Mathematics - 2016-04-30 (page 1 of 3)

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1:02 AM

@Kari There are natural situations (I'm thinking, for instance, $L^2$ spaces) where you'd like to define a norm on a set of function. The $L^2$ norm on functions on $\Bbb R$ is $\|f\|^2 = \int |f|^2$. Ignore for the moment the fact that it doesn't make sense to integrate a random function, and that some functions integrate to zero. (Think the function that's 1 at 0 and zero elsewhere.)\

In these situations, if $f$ is a measurable function, you can perfectly well write $\|f\|$ - it just might be $\infty$. We then restrict attention to functions such that the norm is finite.

Why is this framework helpful? Because maybe you'd like to prove a function $f$ is in $L^2$. Then what you might do is write down a series of inequalities which will finally end up with $\|f\| \leq C$, where $C$ is a finite real number. Thus $f$ is in $L^2$. But it was still helpful to write down the "norm" while doing this, even though you didn't know it was finite ahead of time.

1:22 AM

@BalarkaSen That doesn't look like differential forms. ;) Dune is a bad movie, more or less because of executive meddling. I also think it's a fascinating movie.

@MikeMiller Haha, no that's not differential forms at all.

I have Ted's homework latex'ed up but I don't want to send it until I get more done.

Hmm, now that you mention it I think I have been procrastinating there for half an hour. I should either sleep or get more done.

I haven't done work most of the day. :)

Blame jetlag.

Nah.

Going to go find a place to sit down and work now. I should have gotten more done today.

@MikeMiller re: dune. I see. I haven't actually seen a lot of Lynch movies, but I liked Erasorhead. Not that I do not know of really good directors who messed up quite a few films.

1:30 AM

I love Lynch. I do not blame him for Dune.

@MikeMiller And now that makes me feel a bit guilty too :). But at least I got a positive proportion of Ted's exercises done today.

I'm going to go to a bookstore first. Enjoy your life. Not much matters.

lol

nah, math is life. rest is killing time when one can't do math.

Hi @BalarkaSen

Heya.

1:40 AM

Nice ping with @AlexClark he seems interested in the same things

I thought so.

Mario Kart on Wii is best stress relief

That one was terrible.

what???

you get the wheel and everything

Remind me: what's a positively oriented basis? I feel a bit confused about Ted's comment on orientation on surfaces.

1:43 AM

what is everyone's opinion on "significant" progress on a problem?

"I had to tell my thesis advisor something, so I made up this progress"

If the problem says "does there exist so and so" and you are able to point out some places where it can't exist, is that good progress?

@Krijn you made it up?

@Balarka A positively oriented basis of what?

@ForeverMozart It's a joke

@Mozart: The gamecube and wii u ones were both better.

1:45 AM

I never got the Wii U...is it really better than plain Wii?

If I interpret what Ted says correctly:a parametrized surface $S$ is said to be orientable if there is a parametrization $\vec{g} : U \subset \Bbb R^2 \to \Bbb R^3$ so that $\partial g/\partial x_i, \partial g/\partial x_2$ form a positively oriented basis of $T_p S$ for every $p$

@MikeMiller nevermind, I gotta buy one . They have a new Zelda coming out for it in December

@Balarka Be less specific than "tangent space of a surface". You asked very generally what a positively oriented basis is.

So, very generally: a positively oriented basis of what?

@ForeverMozart I don't have one.

Of a vector space, I presume? I am unsure. A bare vector space cannot in any sense have a notion of orientation given just an arbitrary basis, I would think.

That's where I feel confused. I don't seem to remember if Ted defined this terminology.

Right, that's not a thing that makes sense for a vector space.

1:53 AM

I guess a vector space with a given basis?

with respect to a basis, I can make sense of another basis being positively oriented or not I suppose.

If so it's not clear to me what the choices of a set of "standard basis vectors" are on $T_p S$.

There aren't any. You need an adjective in front of "vector space".

@MikeMiller give me the old n64 one any day

@MikeMiller Sorry, it's not clear to me what that adjective is supposed to be.

is it possible to be addicted to this site?

i like to find questions I can answer

2:15 AM

Hmm.

I see, this is where he's defining it. He just means that the bases given by all of your charts are positively oriented with respect to each other.

Yes, and it's not clear to me what that means. Suppose I have just one chart. What does it mean to say that $\partial g/\partial x_i$ form a positively oriented basis of the tangent space at each point in the chart?

There is only the pair (U,U) so tautologically every pair is positively oriented wrt one another.

Ok, that does make sense (no nonorientable surface can be covered by a single chart). And how are we defining "positively oriented wrt each other" here? The differential of the transition functions have determinant > 0 at the intersections?

