Mathematics - 2011-10-21 (page 1 of 3)

Mike Spivey

00:01

In fact, I'm looking up properties of the Stirling numbers in it right now for edits to our paper, J.M.!

t.b.

@MikeSpivey re: keeping you sharp. Maybe you had this post by Matt E in mind?

Mike Spivey

@t.b.: I'm pretty sure Arturo said it somewhere, too, but, yes, Matt E.'s post expresses what I was trying to say quite well. I'm also glad that mathematicians of his caliber are willing to contribute their knowledge to this site.

J. M.

Reminds me, Mike... this is too little a question to post on the main site, so: do you happen to know of some book or paper with a comprehensive listing of Bell polynomial identities? The identities I know seem to be scattered among Comtet and Riordan...

Jack Schmidt

@tb: :-)

J. M.

("Bell polynomial" being the ones that show up in Faà di Bruno, not the ones that show up from \exp(\exp(t)-1))

Mike Spivey

00:11

@J.M.: No, I don't. There are some on Wikipedia and MathWorld, but they're hardly comprehensive.

J. M.

Also, I sometimes get tripped since I forget those old books use signed Stirling numbers of the first kind, not Stirling cycle numbers...

Jack Schmidt

In invariant theory, there are some minimal generating sets, and the few times I've used them, it seems like those generators are exactly the invariants you need to know in order to find the invariant you want.

Is there something similar for identities? Like is there some formal way in which only the following trig identities are needed (and practically, do you need more)?

(basically is there some way to define comprehensive as complete, or is there always another identity you might need?)

J. M.

For trig at least, I think the "basic set" would be the Pythagorean identity and the addition formulae. I can't think of an identity that can't be built up from those...

Mike Spivey

@J.M.: Lack of a standard notational convention irritates me to no end. Just yesterday there was a question that effectively involved Eulerian numbers, and I had to search for a site that used the same convention as in the post.

J. M.

Mike, if you ever manage a combinatorics textbook, could you make sure you impose the really good notation we like? :D

Mike Spivey

00:19

@Jack: Very interesting idea. Unifying combinatorial identities has got to be on somebody's list of big picture goals for combinatorics. Generating functions and the Zeilberger algorithm do some of that, but not all.

J. M.

(what question was that, BTW? I must have missed it.)

Mike Spivey

@J.M.: If I were ever to write a combinatorics textbook, it would probably be a pale shadow of Concrete Mathematics. The question was this one.

And now I've got to go - the wife and kiddos just got home, and it's time for supper. Nice chatting with all of you; I'll probably do this more often.

J. M.

See you! It was nice to talk!

Jack Schmidt

good night

t.b.

Good night, see you soon here!

J. M.

00:32

Jack: now that you've brought it up, methinks everything is expressible in terms of the sine (or cosine) function anyway. We just have the other functions for the sake of convenience.

(I'm not counting complex exponentials for the time being.)

Jack Schmidt

JM: One result I know is that R[ sin(x), cos(x) ] is isomorphic to R[s,t]/(ss+tt-1), so the only identity needed if you don't change arguments (angles) is the pythagorean identity. I definitely agree that in principle we don't need tan, sec, csc, cot.

JM: but it bothers me that the half-angle identities seem a little far removed from say sine's angle addition formula

I wouldn't call a trig identities page comprehensive if it just had pyth and angle addition, but I'm not sure really where to draw the line.

J. M.

Yeah, I agree with the last sentiment. On the other hand, for inverses, I think arctangent is way more convenient than arcsine.

Asaf Karagila

cough

J. M.

@Asaf: I hear your bread came out splendidly.

Asaf Karagila

Oh it was great.

The garlic made it better, but it was bigger and needed more time in the oven.

J. M.

00:42

So the situation looks somewhat asymmetric to me. I can't imagine defining sine in terms of tangent, due to the inconvenience of removable discontinuities.

robjohn

Again, it is time to perambulate with the pooch.

Jack Schmidt

i think maybe the weierstrass substitution led algebraist to define another algebra that represented trig functions even more easily (in terms of tangent)

J. M.

See you, rob.

I wonder who first called Weierstrass's substitution "magic"? It certainly is...

Jack Schmidt

it is also the same magic used to generate pythagorean triples

(i remember that now that i see the formulas on wikipedia, hehe)

robjohn

I used to use that substitution a lot, I first started using it in connection with the stereographic projection I used when writing a planetarium program (in the 1970s).

Then I saw how it turned a lot of complex integrals into rational functions.

