@BalarkaSen @Danu: There's an Euler sequence for any complex submanifold $M$ of $\Bbb P^n$. If you understand the proof I suggested for $\Bbb P^n$ itself, then you have it. (The vector bundle in the middle is the tangent bundle of the cone over $M$.)

I wondered what had become of 0celo. I assumed he was off gloating because his candidate won.

Hi @TedShifrin

@Danu: I didn't stop to read what you wrote above, but it looks like you're talking about the adjunction formula.

Hi @Brody

@TedShifrin I see. I haven't worked it all out though, but maybe I should bookmark it for later.

@Ted The middle part of that figure (in your picture), there's a name for a similar object

It's a cool exercise, @Balarka. I needed it numerous times in my thesis.

@Brody, um, yes.

I forgot, haha

@TedShifrin i read chapter 3

Also, hi, @Brody

It's a one sheeted hyperboloid.

Yello @Balarka

Balarka beat me to the edit.

OK, meow ...

smug

you know what smugness begets, @Balarka ...

@BalarkaSen Right, and I knew that. Thought there was something specific for twisting a circular "curtain" of lines, but guess not

Ruled surface, @Brody?

If @meow ever reads my chapter on projective geometry, he'll have a proof that the only surfaces in $\Bbb R^3$ that are doubly-ruled (two lines through each point) are projectively equivalent to a hyperboloid of one sheet.

Not sure if that's what you have in mind

@TedShifrin, 0celo7 was banned from chat for 90 days

Wow, @heather. Why?

@TedShifrin A smack?

I mean, lots of people here annoy people a lot.

the hbar is too strict

@BalarkaSen Nope. They're just hyperboloids I guess

but i don't know what happened so

You gonna answer rhetorical questions, huh, @Balarka.

@TedShifrin, no one knows, but the general consensus on the hbar is that the mods were being, uh, nitpicky, to put it politely. 0celo7 contacted a few people off chat and he said he doesn't really know either, the mod message was super-vague.

He'll be back mid february, I think.

hbar is very different without him.

Hmm, ok, @heather. I mean, he was truly annoying a good deal of the time, but so are lots of other people (including me, at times, I'm sure).

Wow, a ban from all SE chats?

@Brody how that?

@TedShifrin should i read the projective geometry chapter first...?

(and how)

@TedShifrin, yeah, the only thing people could think of was a stupid conversation that happened that most certainly was not worthy of a ban.

(::shrugs::)

Most of it is independent of the algebra, other than knowing basics about quotient groups and group actions.

@heather: I mean, we know his politics were not popular amongst most of us chatters, but ...

@meow A word of advice: don't rush.

I have done it before. I had to pay the price.

oh really? I never knew that @TedShifrin, I never saw it come up

@evinda: First of all, it depends on $L$. Second of all, surely your course has proved such things. I have no idea what you know or what I know about this.

@TedShifrin what chapter?

my tolerance for political stuff here depends on my mood

7 or 8?

That should be clear from titles, @meow.

@BalarkaSen I'm postponing our mini-lessons exactly because of that

@heather: I hadn't noticed you here until just yesterday!

sometimes i'm willing to put up with it, and sometimes I just want to say "f* off" to politics

@Brody All the better!

I am not bringing it up either

@Semiclassic: I just committed to a substantial monthly contribution to the ACLU. We're gonna need them. Done politicking.

@TedShifrin, I've come here occasionally before, but I thought you were referencing conversations in the hbar.

Makes sense.

(but, between you and me, that's to avoid Ted's smacks)

$L$ is an elliptic operator. In my notes, we have only the condition $u|_{\partial{\Omega}}=\phi$. Given this, we have proven that the problem has a unique solution. @TedShifrin

No, @heather. I've never been in hbar.

My politics have gone more lefty as time has gone on.

"Annihilating radical left ideals"

Yes, @Evinda, that's one of the standard consequences of ellipticity. So now you have a mixed Neumann-Dirichlet problem. But you've probably proved that the Neumann problem has a unique solution, too, so you have to mix the two proofs together.

lol

@TedShifrin, oh, okay. Nevermind then.

@Balarka: But you seem to be hankering for one or more smacks ...

@BalarkaSen i had my first Ted-smack last night

masochism

Apropos of nothing, I should write up what I grok re: differentials on Riemann surfaces so that I can appreciate my problem better

You guys are going to get me banned for cruelty to children!

burn.

"My politics": do what you want to be done on yourself. period.

I am confused about what my political/ideological stance should be. I don't like rightist politics at all, been exposed to leftism a lot due to my family background, but also am aware of the big, gaping holes in it.

