12:00 AM
@Potato Gerry is answering a different question here, but what he has said looks applicable to you.
(or at least from the third paragraph.)

You mean the joke?

Including the joke. :)
Meanwhile I'm stepping out for a few. Later.

I should be heading out too.

12:48 AM
It's quite quiet in here...

I'm busy doing math!
If you want to chat, perhaps we could talk about study tips for math. I'm at the point where my terrible study habits and focus are starting to hurt me.

Oh, I was just looking around...
Can you expand on "terrible study habits and focus"? Do you mean you're jumping around too much?
Meaning: topic-wise.

No, not topic wise. I'm not too good at concentrating for long periods of time, I guess. My mind wanders.

I see. What does a long period mean? More than 15 minutes, more than an hour, more than two hours?

After maybe 10 minutes I start fiddling with music, chatting in here, that kind of thing. Distracting myself.

12:55 AM
I understand. Why do you need a computer nearby in order to learn? I mean your notes, maybe a book and some sheets of paper should be enough, no?

That is a good point.

I mean: try to reduce the possibilities for distraction: (I'm absolutely the wrong one to say that, but a clean desk works wonders). Where do you work, at home, in a library in your room?

In my room.
A college dorm room.

Here's something that helped for me when I was having similar issues: different places for different things: library for reading books, my desk for reviewing my course notes, a desk in the hallway for writing stuff and a coffee shop for discussing things with my friends. In some sense, different surroundings helped me get into the right mindset for these different activities.

Ok, that sounds good. Here's another question: how do you deal with reading dense books? I've gotten through Ahlfors ok by just reading normally, but I'm also reading Miranda's book on algebraic curves and Riemann surfaces, and it definitely isn't something I can totally comprehend on a first reading.
Definitely it's a good idea to always complete the verifications "left to the reader" and to do the problems, but what else? I'm finding it hard to learn all the new definitions and theorems.
Although perhaps it's just an issue of volume, and I need to go slower?

1:10 AM
Going slower certainly helps. I tend to write a lot when a book or a paper is going too fast for me. I try to summarize the page I just read, formulate things in my own words. I try to work out some examples, come up with my own, and so on. It is hard to say, what exactly I do.

Yeah, I think it's a matter of just being a more active reader.

Probably. Always ask: what are the hypotheses used for? How does this assumption enter the argument? What is the crucial point of the proof, what do I need to remember in order to re-prove that result? What is just standard technique, what is new to me?
Try to think of examples that illustrate the statement of a theorem: can you see what is going on in an easy special case?

Ok, yet another question. How does one study for advanced exams (say, quals)? Specifically, I know the ideal is knowing the statement and proof of every theorem by heart, but there are certain proofs I feel I just can't commit to memory (or simply that they are unimportant.) Two examples: Ahlfors's 6-ish page proof of Sterling's formula, and his proof of Hadamard's factorization theorem.
Of course it goes without saying that once one knows the material by heart that the only way to prepare is to doing problems until you pass out.

Sterling's formula? Is that Stirling, the asymptotics of the Gamma function?

Yes. My apologies for the spelling error.

1:23 AM
Did you have a look at what Remmert does for these two theorems? His two books usually give excellent accounts of various ways of looking at things. Here's volume 1 and here's volume 2. Basically this boils down to: look up the results in other books, think about what points other books emphasize.
If one proof doesn't help me a lot, I need a different angle of looking at things. If I find a place where things are presented the way I like it, I can then go back and see what the other author emphasizes and thus get a more complete picture.

Lurking here finally pays off. Nice to learn from t.b.

@t.b. Well, I guess what I am asking is, is it really necessary to know how to prove those two theorems by heart, especially when it would be ridiculous to ask a student to prove them on an examination?

Well, one could certainly ask for an outline of the proof, I believe. There are a few main ideas that one can try to isolate. I mean these six pages aren't six pages of pure calculation, they certainly are divided in some natural steps. Try to partition the proof in such a way that it looks natural. This takes a few hours to do, but if for some reason you know that this theorem is considered important for your exam, you probably need to really grasp these ideas.
Hi Tim!
Thanks.

1:40 AM
Hi tb. Thanks! Learning from right person is important.
Learning from right books is also important.

