Mathematics - 2011-11-21 (page 1 of 3)

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12:08 AM

I love the smell of basis in the morning.

Henning Makholm

I'm reading Kunen. It surprises me how much its pre-TeX typography bothers me.

2

I know. I hate reading in Kunen for this exact reason.

I always prefer the "Millennium edition" of Jech...

Henning Makholm

Among Knuth's many contributions to mathematics, the cmsy font design may be the most influential of all.

Hah.

Well, I have to sleep. Tomorrow is a long long day.

Ciao!

Henning Makholm

@AsafKaragila Hey, I bought it specifically on the strength of your recommendation of it as "wonderful".

Night.

12:23 AM

Content wise it is :-)

I have learned in my day that many people can go beyond the crummy fonts and actually read this book. I also learned that this book contains many important things, sometimes in a greater detail than Jech.

Which is why I do recommend it to people. Personally, however, I have troubles even with Kanamori's The Higher Infinite, whose font is new but still somewhat annoying.

Anyway. Goodnight and all that.

@Srivatsan Sorry, we went out shopping for things, including new LED Christmas lights for the front of the house.

Just a sec, there is an easy way to handle said series...

12:43 AM

@robjohn Oh, sorry. I didn't realise the OP wanted to find an explicit closed form for the sum of the series.

@Srivatsan I just posted an answer.

I was like: why does the OP ask for the generating function when that series itself is the generating function. =)

Yes, saw the answer.

@robjohn: where does that sign discrepancy between your and joriki's answer come from?

@tb He made a mistake somewhere... I just commented, but I am looking for where it is.

Plug in z=1/2 and you get a positive for the series and negative for his.

Henning Makholm

@Srivatsan Isn't the point of generating functions supposed to be that you find a closed form that you can manipulate algebraically? If you keep the explicit summation around, then why bother?

12:55 AM

@HenningMakholm Well, I understand the point of it. The question does make sense.

What I am trying to say is: If I gave the question to a student, and the student just writes out the same series as the answer, I will award him full points. =)

@tb I think that perhaps he took the derivative of 1/(1-z) and forgot to include the factor of -1 from the chain rule applied to 1-z.

d/dz 1/(1-z) = 1/(1-z)^2

@Srivatsan // It's just that I did not realize that a closed-form expression is what's being asked.

@robjohn: I can't spot the mistake. The derivative seems to be fine. Wolfram agrees.

But I agree with your comment, too -- Guess it's getting late...

@tb No, there's a sign mistake. Check the two denominators: (1-z)^5 vs. (z-1)^5.

@Srivatsan As I said: I guess it's getting late :)

1:11 AM

@tb =) I thought you were going to sleep.

@robjohn I corrected the sign in the post.

1:42 AM

@tb Wolfram agrees with no minus sign is what you're saying, right? The answer you linked to has a minus sign, but (z-1)^5 instead of (1-z)^5.

@Srivatsan Thanks. I hope Joriki is okay with it.

It's pretty quiet here tonight.

2:09 AM

@robjohn Hope so.

Henning Makholm

It's called commutative algebra because it's full of commutators.

Does anyone have their copy of Ahlfors in front on them? I have a quick question.

What's this hatted version vs. hatless version of L_K...?

Henning Makholm

It's from the Dirac formalism for describing angular momentum in quantum mechanics. The hatted L's are Hermitian operators that each define an inner product on the state space such that the norm of a state is the expected angular momentum in a given direction.

I think the non-hatted L's are either the measured values or a classical analogy -- or ought to be, but neither choice ought to lead VVV to a nonzero value for the commutator.

@Potato Shoot

@robjohn Yes, Srivatsan pointed that out to me...

2:22 AM

So, as a mathematician, what is the minimal required knowledge of differential equations I should have (e.g. for say, the Putnam (although the Putnam is not important, but I'm finding it hard to think of when a non-specialist would need DE knowledge))? I feel like a buffoon, because I only know how to separate variables and solve first order ODEs via integration factors (and I fumble my way through even that because I have to re-derive the solution method on the fly...)

@t.b. I figured it out, thanks though.