Yes. I'm sure he did already define this.

2:24 AM

Hey may have. I have never seen it. I also do not see it formulated like "positively oriented wrt each other".

Fine.

What I feel confused about is he mentioned the moebius strip as an example of a nonorientable surface, but that can never appear as a parametrized surface, can it? a parametrized surface is always orientable. however his definition of surface on section 4 is a parametrized surface :S

I guess I am sleep deprived and missing something. Maybe I'll study tomorrow.

2:41 AM

You use multiple charts dude.

Hmm, maybe I am confused. If I take $U_1$ to be an nbhd of the upper hemicircle of the unit circle in $\Bbb R^2$ and $U_2$ to be the lower, define $g_1$ and $g_2$ smoothly embeds $U_1, U_2$ in $\Bbb R^3$ resp so that $g_1(U_1) \cup g_1(U_2)$ covers moebius strip (e.g., $g_2$ adds a "twist" maybe) but $g_1$ and $g_2$ (and all their derivatives) match at the intersections, then $U_1 \cup U_2 \to \Bbb R^3$ given by $g_i$ on $U_i$ gives me a fine parametrization of the moebius strip, not?

Sure.

I am confusing myself. How did I just write down an injective immersion of the annulus into $\Bbb R^3$ so that the image became a moebius strip? seems wrong. I think a better idea is that I just go to sleep.

Uh, why do you think it's an injective immersion of the annulus?

Yes, it's way past your bedtime.

3:29 AM

it annoy anybody else people call SU(2) the unit quaternions

it's like saying U(1) is the set of 2x2 rotation matrices

not really, though I would prefer they said "can canonically be identified with"

2 hours later…

5:02 AM

anyone still ehre?

are you interested in strange topological spaces?

of coyurse.;

:)

I'm, not exactly thinking straight but shoot

which ones do you like?

fiunite ones mistly XD

I'm not too familiar with interesting infinite nonHaud saofsce

spaces8

you can assume Hausdorff

5:04 AM

maky

still lots of strange ones

i spas

spose8

you know indecomposable continuum?

hehe yes

well

that is strange

5:06 AM

I am not aware of any but the Knaster continuum

this is indecomposible right?

yes

mkai

you know explosion point spaces?

is this the thing where like

it's connected but when you remove a point it becomes totally disconnected?

yes

5:07 AM

I've never heard that name used before

I still can't really get my head around this thiung

but people tell me it's real XD

it's easy

actually both of these thigns I learned about at this years JMM

someone gave a talk about pathological suspaces of $\Bbb R^2$.

maybe I wrote about it already

nope :/

yes

What's gotten these monsters on your mind>?

i like to think about them and construct new spaces

cause there are still some open problems around them

5:24 AM

that's neat

anything you can explain quickly?

hmm lets see

one problem I work on: is there a completely metrizable explosion point space in the Cantor fan which is dense?

i am not sure it it can exist or not

Also, what do compactifications of these spaces look like?

and there is also the question I asked here: math.stackexchange.com/questions/1763542/…

lots of interesting questions about these

@EricStucky you see?

For the compactif qquestion

I can;'t imagine what an answer would ook liek

do you have a reaosna to belive it might be somethin ncie

it depends on which compactification you look at

I mean, if you take the closure in the plane you just get the Cantor fan

so it can be nice

but what if you look at bigger compactifications

(closure in the plane is about as small as you can get because there is no one point compactification of such a space)

5:42 AM

the usual one-point compactif is always compact though?

You want your compactifs to embed in $\Bbb R^2$?

yes but the original space has to be locally compact

no they can be different

Does your definition of compact assume Hausdorff?

I'm having a hard time coming up with a space with one point compactification not compact; my usual examples are failing me.

Because at least one of the covering sets has to contain the new point, and then what's left is compact (and closed), and so the cover can be made finite.

Unless there are literally no closed compact subsets of the original space, which I guess is plausible.

Well, in any case, it sounds like a good time :)

I'm going to watch starcraft in bed which probably means I'll fall asleep soon but

was good talking to you :D

thanks

see you around

6:06 AM

ok byebye

1 hour later…

user116211

7:28 AM

Are there any Indians, here?

user116211

@balarka !

user116211

Okay, if there are any one who are studying in Indian institutions, can they tell me when they get the scope to get acquainted with Bourbaki?

8:19 AM

@BalarkaSen you here?

2 hours later…

10:40 AM

Does anybody know the name of the theorem that says that $\left( \mathcal{C} \left([-\tau,\tau];\mathbb{R}^n \right), \| \cdot \|_{\infty} \right)$ is complete?

