Mathematics - 2014-12-24 (page 5 of 7)

@Ted I think I get it. If I take some open set in the basis, I can take it to be equivalent to open basis set in the homeomorphic $\Bbb R^I$ where I order the products so that the ones with open sets not being $\Bbb R$ are at the start, followed by the infinite number of ones that are $\Bbb R$. I can enforce the strict monotonocity only for $x$ in the finite number of products, in the rest it can be constant.

I hope I didn't say too many silly things.

Balarka Sen

7:30 PM

Hrm. I think it holds iff $\{Y_n\}$ is finite, in the box topology.

Ted Shifrin

$\{Y_n\}$? @Balarka

Well, @Studentmath, remember that you have a factor for each $x\in I$. Take a basis element that's an open interval for one particular $x$ and everything everywhere else.

Yes, @Balarka. One reason box topology is not very practical.

Balarka Sen

I'm skeptic.

Ted Shifrin

Go back to my hint.

Balarka Sen

You can see the open sets in the box topology.

Ted Shifrin

Yeah, and they get way too small, @Balarka.

Jasper Loy

7:33 PM

I just sent my dear Lee an email @TedShifrin, lol. Just to ask about his book and wish him happy new year too.

Studentmath

@Balarka you are fond of categories, aren't you? I've heard product topology has to do with it being natural when it comes to them.

Balarka Sen

I am not fond of categories.

Abstract nonsense is BS.

Ted Shifrin

@Studentmath: Still, so when does $f_n\to f$? In the sense of analysis ...

Balarka Sen

Unless, of course, you can do something with them.

Ted Shifrin

Well, all of algebraic topology is about functors, so you have to care a little bit, @Balarka.

Mike Miller

7:34 PM

@Balarka You can, though. That's pretty much the whole point.

Studentmath

$f_n$ being a sequence of functions, right?

Ted Shifrin

Yes, @Studentmath ... In the product topology, what sort of convergence is this?

Balarka Sen

@MikeMiller Eh. I see nonsensical generalization of homotopy theory in categories. What's the point of all that?

Mike Miller

To do math.

user112495

For $0 < x< \infty$, let $\phi (x)$ be positive and continuously twice differentiable satisfying:

(a) $\phi (x+1) = \phi (x)$

(b) $\phi(\frac{x}{2}) \phi(\frac{x+1}{2}) = d\phi(x),$ where $d$ is a constant.

Prove that $\phi$ is a constant.

I'm given the hint let $g(x) = \frac{d^2}{dx^2} \log \phi (x)$

Ted Shifrin

7:36 PM

Well, a lot of stuff is inventing abstraction for inventing's sake, as far as I'm concerned, @Mike. :P

Balarka Sen

You need some Grothendieck-type idea it apply it in practical cases. Grothendieck never generalized much more than he needed, AFAIK.

Mike Miller

One classical problem that's solved using partly of higher-categorical homotopy thwory is the exotic sphere problem: determining what dimensions have exotic spheres.

Balarka Sen

But MacLane, say, ... sheesh.

Mike Miller

If you want to see some applications look at my highest-voted MO question.

Balarka Sen

@MikeMiller what is le exotic sphere?

Studentmath

7:37 PM

Wouldn't it be uniform convergence?

Or, no.

Balarka Sen

tries to recall definition of an exotic sphere from past life

user112495

@MikeMiller @BalarkaSen Do you know how I could go about doing this. I've tried doing what the hint says, but I'm just not getting anywhere.

Jasper Loy

@BalarkaSen You believe in rebirth too now? Seems I have got everyone believing in it.

Ted Shifrin

Is $\log \phi$ going to end up being convex or concave, @user112495?

Studentmath

I would need it for a finite number of $x's$ in $I$ to have that for some $n$, $d(f_n(x),f(x)<\epsilon$ for every $\epsilon>0$

Balarka Sen

7:38 PM

@JasperLoy who else is believing in it?

Jasper Loy

@BalarkaSen Mike seems to have said something similar to what you said.

Ted Shifrin

Your quantifiers are out of order, I think, @Studentmath. So, if for every $x$ and $\epsilon>0$, there is $N$ so that $d(f_n(x),f(x))<\epsilon$ for all $n>N$ ...

user112495

@TedShifrin Wouldn't it be both as we need to show $\phi$ is a constant?