Asaf Karagila

00:51

Wow. It's 3am

robjohn

One of my favorite uses is atan2(x,y)=2arctan(y/(x+r)) where r=sqrt(x^2+y^2)

J. M.

Heh, I've had to fake two-argument arctangent at one point... :D

Jack Schmidt

@robjohn that is crazy

robjohn

@Jack: so I'm a lune.

Jack Schmidt

@JM mine were always messy little case by cases

these days, i bet the extra floating point ops are faster than the ifs

robjohn

00:54

When converting to polar coordinates, you already have the r, so just throw it into that formula and you don't have to do a bunch of casing.

Jack Schmidt

oh yeah, true, so a single addition extra. that might have more or less always been faster

J. M.

@Jack: yeah, times have changed... I never thought I'd see the day when unrolling cases actually gave a performance penalty.

robjohn

I really have to perambulate, before it gets too dark out. BBL

J. M.

I need to leave for work. See you guys.

anon

01:18

In the context of number fields, can I think of ideals as basically "integral (O_k) multiples of certain integral (O_k) elements" (such as even numbers in Z) and fractional ideals as "integral (O_k) multiples of certain arbitrary field (k) elements"?

Jack Schmidt

01:36

@anon: more or less yes. not all (fractional) ideals are principal, but that is the gist

@anon: i think the more standard way is that fractional ideals are k-multiples of integral ideals

anon

Okay, k-multiples of integral ideals is easier to imagine. Is there else to the "more or less" thing I should be aware of, other than ideals may need to be generated by more than one element?

Jack Schmidt

@anon: no, i think that is it. If you allowed two k-elements and took O_k linear combinations of them you would be exactly right

(number rings are nice, all fractional ideals are "1.5 generated", you can choose one nonzero element of the ideal arbitrarily, and then there is always a second element that generates the whole fractional ideal as far as O_k-linear combinations is concerned, that is, as an O_k-module, or in the case of a subset of O_k, as an O_k ideal)

robjohn

02:27

@Mike: The answer by Peter Taylor looks like the Big Bang in reverse. :-)

Mike Spivey

03:29

@robjohn: Yes, it does, partly because of J.M.'s edit. :)

robjohn

03:57

@Mike: I see; he added \left( and \right) which a lot of newer LaTeXers forget about

it also tends to expand

Gortaur

06:34

@robjohn what is the cap?

t.b.

07:16

@Gortaur: Only twenty upvotes a day count, so accepts and bounties aside, you can't get more than 200 rep. a day from upvotes. If you get more than twenty points, you see that you earned upvotes, but you don't earn points for them... Mildly annoying, but as you said, "don't get focused on rep )"

Good morning, by the way!

robjohn

07:58

@t.b.: what happens if you get 21 upvotes and 1 downvote?

t.b.

@robjohn: downvotes still count, I believe. The next upvote then compensates it again

Gortaur

@tb: Good morning, Theo

@robjohn: Good 1 AM, Rob (am I right about your name?)

robjohn

Ah, the t. stands for Theo :-)

t.b.

@robjohn: confirmed by the higher-ups

robjohn

@tb thanks

Gortaur

08:02

@tb thank you for the clarification

robjohn

@Gortaur you can see that from my profile, right?

anon

t.b. used to have his fuller name as his handle

Gortaur

fortunately, there are no maximum number of 'thank you' that you can receive

robjohn

I never know what others can see in my profile, but from looking at others' profiles, I think you can see my real name

Gortaur

@robjohn: no, there is no your full name

as well as nor mine in my profile

at least on MSE - maybe on the other stack website you've put it

@tb: don't you mind that I used your name? I've done it automaticaly, so sorry if shared your secret, seriously

robjohn

08:04

This is the only SE I'm on. I guess it doesn't show for others. I entered it.

Gortaur

@robjohn then I don't know what is your name. I guess, it should be Rob or Robert

@RamanaVenkata: Good morning, Ramanta. Hope everything ok with open balls now

Ramana Venkata

@Gortaur good morning yeah I am fine with balls

robjohn

Since I've given my Flickr site, I think my name is there. But yes, it is Rob for Robert.

t.b.

@Gortaur: my name is no secret. I just tried to reduce the importance of the m.SE account on Google

robjohn

@Gortaur that might be a private matter :-p

t.b.

08:08

@robjohn: in case you're curious this is my MO-account containing my full name

Gortaur

@tb: but there is no link between those two accounts, right?

t.b.

what do you mean?

robjohn

@tb I have seen your full name before. Perhaps on MO. I don't have an MO account though.

Gortaur

so if somebody new on MSE looks for your real name to give you an offence

t.b.

