Mathematics - 2013-09-29 (page 1 of 4)

T. Bongers

00:00

@PedroTamaroff Fall of '08

Charlie

Oh

Pedro Tamaroff

@T.Bongers Oh, OK. Didn't know people started undergrad so early.

@anon Dude.

anon

dude

Pedro Tamaroff

@T.Bongers Do you think you can explain to me a proof of Hahn Banach using Zorn's Lemma?

anon

once I look up what that is maybe

Charlie

00:02

@anon mano...

T. Bongers

@PedroTamaroff I might be able to

anon

man

Pedro Tamaroff

2 hours ago, by Pedro Tamaroff

Charlie

@anon no, dear, mano

Pedro Tamaroff

@T.Bongers I'm curious, you had to quit high school to go to college, right?

anon

00:05

hand?

T. Bongers

@PedroTamaroff I was home-schooled through most of high school, so I started and finished early

Charlie

@anon that's hand in spanish

Pedro Tamaroff

@T.Bongers Oh. And were you specially schooled in math or something?

Charlie

@T.Bongers :D

T. Bongers

@PedroTamaroff No, it was pretty much the usual thing. I just did it online from home

It still took four years to get through

Charlie

00:07

@anon mano is dude, em.paulistish

Pedro Tamaroff

@T.Bongers Oh.

@T.Bongers And what are you going to study in your PhD?

Charlie

@PedroTamaroff relax, Smartpantalones, I still think you're awesome

T. Bongers

@PedroTamaroff Probably some form of harmonic analysis

Pedro Tamaroff

@Charlie Charles, I am just curious!

Many people are all incognito and stuff. It is nice to meet "real" people every now and then.

Charlie

@PedroTamaroff :D

T. Bongers

00:16

The image linked isn't terribly clear about the partial order. I believe that the order is (M', f') <= (M'', f'') means that M' is contained in M'' and f'' and f' agree on M'. If we have a chain (M_alpha, f_alpha), its upper bound will be the pair (M, f) where M is the union of M_alpha, and f is a functional which agrees with each f_alpha. By the choice of the order, f is well-defined. @PedroTamaroff

So every chain has an upper bound, and Zorn applies.

T. Bongers

00:28

@PedroTamaroff What book is that proof from?

Pedro Tamaroff

@T.Bongers Kolmogorov and Fomin.

@T.Bongers Am I right the family $\mathscr F$ should be of linear functionals?

T. Bongers

@PedroTamaroff Yes, I think F refers to functionals with dependence on the subspace that the functional is defined on

Pedro Tamaroff

@T.Bongers I mean linear as in $f(x+y)=f(x)+f(y)$ and $f(\alpha x)=\alpha f(x)$. The claim is that if $f_0$ is a linear functional bounded above by a finite convex functional $p$ on $S$ a subspace of $V$ then it can be extended to a linear functional $f$ on the whole space.

T. Bongers

00:44

@PedroTamaroff Right. So \mathscr F should consist only of linear functionals, I think

Pedro Tamaroff

@T.Bongers Ayup, that is what I think too.

T. Bongers

Sorry, I can't find a copy of that book, so I'm not entirely sure

But that's the only thing that makes sense, so it must be true ;)

Pedro Tamaroff

anon

@PedroTamaroff I never really played the 2D sidescroller zeldas like I did with mario, except for four swords. I might try phantom hourglass and a few others at some point in life. The original that kicked it off was just called The Legend of Zelda.

Kevin Driscoll

01:42

@anon I think most regard A Link To The Past as the pinacle of 2D zeldas

the game has held up surprisingly well both graphically and gameplay-wise

Anyone know if/how the finite part of integrals like $$\int_0^4 \frac{1}{\left|x-1\right|} dx$$ is defined?

Fernando Martin

02:20

@PedroTamaroff: Are you reading about functional analysis?

Pedro Tamaroff

@FernandoMartin Nah, just an intro to Real Analysis.

(And I said I was done with Analysis. I'm a trainwreck!)

Fernando Martin

Any topic in particular?

