Mathematics - 2012-05-07 (page 8 of 9)

@MattN he's phrasing it a bit poorly. The problem is that you beg the question this way: you want to prove that $g \in L^2$ and you start by applying a theorem on $L^2$-functions.

Matt N.

10:18 PM

@tb Eeww. Yes, indeed. Now I'm not sure what I was thinking when I wrote that.

I can't hide my mathematical inability from you, can I? : )

Nico Bellic

General recommendation time: to all the pros out there, I'm soon going to be done with my first linear algebra and differential equations course. What's your recommandation on a good book to read to solidify these topics more in depth?

robjohn

@ClarkKent I'll remember that.

Matt N.

Hence being embarrassed are wasted feelings.

user19161

@NicoBellic I am not a "pro" but seriously it depends a lot on the exact content you wish to cover and your learning style, so you should just go to the library and check them all out.

Jonas Teuwen

Hmm. Krylov's Sobolev Spaces and Rudin's Functional Analysis.

robjohn

10:21 PM

@MattN The older you get, the more you realize that.

t.b.

@MattN it's the right thinking process towards the solution I'd say... You need to re-arrange the argument. You've got a handle on $\int fg$ for all $f$. You want to show that $\sup_f \int fg \lt \infty$. If you have that, then your displayed equation essentially shows what you want.

@MattN you're being too harsh on yourself, as usual... :)

Nico Bellic

@ClarkKent I covered Linear Algebra by Friedberg (4th edition) fairly in depth. But the book seems to focus more on just the theory, and little on examples. So I guess I would love to know of a book that covers these topics with more examples.

t.b.

@robjohn I'm not overcomplicating things here am I? (just making sure I don't miss a slick trick).

user19161

@NicoBellic Are you kidding me? That book is full of examples and theory! It is a great book!

user19161

Why don't you just do all the exercises then, if you need more examples?

Jonas Teuwen

10:25 PM

You also definitely need to read Duistermaat's distribution theory book. It is great.

user19161

@JonasTeuwen Are you answering Nico???

Jonas Teuwen

Yes.

He asked for depth.

user19161

@JonasTeuwen I think you are nuts then, like me.

Peter Tamaroff

OH! How sad is the story of Lamè and FLT.

Jonas Teuwen

You're not nuts. You're superman!

I'm batman.

user19161

10:26 PM

@JonasTeuwen There, now you are a confirmed nutcase, like me. QED.

Nico Bellic

@ClarkKent That's what I thought too for the problems that were worked out. But most of the problems at the end have no answers. I get lost at times!

Jonas Teuwen

What? No I really am Batman!

t.b.

@PeterTamaroff Lamé requires less effort for you and your language :)

Peter Tamaroff

@tb Come again?

user19161

@NicoBellic How do you find the difficulty of the book. If you can understand the text then you are on your way...

Jonas Teuwen

10:27 PM

I'm sure my advisor would also recommend Rudin.

I went to him as a second-years undergrad.

t.b.

@PeterTamaroff Oh, I thought I saw you adding an accent to the e, but I must have hallucinated. It's Lamé, not Lamè

Jonas Teuwen

And said that I really wanted to learn something for a change.

robjohn

@tb you mean in your second comment?

Jonas Teuwen

Then he gave me the books by Stein...

Matt N.

I thought the dual of $L^2$ was $L^2$. The problem is, thought here really means assumed. How do I find a dual space? I remember this horrible computation in one of the exercises that left me clueless.

Nico Bellic

10:28 PM

@ClarkKent Eh.. The vast majority is decent to very understandable.

Jonas Teuwen

@MattN Well, it is quite sucky I must tell you! You define the mapping between the spaces and then you prove it :-).

Peter Tamaroff

@tb Oh. They're next to each other in the kbrd. Do you know anything about Kummer and ideal complex numbers? It is very interesting how Lamé tried to solve the problem.