The first line you just typed is nonsense, @evinda.

how 'bout middlist politics @BalarkaSen =P

The Neumann condition merely says that the boundary component $S_2$ is insulated ...

@Balarka: You cannot expect the real world to be a mathematical universe.

@TedShifrin i assume chapter 8 then

I don't.

Makes sense if you look at the table of contents, @meow.

@BalarkaSen just my experience: any politcians's POINT OF VIEW has BIG holes in it. the reason is how politics is applied.

Any politicians POV is, in general, one that is crafted in order to accomplish a certain agenda

^

You can agree or not agree with it, but its definitely there

@BalarkaSen There's never any real obligation to adhere to a particular "guild's code". Better to take whatever stance you deem fair and rational on each issue, investigating the complexity and various aspects thereof.

I do agree with it. I am just confused about the world in general.

i find Galileo's POV the best: try to proof the opposite pov until the point it makes no sense anymore

To me the thing that gets left out of most political stuff, at least in the US, is class

that is a really openview standpoint imo

@evinda: Have you thought about the maximum principle?

It's such a big deal, and yet both sides treat the phrase 'class warfare' as an obscenity

Do you mean to say classes in the economical sense, or the racial?

The former.

Well, in this election, the class that needed attention apparently swung the election, but are going to be even more screwed by the liars.

^

Which will create even more mayhem and hatred and bullying ...

What a lovely world.

resigns

Throwing a political brick through the window of the establishment is one thing. Throwing a molotov cocktail, on the other hand...

Sounds like that's exactly how rightism appeared in our state.

It's appearing all over the world now, @Balarka ...

except, apparently, for Germany (yet) and Canada.

Back to math(s) ... Actually, I need to eat lunch and go play bridge for the afternoon.

Canada is a weird one, in that they had that Conservative resurgence a number of years ago

But now the Liberal party is back in charge again.

BTW, Balarka, if you're really bored, you'll find in yesterday's transcripts that I actually gave you praise :D

Well, @Semiclassic, it is cyclical almost everywhere ...

Does $X=\mathbb{Q}\wedge Y=\mathbb{Q}$ follow from $\{x+y\vert x\in X\wedge y\in Y\}=\mathbb{Q}$?

Of course, Conservative/Liberal means different things in US politics than Canadian politics

Ah? I gave up on reading yesterday's transcript earlier this morning because it was huge.

What does $\wedge$ mean, @NaCl? Do you mean $\cap$?

I believe you though.

oh boy linear algebra again

@TedShifrin Yeah, right until the system falls off its axis :/

"assume the force on the spring obeys Hooke's law [...] 1. If a ten-pound force stretches an elastic spring one inch, how much work is done in stretching the spring one foot?"

@TedShifrin No, I mean $\wedge$. It means "and".

What does "and" mean with sets?

@meow-mix, I love linear algebra...

he means

Ugh @heather

@TedShifrin what? Linear algebra is wonderful!

$X = \mathbb{Q}$ and $Y = \mathbb{Q}$

@TedShifrin as a member of this "poor" class: i just welcome trump. not because i think he's the solution. i rather think he is the "stone in the shoe", I think unless him nothing will change because the public is hold quiet.

no, @evinda, try to show that if $u$ is not identically $0$, you get a contradiction.

I don't use it with sets. $X=\mathbb{Q}$ is not a set but a statement

@TedShifrin, or did you mean the problem?

I meant the problem, @heather.

ohhhh, I totally misread the syntax. $X$ and $Y$ are both $\Bbb Q$.

@TedShifrin, sorry!

@Null That's sort've what I meant with the 'window' analogy. I get wanting to change things up, but Trump? Yikes.

@Null: I am only going to say this. Regressing socially and economically 50-100 years is not progress.

@NaCl just write $X = \mathbb{Q} = Y$

You mean 1? Because we are given that $u|_{S_1}=1$... @TedShifrin

Though the fact that it was Clinton v. Trump was ugggghhh

my book says that work done by an integral force $f$ is equal to $\int^b_a f(x) \, dx$

@evinda: We're trying just to prove uniqueness? Yes, look at the difference between two solutions.

or, alternatively, $X = \mathbb{Q}, Y = \mathbb{Q}$

@meow-mix Okay, but does it follow from $\{x+y\vert x\in X, y\in Y\}=\mathbb{Q}$?

Sanders v. Trump, now that would've been interesting

And now the Indian government is apparently trying to turn out country into a capitalist state.