(With $\| f \|_{\infty}$ being $\sup_{t \in [-\tau,\tau]} |f(t)|$)

@MikeMiller Thanks for the explanation!

11:20 AM

I posted this browser fluid dynamics simulation once before but it deserves to be posted once more: https://haxiomic.github.io/GPU-Fluid-Experiments/html5/
You need to click and drag the mouse pointer.

2 hours later…

1:00 PM

@MAFIA36790 Who wants to read Bourbaki?

user116211

@BalarkaSen Just wanted to know when they get acquainted with the books.

I don't know what you mean by acquainted. Do you mean when a university uses Bourbaki as textbook? I'd like to believe never, but I don't know.

user116211

@BalarkaSen yep... :(

user116211

Not really as a textbook.... Bourbaki is a monster book of derivations only.

user116211

So, as a reference?

1:05 PM

Which is probably for the good. Bourbaki is a horrible textbook.

@MAFIA36790 Maybe. I have no idea.

user116211

@BalarkaSen o.O

user116211

Don't you guys prefer it?

user116211

Hmmm.... it's quite useful in GR.

Here.

user116211

@BalarkaSen Thanks ;)

1:13 PM

"On 2): Harry Gindi is (or at least was) an undergraduate who likes to learn from Bourbaki." lol I never saw that one.

user116211

@BalarkaSen Thanks ;)

OK, I need to get back to work.

1 hour later…

2:43 PM

hi

2 hours later…

4:53 PM

@BalarkaSen Have you solved any major open problems yet today?

see my last question guys, I have to know if my attempt to solve the problem is correct :D

I have found a sequence that is asymptotic Log factorial.

user116211

in The h Bar, 10 hours ago, by Slereah

user image

and butthead

5:15 PM

$\text{Beavis}^{and}\text{Butthead}!$

I never saw that cartoon, but it seems nice.

user image

lol

6:00 PM

I just got chat banned for a reply to this 2 days old message, which was, word-by-word, "you're a nazi". Contextually, it is clear that "nazi" there really meant "spelling-nazi". That being said, it is not clear to me in any way how is that offensive or inappropriate. Can anyone enlighten me?

German war crimes

The governments of the German Empire and Nazi Germany ordered, organized and condoned a substantial number of war crimes in World War I and World War II respectively. The most notable of these is the Holocaust in which millions of people were systematically murdered or died from abuse and mistreatment, between 95 and 100% of them Jews. Millions also died as a result of other German actions in those two conflicts. The true number of victims may never be known, since much of the evidence was deliberately destroyed by the perpetrators in an attempt to conceal the crimes. == Pre-World War I =...

Are you serious? :P

I mean, I'm being lighthearted about it but yeah

There are other ways to get the point across without dragging in

a history of hate and genocide :/

I strongly do not believe that can be a serious reason for a chat-ban.

but I won't.

6:13 PM

You should.

I think most people find the soup nazi episode of Seinfeld funny.

I haven't watched Seinfeld

6:31 PM

@BalarkaSen Well, Nazis did really nasty stuff.

Perhaps someone thought you were calling him a "genocidal xenophobe, homophobe" and all the other goodies Nazis had.

Fair enough :P

lol

Well, based on Pedro's interpretation, I will stick to my opinion (starred on the panel) that people are being ridiculously oversensitive about things.

I doubt anyone went into your private room and was horrified for Soham's sake that you were accusing him of war crimes.

I think it's more likely that the flag was malicious.

Me too.

Otoh 30m chat bans are harmless.

Especially seeing someone flagged me yesterday on a message for another ridiculous reason.

Akiva Weinberger

6:36 PM

There are way too many homophobes in here. Or is it homomorphisms? I always get the two confused.

@MikeMiller True, but it's mildly annoying.

flag-trolling etc.

Akiva Weinberger

gets smacked by Dr. Shifrin, in all likelihood

Ted hasn't been here for a couple days. Dunno why.

Probably I pissed him off a couple days ago.

Akiva Weinberger

6:41 PM

Do any of you know if there's a way to turn a short-ish exact sequence of chain complexes $0\to C_1\to C_2\to C_3\to C_4\to0$ into a long exact sequence of homology groups?

Like, something similar to the case where there's only three chain complexes.

@MikeMiller @BalarkaSen does my answer make sense? math.stackexchange.com/a/1765080/21137

@AkivaWeinberger Yes.

@MikeMiller I saw that but never thought Ted would be annoyed because of that, since he's generally a kind and not very impatient person. I wish he'd come back :/

Or maybe not.

I leave which of those two is correct as an exercise to the reader.

@ForeverMozart Why not just answer on MO?

