Mathematics - 2012-06-12 (page 1 of 3)

Américo Tavares

00:54

23

MOTIVATION. After having read in detail an article by Alf van der Poorten I read a very short paper by Roger Apéry. I am interested in finding a proof of a series expansion in the latter, which is in not given in it. So I assumed it should be stated or derived from a theorem on the subject. In ...

Eugene

01:05

@robjohn do we use compactness to attain a maximal element?

robjohn

@Eugene yes. You are talking about the diameter thing we were discussing before?

Eugene

@robjohn ah i figured. using the extreme value theorem right?

@robjohn yup

robjohn

@Eugene using the fact that a continuous function maps compact sets to compact sets

Peter Tamaroff

@Eugene Hey there. Look at this. Any opinion?

Eugene

@robjohn yes. then the maximum is attained by the extreme value theorem?

@PeterTamaroff nope. looks good. i told you my opinions on (1) yesterday. seems like you got a lot of pointers from arturo anyway

Peter Tamaroff

01:09

@Eugene Yes. I'm trying to parse his answer.

Eugene

@PeterTamaroff that's good

robjohn

@Eugene if you want to call it that. You could also call it the Heine-Borel Theorem

Eugene

@robjohn yes i see now. thanks for the explanation!

robjohn

That the real image of a compact set is closed and bounded.

Eugene

@robjohn uh-huh. yup.

btw

robjohn

01:11

I have never heard of the Extreme Value Theorem, but I can see that that would be a useful thing to cite.

Eugene

do you pronounce it high-knee

or heighn

robjohn

I've always heard it as high-knee.

user19161

@Eugene I guarantee you there are a million ways to pronounce any name.

user19161

We were talking about Lebesgue remember @rob?

Peter Tamaroff

@Eugene Why not "Heighn"

Eugene

01:12

huh... i've heard it as heighn... high-knee makes me a little uncomfortable.

Peter Tamaroff

@Eugene Doesn't sound European.

Eugene

@PeterTamaroff apparently that's how it's pronounce though.

Peter Tamaroff

@Eugene Ugh.

robjohn

@Eugene like the backside-borel theorem?

Eugene

@robjohn yeah! exactly like that!

user19161

01:16

I reserve "Heine-Borel" for the result that says a subset of Euclidean space is compact if and only if it is closed and bounded.

Eugene

@JasperLoy really? then how about that of complete spaces?

user19161

@Eugene What's the result you are talking about?

Américo Tavares

@Eugene I think he is German. So do you know how to pronounce "deine" (your, feminin) ?

user19161

Anyway, one would realize that a theorem can be given a million names and the same name could refer to a million theorems.

user19161

So don't bother about the names, just remember the theorem in its most useful form for oneself!

robjohn

01:20

Where is Superman when you need him?

user19161

For what?

Eugene

@JasperLoy if a space is complete then all compact sets are closed and bounded. euclidean space is only one such example

robjohn

@JasperLoy and that is just what I was using it for.

Eugene

@JasperLoy and naming things is useful. i don't want to have to repeat the statement of the modularity theorem each time i invoke it

my research papers would be 4x as long and be summarily rejected.

robjohn

I have to take Lilly to the park. BBL

Eugene

01:23

@robjohn see you later!

user19161

@robjohn Bye!

user19161

@Eugene Ah yes. This reminds me of the newspaper conversation we had.

Eugene

?

@PeterTamaroff so close to 2000 but yet so far...

almost every algebra question is answered and those that aren't are on analysis or graph theory

Jordan

partial fractions are way too hard, I am going over a practice test and have been stuck on finding A B C D for 45 minutes, I would have failed that test

Is there some trick when it is A + B + Cx + D?

I got it but I cheated, I guessed and was right

Jordan

01:50

Is there any way to manipulate $\frac{e^x}{e^{2x}}$ to get rid of the top?

Eugene

yup

$u = e^x$

Jordan

but what happens to the two

Eugene

$e^{2x} = u^2$

Jordan

is it u^2?

man that is incredibly hard to visualize

now I have a real problem

$\int \frac{du}{du^2 + 3}$

Eugene

?

can you just tell me the original problem?

Jordan

01:57

ugh I got it I think, I had to have memorized the inverse tan

Peter Tamaroff

@Eugene Check now!

Jordan

$\int \frac{e^x}{e^{2x}+3}dx$

Eugene

ok

Jordan

this teacher is insane

Eugene

so

@Jordan can i help you on the condition that you work hard and complain less?