Ted Shifrin

Well, yes, but I'm suggesting you think about conditions (1) and (2) in that light.

Mike Miller

@Ted I think much of it is. But much also isn't.

Ted Shifrin

7:41 PM

I have no idea, @user112495. That second condition is weird.

Mike Miller

Nobody likes my posts today. It's tragic!

Ted Shifrin

Get used to not being liked, @Mike.

Balarka Sen

I was unsuccessful persuading @Mike to give me rough ideas.

bah.

Mike Miller

You'll learn about them later.

user112495

@TedShifrin If it helps, this is supposed to be the first step in a proof of Euler's reflection formula. So what you said about convexity seems like it's on the right lines.

Jasper Loy

7:42 PM

Merry Christmas all you. I am now looking forward to the new year.

Ted Shifrin

I'm scared of the new year, too, @Jasper :)

I guess I'm starting a new life in it just as you are.

Jasper Loy

@TedShifrin Well, it will be your last semester. Maybe consider postponing your retirement? Lots of students need you.

Ted Shifrin

No, not postponing.

Balarka Sen

@TedShifrin when are you going to retire?

Studentmath

@TedShifrin Right

Ted Shifrin

7:43 PM

If I truly miss it, I can get some sort of temp job in San Diego.

Jasper Loy

@TedShifrin Have you planned what to do for the rest of your life?

Ted Shifrin

So what sort of convergence, @Studentmath?

Hell no, @Jasper.

Balarka Sen

@JasperLoy Of course, hang out in the chat 24/7.

Jasper Loy

@TedShifrin I see. Well, I have!

Ted Shifrin

I know, @Jasper :)

Studentmath

7:44 PM

Uniform, that's the precise definition @Ted

Ted Shifrin

smacks @Studentmath

Jasper Loy

So, I just watched A Walk to Remember again, what a lovely movie.

Ted Shifrin

I don't know that movie, @Jasper.

Jasper Loy

@TedShifrin It's about two high school kids falling in love. The girl eventually dies of cancer, but that happens after they get married.

Ted Shifrin

Oh, that reminds me of the sappy Love Story ...

Mike Miller

7:46 PM

I like it's a wonderful life

's a good Christmas movis

Jasper Loy

My favourite animation movie is The Polar Express. I should really watch it now that it's Christmas, because it is about Christmas.

Studentmath

@Ted D: I'm not sure what we are aiming for here

Balarka Sen

@JasperLoy That was a very nice movie.

Ted Shifrin

A correct definition/answer? :D

Studentmath

Oh, it's unformly.

Ted Shifrin

7:48 PM

NOOO ... you keep saying that.

Studentmath

No, I mean yes - I just mispelled it before, but that's besides the point

Ted Shifrin

You have to watch where you put your quantifiers in analysis. It's not sloppy like discrete math :P

Balarka Sen

@TedShifrin angry

Discrete math is not sloppy.

Ted Shifrin

Quantifiers can be :P

Studentmath

@Ted alright, let me shut up and think

Mike Miller

7:49 PM

Are you missing the days of yore yet, @Ted?

Ted Shifrin

Yes, @Mike. Missing some of the days of Gore, too.

Mike Miller

I don't think he had any days, @Ted. Bush won, unfortunately.

Balarka Sen

Someone upvoted me on the Gal(\bar Q/Q) answer. Now I've got 3 upvotes.

Ted Shifrin

I haven't forgotten.

Mike Miller

I'll go cancel his out, @Balarka.

Balarka Sen

7:52 PM

Well, you'll have to explain why you did that @Mike, if you do.

Jasper Loy

I am wondering if people will come to chat on Christmas day itself. Does that mean they have no life?

Anyway, it's already 25 here.

Mike Miller

No I won't. That's not how votes work.

Ted Shifrin

I'm going to friends' an hour away to help cook dinner tomorrow, @Jasper, but that's not until around this time of day.

Don Larynx

@Jasper: where else are they supposed to be? playing in snow (or sand for southerners)?

Ted Shifrin

BTW, Happy Christmas, @Jasper. May next year be everything you wish.

Jasper Loy

7:54 PM

@DonLarynx They are like supposed to be spending time with family and friends, not in this chat, lol.

Studentmath

@Ted Jeez, point-wise convergence sigh

Jasper Loy

@TedShifrin Thank you. I really hope so.

