You only get major headaches from time to time by offering too generous proof-reading. ;-)
@robjohn: I never quite know how to react when someone changes his question I've answered. Should I just edit in the edit to mark it clearly as such, should I complain or just leave it at that?
I think that since it is marked as an edit and it is apparent that you answered the question without that assumption, things are clear. At least to me they are.
You could forestall anyone commenting "hey idjit, he said no zeroes" by adding a comment that the assumption of no zeroes was added in after your answer.
Ah
Then that is fine, IMHO. However, I have already been shown to have been voted down as far as my opinions this afternoon.
Well, I gave a lazy answer for a lazy question (I could essentially just copy and paste it from an earlier answer where I needed the square root as a minor step), so no harm done. However, I don't think it's proper to rip the answers out of the original context
@t.b.: Well, that those authors found it necessary to use the Lambert projection, it doesn't look as straightforward to split a sphere into six equal-area regions...
Those projection equations they derived are pretty... elaborate.
The reason I brought up the paper anyway is that the derivation they did proceeds to start with the face of a cube whose surface area is the same as that of a sphere...
(To segue: apparently we are now at the question influx rate where it takes me more than half an hour to look at stuff that was posted while I was sleeping...)
@JM: Yes, so I'm happy now requirement (6) is what I was looking for. Thanks. Note also that they're working in Cartesian coordinates instead of polar coordinates (and his angle bit definitely is a bit off)
I certainly won't close the gap to André and Didier is coming closer with great speed, so I guess that's one for my own personal gallery. Pete and Matt were pretty inactive recently, so if there are a few soft questions they want to answer, that was it with the fun in the second row...
@tb Oh, I know what you mean. I have about five or so answers where the (long!) draft took an hour to write, but shortening it and making it understandable to other people took weeks.
@J.M.: You were asking me about a reference for Bell polynomials the other day. Chapter 11 of Charalambides's Enumerative Combinatorics is on partition polynomials, and the Bell polynomials figure prominently. I wouldn't call it a comprehensive reference, but there is a lot in there.
No problem. I picked up that book on a whim at the Joint AMS/MAA meetings several years ago, and I keep finding new and interesting things in it. It's underappreciated.
No, and it won't be. I've given up on it. There was already a lot more out there than I realized when I first wrote up the paper. See, for example, the references on the Wikipedia page on the product integral.
Huh, I just got my second ever downvote on MO, for this answer. I don't think there's anything technically wrong with it, so I'm guessing that someone wanted the combinatorial answer to be bumped higher.
It's fine (and thanks for the upvote!). I suspect generating function arguments and combinatorial arguments are more popular in general than arguments that manipulate binomial coefficient identities. Not sure why I tend to prefer those... although I do like combinatorial arguments, I tend to stay away from generating functions unless I can't think of another way to do the proof.
I'm pretty green to combinatorics, actually. Coming from my experience with manipulating special functions, I'm more at home with fiddling identities than with puzzling out bijections.
But I do see the appeal of finding what a particular entity counts.
Back to your comment about a fractional version of the product calculus stuff... that rings a bell, although I'm not coming up with anything definite right now. If something does come to mind I'll let you know.
Random question: How does the university system in the Philippines work? Are the universities all government-funded? Or are there private universities? Is there a sort of tiered system, with a set of flagship institutions where most of the research is done and then lower tiers where the schools are more teaching-focused?
Not all. I graduated from the state university. There are privately-run universities (mostly run by the religious), and then there are the vocational ones.
So it's a bit of a pickle. If you need a rigorous grounding in theory, you go to U.P. If you need experience with the equipment, there's Ateneo or La Salle.
(That's for chemistry, at least. I have to admit I know little about the mathematics here.)
So far as I can tell, one thing people do is to have their undergraduate degree in one and their graduate degree in another.
Interesting. You said most of the private universities are religiously-affiliated. Are the more prestigious private universities religiously-affiliated as well?