Henning Makholm

Correction, the squared norm of a state is the expected angular momentum. And it's not even really a norm, so just a plain old Hermitian form. Sigh.

Also, this may be specific to my university, but do you think it would be advisable to take advanced measure-theoric probability without a formal undergrad course in the topic? Would everything be retaught from first principles, or would prior knowledge be assumed?

Anyone? These are obviously too localized to be posted as questions.

@Potato I would ask whoever teaches that course at your university and see what he/she says.

Yeah, that's what I was thinking. Any thoughts on DE knowledge?

@Potato I can't tell you how much mathematics I've forgotten that I once knew, and that includes some differential equations. One of the things about being a mathematician, though, is that if you find you need some knowledge that you don't have, you just go teach it to yourself.

So I wouldn't say there's a "minimal" amount of knowledge of DEs that's required.

As far as DEs in other areas, I've seen them used before to give slick solutions to probability problems.

2:35 AM

I see. Let me be more specific. Do you know of any Putnam DE question that has involved more than separation of variables?

@Potato I don't pay much attention to the Putnam, so I can't help you there.

@Potato What does Putnam measure? Isn't that comparable to the Olympiads (to be filed under "nice to know, very slick, I'd be unable to do it, but I'm a mathematician anyway?")

I intensely dislike the Putnam myself. But there are some professors (who believe in a strong correlation between Putnam ability and math ability) who are putting a little pressure on me to do well on it, and I'd like to familiarize myself with its syllabus. Unfortunately, I don't know what exactly they expect in the way of DE knowledge.

For the record, I think the correlation is only there because the people who do well on the Putnam are exactly the people who have had extensive math olympiad training in high school, and of course the people who have put in intense math practice in high school are going to be good at math! The whole idea of innate ability is overrated, in my opinion.

This question is related to the discussion at hand.

@HenningMakholm Oh thanks. I was editing some posts and didn't notice that someone answered my question here.

2:48 AM

@MikeSpivey The example of Perelman winning an IMO gold medal at age 15 is not exactly heartening...

Here's a thread on the IMO. From leafing through a few of the Putnam problem sheets, it seems to me that you should rather focus on having your real analysis and multivariable calculus and basic algebra (mainly linear algebra and modular arithmetic and a little group theory) and a bit of Euclidean geometry ready. Doesn't seem to involve too much knowledge of differential equations, but some basics should be there, I guess.

Gotta run. Good luck on the Putnam, Potato. Catch you later, tb.

See you, Mike!

Bye, Mike.

@Potato: everything helps on the Putnam. Speed helps since it is a timed test. Depth helps because some problems are really helped by a knowledge of specialized methods. Computational accuracy is important because inaccuracies can mess up an answer badly. But rest and relaxation helps me to concentrate and keeps me from freaking out, so get rest and find some way to relax before the exam. Just my views.

3:01 AM

Honestly I just need to work more problems. But that's quite difficult to do with classes going on.

Working problems will definitely improve depth, and the practice will help with speed and accuracy. Do what you can, it will help.

wow, now that's a success story on math.SE

3

@tb Those questions are not even downvoted. :)

@tb I am in the middle of this coffee shop and I am laughing uncontrollably & from time to time, thinking about that user.

Enough to make one blush. :-)

3:17 AM

@tb That is pretty spectacular, not getting any up or downvotes.

@Srivatsan Somehow the user name (Japanese was auto-selected, I know it's Chinese) reminds me of what I got when trying to understand your profile.

Melon seed melon seeds obtained give beans beans (Japanese)

Well, mine is a tongue-twister indeed. // Google does a poor job of translation.

His/her profile could be as well.

Tree, so melon sown (Chinese)

@Srivatsan Indeed, but it is entertaining to see what it comes up with.

What it should come up with is in my profile as well. In case you're interested... :)

3:26 AM

Just out of curiosity: Here is a list of standard complex analysis qual questions: math.princeton.edu/generals/complex.txt . What is a good reference for the Riemann surface/Analytic continuation questions they ask? I don't think Ahlfors covers all of that, and Miranda's book (which I am reading) seems to be more focused on algebraic geometry.