Pedro Tamaroff

@FernandoMartin (I'm in Ch. 4, section 14)

Kevin Driscoll

02:39

@PedroTamaroff Staaaaaahp..... too much math for me to learn

Pedro Tamaroff

@KevinDriscoll There no such thing as too much math, dude.

Kevin Driscoll

@PedroTamaroff Sadly, I have but finite time. And a relatively small free time to learn actual math.

Pedro Tamaroff

@KevinDriscoll We all do. That's the saddest thing of it all.

Fernando Martin

I'd say there's too much math

@PedroTamaroff: Did you read the whole thing up to 4.14 or you just pick some chapters?

Pedro Tamaroff

@FernandoMartin I just mean everything you read is just a tiny bit.

Fernando Martin

02:49

I know, I was just kidding.

YES

Pedro Tamaroff

@FernandoMartin Nah, I went directly to 4.14

Fernando Martin

493 million CpS

Pedro Tamaroff

@FernandoMartin 7 million.

@FernandoMartin If I use my mouse pad as a drum, I can harvest a lot.

Fernando Martin

haha

Kevin Driscoll

03:07

What are you two talking about?

Pedro Tamaroff

@KevinDriscoll You don't wanna know.

Believe me.

T. Bongers

Have a good night, all

Kevin Driscoll

03:25

@PedroTamaroff Well by first instinct was bitcoins but....... I dont know much about all that

Chris's sis

08:09

Greetings noble beings!

user43418

11:12

Hello is anybody here ?

Karl Kronenfeld

11:25

@user43418 Yes.

user43418

@KarlKronenfeld Hello. Could you help me with this:

http://math.stackexchange.com/questions/508726/separable-metric-space-has-a-countable-base

Alizter

How can I say that the area of a circle is $$x^2+y^2=r^2\implies y=\pm\sqrt{r^2-x^2}\implies ? A = 2 \int^1_{-1} \sqrt{r^2-x^2}dx$$

Karl Kronenfeld

@user43418 Hm, that's pretty tough to motivate since it is really what one would expect of a separable space.

On the other hand, I would not be surprised if someone more experienced than me knows a great way to do this.

user43418

@KarlKronenfeld And do you know by any chance of examples of how we can use this property ?

Alizter

11:46

How to prove that $$\sqrt{r^2-1}+r^2\arcsin\frac1r-r^2\arcsin-\frac1r=\frac\pi2$$

Shobhit

Can anybody help me on math.stackexchange.com/questions/508729/…

user43418

meeh

meeeeh

user43418

12:39

@KarlKronenfeld The lindelof lemma was an example proposed to me

@KarlKronenfeld I am still trying to find a way to motivate the problem and a title

Pedro Tamaroff

13:37

@KarlKronenfeld Yao,.

Pedro Tamaroff

13:52

@DanielFischer

Daniel Fischer

Yaaawn, what gives?

Pedro Tamaroff

@DanielFischer Heh, just had to cancel a tennis lesson due to rain.

Daniel Fischer

You should play football, you can play that even when it rains.

Pedro Tamaroff

@DanielFischer Nah. Not a fan.

=D

Daniel Fischer

You don't need to be a fan to play it.

Or, you're Argentinian, aren't you, then Polo?

Pedro Tamaroff

13:56

@DanielFischer This is my linear algebra professor. He has a book on Algebraic Topology (his field, and apparently combinatorics too.

@DanielFischer Hahahha, too expensive!

Daniel Fischer

He's very young, hasn't even learned how to shave.

Pedro Tamaroff

@DanielFischer Don't be mean! =O

@DanielFischer I was told his PhD paper was very cool.

Daniel Fischer

I can believe that. But I don't think I'm qualified to judge.

Pedro Tamaroff

"Algebraic Topology of finite Topological Spaces and Applications."

(And Springer made it into a book! So that's quite rad)

Daniel Fischer

Help an old man, what's "rad"?

Pedro Tamaroff

14:03

@DanielFischer This

Daniel Fischer

Aha, yes, that's rad.