1:58 AM
I agree and also act like this: "I tend to write a lot when a book or a paper is going too fast for me. I try to summarize the page I just read, formulate things in my own words."
I agree but am not able to act like this: "I try to work out some examples, come up with my own, and so on."

2:59 AM
@JM So, I managed to catch you before I leave. It's a quiet day, today. Except for turkeys, it seems.

Back finally. (Sorry about that. =))

Finally?

So it seems, so it seems. This trip of yours, t.b.: business or pleasure?

The Indian restaurant near my school features Tandoori Turkey tonight. =)

Pure pleasure. I'm going to visit an old friend of mine in Vienna. Train leaves in a few hours and I still have to pack my stuff :)

3:03 AM
@Srivatsan ...that is a strange way to treat turkey.
@tb Oh yeah... no passports needed for EU. Nice!

@JM That's how they celebrate Thanksgiving. To each her own...

@JM That's extremely convenient, yes.
Although Switzerland isn't part of EU, but part of the Schengen Agreement, so you don't really need a passport, yes.

I can't believe I haven't asked you this t.b.: does your flat have a good view of the Alps?

@JM Had you not corrected it yourself, I was going to reply no to that. ;)

No, I'm on the wrong side of the Zurichberg, but from the city (especially ETH) there's a beautiful view of the alps. Let me look for a picture.
Hm. Somehow my other browser's hung, it'll take a moment, sorry.

3:13 AM
You don't have to inconvenience yourself for me. :)

You know, you don't remember where you've seen pictures of what you see every day :)

Yeah. If you had asked me about what the view from my place looks like, I'd have a hard time looking for pics too.
Oh well. I'll take your word for it.

So the view from your place, @JM -- is that good?

No. It's a vast expanse of city... I don't live in the province anymore.

That's about the best I could find, it's lousy:

3:22 AM
What's this water body?

Zürichsee = Lake Zurich?

That's "lousy"? :D "Wonderful" must be really good, then.

It's incredible, I just can't find anything good. This postcard looks more like what I was looking for (a photo would have been more convincing, though):

Very nice.

Indeed.

3:29 AM
The other picture is good, but you can't really get an impression of the alps. Above is the view from the Dolder Grand Hotel (the thing in the circle) and it really looks this way.

I'll definitely visit there when I can afford to travel again...

And here's what I was looking for:

Wow! // You took the photo, tb?

Really beautiful.

@tb You're right... the first picture is lousy...

3:35 AM
@JM The buildup was was worth it... =)

@Tim Looks like some filter was applied, but I assure you it can look this way.

Yes, I just wanted to ask if it was photoshoped
But beautiful indeed.

@Srivatsan No, if I had taken it it wouldn't have taken 15 minutes to find it. My hard disk is a mess, but it's not that bad :)

Also feel chilly.

Spring, April or so, I'd guess.

3:39 AM
There is still snow.

Yes, but these mountains are rather high. The famous mountains are always covered in snow.

@tb, How do I flag the moderators about merging two accounts?
See here (the question and the answer): math.stackexchange.com/questions/85131/… // I suppose I can flag the question...

@Sri: Here's what I do: I flag the post, and say something like "Please merge account with (url of the registered user's profile)."

What JM said.

Are you sure the same user wants his accounts merged?

3:43 AM
Same Gravatar... that's a big red flag.

So it is not up to the user but some rule of the sites?

@Sri: it's not worth it if the dupe accounts are all unregistered, though.
@Tim What do you mean? Cookies can always get lost, so no surprise that an unregistered account can be spotty.

I don't know what unregistered means... =)

I mean: can a user have more than one accounts?

@Tim Yep. But if possible, they should be merged.
@Sri: Let's use math.stackexchange.com/users/20023 as the example.

3:47 AM
Oops. I never understood the point of unregistered accounts.
But thanks for pointing it out. I will look for it the next time.

If you're a drive-by asker, I see the point of unregistered accounts...
"I'm not really gonna join, I just need the services of you guys." Something like that.

Um, ok. Then why do we allow unregistered accounts?

Because of that, however, even if you accrue rep from unregistered accounts, there are always things you won't be able to do as an unregistered user, even with the rep. Like upvoting.

Cool. Thanks, JM.

Can somebody illuminate me as to what is being asked and answered here. I can't make head or tails of both question and answer.