Mike Miller

I'm not a southerner, but I don't have any snow...

Balarka Sen

It's 25 here too.

Khallil Benyattou

Merry Christmas, @Balarka!

Balarka Sen

7:55 PM

Merry Christmas.

Ted Shifrin

Yippee @Studentmath. Now you should understand the answer to your original question.

Jasper Loy

By the way, lots of Indians here are vegetarians, @BalarkaSen, even though they are not Buddhist.

Don Larynx

I'm going to be in my own head, as always, @Jasper.

regardless who is around me

Jasper Loy

@DonLarynx I will be with myself and with my mum.

Mike Miller

Go ask a good question, @Ted. I can't find any.

Ted Shifrin

7:56 PM

I'm not here to ask good questions.

Mike Miller

Well, who else am I gonna ask?

Ted Shifrin

OH, you never answered my Lawson question from earlier.

Balarka Sen

I can ask a question @Mike

Jasper Loy

By the way, if you buy a book from amazon, always look and see if every page is printed. Once I got a book with a missing page, and you have to do this within X days.

Mike Miller

A good one, that's hard enough that I won't just say "go do it yourself", @Balarka?

What quesrion was that, @Ted?

Ted Shifrin

7:57 PM

One of my students last summer got a copy of Rudin that was missing 20 pages, @Jasper.

Without any algebraic topology crap, @Mike, prove that a compact hypersurface $X$ in a simply connected manifold is orientable.

Balarka Sen

$X \stackrel{p}{\to} Y \stackrel{q}{\to} Z$ be covering maps such that $X, Y, Z$ are path connected, locally path connected and semilocally path connected.

Jasper Loy

@TedShifrin Yet it is so expensive. I confess that I spent over a thousand dollars on my 12 holy books.

Balarka Sen

Prove that $r = q \circ p$ is a covering map @Mike

Is that good enough?

Ted Shifrin

Hmm, composition of coverings isn't usually a covering ...

Balarka Sen

Sure, @Ted. But I have assumed a hell lot of stuff.

Mike Miller

7:59 PM

No, that's not good enough. There's a counterexample in one of Hatcher's problems.

You assumed jack.

Jasper Loy

@TedShifrin The most expensive book I have gotten so far is Cohn's Classic Algebra, over 200 dollars...

Ted Shifrin

Yeah, I don't think you've assumed enough.

Balarka Sen

That's not semilocally simply connected @Mike

Studentmath

I think I get it now, @Ted! many thanks once again

Ted Shifrin

Well, @Jasper, you and I are at opposite ends ... I've collected math books my whole life and now I'm starting to give them away. In some ways it's sad. In other ways it's very freeing.

Of course, @Studentmath.

Balarka Sen

7:59 PM

A hawaiian earring isn't even locally simply connected.

Mike Miller

Then maybe that should have been in your assumptions.

Jasper Loy

I did not get Apostol's Calculus 1 and 2, but each is also about 200 dollars.

Balarka Sen

oh i said semilocally path connected instead of semilocally simply connected

my bad

Ted Shifrin

I actually have two copies of Apostol I, II. I bought one set and got the other as a desk copy, but I don't remember how.

Mike Miller

And no, the example I have in mind is even locally simply connected.

Jasper Loy

8:00 PM

Of course, I can get the paperbacks at one tenth the price, but I do prefer the hardbacks.

Mike Miller

(locally SC is harder than semilocally.)

Balarka Sen

You're talking about a hawaiian earring covered by hawaiian earring joined with a bunch of lines and that covered by a totally messed up space

Jasper Loy

But I think Springer should do something about its binding. Supposedly many of the hardbacks don't last very long.

Balarka Sen

aren't you?

Mike Miller

No, I'm not.

Balarka Sen

8:01 PM

@MikeMiller oh right

@MikeMiller ok, then i am interested in your example.

Mike Miller

Generally when I say "I'm not doing X", what I mean is "I'm not doing X".

Ok, @Balarka. Then you can find it in Hatcher.

Ted Shifrin

One reason I gave up on publishing, @Jasper ... My linear algebra and multivariable books fall apart. When I went to Wiley for the multivariable, I made a scene about how bindings were crummy and falling apart. My editor said, "Oh that won't happen with us." Well, they've had to replace over 20 books for my students because new books fell apart. I'm fed up.

