteamyRoot

Then, he takes $\epsilon = 1/4$ and claims that $|a-x| < 1/4$ when $x < \delta$

Topologicalife

15:26

Right.

SteamyRoot

which makes no sense given how the function is $\sin(1/x)$, and he's taking the limit to infinity, so he can only say something about large $x$, not small $x$

A bit later, he also makes the implicit assumption that $0 < \delta$ (which is fine) and that $1 < \delta$ (which is not fine).

Topologicalife

Which doesn't make sense to me is the inequality $|a−x|<1/4$

SteamyRoot

The entire proof doesn't make sense, it's blatantly wrong.

Topologicalife

I see, thanks :). Anyway, I got an idea from what you said.

We can do it in a similar way using the negation of the definition of limit.

Balarka Sen

@AlessandroCodenotti It was more of an analytic workshop than either geometry and topology

but yeah for example I want to learn complex geometry

in 1 dimension :p

Kirill

15:38

@Topologicalife the main thing I do not like there is that $k$. I see no reason why such one should exist.

Astyx

The Balarka is back

Balarka Sen

I have come to chew bubblegum and kick ass.

I am all out of bubblegum.

Semiclassical

rip Rowdy Roddy Piper

2

Alessandro Codenotti

@BalarkaSen what kind of analysis?

BAYMAX

15:54

RIP

Balarka Sen

@Alessandro Of various sorts. Complex analysis/measure theory/functional analysis.

But the geometric content was very good.

I can tell you about the dynamics I learnt when Daminark comes in

Alessandro Codenotti

I see, sounds cool

I'm going to have an exam on measure theory and other analysis stuff in September

Balarka Sen

maybe you could teach me

i only know it vaguely/as a blackbox

Alessandro Codenotti

I don't know much measure theory to be honest, just the one needed to get a sensible theory of integration

Balarka Sen

mhm

Paul Plummer

16:02

@BalarkaSen Hello

Balarka Sen

ohi

Alessandro Codenotti

The aim of that part of the course was defining what it means to integrate wrt a measure, to focus on the Lebesgue/Hausdorff ones later

hi @Paul

Paul Plummer

Hi @AlessandroCodenotti

Balarka Sen

16:15

@PaulPlummer Learnt about Teichmuller spaces and it's relation to mapping class groups

Paul Plummer

Cool anything particularly cool? What was this workshop?

Balarka Sen

@PaulPlummer It was a differential geometry workshop. We did a bunch of things, hyperbolic geometry & Riemann surfaces was one of the things

One of the upshots was that Teichmuller space $\mathcal{T}_g$ is homeomorphic to the space of quadratic differentials on $\Sigma_g$

Paul Plummer

16:32

@BalarkaSen Well it is homeomorphic, but isn't it just "naturally" embedded in the space of qd's. I am not to familiar with this perspective, or how it is used.

Abcd

I need help with the above questions since I am new to conditional identities

My attempts:

Balarka Sen

@PaulPlummer Well, the dimensions are clearly the same (6g - 6). I think the correspondence is given by taking a conformal structure $\Sigma_g \to (X, \lambda dz \wedge d\bar{z})$, finding a unique harmonic representative of the homotopy class of that map, pulling the metric by the map and ...

... taking the (2, 0)-part or something

Abcd

For 1st question: Tried to simplify it using sin A + sin B formula , tried to change the stuff inside the parentheses but couldn't

Balarka Sen

I should not use the word clearly.

Abcd

For 2nd question: I tried to use the property: Arithmetic mean >= Geometric mean >= Harmonic mean (using their formulas)

Balarka Sen

16:37

To compute dimension of T_g one needs Finchel-Nielsen parametrizations and to compute dimension of $H^0(X, K_X^{\otimes 2})$ one needs Riemann-Roch :3

so #rekt

Abcd

No one?

Paul Plummer

Well I was just saying that being homeomorphic isn't necessarily interesting, since T_g is also homeomorphic to R^n @BalarkaSen

Balarka Sen

Oh, I see. Yeah, they are naturally homeomorphic.

Ted Shifrin

Oh look who's back from his stay in solitary confinement :P

Balarka Sen

It is I.

Ted Shifrin

16:44

Did you have a good time?

Balarka Sen

I look like this guy now.

Ted Shifrin

Ah :)

@Abcd: Did you try using $A+B+C=\pi$ to set $C=\pi-(A+B)$ and rewrite everything on both sides just in terms of $A$ and $B$?

BAYMAX

Here@Abcd

Ted Shifrin

@Alessandro: As often occurs, Frobenius has his name on a theorem in manifold theory, along with his name on various things in algebra. And hi.

Abcd

@TedShifrin Nope. I will try that now.

What about second question?

Ted Shifrin

16:56

Cool, @Abcd. :)

Alessandro Codenotti

Hi @Ted I was thinking about his theorem classifying real division algebras

Balarka Sen

Frobenius <3

Ted Shifrin

I don't usually think about these sorts of questions without calculus. But since I'll be teaching precalculus soon, I guess I had better learn to. Let me think a bit. @Abcd

@Alessandro: Nope. This was about integrable distributions.

Alessandro Codenotti

I have no idea what you're talking about :P

Abcd

@TedShifrin Okay :). I know basic differentiation and basic integration. If it can be done using that then let me know.

Transcript for

$Mathematics$ Mathematics