Give me an offence? I dont understand

Gortaur

08:10

you never know (

Rajesh D

hi folks

Gortaur

hi

robjohn

Hey Rajesh!

t.b.

hi

robjohn

@RajeshD: what time is it there?

Gortaur

08:11

@tb ok, my point was that if somebody knows only about your MSE account - he may not know which account on MO do you have

robjohn

I like to keep track of what time it is for different people.

Gortaur

so he have no ideas about your real name

Rajesh D

@rob its 1:40 pm here in India

Gortaur

wow

exactly 12 hrs man

you're almost antipodes )

@tb anyway - sorry for being confusing. I'd like to say btw that I really enjoyed your discussion with Jack tonight: question you asked and answers he replied

I know not so much about this topic but papers he referred to are interesting.

t.b.

They definitely are.

I'm looking forward to reading them more closely :)

Gortaur

08:15

could I save your discussion in doc-file to read it when I will understand more about it? )

I'm afraid the time I will understand more I won't be able to find it in chat

@robjohn do you really think of me is such a way? 0_o7

t.b.

@Gortaur: was that a serious question? Just grab it, it's out on the internet

robjohn

@Gortaur I wonder if it would be considered 12 and a half or 11 and a half hours difference?

t.b.

@Gortaur: Now you have to explain this James Bond emoticon to me :)

Ramana Venkata

@Gortaur I came across this theorem: Let Y is subset of metric space X. E subset is a Y, is open relative to Y if and only if E = Y intersection G where G is some open set in X. My question is can I say X is open relative to itself by this statement??

robjohn

@Gortaur in what way?

Gortaur

08:17

@tb that was serious because I try not to lose my politeness

guys please, not all together )))

@robjohn private matter I mean, just open metric balls were meant

@RamanaVenkata just a second

robjohn

@Gortaur it was a joke :-)

Gortaur

@robjohn 11.30

@tb why it is a James Bond emoticon?

phew

robjohn

007

t.b.

007. So 0_o7 means serious?

Ramana Venkata

@Gortaur I have a class Now I have to go I'll cehck your ans later bye

robjohn

08:18

0_o7

Gortaur

@RamanaVenkata X is open by another reason

robjohn

looks like a wink to me

Gortaur

@RamanaVenkata Y is open by this statement if you take E = Y and G=X

@tb I never though about 007 in fact ))

so, first emoticon was 'amazed and surprised so that one of my eyes is bigger': 0_o

or its brother o_0

but then there is an emoticon where you're surprised, amazed and scratch your head with the hand trying to realize why: 0_o7

t.b.

aaah :D

robjohn

@Gortaur I didn't mean to put you in a bad light.

Gortaur

08:22

@tb yeah, let me show you one thing if I manage to find it

robjohn

0_o? looks similar but the ? marks the sense of query

Gortaur

@robjohn sorry for being paranoic

<_<

>_>

that were paranoic emoticons

robjohn

My son uses -_- to denote non-amusement

Gortaur

@robjohn but then you can think of ? as a part of the sentence, not of the emoticon.

@robjohn just one second....

xxx: hi, man
yyy: hi
xxx: I will tell you one secret now >_>
yyy: <_<
yyy: nobody's there, go on

t.b.

)))

robjohn

08:26

pair-a-noid when there are two of them :-)

Gortaur

)

there was also a sequence of waking up emotions, but cannot find the original message, smth like this

-_-

^_-

o_-

-_-

-_^

-_o

o_o

anon

This convo takes me back to grade school.

Gortaur

ah, that trial was lame

robjohn

trial?

t.b.

Some users fill their posts with emoticons I can't quite decipher...

Gortaur

08:29

@robjohn Did I miss some discussion about my nick this night?

robjohn

@Gortaur I don't think so, unless I missed it, too.

Gortaur

@tb yeah, that's annoying in the questions/answers

t.b.

@robjohn: trial : attempt, try, I guess

robjohn

why?

t.b.

I think you and Mike had a short exchange on Gorthaur/Sauron, LotR

Gortaur

08:30

))) so either you forgot it, or you don't want me to know it

robjohn

@t.b.: I thought that Gortaur was here, but maybe not.

Gortaur

@robjohn that's the drawback of the Ring

now people think that I'm there where I'm not

reverse of invisibility

robjohn

We were talking about the Gortaur conjecture and things went from there.

Gortaur

ah I see. I liked your interpretation, but the Ring surely was not commutative

t.b.

Well, robjohn has this habit of thinking people are here watching us (<_< >_> ? ;-P)

robjohn

08:32

yes, but that was why the conjecture was disproven

Gortaur

@robjohn nice try ) he was a real mathematician, that hobbit. Sacrificed his finger for the proof. There are not so many such persons now

robjohn

@t.b.: well the cheshire Mariano was there... he was I tell you!