Nico Bellic

@ClarkKent I guess Friedberg just doesn't do any applications though. It's all theory.

user19161

@NicoBellic Like I said I think you should just check them all out.

t.b.

@robjohn yes, first Riesz to find another function (an a priori different one), then fundamental lemma of calculus of variations to identify the two functions a.e..

Matt N.

10:29 PM

@JonasTeuwen Meaning you have to have a clue first?

Jonas Teuwen

@MattN Yup.

t.b.

@PeterTamaroff not very much, honestly.

Nico Bellic

@ClarkKent Alright sounds good. What about D.E?

Matt N.

: (

user19161

@NicoBellic It depends on what you mean by applications. For example he does apply them to differential equations or geometry in n-space

user19161

10:29 PM

@NicoBellic What text did you use?

Peter Tamaroff

@tb Well, in my inmature maths knowledge, it seems interesting. Do you know anything about Kummer and ideal complex numbers?

Jonas Teuwen

@MattN But $(L^p)^* = L^q$ with $p^{-1} + q^{-1} = 1$ and $$g \mapsto \int fg.$$

(No conjugates, bro!)

Peter Tamaroff

Maybe I should start with cycltomic polynomials, but whatever....

Nico Bellic

@ClarkKent I mean as in real life applications. Comp sci, etc. I use D.E. With Boundary Value Problems by Polking.

user19161

@NicoBellic Well, if you understand the theory you can apply it to anything. For specific applications look for specific books.

t.b.

10:31 PM

@PeterTamaroff It certainly is interesting to many people, so follow it, but don't ask me, because I know virtually nothing about number theory.

Matt N.

@JonasTeuwen Because they're real-valued? Or what do you mean by no conjugates, bro?

Nico Bellic

@ClarkKent is Strang's L.A. book any good?

user19161

@NicoBellic Ah one of those fat books with lots of pictures. Well, I am sure it is full of examples already, maybe a bit too many

Jonas Teuwen

@MattN Because you could be confused with the inner product on $L^2$.

Peter Tamaroff

@tb I see. Why don'0t you find it interesting?

Jonas Teuwen

10:31 PM

Which does have a conjugate.

user19161

@NicoBellic I don't mean to not answer you, but like I said you should use what suits you. What I like and what you like could be very different.

t.b.

@PeterTamaroff I do find it interesting, but some things are more interesting to me :)

Peter Tamaroff

@tb Oh. Well, that is certainly true!

Bill Dubuque

@robjohn Hey Mean Square, are you ready to rumble with the 66.6 Beast? Where are the numerologists when we need them....

Jonas Teuwen

Well, actually those things are more interesting, not only to you. It is a general truth 8-)))).

Nico Bellic

10:33 PM

@ClarkKent Haha, alright I gotchu. Thanks man. I'll keep that in mind.

Matt N.

@JonasTeuwen Thank you. I made exactly that mistake! (among others, of course)

Ha. Nice. : )

Peter Tamaroff

@JonasTeuwen Who are you adressing?

Jonas Teuwen

@PeterTamaroff tb.

t.b.

bleh got myself confused...

user19161

@NicoBellic Just to name you a few of what I like. For linear algebra, try Hoffman/Kunze's linear algebra, Axler's Linear Algebra done right or Halmos's finite-dimensional vector spaces.

Matt N.

10:34 PM

Now let's see if I can fix my post.

Jonas Teuwen

@MattN I think it just reduces to the fact that $\int fg = 0$ for all $g$. Then $f = 0$.

Which is pretty bloody monkey I'd say.

user19161

@nico Again to name a few I like, for ordinary differential equations look at wolfgang walter's ordinary differential equations, corduneanu's principles of differntial and integral equations, teschl's ordinary differential equations and dynamical systems, vrabie's differential equations.

Jonas Teuwen

Walter!

@ClarkKent But where is Krylov?

user19161

@JonasTeuwen Yes, typical German thoroughness.

