We are not amused...

more Pythonisms :-)

@t.b.: I think it makes a good bored emoticon, too.

Not that I am bored of MSE.

More like , ho hum, where else would I be...

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 would append the answer to mention that the question was changed and perhaps how the question was different.

So that your answer still makes sense.

see here for the solution I chose

so he added the "without zero points" in response to your answer?

Yes.

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.

Actually, I edited the edit in: see here

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

as long as you don't write that upside down...

the upside down site is down or slow :-(

lmao

good for me, because that would cause *me* headaches

Well, I have to go AFK for a while to walk the dog, get dinner, etc. be back in a few hours.

see ya

See you, catch some fresh air!

Then I get to finish proofreading the answer.

@t.b.: fresh air, what is that?

Oh, is that German? Air from outside, nice clean air from the woods, and so on.

Or don't you have that in CA?

Helps preventing headaches

@JM Are you around?

morning!

@t.b. Now I am.

morning guys!

@JM I was a bit confused about your comment on that cube question.

@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.

Really, but it can't be that hard to write down a great circle whose endpoints you know.

Oh wait, which question?

bobobobo has two sphere questions.

that one

the answered one

Ah. The tricky part is the "equal area" constraint, I'd say.

Isn't that area-preserving thing they do much stricter? (I have only read the abstract)

They actually did the more complicated matter of mapping a grid into a sphere.

I think what they want is a map that preserves the area of all subsets of the faces

@tb ...

tease!

oh well, i guess i'll read that one in the next life :D

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...

( Here's the preprint version.)

:D

(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)

@JM: seen that one for the gallery? As I told Asaf already, robjohn is definitely missing.

That was quick! It took Asaf months to get that. People just like you I guess. :)

Apparently! :)

With rob, that may be because he may have a number of starred comments but only starred by relatively few people.

(i.e. < 10 users)

That's possible. But I think "glad my wife doesn't read that" was upvoted by 6 people...

So I figured...

If it's those same six people who kept starring his lines, then no, he's not qualified yet.

Sure.

Hey, I think I've reached my personal culmination point rank-wise on the all users lists. Let's see how long that lasts...

What did you mean by "culmination point"?

Wait, wow. You've overtaken Pete?

yeah turns off bragging mode

The rep spacing between users seems to become logarithmic as you go higher...

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...

Ah, Didier has never lost inertia, I would say.

Well, he was not so active during summer.

Okay, there's that... but still, he almost always has a good haul the past few days (not counting his blockbuster).

Sure, his answers are almost always pretty good. I've been pretty slow recently. Answering the wrong questions, I guess.

I wouldn't want to say "wrong". Remember our talk about the bike shed?

Yeah, sure.

wow, the fundamental theorem of algebra DOES have a purely algebraic proof! 8)

Most of it was pretty easy stuff and nothing particularly inspiring. I'm having some nuclear plants prepared, but they need some further ripening.

now i just have to double time lang!

@Alexei: what do you mean

which proof, Alexei?

well, i was told that the fundamental theorem of algebra cannot be proven using only algebra w/o topology or analysis

My history is failing me now... how did Gauss do it, then?

@JM i know it uses galois theory, i'll have to read lang some more to describe it any more precisely

wiki says gauss's proof was mostly geometric in nature

@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.

@Alexei: the closest I know to an algebraic proof is in Milne's notes on field theory. I'm pretty sure Lang does use something along similar lines.

what do you mean by 'the closest'?

unless we're actually using something non-algebraic apart from the definition of R, it counts as algebraic in my book :)

Milne uses that positive real numbers have square roots and that a polynomial of odd order has a real root

oh

the latter cannot be proved w/o MVT, i take it

Yes, but if I remember correctly Lang uses that as well. It's a classic.

So: Aren't we talking about this proof? It's the same as the one in Milne

(I'm fresh out of drinking water. I'm stepping out to buy some, so see you two in a few.)

@JM See you later!

yep, that's the one

later!

well, i haven't read the proof yet, i'm just excited that it exists :)

@alexei: seen this thread on MO?

no

@Alexei: As I said Lang uses the order structure, square roots of positive numbers and existence of real roots of odd degree polynomials

the thread is great 8)

yeah it contains some awesome proofs

Oh, that thread. The proof Donu posted is a nuke, but I still like it... :)

@JM no longer thirsty, I hope :)

people are strange...

Hot day+weekend = long queue.

Unfortunately I live in that section of the city where drinking water has to be bought instead of being obtained from the tap.

By Jove... he can't even claim that he wrote at about the same time...

I thought so. Can the tap be used for cooking or only for cleaning and showering?

Cleaning and showering, yes. For clothes, I have to use detergent instead of soap. I've lost track of what the minerals in the water are...