I'm not mad, I just know it.

t.b.

his grin was there, I know :)

robjohn

though I do have a hat, somewhere :-)

Gortaur

anyway, guys - I need to submit two papers today

also my supervisor will give a talk on a secret topic

robjohn

08:34

two papers? that seems steep.

Gortaur

he told us that it will be a surprise - and that's 2 months before the Christmass

robjohn

@Gortaur Perhaps he has made a big discovery

Gortaur

@robjohn on is for the conference, so that's not that steep

robjohn

@Gortaur what is the conference about?

Gortaur

Hybrid Systems: Computation and Control

I do neither computation nor control nor hybrid stuff )

but my work is highly related, so I also put hybrid case study to make these guys feel comfortable

to be honest, I do computation - but not like the complexity of algorithms

I rather focused on finding really strict and controllable bounds on the errors of infinite iterations/fixpoint equations

robjohn

08:40

more theoretical it sounds

Gortaur

I was a bit dissapointed when he told me a year ago that our results should be computable

but now I found some nice connections (I hope) with dynamical properties and the journal paper should be quite theoretical

robjohn

@Gortaur: is this for the 2011 or 2012 conference?

Gortaur

2012 - I just will submit it today

then I will submit journal paper

robjohn

So are you going to Beijing or just your paper?

Gortaur

if they will admit it, I think I will

then I will have to finish a paper on applications in finance, find a journal, submit it and finally focus on research and self-education

robjohn

08:44

Cool. I have never left North America (if Hawaii is considered North America)

Gortaur

Cool ) I never been to Americas

I mean all of them

so this december I go to Orlando to CDC'11

robjohn

We are thinking of going to Australia for an eclipse cruise.

Orlando is quite a ways from here. Too bad.

Gortaur

I know, but maybe I will go in CA to spent 2-3 months in the next year

or in Switzerland where @tb , I hope, is not computing ext and tor for food

anyway, we'll stay in touch

Rajesh D

does any of read/know of the book, The Geometry of Physics by Theodore Frankel ?

robjohn

@RajeshD don't think I have seen it.

t.b.

08:50

Me neither.

robjohn

@Gortaur If you stop in CA, let me know.

if you get to L.A. I can show you Mulholland (that seems to be what everyone knows about on MSE)

t.b.

@Gortaur: I should probably get rid of all those inside jokes on my old webpages :)

robjohn

@t.b.: interior jokes...

on an open webpage.

Rajesh D

ok....i am interested in learning Diff Geom...i have been tryin to read books in my spare time...but i am somehow not able to pass through the first chapter, beyond tangent vectors and covectors....i find the notation to be obscure and different authours use different notations and they keep changing within the same book.........hence i need a good down to earth book which introduces all the concepts of Diff. Geom. slowly but steadily

robjohn

@Gortaur I was in Chicago just a couple months after the 2011 conference it seems.

@RajeshD the d/dt and dot notations?

t.b.

08:55

@RajeshD: I would start with a book that treats curves and surfaces thoroughly. do Carmo is pretty good, I think. Once you've absorbed that you can move on and the notation shouldn't be that mysterious anymore.

Rajesh D

co-ordinate charts...change of co-ordinates, change of tangent vectors from chart to another...such things

robjohn

what notation changes? Perhaps I have not encountered it or I just have gotten accustomed to various notations.

Gortaur

@robjohn thanks, I certainly will ) thought next CDC'12 will be on Hawaii, pretty cool

Rajesh D

@t.b. : is it this ? amazon.com/Differential-Geometry-Curves-Surfaces-Manfredo/dp/…

robjohn

@Gortaur December in Hawaii is good because it is the off season. Not as many tourists.

Gortaur

08:59

@robjohn I've never been there, so I think I would enjoy it anyway

Rajesh D

I've tried reading books which start with a manifold without first reading and getting accostomed to books that start with curves and surfaces......may be thats why i am having nightmares !

t.b.

@RajeshD: That's the book, exactly.

Rajesh D

@t.b. : does it need any special knowledge of multivariable calculus....it can be readable without too much of it ?

t.b.

You definitely need to be comfortable with the mechanics of calculus, but the point of the book is that it doesn't assume much more and gives the geometric background you need to understand before delving into abstraction.

robjohn

@Gortaur The 2012 conference is just up the road from where we went on our honeymoon and several anniversaries. It is a nice area.

I have been in that resort.

The Grand Wailea

t.b.