Jonas Teuwen

The Sobolov space book is pretty kickass.

user19161

10:36 PM

@JonasTeuwen Is he still alive? Rudin is dead.

Nico Bellic

@ClarkKent I have Axler! It's fantastic too.

Jonas Teuwen

@ClarkKent Not sure! The books are pretty recent.

Otherwise, Evans could do too...

But Evans is not so rigorous.

Peter Tamaroff

I'll go play the guitar for a while. Let me know if there is any important news!

user19161

@PeterTamaroff All gossip in this room is important.

t.b.

@JonasTeuwen he's at Bogolyubov's place. They drink Vodka

Jonas Teuwen

10:37 PM

That sounds Russian! So it must be good.

Yes.

Peter Tamaroff

Oh, @BillDubuque if you could spare some time to read this, it will be much appreciated!

user19161

@tb That is also a chess player

Peter Tamaroff

@ClarkKent News $\neq$ gossip. QED.

Matt N.

I think catty purring is the sweetest sound.

user19161

@PeterTamaroff maybe i'll start typing without proper spelling and punctuation, it's faster

Peter Tamaroff

10:39 PM

@ClarkKent What? Let me go!

user19161

@PeterTamaroff sorry i did not mean to ping you there, accident

t.b.

@ClarkKent Nikolay ≠ Efim

user19161

the combinatorialist imre leader is also a go champion

Brian M. Scott

@ClarkKent When I play white, I win because I play white. When I play black, I win because I am Bogolyubov. Or something like that.

Bill Dubuque

@PeterTamaroff I'm not going to have much spare time now with mod candidacy. Why not post questions to MSE?

Jonas Teuwen

10:41 PM

@MattN How many dB?

user19161

@BrianMScott this reminds me since we are on chess that the most common mistake is rotating the chessboard by 90 degrees and also alternating the king and queen position

t.b.

@ClarkKent I was alluding to this cool theorem.

Matt N.

@JonasTeuwen Not sure : )

Jonas Teuwen

Invariant measures are pretty kickass.

I used that theorem today to prove the contractivity of some semigroup related to Mister Ornstein-Uhlenbeck.

t.b.

@JonasTeuwen it all boils down to amenability :) remember this uniform ergodic theorem answer? one of our first interactions here :)

Brian M. Scott

10:42 PM

@ClarkKent The first is not a problem when you're playing on a cheap board that folds in the middle!

user19161

@BrianMScott There is nothing to stop one from having the line vertically.

Jonas Teuwen

The coolest operator ever: The Ornstein-Uhlenbeck operator.

@tb Yes!

@tb I was the only one finishing the course on ergodic theory, btw 8-)).

Matt N.

Oh, it's late. I have to force myself into bed and do Riesz tomorrow. Have to get up quite early (9 o'clock).

Brian M. Scott

@ClarkKent Except the knowledge that it's intended to go horizontally.

Cowbell

May I ask about a formula for a $1$-form?

Matt N.

10:43 PM

Good night!

user19161

@BrianMScott Well, the manufacturers could fold it wrongly too.

Brian M. Scott

G'night, Matt.

user19161

@MattN Good night.

Brian M. Scott

@ClarkKent They could, but I've never seen an actual example. (There probably is at least one somewhere, though.)

t.b.

@MattN Matt leaves, the cowbells arrive. Coincidence? Good night my favorite peasant in this chat room!

Jonas Teuwen

10:44 PM

@MattN Night! Ding ding!

Cowbell

Sorry, I'm not much of a substitute.

Matt N.

@tb Haha, likewise : )

Jonas Teuwen

@MattN Facial hair looks pretty good on monkeys btw.

user19161

What does cowbell mean? I am not familiar with all these internet terms. I need a copy of urban dictionary or something

user19161

@JonasTeuwen not on female monkeys

Jonas Teuwen

10:45 PM

That's... A bell for a cow.

t.b.

