idk the full form but theres like RRAM,LRAM and MRAM

right like to calculate the area under curv

like u know when using the sigma letting n go to infinity

I tried finding a video on it but I can't find the one I want like examples like the one I was given

hi @Daminark

would yu like to discuss functional analysis question ?

Yo Adeek!

hey @Daminark

Let $H = L_2([-pi,pi])$. Let f,g in H defined by$ f(t) = t^2$ and $g(t) = t$ for every t in $[-pi,pi]$. Let E be subspace of H which is defined as the span of f and g. Find the distance between h = sin(t) and E.

So let's hear it

I think I have it

so $dist(h,E) = inf_{e \in E} ||h - e|| = inf_{c_1,c_2} ||h - c_1 f - c_2 g||$

and we just integrate to find such minimum

Yeah

Although

I find it to be $\sqrt{\pi}$

Noting that $L_2$ is a Hilbert space though

oh man I have to review some integration skills

Yeah so ?

You can play with the orthogonal complement, I believe?

hmm

wait so what is the point ?

yeah so we have $H = E \oplus E^{\perp}$

sure

Okay so the span of $f$ and $g$ is gonna be a closed subspace of $L_2([-\pi,\pi])$, yeah?

Closed because finite dimensional

yeah

yeah sure

I agree

So call that subspace $M$

Yeah

we have $H = M \oplus M^{\perp}$

The distance between $\sin(t)$ and $M$ is achieved by some $y\in M$ which is the unique one such that $\sin(t) - y\in M^{\perp}$

yeah sure

So you want to find that $y$

yeah

oh I see that is much easier

let me see doing it that way

1 sec

And that, you can do by solving $\int_{-\pi}^{\pi} (\sin(t) - y(t))(\lambda_1t + \lambda_2t^2)dt = 0$

sorry I had to make tea

It's fine

wait

so we know given any element in Y we know that

$sin(t) - y(t) \perp M$

I don't remember what was the inner product space again on L_2

?

was it just the $int(fg du)$?

yeah that is probably it

Yeah, $\langle f,g\rangle = \int_{-\pi}^{\pi} f(t)g(t)dt$

So now the idea is this, you want to find some $y(t) \in M$ such that $\sin(t) - y(t) \perp M$

yeah

And since $y(t)\in M$ you can write $y(t) = \gamma_1t^2 + \gamma_2t$

So now, given some arbitrary $\lambda_1t^2 + \lambda_2t$

right

we can use $\beta$

You're trying to solve for $\int_{-\pi}^{\pi} (\sin(t) - y(t))(\lambda_1t^2 + \lambda_2t) dt$

instead

hm

Now, some stuff you can just solve for outright

let us see

Like, that expression will simplify to $2\pi\lambda_2 - \lambda_1\int_{-\pi}^{\pi} y(t)t^2 dt - \lambda_2\int_{-\pi}^{\pi}y(t)t dt$

I think in this case y(t) = 0

for all t

Nope

right and we can put $y(t) = \gamma_1t^2 + \gamma_2t$

So, we want that equation I last wrote to be $0$ for any choice of $\lambda_1$ and $\lambda_2$

yeah

So in particular

We can let $\lambda_2 = 0$

right aha

And that equation has to be solved, meaning $\int_{-\pi}^{\pi} y(t)t^2dt = 0$

wait what no ?

And if we let $\lambda_1 = 0$, then we see that $\int_{-\pi}^{\pi} y(t)t dt = 2\pi$

oh right

right nvm

yeah ok

So, here you can just try to compute, knowing that $y(t) = \gamma_1t^2 + \gamma_2t$ and solve for $\gamma_1$ and $\gamma_2$

Make sense?

oh ok I see

yeah we have two equations and two unknowns

yeah makes sense

thanks a lo

Nice

No problem! Thanks for some analysis practice :P

good I am loving functional analysis

np

Yeah honestly the stuff is dank, I'm gonna do it next quarter actually

@Daminark nice

Also just to be sure that I still know how to integrate, did you get that $y(t) = \frac{3}{\pi^2}t$?