Peter Tamaroff

01:58

@Jordan Let $e^x = u$

Jordan

I got the answer :(

I just didn't like the solution

Peter Tamaroff

@Jordan "Didn't like"? What is that suppose to mean?

Is there a better one?

Eugene

@Jordan it's ok if you keep a positive outlook. ;)

anyway

using the trick i showed you

$\int \dfrac{e^x}{e^{2x}+3} = \int \dfrac{du}{u^2+3}$

Jordan

I don't like solutions that require you to have memorized one off formulas or identities like inverse tan, like I will ever need to use that in real life

Eugene

@PeterTamaroff oh thanks peter!

@Jordan so we have that $u = \sqrt{3} \tan \theta$.

Peter Tamaroff

02:02

@Eugene You mean $\sqrt 3$ there.

Eugene

@PeterTamaroff you're right thanks!

so $du = \sec^2 \theta d \theta$

so we have that $\int \dfrac{du}{u^2 + 3} = \int \dfrac{\sec^2 \theta d \theta}{3 \tan^2 \theta + 3}$

can you solve it from here?

Jordan

I had to use inverse tan

but yeah I think so

is that just 1/3?

Eugene

@Jordan yes

@PeterTamaroff i had to do so much grading today!

Dylan Moreland

Grading is truly the worst.

Peter Tamaroff

@Eugene What was the best mark you gave? And worst¿

Eugene

02:07

@DylanMoreland it is... 250 papers what's more...

@PeterTamaroff 100 and 0

Peter Tamaroff

@Eugene LOL

Eugene

@PeterTamaroff it's true though.

Peter Tamaroff

@Eugene Do you have a copy of the exam?

Eugene

@PeterTamaroff goodness no. i want to forget about it as quickly as possible.

Peter Tamaroff

@Eugene But I want to see it!

What was it about?

Eugene

02:09

@PeterTamaroff calculus 2

Peter Tamaroff

@Eugene Let me see, c'mon, dude!

Eugene

@PeterTamaroff i really don't have it! =D

Peter Tamaroff

@Eugene Oh! That's OK then.

What was the toughest question?

Eugene

@PeterTamaroff beats me. they all looked pretty simple to me

Peter Tamaroff

@Eugene Hahaha damn you.

Eugene

02:12

@PeterTamaroff it's calc 2. it's probably really simple to you as well.

Peter Tamaroff

@Eugene At least you remember the topics?

Eugene

@PeterTamaroff well first indefinite integrals, second improper integrals, third ODEs, fourth monotone convergence theorem and the fifth is series.

@PeterTamaroff only 5 questions

Peter Tamaroff

@Eugene Oh, OK. I only miss monotone convergence. What was that about?

Mark Dominus

a bounded monotonic sequence converges.

Eugene

there you go

Peter Tamaroff

02:16

@MarkDominus That? Bah.

I thought it had to do with dominated convergence, lol!

Yeah, I handle that too.

Eugene

@PeterTamaroff see it should be easy for you

Peter Tamaroff

@Eugene I'm gluing Apostol's book now!

It should be done by tomorrow morning.

Eugene

@PeterTamaroff that's good. you seem really intent on learning anal

Peter Tamaroff

@Eugene Hahahahhaha f**k you.

@Bill I'd really appreaciate your thoughts on this, sire.

@Eugene Though I'm desperate about getting Landau's book on Number Theory.

Eugene

@PeterTamaroff hahah. it's nice that you have such an interest in landau i guess. even though i still don't know who he is.

Jaydon Zhao

02:22

hey, quick question, why doesn't $\int_{-\infty}^{\infty} \frac{x}{1 + x^2} \mathrm{dx}$ converge?

Peter Tamaroff

@Eugene I'll facilitate you a short entry about him , from Hardy. It is very emotive IMO.

Eugene

@JaydonZhao why would it?

@PeterTamaroff from apologies?

Peter Tamaroff

@Eugene No no.

@JaydonZhao Hint The function is odd. Consider it's principal value.

Stricltly speaking, it doesn't converge.

Since it would converge if $\int_0^\infty f(x) dx$ and $\int_{-\infty}^0 f(x) dx$ did.

Eugene

@JaydonZhao it integrates to $ln | 1 + x^2|$ use the integral test.

Peter Tamaroff

@Eugene Read this

Jaydon Zhao

02:28

oh, right, thanks. just realised I integrated it totally wrong initially, no wonder.

Eugene

@PeterTamaroff omg it's so long!!!

Peter Tamaroff

@Eugene 8 pages!!!

Eugene

@PeterTamaroff summary landu is great. all worship landau!

Peter Tamaroff

I have to read 150 for my midterm on Monday!