Carpediem

Can someone help me with this? math.stackexchange.com/questions/508726/…

Pedro Tamaroff

@Carpediem ?????

Carpediem

@PedroTamaroff what

Pedro Tamaroff

@Carpediem What do you need help with?

You linked to someone else's question.

Carpediem

14:18

@PedroTamaroff Yes we are working together on this. We need a way to introduce this problem. How to motivate it

Pedro Tamaroff

@Carpediem Well, separability is used to prove Arzelá Ascoli in a great manner.

The countable has many advantages.

Carpediem

@PedroTamaroff That's what we've come up with: "The property derived from this problem is particularly useful in the context of Lindelöf spaces, for example. A Lindelöf space is a topological space in which every open cover has a countable subcover.
In particular, the Lindelöf lemma states that every separable metric space which has a countable base is a Lindelöf space. To prove this lemma, we use the fact that a countably compact metric space is compact and the property which we have proven above. "

Pedro Tamaroff

Of course it is usually more boring.

Carpediem

@PedroTamaroff We wrote this in order to provide some form of extension to the problem

@PedroTamaroff What do you think ? and is that the theorem you were referring to ? en.wikipedia.org/wiki/Arzelà;–Ascoli_theorem

@PedroTamaroff Do you have an idea for an introduction

Pedro Tamaroff

@Carpediem I don't really know. I never thought about doing such a thing. Maybe you should get some books that treat separable spaces, see what they have to say.

@Carpediem Funny, I am reading that $n$-manifolds are by definition separable Hausdorff spaces locally homo to $\Bbb R^n$ or $\Bbb H^n$.

Daniel Fischer

14:39

@PedroTamaroff Cool fact: For Riemann surfaces, you don't need to require separability/second countability in the definition. It follows.

Pedro Tamaroff

@DanielFischer Certified.

@DanielFischer I also know that if a complex function admits a derivative on an open set, the derivative is automatically continuous.

Daniel Fischer

Yes, but that's quite elementary. Rado's theorem is deep.

Pedro Tamaroff

@DanielFischer Well, Daniel, don't expect me to come up with deep facts! =D

Daniel Fischer

Sooner or later, you'll find your very own deep result ;)

Pedro Tamaroff

@DanielFischer I hope to, yes!

Pedro Tamaroff

15:09

@TedShifrin

Ted Shifrin

@Pedro: Greetings. Well, I'm embroiled in two bickering fights -- one thanks to you (the guy who thinks differential forms are infinitesimals), the other with someone who thinks the theory of Taylor polynomials is the same as L'Hôpital's rule, apparently.

I was trying to figure out how to include a link in chat.

Hay to you too :P

Pedro Tamaroff

@TedShifrin Awww, you stood up for me!

@TedShifrin =D

@TedShifrin [text](link)?

Ted Shifrin

Yes, I just looked it up in meta.

Pedro Tamaroff

@TedShifrin Don't worry that much about crankery Ted.

You cannot expect everyone to be sane!

Daniel Rust

g'dafternoon all

Pedro Tamaroff

15:13

Well, not really crankery, but just people that study things they cannot groke, maybe?

Daniel Rust

what's this about drama?

Pedro Tamaroff

@DanielRust

Daniel Rust

was that a picture of sap?

Pedro Tamaroff

Darn.

Ted Shifrin

Here's the link @Pedro. I'm impressed by the tour de force solution that Glen gave after I went to tennis. But it irks me that on philosophical grounds people think of Taylor polynomials as equivalent to L'Hôpital. I don't think it's crankery or not groking. This guy ABC has been around. He and I just disagree about mathematics here.

Daniel Rust

15:14

@ted that is rather strange

Pedro Tamaroff

@TedShifrin I was rather talking about the guy talking about infinitesimals.

@TedShifrin Yeah, that doubling tripling the angle and fooling around is a known trick, which I will never ever try to do.

@robjohn is known for such stunts.

Ted Shifrin

Yeah, @Pedro, that guy has his own intuition, which is fine, but it's not mainstream mathematics.

Pedro Tamaroff

@TedShifrin What do you think about Zorich's book on Analysis? Looks pretty long.