@Daminark I am gonna compute it now

1 sec

(I'd also like to mention that I don't know shit in this subject so it's possible I led you through the most inefficient way out there, it's just the only systematic way I know)

let me go through it again

(Double check with people who know their shit)

yeah I will go tomorrow and ask as well

Doing out the integral I got $\frac{\pi^2 - 6}{\pi}$, which disagrees with your $\sqrt{\pi}$ so it's possible I fucked up

:thonk:

But yeah double check things, but I think that's at least one systematic way to find the distance between a point $x$ and a closed subspace $M$ when you're working in Hilbert spaces

how do you know $sin(t) - y \in M^{\perp}$ ?

Well, I solved for it, try computing it and see

Wolfram alpha agrees with me here

@Daminark how did you get $sin(t) - y \in M^{\perp}$

That's just my checking the answer once I had it

The way I got it was by solving the equations I had above

$\int_{-\pi}^{\pi} \gamma_1 t^4 + \gamma_2t^3 dt = 0$

No I mean inititally how do we know that it is is the unique y such that $sin(t) - y \in M^{\perp}$?

That says $\gamma_1\frac{\pi^5}{5} + \gamma_2\frac{\pi^4}{4} + \gamma_1\frac{\pi^5}{5} - \gamma_2\frac{\pi^4}{4} = 0$

And that requires $\gamma_1 = 0$

It is probably a theorem

Oh, it's a theorem in Hilbert spaces, it's how you prove the existence of orthogonal projection

Hey @Ted!

Hi Demonark

hi @TedShifrin

yeah

hi Karim

But yeah so given $x\in H$ and $M\le H$, you know that $d(x,M)$ is achieved by the unique $y$ such that $x-y\in M^{\perp}$

Assume $M$ closed :P

Do you want me to try to prove it? It's been a while but I recall it being pretty simple

@TedShifrin I love functional analysis

Lol right, yeah. In this case we're good because we had a finite dimensional subspace

I love analysis in general

The importance of understanding linear algebra ... yet again.

This is true

@TedShifrin I didn't know about how important linear algebra is until I studied more advanced math

@Adeek so I believe the way you'd want to prove it is this

it has its roots everywhere

I believe you'll find it in the record that I kept telling you ...

yeah

You let $d = d(x,M) = \inf_{y\in M} \|x-y\|$. Now, general rule in life, when you see an infimum, take a minimizing sequence first and ask if it helps letter. So choose $y_n$ such that $\|x-y_n\| \to d$

yeah

So for any $\epsilon$, there's some $N$ such that $\|x-y_n\| \le d + \epsilon$ where $n \ge N$.

You want that the $y_n$ is Cauchy, then because we're in a Hilbert space we're good

But we can do parallelogram law

The orthogonality issue is just the Pythagorean Theorem. Are you getting there?

Yeah

yeah

$\|y_n - y_m\|^2 + \|2x - y_m - y_n\|^2 = 2\|x-y_m\|^2 + 2\|x-y_n\|^2$ if I got it right

hi Kevin

But $\|2x - y_m - y_n\|^2 \ge 4d^2$

oh good grief ... Mathei is awake at ridiculous hours

Because $\frac{y_m - y_n}{2} \in M$

whose Mathei

And that should do it, for $m,n \ge N$, you have $\|y_n - y_m\|^2 \le 4(d+\epsilon)^2 - 4d^2$

hey everyone :P

Which is $4\epsilon(2d + \epsilon)$. Let $\epsilon$ die off and you're good, the sequence is Cauchy. So it converges to some $y$

@Mathein I see you have a hat

Oh, the hat season has returned?

I'm immune.

@MatheinBoulomenos hi

But yeah @Adeek you know that the limiting $y$ will satisfy $\|x-y\| = d$ by continuity of the norm. Now you want to show that $x-y \in M^{\perp}$

yeah

So, you know for any $z\in M$, you have $\|x - y\|^2 \le \|x - y - \lambda z\|^2$, yeah?

@Daminark, just by curiosity: do you participate of "I don't think that's math" or any kind of math group on Facebook?

heya @Lucas

yeah

hey @Ted :)

@Lucas I don't recall the one you mention, though I do follow some pages for math memes. Mathematical Mathematics Memes, Derived Memes for Spectral Schemes if I want to hear some words fly over my head because come on we all have that desire sometimes

Stuff like that

Anyway @Adeek, so you're prob gonna expand the right hand side of that inequality, and I will let you know what that comes out to once I get some paper

hi @Antonios

okay

hi @TedShifrin! Got my exams tomorrow. Wooo

Well, that's exciting.