Also, does anyone have experience with this book: amazon.com/Complex-Analysis-Variable-Raghavan-Narasimhan/dp/… ? It seems to contain a lot of exercises and generally be cool. I'm thinking about asking for it for Christmas.

3:41 AM

@Potato I don't know a good source in English for that, sorry. Narasimhan's writing is usually great, but I don't know that specific book. Concerning your previous questions on Putnam: did you see this thread?

@t.b. Yes, but I was looking more for a reference than additional questions.

@tb The streak is gone. :(

@Potato I think most of this is in Ahlfors. But you may want to look at Remmert's books vol 1 and vol 2 (for the mapping stuff vol. 2 is better, for analytic continuation, I think it's mainly in the first part).

Some people also like Conway's books on complex analysis (also Springer GTM).

@Srivatsan I guess the seeds grew in the wrong direction :)

@t.b. @Srivatsan I zeroed him/her out again, for my amusement.

Him or her? =)

3:49 AM

@Potato Ahlfors is where I think I learned that stuff.

I don't remember what generals questions I had.

So my proposal wasn't good enough?

@robjohn Ahlfors is such a horrible book to learn from though :(

@Potato I guess it depends on the reader. I thought it was okay.

@tb The present title doesn't quite make much sense, does it?

@Potato I think it's not easy to read, but definitely worth the effort. I guess you've seen this thread on textbooks, too?

3:53 AM

@robjohn You are probably the only person I know who has expressed that opinion (this surprises me!). The chair of my department likened it to "a long walk through Siberia."

@t.b. Stein, Shakarchi and Lang are what help me through Ahlfors when I get stuck, but they both lack Riemann surface/analytic continuation(I think) stuff.

(Nullology) tag is sorely missed. =)

@Potato Oh, the chair can't have been serious. Ahlfors is a gem! Challenging, but, as I said, definitely worth the effort.

@Srivatsan Escape brackets in links, like so: [\[asdf\] jkl;](http://...) this gives [asdf] jkl;

Ok. Thanks... // This time, I was afraid of missing the edit window, so I wanted to get something right.

t.b. Oh yes, and there are definitely views and approaches in there you can't find elsewhere (that hideous proof of Sterling's formula, for example!). But if I had to use it as my sole source I would be hopelessly lost.

I have not read this book but I have heard good things and I had Krantz for real and complex analysis. He is an excellent teacher and I can't imagine him writing a book that is less than excellent.

3:59 AM

robjohn - You're a WashU grad? PhD or undergrad? We can do this more privately if you wish, but uh, there is some connection here...

@Potato Nope, he taught at UCLA when I was an undergrad there.

Ah, I see.

@robjohn I haven't read it either, but if it's just a little bit like the book on the implicit function theorem then it certainly is a very good recommendation.

@robjohn Um, that must be a little while back. =)

@Srivatsan somewhere between 1977 and 1981 :-)

4:02 AM

@robjohn Nope, I don't know these years existed. :)

:)

@tb drat, I missed it :-(

@robjohn I just was amused for a second that there's a book dedicated to a theorem. But on second thought, I said I wasn't too surprised.

Too bad I don't know why this theorem is considered a big deal.

@Srivatsan Thanks. I saw tb smile and it's nice to know what I missed.

@Srivatsan I only started to seriously appreciate it when I started to learn about differential geometry. But how about this: suppose f: R^n --> R^n is smooth, suppose |f(x)| -> \infty as |x| -> \infty and df(x) is everywhere invertible. Then f has a smooth global inverse.

When I first learned about the implicit function theorem in my first year analysis course it seemed sort of ... contrived.

4:15 AM

@tb is this the inverse function theorem?

or a cousin of it

Are there folklore stories of proofs with bugs/gaps associated with inverse/implicit function theorems?

@Srivatsan It's called the Hadamard-Cacciopoli global inverse theorem. The basic inverse function theorem only tells you that an inverse exists locally, so this is much harder. Note also that z -> e^z on C is onto C \ {0} and satisfies the conditions, but there is no global inverse.