3:59 AM
=) no clue

That I can't visualize what OP is saying is hampering me...
What would the measure look like, then?

I don't know, it could be that the OP was asking for different Riemannian metrics and the associated arclength or volume notions (that's probably what gary interpreted, but I can't follow the answer either). Oh, well...
Do you guys agree that this is a dupe of this? Changing the integrand from x^2 to x^3 doesn't change anything important, and there are two very good answers to the x^2 case.

bloody early 6_6

Hi Ilya

Hi Ilya.

4:12 AM
Good morning, Ilya.

@Matt didn't get you
Hi guys )

@tb I agree.

@Srivatsan second it

Yeah, Panagiotis can pick things up from the answers to the other thread.

In fact, they might even be homeworks for the same course.

4:14 AM
Robert and Chris gave sufficient detail for coding an algorithm.

I need to fry some sausages and drink some tea. Crazy Russian breakfast without wolf milk

@JM Chris's answer is awesome, I think.

But one second. What's the "opposite variables" method? @tb

@tb Agreed. Now if it were a cartoon... :D

@Srivatsan I'm pretty sure they are. Kostas keeps suggesting edits adding a (homework) tag for Panagiotis's questions :)

4:16 AM
@Srivatsan Not standard terminology, I'd say.
For "hit and miss", I've also seen "method of darts"... :)

@JM google finds hit and miss in the right way. It does not work with 'opposite variables'
I bought a green banana 2-3 weeks ago, no it seems to dye in my fridge without getting yellow. Can it happen?

@tb What makes you say so? I checked the three questions of Panagiotis: 1, 2 and 3. Not touched by Kostas.

@Srivatsan Well, they were rejected suggestions.

Oh, ok. =) I forgot about them.

Let me grab from the mod panel...
@Ilya Yeah. Depends on the banana.

4:21 AM
@tb so it's not true? (((((

@Srivatsan there should be links to the other suggestions here at the bottom of the page

(By the way, I found this old unanswered question, tb. I was reminded of what I was telling you a few days back.)

@Ilya Well it is true )))))

@Srivatsan Here is one.

@JM Hilarious =)
@Ilya Banana dye?

4:24 AM
@Srivatsan indeed it did

@Ilya: A green banana should be kept near ripe bananas if you want a quick ripening.

@JM: here's another thing I'm scratching my head about: this suggested edit, the deleted answer by the same user here. Vote the existing answer down and vote for deletion? Flag for the mods?

@tb I still believe that Switzerland is like in Milka chocolate commerical: with green fields, violet cows and funny beavers. I think, Matt would comment on cows
@JM so how should I obtain the first ripe banana to start the process?

@tb I'm not quite sure why Qiaochu didn't already do a merge of the question and answer...
@Ilya That's why you're supposed to get bananas in bunches. :)
Not all the fruits ripen at the same rate.

@JM in the shops there are bunches of green bananas. In the university they feed us with ripe ones

4:29 AM
The fruits help each other ripen through the gas they exude.

@JM Me neither. But apparently OP wants to get rid of what's written in the answer, no?

A bunch of greens is alright. That's how I buy my bananas.
@tb I'd flag if a second vandalism attempt happens.

@JM football fans?

@JM Okay, agreed.

No, that post t.b. was showing above.

4:32 AM
@JM my old laptop almost failed to compile those formulas )

@Srivatsan Well, I guess that's why there are books about it :) Seriously, though: Does it still seem impenetrable?

yeah, that was odd a bit

@tb Not impenetrable. I was just reminded of that conversation, that's all. =)
I haven't thought about it since.

my trip begins, so see you all someday

So Ilya and t.b. are off to their respective breaks...

4:35 AM
See you, @Ilya. Have fun.

@Srivatsan thanks! )

@Ilya See you! Enjoy

See you, Ilya!
@tb Wow, Banach is actually quite handsome...

@JM I recently found that a homepage for him is being developed. I have absolutely nothing against smoke, but the cigars smell through his pictures 75 years later
2
@Srivatsan did you do the flagging after all?
(answer's now deleted, but accounts aren't merged)

4:45 AM
Oh, I did. Waiting for review.

Okay. I see.