@heather: why integral force?

@Null No change is better than what Trump is promising ...

I'd say that the work done by a (possibly position-dependent) force $f(x)$ is $\int_a^b f(x)\,dx$

@Astyx i am thinkin a bit further: IF trump says something that is unbearable then protests(until execution) will follow

@NaCl You have a conditional statement. Do you know a counter-example?

@Semiclassical, I think that's what it is saying, yeah.

Anyhow, @NaCl: You're asking if $x+y$ is always rational, must $x$ and $y$ be rational?

It's not just what he -says- that worries me, it's what he (and the people he appoints) will do.

@TedShifrin Yes

So then we get the problem $Lw=0 \text{ in } \Omega \\ w=0 \text{ in } S_1 \\ \frac{\partial{w}}{\partial{v}}=0 \text{ in } S_2$. Do we use now a theorem to show that w=0? @TedShifrin

@NaCl well, consider $\sqrt{2}$ and $2-\sqrt{2}$

That clause is a crucial part of it, btw.

I assume $b = 1$ and $a = 0$ because it is moving 1 inch so we are looking for the force over that interval

their addition is $2$ which is rational

@Semiclassical And his followers who think they now have permission to act illegally and brutally.

but neither is rational

Yeah. What he says matters, it's just not the only thing

The people he's appointing make my skin crawl

How bad is Trump? Has he started doing any of the terrible stuff he said he will do?

@meow-mix but if there is any rational $y$ in $Y$, $\sqrt2+y\notin\Bbb Q$.

$X\subseteq\mathbb{Q}$ and $Y\subseteq\mathbb{Q}$ is known beforehand

He's not in office, yet

ah

@NaCl: It would help if you stated the entire problem clearly and correctly.

@NaCl $(x+y)\in\mathbb{Q}\neg\iff $x\on\mathbb{Q}$ or y...

the book also says that the force f(x) needed to stretch a distance x beyond its natural length is proportional to x, which is apparently Hooke's Law

He's the president-elect right now. So he'll be President come January when Obama formally ends his term.

@BalarkaSen Right now, people are just freaking out over the potential Cabinet members and White House staff

@heather: only for small $x$.

@heather I always hate that formulation of Hooke's law.

Also, Supreme Court appointees

Well, @Semiclassic, it's analogous to the assumptions made in deriving the wave equation ... which always bug me no end.

@Null ty

@NaCl lol^^ i just wanted to say: you can, with some time think of a counterexample

@TedShifrin, so when $x$ is large that isn't true? huh.

@Brody I see.

@NaCl: Your statement is false.

The version I prefer: If I change the length of a spring by $\Delta x$, then the change in the restoring force $\Delta F$ is proportional.

anyway, the problem said to assume that was true, so.

I don't like versions that are in terms of the natural length, because you really don't need it.

no, wait, it would be b = 12 a = 0, sorry. I was looking at the example.

@heather: it's important to think through all applications problems with integrals in terms of approximating a little bit of the process, and then adding all the little bits up. Books tend to just plop down formulas (e.g., more complicated volume, work, water pressure problems) ... I hate that.

OK, I need to get going. Have fun, all.

and then 10 pounds of force = 1 inch of movement

Here's an example. Suppose I hang a spring off the top of a ceiling, and connect a weight to that spring by a string.

@TedShifrin, okay, I will keep that in mind. Thank you, have a good day!

@BalarkaSen Remember when I had trouble making intuitive sense of the integral formula for arclength? (via magnitudes of tangent vectors)

so then I think it would be f(x) = 10x for the force

so then integrate that over b = 12, a = 0?

@Brody Mhm?

I wait for an idiot-proof argument, until 10 minutes later see ya ;)

If you talk about it in terms of natural length, that makes it seem like it being connected to the ceiling matters, and that I'd need to know that length to measure the spring constant.

But I don't. All I need to know is how much the string moves when I add some amount of weight.

I don't remember anything from linear algebra

@heather @Semiclassic: I'll leave you with one of my favorite problems. There are two identical gymnasts and two ropes hanging from the ceiling. One rope is just a standard rope. The other is elasticized, and it is precisely such that when the gymnast jumps onto it, the gymnast descends to the ground level. How much work does the second gymnast do climbing to the ceiling compared to the first gymnast?

(that was hyperbole)

hm.

@meow: Linear algebra is about the most important part of mathematics. You can review it with my YouTube lectures when you need to.

I still have Babai notes to read sooner or later @Ted

Oh yeah, @Alessandro ... those are magnificent.

Should be doable from just energy considerations, i would think.