Tobias Kildetoft

@AkivaWeinberger You can, but it will involve some more stuff (write it as a couple of short exact sequences)

6:45 PM

@TobiasKildetoft Ah yeah I see what you mean.

You can break a exact sequence up into short exact sequences.

And can compute homology LES of those individual pieces.

I remember being envious of carpenters and artists, who after an achievement can point at what they've done, its value obvious; while in mathematics our very achievements can undermine our perception of their value. – Joel David Hamkins Apr 25 at 21:44

But I am not sure how to "glue" those two individual LES's back.

But then I have calculus to do.

"[...] as though with the work of a favorite student: pleased to have engendered a kindred mind, a little worried perhaps at his precocious possession of thoughts I cherished as my own, more ruthless in condemnation and correction than when less involved, and yet as meticulous as affection demands."

@AkivaWeinberger So, you want people to call you a pain-in-the-arse instead?

Akiva Weinberger

6:53 PM

@PedroTamaroff Something something hyphens something

Yeah.

Hold your horses.

Akiva Weinberger

I wasn't really being serious, anyway. People were talking about grammar Nazis, so, I thought, why not

7:22 PM

So, given a smooth map $g : \Bbb R^m \to \Bbb R^n$, I have to show that $g^*(dx_1 \wedge \cdots \wedge dx_k)(a)(v_1, \cdots, v_k) = (dx_1 \wedge \cdots \wedge dx_k)(Dg(a)v_1, \cdots, Dg(a)v_k)$ where $a \in \Bbb R^n$ and $v_i$ are vectors in $\Bbb R^m$. By multilinearity, it is sufficient to show that this holds for $v_i = e_i$, the standard basis vectors, nope?

And for that it is obvious since both sides evaluate to the determinant of $[\partial g_i/\partial x_j]$, $i, j$ running through $1$ to $k$. Right?

Akiva Weinberger

7:44 PM

@MikeMiller

> By construction, each codimension 1 face in the resulting $\Delta$-complex is contained in precisely two (possibly the same) simplices.

"Precisely two"? Can't it be any even number?

Or, by gluing them one-by-one, does every place where more than two faces meet get split up into pairs?

What is the context?

Akiva Weinberger

assorteddetails.wordpress.com/2016/04/27/what-is-homology

@Ramanewbie what is 42 for?

Hello!!

Does it hold that $(p_1+p_2)\circ f=p_1\circ f+p_2\circ f$ ?

@Akiva You choose to identify things in pairs.

Akiva Weinberger

7:49 PM

Got it

Is that David Bowie in the image at the top?

Tobias Kildetoft

@MaryStar Depends on what everything is.

i assume those are constant functions

the $p$

$p_1, p_2, f$ are polynomials.
Especially $f(x)=ax+b$. @TobiasKildetoft

haha yeah that is David Bowie

Akiva Weinberger

@MaryStar I believe so

Tobias Kildetoft

7:52 PM

@MaryStar And you are composing them? In that case sure, as the addition is just defined pointwise when they are seen as functions. Just evaluate at a point to see that they agree

Ah ok... Thank you!! :-)

@Semiclassical Hey, you're around?

yeah

So, idle wondering: suppose I have a vector field $X$ on $\Bbb R^2$.

$x_0$ be some arbitrary point, and suppose furthermore that the integral curve $\gamma$ of $X$ starting from $x_0$ (i.e. $\gamma'(t) = X(\gamma(t))$) is closed, i.e., $\gamma(0) = \gamma(1) = x_0$.

Is $\int_C X$ is minimized among the class of all closed curves $C$ starting at $x_0$ of the same length as $\gamma$, exactly when $C = \gamma$, @Semiclassical?

Tobias Kildetoft

@user1618033 Just go away will you?

None of the people who care about your integrals are around right now

Akiva Weinberger

8:02 PM

@user1618033 What about me?

I mean intuition says it does, but I'm not sure.

@balarka hmm

not sure i have intuition for this, though i presumably should

@Akiva Yes, it's a still from "The man who fell to earth", one of my top 5 favorite movies. And yes, you could glue together more faces along that edge... but you have no reason to, so you shouldn't :)

Akiva Weinberger

Arright. :) This makes there be more than one possible surface corresponding to a single chain, though

yeah, i probably can't help @balarka

8:04 PM

@TobiasKildetoft You should only talk in your name, you're not qualified to talk about others.

Yes, that's clear.

Tobias Kildetoft

@user1618033 You mean I am unable to parse the list of people currently active in this room?

more or less an integral curve should be the curve where, if you step onto it, the force field would move you through. so something along that should be minimized. it's not clear what.