@ClarkKent a bell hung around a cow's neck?

user19161

@tb ah i thought it was another term for cow dung

Cowbell

:(

Jonas Teuwen

Ahhh, that's the smell 8-)).

user19161

@Cowbell cheer up, have a kit kat

t.b.

10:46 PM

@Cowbell Oh, no worries. You know: We're from the country full of mountains, chocolate, watches and banks.

Jonas Teuwen

In the movie "Good Will Hunting" you can see Parseval on the first blackboard!

user19161

In the movie a beautiful mind they play go.

Jonas Teuwen

@tb And incomprehensible German?

user19161

So we just mentioned my two fave movies.

Cowbell

I'll just throw this out there. If $\alpha=fdx+gdy+hdz$ is a $1$-form, what is the formula for $d\alpha$?

Jonas Teuwen

10:47 PM

It is: $d(f dx + g dy + h dz)$.

t.b.

@Cowbell it should be awfully reminiscent of the curl...

Cowbell

@tb I like Switzerland. I went to Italy once, and found the cleanliness of Switzerland refreshing. There was no pigeon poo on the ground.

Jonas Teuwen

@Cowbell That's because they ate all the pigeons...

Cowbell

Those neutral monsters!

Jonas Teuwen

Get in my belly, you flying monster!

robjohn

10:49 PM

@BillDubuque Oh, no... The Beast!

t.b.

@Cowbell I like Italy a lot, too. Despite the pigeon poo.

Jonas Teuwen

We also have pigeon poo!

t.b.

@Cowbell Can you do $d(f(x,y,z)\,dx)$?

Jonas Teuwen

And a right light district.

"Oh man, this is the hell on earth!"

robjohn

@JonasTeuwen I've heard of a "red light district", but not a "right light district"

Bill Dubuque

10:51 PM

@robjohn Someone just pointed that out, and I couldn't resist milking it for a chuckle. Bye the way, happy B-day. Now I can feel younger than you temporarily.

Jonas Teuwen

@robjohn Typo :-).

Cowbell

@tb I think altogether, it's should be something like $d\alpha=dfdx+dgdy+dhdz$ where $df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz$, and similarly for $dg$ and $dh$?

Jonas Teuwen

Old Man With Wings.jpg

robjohn

@BillDubuque I had no idea we were that close in age.

t.b.

@Cowbell $d(f \,dx) = f_x \,dx \wedge dx + f_y \,dy \wedge dx + f_z \,dz \wedge dx$.

Now you can simplify this a bit. (you're right with what you say)

robjohn

10:53 PM

@JonasTeuwen I was thinking it was where the Republicans (the "Right") went for their "fun & games".

t.b.

yes

Cowbell

@tb Are you referring to me in that last message?

robjohn

@Cowbell remember that $dx\wedge dx=-dx\wedge dx$ (swapping dx and dx)

Jonas Teuwen

@robjohn Oh... 8-).

t.b.

@Cowbell yes, I am.

Cowbell

10:55 PM

@robjohn so $dxdx=0$. Thanks.

t.b.

Next $dy\,dx = - dx \,dy$, etc.

Then re-group the entire thing.

Gigili

@tb Does "viel spaß Ausstellungsdauer" make sense to you?

robjohn

@Cowbell Another way of looking at it: the outer $dx$ integration holds the inner $x$ constant so that the inner $dx=0$.

t.b.

@Gigili uhm. no, sorry. "Have fun, duration of the exhibition."

Gigili

@tb During the exhibition?

Jonas Teuwen

10:57 PM

Wunderbarrrrrrrrrrrrrrrr!!!!

t.b.

aaah. :)

Bill Dubuque

@robjohn Maybe I was misrembering. If you're middle name is Wm then we're not.

t.b.

@Gigili "Viel Spaß an der Ausstellung" is what I'd say. During is während, but "Viel Spaß während der Ausstellung", sounds a bit murky, but I can't say exactly why.

robjohn

@BillDubuque nope, my middle name is Alan

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$Mathematics$ Mathematics