Balarka Sen

Ugh that book Mike.

It'll take me ages to find it.

It's so huge.

Mike Miller

It's somewhere in the section 1.3 problems.

Jasper Loy

I think Lee is not checking his email on Christmas Eve. I don't think he would ignore me, lol, considering I am such a big fan.

Mike Miller

8:03 PM

Which is the section on covering spaces, which is why you should look there.

Ted Shifrin

He deserves a vacation, @Jasper.

Jasper Loy

@TedShifrin I think you can spend your next few years writing even more books, like Lang.

Ted Shifrin

No, @Jasper, not going to happen. I may LaTeX up my graduate diff geo courses, but I probably won't, because hardly anyone would care.

Jasper Loy

One good thing Lang did before he died is that he got all his books to be done by Springer. That makes sure they stay in print.

Balarka Sen

Lang's book are... eh.

Jasper Loy

8:05 PM

Next time, I will publish my books with either Springer or the AMS.

Balarka Sen

Wiley's books always fall apart @Ted

Jasper Loy

Wiley books are very very expensive.

Mike Miller

@Ted Interesting question. Orientability of a hypersurface is the same as having trivial normal bundle. So look at the associated sphere bundle of the normal bundle. If it wasn't trivial there'd be a loop on $M$ (the hypersurface) that lifted to something nontrivial in the sphere bundle. I'm not sure how to finish yet.

Balarka Sen

You'll feel sorry for my copy of Dummit-Foote.

Ted Shifrin

That question was Lawson's gift to me when I first got to know him, @Mike.

Jasper Loy

8:06 PM

@BalarkaSen Is that also Wiley?

Mike Miller

I gotta go, so I'll finish it before next we meet.

Balarka Sen

@PedroTamaroff Congratulations. You're a mod now.

Ted Shifrin

Happy holidays to your family, @Mike.

oh, @Pedro is here? Merry Christmas, mod-in-training.

Jasper Loy

@BalarkaSen I once got a copy of Dummit and Foote but I sold it to someone else because I did not like it.

Balarka Sen

Dummit-Foote is not quite that bad.

The theories are crummy, but exercises are good.

Jasper Loy

8:08 PM

@BalarkaSen Well, I have a weird habit here. I only like reading algebra books where rings are defined with 1, lol.

Balarka Sen

Well that's nothing compared to me not pretending even existence of noncommutative rings.

Pedro Tamaroff

@TedShifrin Hello!

Balarka Sen

Merry Christmas @Pedro the mod.

Pedro Tamaroff

@BalarkaSen You're one observant feller, eh? =)

Jasper Loy

@BalarkaSen Well, even matrix rings are noncommutative. You definitely need them.

Ted Shifrin

8:10 PM

Ah, @Pedro and @DanielF finally all in blue.

Balarka Sen

Yes, two more mods join the chat.

Ted Shifrin

@Pedro, you still gonna talk with us li'l people?

Balarka Sen

He's going to forget us.

Ted Shifrin

That might be well-advised.

Lord_Farin

Does anyone know since when the "Related" section on meta has disappeared? It's a stupid change IMO.

Balarka Sen

8:13 PM

Gah I want more exercises for alg top. Munkres's exercise sections are puny.

Ted Shifrin

That's what Hatcher is for, @Balarka. Or you can look at Massey's book.

Jasper Loy

@BalarkaSen You got the latest edition for Munkres's Topology?

Balarka Sen

2nd edition, sure.

Jasper Loy

If you get Massey, make sure it's the latest edition too. He wrote 2 books and then combined them into 1.

Balarka Sen

@TedShifrin Hatcher is too huge and presumably, from what I see, is too advanced for me.

Jasper Loy

8:14 PM

@BalarkaSen How can it be? You are a genius?

Balarka Sen

I am not a genius, I keep saying that.

Ted Shifrin

Our grad students say the same thing, @Balarka.

Balarka Sen

I am a slow guy. Can't grasp that much stuff so quickly.

Pedro Tamaroff

@TedShifrin Well, of course.

Jasper Loy

Bredon does more things with manifolds than Hatcher does.

Pedro Tamaroff

8:16 PM

But now I'll be a Ted busier with flags and stuff.

Jasper Loy

But neither does classification of curves and surfaces, which is done in Lee.