09:06

@RajeshD: What you really do need to understand is what the existence and uniqueness theorem for ODEs and what the implicit and inverse function theorems tell you geometrically. If I remember correctly, do Carmo does a great job when explaining these, and he has lots of worked examples and good exercises.

robjohn

@RajeshD though I am sure multivariable calculus will help as you go further in Differential Geometry.

I have never read doCarmo. I should get a copy since I have heard a lot of good things about it.

t.b.

Both, the curves and surfaces and Riemannian Geometry books by do Carmo are really good.

John M. Lee's trilogy starting with smooth manifolds is excellent, too.

Oh no, Didier's answer wasn't accepted :o

Rajesh D

ok.

anon

hehe, you mean "W" ?

robjohn

@RajeshD: I think that the Theorema Egregium is worth any pains to get to

Theorema Egregium

Gauss's Theorema Egregium (Latin: "Remarkable Theorem") is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces. The theorem says that the Gaussian curvature of a surface can be determined entirely by measuring angles, distances and their rates on the surface itself, without further reference to the particular way in which the surface is embedded in the ambient 3-dimensional Euclidean space. Thus the Gaussian curvature is an intrinsic invariant of a surface. Gauss presented the theorem in this way (translated from Latin): ...

t.b.

09:12

@anon: exactly

robjohn

I liked the W answer. It didn't get accepted?

Gortaur

@robjohn nice to know )

@robjohn what are you guys talking about?

t.b.

@Gortaur: this one letter answer

or are you asking about Rajesh's query?

robjohn

this

what t.b. said :-)

Gortaur

@tb no, about W

thanks

robjohn

09:17

Well Didier capped today, too. 20 points were yesterday and 20 today

on that one answer.

t.b.

well-deserved, I would say

I probably wouldn't dare

robjohn

Actually, I see that only 18 were today, but 3 for other answers capped the 18th

Perhaps W = |x-2|-|x-3|+|x-4| was too close to |x-2|+|x-3|+|x-4| which was the accepted answer.

Gortaur

09:38

@robjohn: oh yeah! I almost forgot - I have to fix my answer for the recursive functions which you put so much effort on. Feel guilty for skipping this duty: I'm so focused on these papers now.

robjohn

@Gortaur I would attend to the RL papers before the MSE answers.

Gortaur

sure, but I'm also guilty for putting lame questions/answers on MSE for first 7 months being here - so I have to do my best to break that habbit

that has already brought some minus points to my name here in eyes of other members

robjohn

Well, in that case... :-D

Gortaur

fortunately, not to my name, only to my nick )

but still

robjohn

I assume that people here who were on sci.math would connect me here to there. Perhaps I should be more paranoid and change my nick <_<

Gortaur

09:44

dark past ) may I know what happened there in a couple of words? Or maybe you should't tell me

anon

ironically, my lamest question (that I wish I could delete) got 4 upvotes.

robjohn

Nothing happened there, I just stopped reading sci.math because it was so noisy.

I was just feigning paranoia

Gortaur

ah, I see )

robjohn

@anon and which q was that?

anon

4

Q: Does there exist a self-adjoint operator whose spectrum consists wholly of prime numbers?

The zeros of the canonical Riemann zeta function have been compared to the prime numbers, and they have a number of special, definite connections. The infamous zeros have also been conjectured to be the spectrum of some Hermitian operator given certain distributional similarities that have been e...

Gortaur

09:47

not so lame - at least you didn't happen to ask it three times in different formulations. Though to be honest the time I did it I didn't know that these questions were the same (

but shame on me anyway, that overloaded Didier's patience

Rajesh D

@t.b. :"implicit and inverse function theorems"....i could'nt find them on the contents page...could they present under a different nam ?

robjohn

@Gortaur what?

Gortaur

I asked the same textbook question three times in different formulations

anon

Jesus Christ, you've asked over a hundred questions

Gortaur

is it that bad?

anon

09:54

Of course not!

Gortaur

what then surprised you?

t.b.

@Rajesh: the geometric meaning of the implicit function theorem is essentially covered in chapter 2 on regular surfaces and the inverse function theorem is definitely discussed. I stand by my recommendation

robjohn

@anon He's on here? I never knew :-)

Gortaur

@robjohn he didn't address with @ - haven't you mentioned? )

robjohn

and He's asked a lot of questions, too :-p

anon

09:57

I didn't get it until you capitalized He after lowercase and.

robjohn

@Gortaur I wonder if it gets His attention if you use the @?

Gortaur

@robjohn different religions give different answers for this question

maybe, one of the most important questions in religion

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$Mathematics$ Mathematics