Not sure if I thanked you and @JM for your input. Thank you both. =)

i.e., Mariano was reading questions before checking for mod flags...

pedja is quite a fan of Bill's... I distinctly remember that anon's answer was accepted at some point.

I think so. Unfortunately the new interface doesn't seem to allow for seeing unaccept activity...

4:55 AM
@JM It does. But I haven't figured out under what circumstances.
(maybe if the accept happened on one day and the unaccept on another?)

Odd. One of my answers was unaccepted yesterday (but was accepted after some quick back-and-forth comments). I didn't see the -15 on my profile, though.

@JM Would you mind linking to it? I'd like to check something.

@tb Here.

@tb That answer was a bit heavy for this question =)

@JM So you can see the unaccept here, but there's none recorded in pedja's question.
so probably I misremembered

5:04 AM
Ah, I forgot that timeline thing. You'd think there'd be a quick way to access it from questions...
@Srivatsan Well, Bill insists on "pedagogical correctness"... (whatever he means.)

That's one reason why I use the SE modifications. You get two additional links under every question: timeline and history.

@JM Usually it's ok, but to say "divide by 3"... =)

I was quite amused by the back and forth here.
OP accepted instantly, I intervened, then JDH posted a cool answer, but after all, I seem to have addressed what OP was really interested in :)

@Sri: I just noticed, you might be able to join the 10k-club in a week or so...

@JM No, 1 week is too short a time for me. =)
Plus I am not too keen on pushing it.

5:12 AM
It's high time for that :)

@tb Why? Am I troubling the 10k+ users with requests? =)
But I will be there. By next year hopefully.

@Srivatsan I appreciate your contributions very much, that's all

@tb Er, to clarify: that was just an weird attempt at humor.
In any case, let's see how much time the climb takes.

If you keep your current pace, you'll be there in two or three weeks, I'd say.

We should start a betting pool on that. // What's JM's estimate?
@tb On a serious note, that estimate seems fair enough. I have a similar guess.

5:25 AM
Wow! Henning already has 338 answers. In two weeks or so he will have overtaken me...
And another Wow! Wouldn't have guessed that!

Henning has this propensity for blockbusters... how does he do it... :D
@Srivatsan Okay, my pessimistic estimate is ten days.

@tb Not clear. Who is this user?
Certainly doesn't appear too involved in MO/MSE.

@Srivatsan Well, I guess I've confused him with this famous guy.

Tiny question about the chain rule proof that came up in my reading: Suppose we want to show (for complex functions) that (g(f(z))'=g'(f(z)*f'(z). Use the definition of derivative and write the limit as w goes to z of (g(f(z))-g(f(w)))/(z-w) = ((g(f(z))-g(f(w)))/(f(z)-f(w)))*(f(z)-f(w)/(z-w) and let z go to w. But it could happen that f(z)=f(w) for w near z, in which case we are dividing by zero. How do we fix this?

@tb How probable is that. Very same name and all... =)

5:38 AM
@Potato ...which makes (g(f(z))-g(f(w)))/(f(z)-f(w))) indeterminate, no?
So you have the product of two indeterminates.

Indeed. The text notes there is some way around this but I am not sure what it is.

I'd say if z\to w, then f(z)\to f(w) ... assuming continuity of course.

Right, but the point is that, for example f(z) could be constant, or it could be that f(z)=f(w) an infinite number of times in each neighborhood, or something.
in which case dividing by f(z)-f(w) is bad

What do you think is the difference of this from dividing by z-w in the other indeterminate factor?

What do you mean?

5:43 AM
I mean, why do you not have the same complaint about dividing by z-w?

Because z-w is never zero when you take the limit
but it is possible that f(w)-f(z) is zero for a point in each neighborhood around z

"z-w is never zero when you take the limit" - wait... if z\to w, why are you saying z-w can't be zero in the limit?

I'm saying that when you take the limit you never actually divide by zero - you just divide by arbitrarily small epsilon>0.

@Potato: Don't divide then, use the definition via best linear approximation instead |f(z+h) - f(z) - df(z)h| = o(|h|)

@t.b. I'm aware that's the standard way, but this book says there's something way around this technical issue and I'm curious what it is.

5:49 AM
That's what I'm saying. The other thing is hand-wavy and will involve something a la de l'Hôpital (so assume what is to be proved).