Ted Shifrin

@Pedro, yeah, I've seen it before, although I would never teach my students to spend hours with such trickery. I want them believing in Taylor polynomials :) I HATE it that students will do L'Hôpital's rule for things like $\lim_{x\to 1} \frac{\log x}{x-1}$ when it's just the definition of the derivative. I rant about that to my students.

I don't know it, @Pedro, sorry.

Pedro Tamaroff

@TedShifrin Yeah, I agree with you.

Daniel Rust

15:18

I've always wondered in such questions where you 'can't use x', what would the marker think if you deduced x in your answer and then used it?

Ted Shifrin

Well, everyone comes out of high school "knowing" L'Hôpital's rule and then takes college calculus where we make them derive limits from first principles. But if a student gave me the Cauchy Mean Value Theorem and proved L'Hôpital's rule, I would allow it. :)

Pedro Tamaroff

@TedShifrin Here is its table of contents. Quite humongous!

Ted Shifrin

The problem is that we train students like monkeys in high school and they want to use all their formulas rather than learning to understand what they're doing.

Pedro Tamaroff

@TedShifrin Yeah, that is sad.

Daniel Rust

yeah it's a shame

@TedShifrin although I've often found myself going 'I wonder if I can use X's theorem to prove this' which is similar to wanting to use a formula.

Ted Shifrin

15:21

Come to think of it, we still train them for the most part like monkeys in university calculus, too.

@Pedro: The table of contents is impressive. It depends how well he writes and what kinds of exercises he includes. Plus, although you will disagree with me, Rudin fails to handle multivariable analysis with the skill I would like, and that happens a lot in mathematics.

Pedro Tamaroff

@TedShifrin Hmm... I must say I am not a total fan of Rudin, but some of his proofs are quite awesome. For example, I did like the proof he gave of the Inverse Function Theorem.

Ted Shifrin

Well, @DanielRust, it's always an issue if someone finds what you want proved as a theorem in a book and quotes the theorem, it's obviously not meeting the point of the exercises/exam question. But for doing research, it's fair game :P

Daniel Rust

@TedShifrin haha true

Ted Shifrin

He does the standard contraction mapping proof, right? I have always liked the Banach space proof, even in finite dimensions, and did that in my book. I do not like Spivak's proof that uses the maximum value theorem to prove open mapping.

Pedro Tamaroff

@TedShifrin Kinda, yes, but he also proves the open mapping theorem, and the use of matrix norms and continuity of inversion makes things neat. What is the "Banach space proof"?

Ted Shifrin

15:27

That's the Banach space proof. It works in Banach spaces verbatim. See Chapter 6.2 of my book.

Pedro Tamaroff

@TedShifrin Ah, yes. I did see that chapter! I even copied the proof in my notes and all. =)

I have a question.

It's about the Hahn-Banach theorem.

Ted Shifrin

Wow, I'm impressed :D I learned the proof years ago from Lang's Analysis book.

I'm not an expert on that sort of stuff, @Pedro, but you can ask.

Pedro Tamaroff

@TedShifrin Well, the statement is this. Let $V$ be a real vector space, and suppose we have a linear functional $f_0:S\to\Bbb R$ defined on a subspace $S$ of $V$ and a convex finite functional $p:V\to\Bbb R$, and we have $f_0\leqslant p$. Then there exists an extension $f$ of $f_0$ to all of $V$ satisfying $f\leqslant p$.

Ted Shifrin

That sounds standard, @Pedro, but I haven't thought about this sort of stuff since grad school :

Pedro Tamaroff

A convex functional is said to be finite if it doesn't attain the value $+\infty$ which the authors sometimes allow.

@TedShifrin Well, I get most of the proof. I just need some help with the part that uses Zorn's Lemma.

Ted Shifrin

15:36

I'm not in a frame of mind to help on that now ... but maybe later I'll think about it :P

Pedro Tamaroff

Ah, OK.

@TedShifrin Look, that's some exercises in Zorich.