Nearly all of the great private institutions in the U.S. started out as religiously-affiliated (in fact, they were originally founded to train ministers and preachers), but a few hundred years later have pretty much completely separated themselves from their religious roots.
Anyway, thanks for indulging my questions. I don't know much about but am curious about how higher education works in countries other than my own - how it's similar to and different from the way we do things here.
Now that you have me thinking... indeed, the secular universities are in fact the state universities. The religious ones are precisely the private ones.
There aren't very many religiously-affiliated institutions in the U.S. that have a heavy research component. Notre Dame is the only one I can think of that does.
Not as well as I should be. I understand they constitute a fairly general method for proving combinatorial identities - maybe not as general as Zeilberger's algorithm, but easier to understand, and I think they cover some cases Zeilberger's does not. Why do you ask?
Two not-very-related things: that other paper of yours that got accepted, and noticing that Bell polynomials can be specialized to Stirling numbers, binomial coefficients, Lah numbers... is there anything Bell polynomials can't do?
@anon: it's a (highly modified) clothoid, actually. :)
I don't know as much about Bell polynomials as I probably should, either... but check out Chapter 11 of Charalambides and watch how he makes them dance. :)
Tomorrow is going to be a long wait. :) Somehow, maybe I didn't quite fully understand Riordan's book, but it seems his Riordan arrays translate naturally to Bell identities. Now if I can only figure out how to translate...
@anon: do you want the parametric equations, by any chance?
The Sprugnoli reference (I think) attempts to be an exposition on Riordan arrays, so it might be worth reading if you're trying to learn more about them.
@Ramana: With induction the logic is "P(0) is true" plus "P(k) -> P(k+1) for k >= 0" yields "P(k) is true for all k >= 0." With complete (or "strong") induction the logic is "P(0) is true" plus "P(0), P(1),..., P(k) -> P(k+1)" yields "P(k) is true for all k >= 0."
Sometimes you need the latter when the induction step requires you to use the fact that P(i) is true for some i < k.
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@Ramana: The classic example is the proof that every integer larger than 1 is a product of primes. (Paraphrasing from Wikipedia) If m is prime then it is a product of primes. If m is not prime, then it is a product of two other numbers n1 and n2, both of which are smaller than m. If we know that every integer from 2 to n-1 is the product of primes (the strong induction part), then so are n1 and n2, and thus so is m.
If we only know that n-1 is the product of primes (if we were using regular induction) then we couldn't conclude that n1 and n2 are, and so we couldn't conclude that m is.
I hope that helps. Unfortunately, it's late here, and time for bed. If you have more questions maybe someone else can help out.
So you can see it at once in Mathematica : `{Cos[(Pi*(-FresnelC[t] + FresnelS[t]))/2]* Cos[(Pi*(FresnelC[t] + FresnelS[t]))/2], Cos[(Pi*(-FresnelC[t] + FresnelS[t]))/2]* Sin[(Pi*(FresnelC[t] + FresnelS[t]))/2], Sin[(Pi*(-FresnelC[t] + FresnelS[t]))/2]}`
It's a clothoid, which I tried to embed into a sphere. Turned out pretty good, I think.
@rob: I don't really know if it's something's attractor. All I wanted back then was to get the space curve that you'd get if you drew a clothoid on a piece of tape and taped it on a sphere.
not so deep for each of the subjects, but it's worth to read before taking the seriuos deep book
I beleive it's important to start with something nicely written, though not so focused. After you digest that information, proceed with more focused book. Obvoius advice but still )
@RamanaVenkata I've just recently saw a interview with Lynch made in Cannes in 2001 where he showed MD. He told that's very important to write all your ideas - otherwise when you forget one of them you want to commit a suicide
and I though that he is right. Making notes in notebooks is useless - I finish one notebook in 1-2 weeks and use 2-3 notebooks per time
The new album is three years old? I guess they don't need more money anymore. If it were not for the constant belching into the microphone a live show of them must be pretty entertaining, at least if you're into fireworks (which I'm not).