@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.

Aha. Thanks Mike. I might be able to grab it from the nearest library this week.

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.

BTW, that product calculus bit is terribly interesting. It hasn't been accepted by a journal yet?

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.

Oh well. I'm now wondering if there's a fractional version or a q-version. So much math, so little time...

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.

I'd have done it the same way (and sadly I've already upvoted it, so I can't give another one).

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.

Not sure why.

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.

I was planning on fiddling with it anyway after dealing with our paper. Still, thanks. :)

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.

"The" state university - there's only one?

And yeah, there are "lower-tier" (whatever they mean by that) state universities.

"the" would be the University of the Philippines. Actually, it's sort of a (rather silly) idiom: there's the "state university", and the "others".

What primarily distinguishes the "others" from the University of Philippines?

So far as I can tell, the research in the other places isn't as extensive as what's being done in the University of the Philippines.

Most of the science-related funding by the government (which is rather miniscule to begin with) is poured into there.

Oh, sure. And they tend to have better equipment, too. :)

(Probably the high tuition fees at work.)

Better funded, I guess? :)

From the high tuition, I suppose. :)

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?

Or they go to Europe or America for their graduate degrees. ;)

Yes, they are.

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.

No worries. :)

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.

Getting back to math: Mike, are you familiar with Riordan arrays?

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?

ooh, a loxodrome

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?

uh, sure, I guess..

Check this out for references on Riordan arrays, especially #54: Sprugnoli, "Riordan arrays and combinatorial sums."

Again: "so much math, so little time." :D

Sure. :)

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.

What's difference between mathematical induction and complete mathematical induction ??

@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.

@MikeSpivey I feel that both are essential same I can't see the difference through the definitions

@MikeSpivey Can you give an example which demonstrates??

@anon: I see Victor's recent edit somehow made your answer look out of place... :(

do we really have appleboys' stack? :)

i think it's clinical :D

@Alexei: You see that ad, too? :) I don't have an Apple so I just imagine those things...

i have about 7 apples

mom says they're good for me

seriously, i don't feel like relearning how to copy and paste stuff and in what direction <del> works just to feel superior for no reason

@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.

nice observation: if we pull back $1_Y: Y \to Y$ by $f: X \to Y$, we get a projection from the graph of $f$ to $X$

Anybody with group theory expertise around?

Did you say set theory without the axiom of choice?

I should read what's gone on tonight. I had a nap after dinner, and then finished proofreading quantumelixir's answer.

Because if you did, that's the strangest name I have heard for it so far.

@J.M.: nice gravatar!

@Asaf: You're feeding me words, man. Feed me chicken instead. :P

@J.M.: Sure, come to Israel and I'll make some chicken.

@rob: thanks. You want the parametric equations?

@J.M.: watch out, he may shower you with the chicken.

I'll be prepared for that possibility.

@J.M.: I see, they are orbits.

@rob: That's a dirty thought. Besides, I am not showering guys... only chicks :-)

sure, what are the equations?

It's a clothoid, which I tried to embed into a sphere. Turned out pretty good, I think.

So I should use Plot3D?

ParametricPlot3D[] :)

ParametricPlot3D

yes :-)

what range of t?

Actually, any symmetric range ought to do. That one, I used {t, -5, 5} .

It looks like a face in the sphere :-)

Is it an attractor for some process?

Snazzy, eh? One of those things I did in Mathematica during my teens.

And I thought I was a nerd :-)

I did all sort of reverse engineering hacks :P

@AsafKaragila you hang out on MSE and expect to be the only nerd?

@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.

with the accumulation points antipodal.

@rob: Of course not, but I didn't expect for someone who's older than 25 (I think) to be playing with Mathematica in his teen years... :-)

I'd be lucky to have had Mathematica in my teen years. Personal computers were still a few years off.

Okay, looking from the timestamp of my file, I apparently did this in '98.

I think I did it in school.

(It wouldn't be until I was out of the university that I'd get my own computer.)

@J.M.: in any case, it is pretty cool.

It's hard to see it is a curve in a 128x128 icon.

:D It does look illusory when shrunk, no?

You can't tell where the antipodal points are.

Ooh, I just looked at your profile to see the 128x128 version. That is cool.

I was just looking at the 32x32 one on the chat screen

gives me an idea...

If this idea is to kill your pets, then it is probably not a good one.

No, just a curve similar to J.M.'s but with simpler functions.

Simple functions?

Are you gonna integrate it?

I think I have a "fake spherical clothoid" in my files somewhere. Lemme check...

Ah, turns out it's in here.

The only thing is that you need a bigger range for smaller values of a.