I am essentially wondering if this theorem is subtle/difficult.

@tb Huh, now it doesn't make sense anymore :)

- which is good, I hope.

The difference is in converting R to C?

No, I removed a point from the image.

and I made a terrible mistake |z| \to infty doesn't imply e^z \to infty of course

(but too late)

:)

I am not checking either :)

may be, we should go to sleep. :)

@Srivatsan It's not terribly difficult, at least not in its basic version. It doesn't have a history of horribly flawed proofs, anyway certainly not comparable with the fundamental theorem of algebra or the Jordan curve theorem.

4:24 AM

Um, actually, I just stopped short of these two theorems in my recent reading.

I will finish it and then come back.

You should definitely have a look at them, they are nice.

The trouble is I do understand that these are quite non-trivial theorems, and I do follow the proof somewhat. But other than the obvious exercises, I do not see where they might be applicable.

It's just another theorem for me. That's why I guess I am intrigued when it's talked about.

@Srivatsan One standard application of it is the Lagrange multiplier theorem.

Um, I will look at it. Never seen a proof of it before.

4 hours later…

8:20 AM

@Srivatsan: Another amusing typo: "explaination" : ) It's kind of cute.

But it only appears 7 times.

7 times where?

on mathSE

8:31 AM

Hello

9:21 AM

Please vote to reopen here

@AsafKaragila: please, please, please start reading the comments before voting to close. I find this over-hasted voting to close seriously annoying, and it's not the first time that you don't check.

Ah ! I am pleasantly surprised and cheerful after reading this math.stackexchange.com/questions/4846/…

@tb The Lp or the dense subset thingie?

@AsafKaragila The product measure "thingie"

9:36 AM

@tb Ah. Very well. I voted to reopen.

I'm almost tempted to replace the brackets...

How do you add a link to a text in chat @J.M.

@tb The last of the jury has spoken

@robjohn Thanks.

@tb It's now up to the executioner ;-)

9:56 AM

@RajeshD [text](url)

I already did jury duty; I don't see any other executioner...

10:17 AM

@robjohn What has to be executed? :-).

@JonasTeuwen Too late for the party, as usual :)

:(.

fedja definitely is in a combative mood, these days. I had quite a chuckle at Communication styles are different. I prefer to cut things short sometimes, that's right.

(last post)

@tb Hah! :-).

@JonasTeuwen But here's one waiting for its execution

10:31 AM

"Why submit to the top journal if they are going to be snotty?" - why, indeed...

and another one

Done!

imgur.com/gallery/fwigs

user image

(for convenience ;) )

Oh, neato :]

10:38 AM

@JM Yes, it isn't nice to get snotty reviews. On the other hand, as a reviewer it's sometimes very hard not to be snotty. Especially if the author is a medium sized or even a big shot and the contents of the paper is completely trivial but the thing was submitted to a top journal.

For some definition of "trivial", of course.

Would something like: "The result is nice, but it is not suitable for this journal" suffice?

Sounds nice; even better when appended with "because..." plus a reference.

I wish I could once quote Littlewood in a review: "The object of this paper is to prove (something very important).' It transpired with great difficulty, and not till near the end, that the `object' was an unachieved one."

Cool! :D.

10:53 AM

There's also the famous review by Truesdell "In this paper are presented incorrect solutions to trivial problems. The basic error, however, is not new."

Wow! I should check that out.

Jonas M. posted the MR-link in the MO thread.

And of course I copy-pasted the wrong quotation here's the correct one: "This paper, whose intent is stated in its title, gives wrong solutions to trivial problems. The basic error, however, is not new [...]"

There's that variant of Murphy's: "any conversation involving errors will eventually have the participants committing some."

@JM especially when it comes to ortografy and grammer

I would really love these chat places to be filled with more people!

11:18 AM

hello

Are we supposed to talk only about mathematics or we can chat on various topics too?

anything you want

Bring something up. If it's interesting enough, we'll reply.

I don't really have any interesting topics on my hands, but I am kinda bored atm, so I'll ask: what music do you usually listen to whilst working on mathematical problems (or programming, if you are a programmer)?