Ted Shifrin

Ah, looks good :)

Dieudonné also has excellent exercises in his 4- or 5-volume analysis treatise. I love that book.

Pedro Tamaroff

@TedShifrin 5 volume. Woah.

Ted Shifrin

Would be terrible to learn from for the first time or to teach out of, but it's great.

Pedro Tamaroff

@TedShifrin You didn't get my "Sap?" salute to Daniel.

Ted Shifrin

15:39

You didn't respond to my Hay :P

Google Dieudonné Treatise on Analysis.

Daniel Rust

@PedroTamaroff oh I thought you were calling me a sap :P

Pedro Tamaroff

@TedShifrin But I obviously did.

Ted Shifrin

No, I said hay after your pic :P

Daniel Rust

@PedroTamaroff i have 200 heavenly ships now :>

Pedro Tamaroff

@DanielRust What are those?

I have 25 ACs.

Ted Shifrin

15:41

what are ACs?

Pedro Tamaroff

@TedShifrin You don't wanna know.

Daniel Rust

(Note : each heavenly chip grants you +2% CpS multiplier. You can gain more chips by resetting with a lot of cookies.)

you gain a chip for every 50Tril that you've baked

Ted Shifrin

You're probably right. I don't wanna know much.

Pedro Tamaroff

@TedShifrin It's a cookie game. Quite silly, but addictive.

Daniel Rust

oh actually, it's 100Tril

Ted Shifrin

15:43

If you eat all those cookies, you'll get a lot fatter.

Pedro Tamaroff

@DanielRust How many zeroes after the 1?

Daniel Rust

12

well, $100\times 10^{12}$

Pedro Tamaroff

Daniel Rust

Ted Shifrin

I leave you two to your challenges :P

Daniel Rust

15:47

haha

@TedShifrin sorry ted :P

you should give it a try

Ted Shifrin

NO.

2

Pedro Tamaroff

@TedShifrin You're a wise man.

Ted Shifrin

I have enough computer addictions, thank you.

Daniel Rust

You know you want to ;)

Ted Shifrin

Cutting back on MSE too ...

Daniel Rust

15:49

@TedShifrin probably not a bad idea

Pedro Tamaroff

@DanielRust I am a lousy player.

Daniel Rust

@PedroTamaroff I just have my screen split-screened and click any golden cookies I see :P

Pedro Tamaroff

@DanielRust LOL.

Do they appear that often?

Daniel Rust

with upgrades about once a minute

Pedro Tamaroff

I clicked less than 10, you have over 1000!

What ups?

Daniel Rust

15:52

and if you save up, a well timed one can give you 100Tril

Pedro Tamaroff

Do you lose them when you reset?

Daniel Rust

you lose everything when you reset except your achis

Twink

good morning everyone

Daniel Rust

good afternoon

Twink

look academia.stackexchange.com/questions/10266/…

what I told you was true

she is a he

Pedro Tamaroff

15:56

@DanielRust Do cookies dissappear if I don't click them?

Daniel Rust

yeah

Twink

did you see?

I was right

Daniel Rust

@Twink No, she is a she, as she chooses to self-identify as female. You have no right to choose her gender for her and should respect the pronouns she wishes to use for herself.

Pedro Tamaroff

@Twink So she is a guy at college but dresses and acts as a woman outside? I didn't understand.

Twink

No I was just pointing out that she's a trans woman

Daniel Rust

16:00

Good for her

Twink

math.stackexchange.com/users

and she's top 8

of the month

and she has been a user for 35 days only

in MSE

she's amazing

Daniel Rust

217 answers in a month is a lot of hard work

Amzoti

16:20

How is it possible that a question that had an answer can be deleted? math.stackexchange.com/questions/508896/…. I thought that was not allowed

Twink

if the answer doesn't have any upvotes, it's possible for the question's author to delete it

Amzoti

16:51

@robjohn: Can you look at this? math.stackexchange.com/questions/508896/… and this META post meta.math.stackexchange.com/questions/11102/…

Mats Granvik

17:01

Booyakasha.

Transcript for

$Mathematics$ Mathematics