If a is large, the whirls are too shrunk to enjoy.

a=1/4 and t from -50 to 50 looks fine when I tried it out again.

{Cos[2 t], Sin[2 t], t}/Sqrt[t^2 + 1]

It is a spiral on a cylinder projected back onto a sphere.

Ah, there you go. :) I remember finding the accumulation points too wound up...

Again, I come up with something only to be beat to it by Wolfram.

I tried Sinh[t] instead of t to spread out the accumulation point, but it was too spread out.

Okay, I need to step out for a while. See you guys in a few!

{Cos[2 t], Sin[2 t], Sinh[t]}/Cosh[t]

@j.M. later

I am going to the office now, so I too will be back later.

have a good walk?

drive?

Walk. I don't have a driving license ;-)

ride?

okay have a nice walk then.

how far is it?

Also I live about an 8 minutes walk from the campus.

I used to live about a mile from Fine Hall when I was a grad student

So I used to walk about 4 miles a day

My previous apartment is just the other side of the road from the campus, I would go 7 minutes from my office to the apartment.

This one is somewhat further, so it takes me about 10-15 minutes.

It would usually take me 20-30 minutes to get from apartment to department.

Fun.

Anyhow, I gotta scram. Ciao!

later

{Cos[2 t], Sin[2 t], t/(t^2 + 1)^(1/4)}/Sqrt[t^2/(t^2 + 1)^(1/2) + 1]

@Gortaur Hi

@Ramana: hello

@RamanaVenkata: hi )

@robjohn good night )

@Gortaur:sup?

in the sense of good morning

@robjohn sup? what is it

How are you guys??

I finally finished proofreading quantumelixir's answer. Hard to follow.

What time it is there

It is 3:21 AM here

comsi comsa. I just realized that I cannot login on my laptop from home if I turned off it

and I'm also not sure that login within the university domain will help

@Gortaur: a shortcoming of remote access.

@robjohn: hz

@RamanaVenkata What time is it there?

3:52 PM

@Gortaur Hertz?

one more antipod )

12.5 hours off

@robjohn oh, sorry - I was full of thoughts. hz - shorthand for Russian 'somebody knows, but not me'

Have you heard of the book Topology James Munkres

aha

I have not

I have heard of it, and even seen it

I have not looked inside of it though

)))

were you scared?

I was

we fear Topology

:-)

I am trying to read it

not really

:)

Is it easy to read?

Not so easy for me

@robjohn who are that 'we'? )

@RamanaVenkata who are you as a student and what are you studying?

@Gortaur It is a "royal" we. "We fear change" is used often, especially by Garth in Wayne's World

I am 1st year undergrad

I want to take Math as my major

Hey, what a coincidence, I took math as my major, too! :-)

I have to take my major in 4th semester

@robjohn don't pick on her )

@RamanaVenkata: check it out.math.ucdavis.edu/~hunter/book/pdfbook.html

I apologize, just a bit punchy at 3:30 AM :-)

Evidently, I've already looked at Chapter 8. It is marked as downloaded.

@RamanaVenkata: for the first time I've studied a bit topology from that book

perfectly written I would say

thank you I will read it for sure

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 )

Yeah its absolutely true

@Gortaur Sage advice. Harder topics are most easily learned when you have someplace solid from which to hang new information.

merci

don't know how that edit got messed up.

I even read all the history to understand what has changed

btw

@Gortaur history on what?

on your message

ah

one more advice. Yesterday I've decided to make notes it latex not to forget good ideas. Again, maybe everyone do it but me

so, I've created a file say 'read.Lee-topology.tex'

and there I will write my questions and things which I need to check and clarify in other books or MSE

I downloaded the book. I plan to read it.

otherwise it's hard to bear all of the in the head

I too downloaded it just now

@RamanaVenkata: Lee Introduction to Topological Manifolds?

I even cannot reach my e-mail, so I will send a letter about typo to Lee on Monday

I just downloaded a photocopied PDF of Munkres.

@Gortaur I believe that that is a definite typo.

@Gortaur Absolutely true keeping good questions on mind is very difficult

@Gortaur No The pdfs from link which you gave

ok, I see

@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

I do all work on loose sheets it is much more complex to keep track of them

G'day from central Europe!

@robjohn: how's your headache?

@tb hi, how is the weather?

Freezing but the sun is shining. So: more or less acceptable

add 3-4 degrees and you obtain Leiden )

Leiden in German means suffering...

Das alte leid

from my Rammstein past ;-[

Oh, Rammstein. Do they still exist?

@tb I finished proofreading quantumelixir's answer. My head hurts much less, now, thank you :-)

they do. last album from 2009

I even liked it

it means that mb I'm not so old

as I used to think of me

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).

@robjohn: glad to hear that.