11:38 AM

Unless I'm doing computational problems, I rarely listen to music while solving problems. Heh, I have a hard enough time doing them when it's quiet, so 80s hardcore doesn't help.

What about you, Daniil?

Music... depends. Sometimes I want background, and then there are those times where the only things I want to be hearing are my thoughts.

I mostly listen to simple piano music like Satie. I tried working while listening to "aggressive" music, but it just did not work out.

12:36 PM

You call Satie simple?

1:04 PM

@AsafKaragila Which SMBC comics did you have on your office door?

1:40 PM

What is the 2d thorus without one point homeomorphic to?

Henning Makholm

Another torus without a point? An infinite plane with a handle? Two cyclic ribbons glued together to meet each other in a flat crossing?

I distinctly recall somebody making an animation of turning inside out a torus with a missing point...

@JM Do you mean this:

user image

Henning Makholm

Yeah, but that is homotopy. Mere homeomorphism to itself inside out doesn't even need a missing point.

Hmm, interesting. If the surface has no cuts, how does the inside-out operation go?

Henning Makholm

1:54 PM

A homeomorphism is just a map -- it doesn't come with any intermediate stages.

So... it's a trivial map, then... :)

@JM It's the abstract torus S^1 x S^1. Just reverse the orientation of one of the circles. It doesn't extend to an ambient homeomorphism if the torus is embedded in R^3

I can imagine that. Thanks.

@Matt Thanks. That's a cousin of "pronounciation"... :)

so did you ever use homotopy in number theory?

Henning Makholm

2:00 PM

Note that the animation exchanges the two circles rather than reverse one of them in-place.

@QED I didn't, but for example Voevodsky did.

Eat the bitter pill and say "I don't know" :)

Now I think that the "real" question is part 2.

The real question is always the one you haven't managed to solve.

Those brackets are floors, no?

yes

@Henning: have you finished reading Dudley?

2:12 PM

@HenningMakholm ok, thanks. I thought it would be something more beautiful

Henning Makholm

@Gortaur I'm not claiming my answer is exhaustive :-)

@JM Yup.

What was the chapter you liked the most? :)

Which one's the Dudley?

@HenningMakholm is it like your comment about the box topology? )

And your torus reminds me of chocolate :), @JM.

2:19 PM

@Srivatsan I chuffed to bit your torus )

@Srivatsan: this book

@Srivatsan Exactly my intention... :) (It took me a while to figure out how to color the torus realistically...)

@JM It looks very nice.

Hi Gortaur

@Srivatsan Hi

Henning Makholm

@JM I didn't compile a ranking, but one I found most heartbreaking was "Money to be made from mathematics, lack of".

@Gortaur It's like everything I write that's not about core computer science topics.

@JM I thought you guys were up to something serious. Like this one. // I should've known better ;)

2:23 PM

@HenningMakholm Oh, that was a downer... apparently he didn't realize that Gauss was sponsored.

I have -> one has

Henning Makholm

I was about to reject that one, but came too late.

@Srivatsan even stricter than M. Hardy

@HenningMakholm The other correction was good: indeterminant -> indeterminate.

Sometimes I change formatting to that which (in my non-perfect opinion) is more readable. Is it also too much? Usually it goes together with necessary corrections

2:29 PM

@Gortaur I don't think it is too much but that's my opinion again.

I usually "correct" the formatting of any page that I touch for other reasons.

Henning Makholm

Hey @Gortaur, your gravatar is back to the coffee splotch.

These people are jobless or what?

This question came up here: math.stackexchange.com/questions/84237.

@Srivatsan that one even new at MSE

@HenningMakholm give me two days

I think Henning must give an answer that covers all possible 4-tuples =)

2

should I use [algebraic-topology] tag here?

Henning Makholm

2:32 PM

Well, I like the coffee splotch.

@HenningMakholm nice ) though I don't (

meaningful sentence with brackets put in a wrong order. I hope, meaningful

Henning Makholm

@Srivatsan Well, the image is the same old, so I didn't even read the percentages in the text.