The canonical map is from my group to singular homology, not the other way around. The post is more or less a sketch that this map is an isomorphism.

Akiva Weinberger

@user1618033 Wait, were you being serious?

8:06 PM

@Semiclassical ah, ok. no problem.

Akiva Weinberger

Ah, OK. @ Mike

this is an idle wondering anyway, so not that it's important.

@balarka my main issue, i suspect, is that i don't know what $\int_C X$ means. presumably it's something that can be written in regular vector calculus in an obvious way

@Semiclassical did you continue the work on my last series? :-)

Andrew Thompson

@MikeMiller Who wrote this?

8:07 PM

nope, tied up with research @user1618033

@Semiclassical You seemed so interested in that stuff - hence my insistence.

Can't follow arrows, what is "this"?

Oh by $\int_C X$ I mean $\int_C (fdx + gdy)$ where $X = (f, g)$ @Semiclassical

i was, but my interest tends to be a flickering thing. right now i'm trying to get mathematica to answer some questions for me @user1618033

Akiva Weinberger

1 hour ago, by Mike Miller

"[...] as though with the work of a favorite student: pleased to have engendered a kindred mind, a little worried perhaps at his precocious possession of thoughts I cherished as my own, more ruthless in condemnation and correction than when less involved, and yet as meticulous as affection demands."

8:08 PM

@Semiclassical I extended that series to more tails! :-)

@user1618033 nice!

Akiva Weinberger

@MikeMiller That's Anders's "this"

Rudolf Arnheim, discussing himself viewing his old work, preface to "film as art".

Akiva Weinberger

(Typo, but I'm keeping it)

Tobias Kildetoft

@user1618033 Ahh, I was wrong. I was not aware @Semiclassical was among the interested

Andrew Thompson

8:09 PM

Thanks.

@balarka ah, gotcha. i see what you mean by force now. in physics terms, one can view said integral as the work required to move a particle along the path

yeah

Akiva Weinberger

My conjecture is that @Balarka's conjecture is wrong

i'd have to spend a bit of time converting it into purely physics terms to have an opinion

I am not making any conjectures. Just a question ;)

8:10 PM

and, well, i probably shouldn't right now

Something should be minimized along the integral curve nonetheless, if not that (probably not that).

I'll think it through carefully later.

@balarka agreed. something something lagrangian mechanics something something action.

@Semiclassic If you're busy, no problem. :)

@TobiasKildetoft Now you can also ask some apologies to be(seem) even nicer.

right now i'm working on a university pc instead of my laptop, so mathematica is doing these computations reasonably fast.

Tobias Kildetoft

8:12 PM

@user1618033 I would have if you had not started by calling for me being banned.

@Semiclassical What Mathematica version do you use?

Mathematica 10? not sure which specific version on my laptop, but the university PCs have 10.2 @user1618033

@Akiva: You should try to prove the abelianization claim, as mentioned in the post.

@TobiasKildetoft Trust me, being banned for one year or so would help you a lot (also mathematically speaking).

I encourage @Akiva on that. It wouldn't be too hard for him, if he starts working it out.

Tobias Kildetoft

8:13 PM

@user1618033 Well, you should know if anyone can.

@Semiclassical I noticed some issued in Mathematica 10 ... (issues that are not present in Mathematica 9)

@MikeMiller: If you're not busy, would you mind having a look at this?

I am a bit worried about my logic there.

@TobiasKildetoft To close this discussion you opened about integrals, here is the difference between me and some of you that tell things about a part of mathematics: when I meet something I don't know and I'm interested in, I began to study and learn, but you have a different solution, you label mathematical areas as being unimportant to you, it's easier to do so because you know what? You have to study some many years, like 16-18 hours a day, every day, to get resonable results in my mathematics.

Tobias Kildetoft

@user1618033 I do the same thing. I am just not interested in integrals

@TobiasKildetoft OK, but don't say my mathematics is not important and I'm fine with that.

I'm out to some work.

@Semiclassical hope you will be less busy in the next period of time.

:D

8:23 PM

we'll see

8:51 PM

@Karl Hi.

Did you finish deciding?

Karl Kronenfeld

@BalarkaSen Some idealized version of me perhaps.

What would that version of you be doing instead then?

If he's doing some math, I'd like to know what ;)

Karl Kronenfeld

That's the problem (hence why I am deciding)! Also, there may be multiple idealized versions.

You're hundred times better than me at trolling.

Karl Kronenfeld

Why thank you, I guess.

8:59 PM

I am actually not sure if this classifies as troll. In where I am, it has a name, but the literal translation has a vastly different meaning which can probably be interpreted inappropriately.

flagged

totally flagged

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