Can't wait for 24 hours from now.

I bet.

When you teach classes, then you'll appreciate how your students feel. :P

@Antonios-AlexandrosRobotis are you going to factor some prime ideals?

Presumably. @MatheinBoulomenos. That stuff is not too bad (at least the quadratic extension problems). Actually I'm not really that worried about the tests.

@TedShifrin the only frustrating thing at this point is that we're asked to "learn" some fairly obscure theorem which we had no homework on and mentioned only during the last lecture.

I'm not a fan of that for final exams.

Shows bad teaching planning.

Seems like the only feasible problem would be "state the theorem," which seems like a pointless idea. People just memorize it, and then forget it. Sorta inevitably.

Yeah I've always found this super dumb

What kind of obscure theorem?

@TedShifrin I plan for my lectures pretty well by looking at things 5 min before.

But it is calculus lol

The "Duality Theorem." It's related to Von Neuman's minmax theorem in math. econ, I guess.

@TedShifrin I keep hearing the buzz word index theory btw

it's not that hard, Karim — look it up.

Okay @Adeek so I did out the stuff

And $\|x-y\|^2 \le \|x-y-\lambda z\|^2 \implies \lambda^2\|z\|^2 \ge 2\lambda \langle x-y, z\rangle$

@Daminark yes

$\lambda = 0$ is whatever so let's say it's not, then divide out

$\lambda \|z\|^2 \ge 2\langle x-y,z\rangle$

This holds for all $\lambda$ and for all $z\in M$

So given $z$, you choose $0 < \lambda < \frac{2\epsilon}{\|z\|^2}$

Anyways, I guess I should do my final review.

Good night, all.

Good ruck, Antonios.

Could someone help me?

Thanks :)

yeah

Demonark: That seems silly. Why not just set $\lambda=0$?

Well, you can't divide out by $\lambda$ then

Lucas, are we supposed to be mindreaders?

I want to prove that problem 1c of the quiz has "the binary expansion of N" as a solution. But I have no idea.

So you have to make the thing less than $\epsilon$

Too much reading and thinking for me, Lucas. Sorry.

@Adeek so does that sound good?

yes

yeah makes sense

@TedShifrin I always ask before, just cuz I'm educated. :P

thanks a lot @Daminark

Also this is just something I find to be kinda dank but if you look in the proof of the $d(x,M)$ being attained

There was only one place where I used that $M$ was a subspace

Which was to say that $\frac{y_m + y_n}{2} \in M$

But subspace is not necessary, convexity is enough for that. So $d(x,M)$ is achieved whenever $M$ is a closed, convex subset of $H$. I don't think you'll use that for anything but I dunno, it's kinda cute or something

yeah it is very nice theorem

Anyway yeah no problem!

Well, without convexity you can have as many closest points as you want.

yeah

both convexity and closedness are important

Oh I guess you can take the set of things with norm at least one and the origin

Then you've got a unit circle's worth of closest points

I was thinking of something different, with a large finite number, but sure.

You can also lift your origin out of the plane.

Oh yeah lol, even when I'm talking in infinite dimensions I never imagine more than 2 for some reason

@Ted can you recover some classical theorems from complex analysis from theorem B for Stein spaces? It seems like a very powerful theorem

I don't recall which theorem that is.

@TedShifrin given the sequence $1,\ldots,100$, we tag all numbers with zeroes. Let, at any moment, $S_0$ denote the set of numbers tagged with zeroes with analogy to $S_1$. We'll always tag $\min S_0$. If we tag some number $x$ such that $\exists y \in S_1: x>y$, we put $y$ in $S_0$.

Presumably the one after theorem A

any ideas for changes or should what I have be fine?

@Jasch1 if Calc 3 means multi, I'd recommend doing linear algebra first

Demonark: Mostly because you haven't been made to think about 3D or 4D or ... again, my complaints.

So the sequence goes like $1-2-1-3-1-2-1-4-1-2-3-\dots$

yup its multi

Wait 4D?

Let $X$ be a Stein space and $\mathcal F$ be a coherent sheaf, then for every $q>0$ $H^q(X,\mathcal F)=0$

i've done a bit of lin alg but not enough

Hold on a second is that even possible?