You really don't see an elementary way around this? The book strongly hints there is one.

This is still Ahlfors, right Potato?

I'm flipping through the sample pages of "Complex Made Easy" now, on a whim

What's not elementary about writing it the other way? If you're talking analytic functions, you can of course exclude the accumulation points by the identity theorem.

I fully understand that's the "right" way to do it - that's the proof of the chain rule I originally learned in Rudin. But it seems there's some trick I'm missing here that I would like to learn, for educational purposes.

5:53 AM

(http://books.google.com/books?id=-eNiTAAioRUC&pg=PA7 is a nice direct link.)

Also, another question about the problem at the bottom of the page: I've never formally taken a course in multivariable analysis (I'm correcting this next semester!) but are the partial derivatives at 0,0 of that function just 0 and 0, which are constant functions so obviously continuous? But this would imply it's complex differentiable, which it's obviously not.

I guess the intention is this: if f(z) = f(w) infinitely often then the numerator of the left hand side will be zero infinitely often, so the derivative of the composition must be zero.

@t.b. Ok, that makes sense. Thanks.

6:02 AM
@all hi all

@t.b. Could you give me a precise explanation of why the partials of that function (bottom of page 7) fail to be continuous? I can only see it intuitively.

If you apply Cauchy-Riemann directly, you end up with differentiating constants, no?

You get 0 for everything.

@Potato The precise statement is this: If the partial differentials exist in a neighborhood of the point 0 and are continuous at 0 then the function is differentiable in 0. On the x-axis the partials in y-direction don't even exist, similarly the partials in the x-direction don't exist on the y-axis.

@t.b. Ah ok, that makes much much more sense.

6:09 AM
hi @tb : i have a question requesting some reference

Another easy thing I'm missing: Why are the only maps T(z) such that T(z+w)=T(z)+T(w) and T(zw)=zT(w) for z,w in C multiplications by a complex number?

@RajeshD: hi, shoot
@potato: well, it's a C-linear map and thus representable by a 1x1-matrix

Consider a unit circle in a complex plane (Argand plane). Any point on it with phase $\phi$ has real and imaginary parts as cos(phi) and sin(phi) which are projections on to real and imaginary axes respectively.
I need a similar interpretation for a unit circle in Quaternion plane....(i guess its a 4-sphere)......i wonder how it traversing along on the 4-sphere and how are the projections on the real and hyper complex axes look like
I ned a good reference explaining all this
perhaps a reference with down to earth explanation is good for me
@tb

@RajeshD You know how to take the norm of a quaternion?

Well, write (x,y,z,w) then (x,y) and (z,w) are points satisfying x^2+y^2 \leq 1 and z^2+w^2 \leq 1, so (x,y) and (z,w) are points on the unit disks of the (x,y)-plane and of the (z,w)-plane, respectively, no? The interval [-1,1] = {t : |t| \leq 1} in the reals is replaced by the unit disk {(x,y) : |(x,y)| \leq 1}.

6:18 AM
I am very new to this notion.......but i guess i need a reference for it...i can go through it without problem

Think of the 3-sphere and its projection to the (x,y)-plane if that helps.

@RajeshD so yes, |a+ib+jc+kd|=1 would be your usual hypersphere.

ok

As for references, I must say that I don't know of any good ones (at least not in English). Maybe J. M. can help.

what about phase of a point on it

6:20 AM
Unfortunately, there isn't a neat equivalent of phase/argument for quaternions.

Oh
@JM and @tb I am reading Quaternion Fourier series/transform and i need a good visualization for analog of $e^(-jwt)$ in quaternion plane

The pure quaternion part (the ones with i,j,k), you can treat as equivalent to usual three dimensional space. (That's actually how vectors started out.)

i guess there will be two such phasors

@RajeshD There isn't a "plane" for quaternions. 4-space, sure.

ok

6:23 AM
(and I hope you appreciate the difficulty of visualizing in > 3 dimensions)

yes impossible

I didn't say "impossible". I just said "difficult".

i mean mathematical visualization.in terms of equations and not physical
@JM that was for me

Thanks for the help J.M.!

Do I get any useful material on this on clifford algebra books ?

6:26 AM
@Potato and t.b. too. :)

Yes and t.b.