Ted Shifrin

Don't forget to keep learning math, @Pedro. This is what worried me.

@PedroTamaroff a Ted busier, huh

Balarka Sen

haha interesting pun

Jasper Loy

I don't like puns. I only like stupid jokes that nobody else likes, lol.

Pedro Tamaroff

@TedShifrin I am reading math right now,

Unworry!

Ted Shifrin

8:17 PM

LOL

Balarka Sen

@TedShifrin Well I am not a grad student plus Algebraic topology is way too advanced.

Ted Shifrin

Then go back and do more basic mathematics, @Balarka. Duh.

Balarka Sen

I have done algebra...

Jasper Loy

@TedShifrin Lang wrote a book called Basic Mathematics, lol.

Balarka Sen

the problem is that i just can't handwave in algebraic topology

Ted Shifrin

8:18 PM

OK, I'm outta here ...

Jayesh Badwaik

@PedroTamaroff Congratulations. Glad to see one more mod in the chat. :-)

Pedro Tamaroff

@TedShifrin Bohoo.

Studentmath

8:35 PM

@Pedro! You're blue!

Pedro Tamaroff

@Studentmath Yes. Modding makes you sad.

teadawg1337

Being sad makes me sad.

Okay, I REALLY should not try to type when I'm extremely sleep-deprived....

Alizter

Go to bed

teadawg1337

@Alizter But it's 3:00 in the afternoon...

Don Larynx

@teadawg1337 drink

teadawg1337

8:43 PM

@DonLarynx But I'm not 21 yet....

Alizter

@teadawg1337 Sleep who cares?

Ted Shifrin

@Pedro Sigh.

Don Larynx

@teadawg1337 oh ok, have fun with that

Balarka Sen

Feynman wearing MSE icon!!

user153330

@Venus still nothing :(

teadawg1337

8:49 PM

I created a relatively simple sequence today

3, 15, 13, 9, 12, 10, 18, 7, 14, 19

Jasper Loy

What is the pattern?

Don Larynx

no

don't say it

teadawg1337

Why would I share that?

Don Larynx

please

thx you

Jasper Loy

Well, I am not really into patterns.

Don Larynx

8:50 PM

smacks jaspee

@Jasper why are you here?

Jasper Loy

@DonLarynx I am here to celebrate Christmas.

Daniel Fischer

@Lord_Farin It didn't disappear, it just takes some time to populate. After five minutes or so, it should be there.

Don Larynx

Oh. Merry Christmas @Jasper, and may a bunch of Ho Ho Hos be in your favor!

teadawg1337

Anyone who can figure out the pattern to that sequence gets a cookie, and anyone who can find the 1257th term of it gets a free hug

Jasper Loy

@DonLarynx So, are you going to apply for grad school?

Lord_Farin

8:52 PM

@DanielFischer Oh, I see now. Thanks :).

Ted Shifrin

Do you get a hug, @teadawg?

Daniel Fischer

@Lord_Farin De nada.

Jasper Loy

Someone just sent me a very encouraging email. It means a lot to me.

Don Larynx

@Jasper: No, I want to program

Jasper Loy

@DonLarynx I see. Going to become a programmer then?

Ted Shifrin

8:53 PM

JL, @Jasper

Jasper Loy

@TedShifrin No, lol. JL is not exactly a friend for now, lol.

teadawg1337

@TedShifrin Of course, I don't post this sort of thing unsolved. I came up with that sequence myself and can name the 432607th term

Ted Shifrin

you just said "someone."

Jasper Loy

@teadawg1337 You sure it is correct? Don't waste our time if it is wrong.

teadawg1337

@JasperLoy I am sure

Jasper Loy

8:55 PM

Anyway, I had KFC for dinner just now.

Ted Shifrin

I'm always bugged by these IQ test sequence questions, @teadawg ...

Don Larynx

@Jasper: No

@Jasper: I am becoming a programmer.

Jasper Loy

@DonLarynx I see. Then you should not be in this room, lol.

Don Larynx

@Jasper: I am completing Project Euler, and I am taking 5 math classes next semester.

Ted Shifrin

More than 2-3 math classes at one time is usually a mistake.

Robert Cardona

8:58 PM

@DonLarynx, are you computer science major?

Don Larynx

@Ted: If you recall, last year I took 5 math classes.

Transcript for

$Mathematics$ Mathematics