Well, mostly for one cx variable that's going to get you Runge's Theorem ... and probably Mittag-Leffler and similar, @Mathei.

Like, I'm not exactly a pictures person in any event but like, I can get a vague picture of 3 dimensions even though I can't work with it, but I thought visualizing 4D is physically impossible

You can watch a few of my lectures, Demonark, thereupon.

This is some black magic we got here

@Jasch1 yeah I mean, the vibe I get is that you basically need linear algebra for everything in life

Thanks @Ted I don't really see how that works, though

And it's also just dank, you know?

how far into should i get before multi? any resources you could recommend that would give me a good start

@Daminark I used some of the 3Blue1Brown vids and they were great

So... If you're gonna do multi afterwards it'd probably be helpful to do linear algebra using a more geometric/pictorial eye

A lot of those standard theorems can be rephrased as $H^1(X,\mathscr O^*) = 0$, @Mathei. ...

I don't know too many resources for that since I sorta learned it with a discrete math/algebra focus

Do ask Ted!

Ted what are your thoughts? I have a decent background in linear w/ vectors, matrices as transformations, cross and dot products but thats about it

Thoughts for what? ... If you want to get an idea of the interplay between linear algebra and multivariable calculus done right, look at some of my lectures on YouTube (see my profile). The standard multivariable calculus courses in the US don't use any linear algebra at all, but in Europe they do.

In differential equations, a point x is stable if all the eigenvalues have negative real part. If at least one eigenvalue have positive real part, then x is unstable. What happens when the equilibrium point x doesn't have real part? Is it stable or unstable or nothing?

@TedShifrin I watched all of your lectures btw

You just get periodic motion around the singular point, @Isabella. Write it out.

And you're not dead yet, Karim?

@TedShifrin will take a look, thanks for the pointer

@TedShifrin I am getting back problem haha

back problems *

@Adeek wait are you saying this ironically or...?

yeah @Daminark haha

1am here.

I should get more exercise.

goodnight, guys. :)

Like someone in analysis said that he/she got carpal tunnel, supposedly from the psets, and I'm not sure that this is ironic

Night, Lucas. How did it get 6 hours ahead of my time zone?

what is psets?

Problem sets, like homework

Anyway

What are you guys up to?

I am just working as usual on my research

periodic motion around the singular point? I don't understand.

I want to understand as much as possible so I have good base knowledge for phd

points will move in circles centered at the singular point: pure imaginary eigenvalues means you get $x-x_0 =ce^{ikt}$.

@Adeek that makes sense

I don't see how Mittag-Leffler is $H^1(X,\mathscr O ^*)=0$. Do we use some kind of exponential sequence of sheaves?

I know that when it's about imaginary eigenvalues, then circles (or something similar to circles ) will appear, but what can be say about their stability?

I lied, Matthei. We need the sheaf of principal parts for Mittag-Leffler ... That probably isn't coherent. I don't remember. Weierstrass product theorem probably works. But Cousin problems are all about gluing, so is Runge theorem.

not stable, @Isabella, because if you perturb slightly you can get an eigenvalue with a positive real part and ... away it goes.

bye all

bye @Ted

See you @Ted!

@Mathein are you doing complex geometry now?

but I also could get an eigenvalue with negative real part, or not

?

@Daminark just a little bit, just saw some interesting stuff and thought about it

@TedShifrin?

@MatheinBoulomenos your plan is to work with general algebraic geometry ?

later or complex geometry ?

I would like to do arithmetic geometry

I see

yeah that is interesting.

@Mathein aight phew, if enough people here start doing it the peer pressure will reach critical mass and I may be sucked in

@Daminark I am gonna start taking couple of physics classes next year btw

I want to get knowledge to be able to do algebraic geometry with physics

that would be cool

Sick

Howdy folks

Hey @Eric and @Kevin!

sup dog

How's it going?

p good

been doing some complex geo

it's been pretty great honestly

Fricc

Must resist

Lol jk I've heard the stuff is cool

I would love anyone's advice on this problem I'm working. I'm trying to show that for any non-orientable n-manifold there is a surjective immersion $f: O \to N$ where $O$ is some oriented n-manifold

I really don't have nay idea how to start even

double cover

@EricSilva which book are you using ?