That stuff is dark magic

@Semiclassic: Don't ruin heather's fun. I actually tried to work it once modeling a rope as a bunch of tiny springs stuck together.

Psh.

Fair enough.

LOL

Bye, all. :) Hi/bye Alessandro.

later

Bye Ted

Bye

Bye @Ted

I have seen that tiny spring model before

Depends on your proof.

Sorry, no, I meant in deriving the wave equation.

oh, sure.

@BalarkaSen When discussing how norm is defined for surfaces. The arclength is given by integrating $||\gamma'(t)||$ over the parameter domain. Geometrically, one can visualize that though the number of vectors summed is taken arbitrarily large, each vector is also scaled down toward zero, and this sum approximates the curve's length.

@Brody Right.

Let $X\subseteq\mathbb{Q}$ and $Y\subseteq\mathbb{Q}$. If $\mathbb{Q}\setminus\{x+y\vert x\in X,y\in Y\}=\emptyset$, then $\{x+y\vert x\in X,y\in Y\}=\mathbb{Q}$. Then $X\subseteq\mathbb{Q},Y\subseteq\mathbb{Q}$ follows, doesn't it?

$X, Y \subseteq \mathbb{Q}$ was given...

@NaCl You assumed the relations $X,Y\subseteq \mathbb{Q}$ from the beginning, not considering the rest of your comment (meaning it may lead to a contradiction)

@Brody I want to contradict

Is that a unit vector in the z direction?

@NaCl What statement are you trying to disprove exactly?

I'll try myself

lemme see

Hey guys

What do you call a distance preserving chew?

An I-JAW-Metry!

ow

yes, i just made that joke

@trilolil looks like they're assuming the current points straight up, so that $\overline{I}(\overline{r}')\,dl'=I\hat{a}_z\,dz'$

so the current points in the z direction?

(do you have chatjax enabled? if not, look at the "Latex in chat" link in this room's description)

yes

Right.

I do have it.

Why do you think that they assume this?

Because that's the situation they're modelling.

More generally, suppose you had a current source like that, pointing in some direction

Then you could perfectly well -define- your coordinate system so that the z-axis points along that direction.

@TedShifrin is that similar to the looping problem (the gymast/rope thing)

@Brody Did you have a question? The infinitisimal weight associated to the norm of the tangent vector at the point of the curve indicates it's contribution as a whole in the integral.

The orientation of the coordinate system isn't god-given, after all. You can pick it yourself, and in the present problem it makes a lot of sense to pick the axis to coincide with that of the current

You could instead assume your coordinates are oriented so that, say, the current points along the x-axis.

But that'd make the computation waaay more of a hassle.

@Semiclassical

what's that in regard to?

@Semiclassical "... isnt godgiven, you got to choose..."

@Semiclassical

ah.

Trump vs Clinton? :P

I was wondering about that, heh @BalarkaSen

So it works in two different ways

You have to choose but a good choice for the orientation of the axis does make calculations much easier espacially for attitude determination problems or so.

Right.

If anyone can complete this: a good choice makes calculating the integral MUCH easier...

@DanielFischer Hello?

@trilolil Not seeing what you mean. The form of (15) follows from (11)

Plus, the vector potential $A$ points entirely in the $z$ direction (eq 5) and depends only on $r$

@BalarkaSen No, just wanted confirmation. My bad

@Brody Ah, Ok. Great.

equation 15.

@BalarkaSen Same idea with the rectangles in Riemann sums, just had to apply it to a different set of objects

Do you think that setting limits encourages the will to exceed the limits?

Well, the integral can be defined via Riemann sums anyhow, so... :)

It's of the form $\overline{H}(\overline{r})=\hat{a}_\phi H_{\phi}$

@Semiclassical exactlty! That is what I mean. How do you know it is in that form?

I don't follow. Do you mean why it doesn't have $a_r$ or $a_\theta$?

@Brody Correct. You'll see all this in chapter 7.

Ah.

Well, suppose you're in Cartesian coordinates. Then you would write H=H_x a_x+H_y a_y+H_z a_z

yes.

i.e. H_x would be the coefficient of the a_x unit vector.

oh ok yes I see

it is like polar coordinates.

Yeah. In spherical coordinates you'd write H=H_r a_r+H_phi a_phi + H_theta a_theta

hi @BalarkaSen

Hi @Adeek

well, the x,y,z coordinates are in here. They're just expressed in terms of r, phi,theta instead

can I discuss with you the proof about jordan holder group composition proof ?

there is some stuff I don't understand