@Srivatsan closed

Thanks, boys.

It took a while, though :)

Henning Makholm

2:38 PM

Hey, when you close these duplicates, remember to upvote the dupe such that the OP gets privilege enough to upvote the excellent answer he's being referred to :-)

@tb but it was only by guys from this chat

2

Q: If you choose an answer to this question at random, what is the probability that you will be correct?

Possible Duplicate: Does this question even have an answer? http://i.imgur.com/qvzU4.jpg Question: If you choose an answer to this question at random, what is the probability that you will be correct? What is correct answer for: A) 20% B) 40% C) 80% D) 100% E) 40%

falsely closed

the problem he posted requires an essentially new idea

to solve it

B and E are the same thing, no?

@QED A tip in case you didn't know: Typing a bare link in the chat window produces a preview. Adding some text before the link suppresses the preview. Good for the eyes =)

E.g.: math.stackexchange.com/questions/84237/…

Sometimes I wish there was an option to shut off the auto-expansion...

2:42 PM

I posted "None of the answers are correct." before reading the question

I wonder why it says "1 answer" even though I deleted it

Ah, the software's syncing can be weird sometimes...

What's the new idea required? // What's the final answer?

Henning Makholm

@QED The number shown in the "___ answers" heading is exactly the number of answers the software shows to you, whether or not they are deleted.

@QED I don't see a new idea required here and how it is "different" in an interesting way. Can you elaborate?

Henning Makholm

@QED It seems to be essentially isomorphic to the "is the answer to this question 'yes'" archetype that I discuss in my answer.

2:46 PM

@HenningMakholm, that's what I thought at first but if you try to solve it using the same method it breaks down

Henning Makholm

Which "same method"? 0, 20 and 40 are all internally consistent answers, and intuitive logic offers no way to choose between them, except perhaps to exclude 0.

On the other hand, if 20 and 40 are both correct, then the only true answer must be 60, which is not among the options, so the answer must be 0 ... and, ta-dah!, that is truly consistent. Blah blah blah.

I can't even formalize the question

let R be a random variable that uniformly chooses between A,B,C,D,E. Then the question Q is compute P(R=correct answer to Q).

Q = <...Q...> isn't valid recursion

Henning Makholm

@QED Which is exactly the same as the 25-40-60-25 question, right?

no

Henning Makholm

What's the difference?

2:50 PM

25-40-60-25 are excluded at the first step

I am wondering if this question is meaningful or not

i.e. can we express it formally

Henning Makholm

How are they "excluded" in a way that doesn't apply just as well here?

84

Q: Does this question even have an answer?

I found this math "problem" on the internet, and I'm wondering if it has an answer: Question: If you choose an answer to this question at random, what is the probability that you will be correct? a. 25% b. 50% c. 0% d. 25% Does this question have a correct answer?

probability faq

@tb This might have come up earlier. How to get a link to a particular comment?

Woohoo, my office mate is a master of inequalities!

He says "It's a multiple-choice variant (with bells and whistles) of the classical liar paradox" but something like Godels liar statement is actually formal

Can I ask this question on the site: How do I formalize this <the multiple choice thing>?

Henning Makholm

2:55 PM

The classical liar paradox is not the same thing as Gödel's statement.

@Srivatsan I use the SE modifications, then you can link to a comment by right-clicking on the time stamp and copying the address.

Henning Makholm

@QED You can ask, but I predict it will be closed as a duplicate as the existing question that asks the same thing.

where is the question that asks the same thing?

Henning Makholm

The one you just linked to.

that's different

Henning Makholm

2:56 PM

And the answer is: No, such things cannot be formalized.

IT'S THE FREAKING SAME THING

You seem annoyed again

Henning Makholm

Damn right I am.

I would appreciate you to be a bit more patient with slow people like myself

@tb Thanks tb. Quite cool.

Henning Makholm

Repeatedly asserting they are different without actually showing any difference is not going to make it so.

2:58 PM

Okay. QED, which parts of the new question are not covered by Henning's answer?

An enumeration might be easier to address.

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