griffiths little book on algebraic curves

Oh Im sorry and I should also say, the definition of orientable I have to use here is that theres an atlas whose transition functions have positive determinant

nice

i want to pick up griffiths and harris but it costs so much money

@KevinDriscoll look up the orientable double cover

@EricSilva Thanks wasnt sure you were talking to me

ya sorry lol

@EricSilva I bought it

it is worh the money

worth*

@Adeek pdfs tho

/alternative books

Yeah

i imagine but i dont have any money

but the pdf version is bad for that book @Daminark

it is only available in DJVU

I have an addiction of buying books lol

my wife cut my credit card

lol

Lol I buy almost nothing

i like physical books personally

i stay away from my computer when im doing math

that is good @EricSilva

I have 5 physical books. One called second year calculus that I bought off someone when I was dying in physics, Spivak because since only the third edition was up as a pdf, Sally because I didn't find a pdf for quite some time, one on hyperbolic manifolds that I found in a free books pile, and one a friend gave me from complex (with Schlag), that I ended up using in my own complex class

im really liking griffiths little book a lot thought so i might bite the bullet and just save up for G&H

lol i have like hundreds

And the best part is, I might sell 3 of them

is the one on hyperbolic manifolds the one by those italian dudes

if so sell it to me

That one I got for free, you can have it if you like

gimme gimme gimme

And yeah it's Benedetti

yeah that's the one

ive been looking to acquire that one for a while

The ones I might sell are second year calculus to some physics major, then Spivak and Sally to the bookstore

but im a poor lad

btw @Daminark do you know what reus ur applying to

Ah, I haven't thought about that yet actually, I should figure this out

yeah ive decided on mine i think

Nice, which?

williams, the one at brown, this geotop thing at berkeley, maybe one at csusb and the one back at chicago

Nice. I may apply here but ideally I'm looking for one which will help pay for tuition next year and Chicago's is... not exactly the best for that

Do you know what's happening with the bootcamp?

no idea

may find out at the beginning of winter quarter

I was previously under the impression that we were gonna press F to pay respects to it but Soug told the second years that it might be a thing

depending on whether i get invited to be involved i might

idk

i might wanna leave

If $\omega$ is an integrable 1-form on a 3-manifold, how can I show that $d\omega = \omega \wedge \alpha$ with $\alpha$ a 1-form? Locally I know why it cannot be that if $\omega \mapsto dx$, then $d \omega \mapsto dy \wedge dz$, but I'm not sure how to show that $d \omega \neq dx \wedge dx = 0$

Yeah I'm inclined to leave as well. I think I'd rather do something along the lines of number theory/algebra/discrete next, that's sorta my next project. For now I don't think I can do much but after Marianna and Emerton I'll hopefully be able to start doing some stuff

the icerm reu at brown looks really up my alley

Ah that looks nifty

One which might be overshooting a lot but which might be a lot of fun is Emory

Also Twin Cities

why overshooting

@robjohn I have recently asked about a duplicate closure in another chatroom. Since you have answer on one of the posts involved, maybe you would care to have a look. Thanks!

Morning everyone!

Yo @Dami

Hey @Perturbative!

@Daminark coooome

Ah yeah we'd finally get to meet, that could be really fun

Take my star @Semiclassical

lol

If Twin Cities takes me that'd be high on my list, I really like that one

kk

@Eric Emory is like, number theory/arithmetic geometry

the Twin Cities campus for the UMN is mostly minneapolis

the Saint Paul part of it isn't where the math department is

that said, worth checking out both cities if you visit

Who is running the UMN reu?

Vic Reiner

neat

Do you know him?

not myself, no

though I've actually been planning to stop by his office myself some time; I'm trying to figure out if there's anyone in the math department with expertise on something I'm working on, and his stuff is close enough that I'm hoping he'll have a suggestion (of who to talk with if nothing else)

I see, that's nifty

But yeah he often co-mentors with people, and the coordinator is Gregg Musiker

gotcha

one of the guys I knew in the math department worked with Gregg

@Daminark why is that overshooting tho

I only know like, very basic number theory

By the time that thing starts, I'll probably know Marianna's stuff + whatever is gonna come up in algebra, maybe plus some self-study

ohh ok

If they plan to hit the ground running with arithmetic geometry, I probably won't be able to keep up

diophantine equations came up as an aside in one of the books on alg geo im reading this break

was p cool