@RajeshD Unfortunately, Clifford algebras are a bit beyond my expertise.
Quaternions, however... let me check my lists.

I wouldn't recommend looking at Clifford algebras to be honest.

ok

(Sure, it's a beautiful thing, but somewhat technical overkill for what you seem to be looking for)
One thing you need to be aware of: quaternions are a four dimensional real algebra not a 2-dimensional complex algebra: multiplication of quaternions doesn't interact well with complex numbers.

6:29 AM
yes i do not intend to use tham alongside complex numbers

@tb There was someone on sci.math who swore by Clifford algebras.

@Raj: Ah. Here's something recent: this one.
Hey rob!

Howdy!
@JM we are back from our Thanksgiving dinner.

@robjohn Oh, I like Clifford algebras, very much so, in fact :)

You aren't sleepy? :)

6:31 AM
f(z)=z^2 * sin(1/z) at z not equal to 0, f(z)=0 at 0 is not holomorphic, yes?

@JM actually, I had a small nap when we got back. That was a couple of hours ago :-)

my guess is that the projections would look like sin(theta)cos(phi), sin(phi)cos(theta), sin(theta)sin(phi) and cos(theta)cos(phi)....am i right

@RajeshD That would be somewhat like the equivalence of rotating with quaternions and rotating with Euler angles...

Unless you say what phi and theta are, it's difficult to tell.

@Potato It is meromorphic.

6:33 AM
@tb (I think he's talking spherical)

Why does it fail to be holo at 0? I'm not so good with my sin values off the real axis...

@tb i do not know what phi and theta are ....thats infact what i am looking for

Err nevermind.

@RajeshD That book I linked you to? There's an entire chapter on rotations and angles. You might want to see what you can glean from there and look into what's applicable for your quaternion Fourier analysis.

@Potato It is like e^{-1/z} on the imaginary axis

6:35 AM
@JM thanks very much

@robjohn Isn't that function actually going to have an essential singularity at 0? Because the laurent series is going to have an infinite number of power of (1/z)?

i hope i could find those books online somewhere
without paying ofcourse

As I just said (in correction to my previous erroneous statement)

Happy Thanksgiving, @robjohn. // Guess it's a little late, but...

@Raj ...or check your nearest library.

6:37 AM
@robjohn sorry, didn't catch that.

@Srivatsan Not late at all. It is still T-Day here for another 83 minutes.
1 min ago, by robjohn
@Potato It is like e^{-1/z} on the imaginary axis

my university library hasn't got any ...i need to check at adjacent ones........i am anyway excited to know about the existence of such concepts like quaternions

@Srivatsan It was very nice, I avoided turkey altogehter

Me too =)

6:39 AM
@Srivatsan Sure you do... :D

We had a very nice dinner in a British Inn atmosphere (Scottish Inn actually)

6:50 AM
@tb @JM i ran into quaternions when i was studying Fourier series/transform for functions(signals) of 2 dimensions............i came to the opinion that the formulation of 2-d Fourier series should be done in terms of quaternions and not in terms of complex numbers...using complex numbers in this case is a waste of time
there were some papers on QFT in signal processing literature

7:22 AM
Morning!
@Ilya I assumed you were trying to tell me that it was not nice to wall you (or anybody else in this room). Must've read too much into it then.
Nothing better to start your day with than having to clean up cat puke.
But hey: after tomorrow (once the dog is here) the problem will be sorted.
Not looking forward to 10 hours of driving tomorrow... : (

7:44 AM
10 hours?

Yes : ( We're picking her up from Germany.
The place is in Bavaria, right next to the Czech republic.

Sounds painful :[

Yes : ( Especially since I'm scared of driving : (
I'd rather fly : (

You can't take a train? // 10 hrs of driving sounds painful.

The train would take even longer. We were considering taking the train there and hiring the car just on the way back. But it doesn't make that much of a difference...

7:48 AM
@Matt "Her" refers to the dog, right? What do you call her?

The thing is, she's just a puppy and on the train we can't take a break
@Srivatsan: Well, we're still arguing. : ) I'll make sure though she won't be called "Aleph Naught".

It's Scylla and Charybdis. Long trip, or your driving discomfort?

Or both : )
It's all set now, we're going both ways by car.
10 hours without math.SE T_T