Eugene

@PeterTamaroff some papers i read are only 8 pages long and yet contain super important results

Peter Tamaroff

02:29

@Eugene That's true!

Eugene

@PeterTamaroff so it's really long!

Peter Tamaroff

@Eugene Bah! Mah! Pah!

Eugene

but i get the message

landau is great. all worship landau!

Peter Tamaroff

The AlgebraischeZahlen gives a short and self-containedaccount
(pp. 1-54)of the theory of algebraic numbers and ideals, intended as an
introduction to the proofs of the prime ideal theoremand its refinements
which occupy the remainder of the book.He does not go so deeply into
the algebraic theory as, for example.. Heckc. being content with what is
required for his applications.

Eugene

i see.

so an analNT lover to the end.

Peter Tamaroff

02:31

Landau wrote over 250 papers, but it is possible tha t he will b,e remem-
bered first for his books, of which he wrote seven.

Eugene

@PeterTamaroff wow.

Peter Tamaroff

Landau was the complete master of a most individual style, which it
is easy to caricature(as some of his pupils sometimes did in an amusing
way*),but whose merits are rar eindeed.I thas two variations,the"old
Landau' style", best illustrated by the Handbuch, which sweeps on
majesticallywithout regard to space, and the "new Landau style" of his
post-war days, in which there is an incessant striving for compression.
Each of these styles is a model of its kind.There are nomistakes—for
Landautook endless trouble, and was one of the most accuratethinkers

@Eugene This is what I like most of him

Eugene

@PeterTamaroff that is a glowing recommendation indeed.

i try not to admire mathematician lest i be intimidated by them

Peter Tamaroff

@Eugene So true. All hail Landau!

(don't get me wrong there, ay?)

Eugene

@PeterTamaroff hahah. wait until you get to modern research. you'll see a whole new landscape

and next you'll be telling me about this guy

Peter Tamaroff

02:36

@Eugene He got a fields medal right? Or something of the sort.

Yeah, Fields.

Amazing.

Eugene

@PeterTamaroff he is simply remarkable.

Peter Tamaroff

This is directly from Landau's Book n Differential and Integral Calc:

2. The Decimal System
*Definition 4*: 3 = 2 + 1, 4 = 3 + 1, 5 = 4 + 1,
6 = 5 + 1, 7 = 6 + 1, 8 = 7 + 1, 9 = 8 + 1, 10 = 9 + 1.

He previously defined $2=1+1$

"*Definition 101*: C is called Euler's constant.
I do not know whether it is rational or irrational. "

@Eugene What is his greatest achievement? The one on the arithmetic progressions?

Eugene

@PeterTamaroff tao? well that is one of his great achievements. the man has done too much!

J. M.

@PeterTamaroff Speaking of which... (see comments)

Peter Tamaroff

@JM lol

That is simply awesome. Internet is awesome. Math is awesome.

Dan M. Katz

02:46

Hey guys...I have what I believe is a relatively simple group theory question. Let me know if it would be more appropriate to ask as a full question.

Given elements a,b in a group such that (x^-1)a(x)(y^-1)b(y)=c, is it necessarily true that there exists an element z such that (z^-1)ab(z)=c ?

Mark Dominus

What if you take y=x?

Dan M. Katz

By right cancellation law, if (y^-1)b(y) not equal to (x^-1)b(x), then this won't be true, correct? I'm not interested in constructing such an element, simply knowing whether is true that one exists.

I'm working with a group right now where I suspect this identity holds, but before I go about trying to analyze it further or eventually prove it, I wanted to check whether this was a general group theorem that I hadn't heard of (or had forgotten about).

and...err...I guess it's not really an identity, but you know what I mean =) "Implication" might be a more proper term.

Eugene

03:07

@DanMKatz they are just asking if there exists such an element $z$. if you take x = y, then there is.

Dan M. Katz

Oh, of course. Sorry, I'm asking for the case in which x doesn't equal y. And there's no "they", just me =)

Let me put it another way. We are given two elements x,y, and we wish to determine if they are conjugate. Furthermore, we identify that x can be expressed as a product of two elements, say x=a*c. If we find that the product of a conjugate of a and a conjugate of c are equal to y, does it necessarily follow that x is a conjugate of y?

Eugene

if $G$ is abelian certainly.

Dan M. Katz

Sure. Otherwise, this is probably not true in the general case, correct?

Eugene

@DanMKatz is this homework?

Dan M. Katz

No, this is a question that, if so in the group I am working in, will allow me to complete an algorithm I'm working on for research.

Eugene

03:17

@DanMKatz oo. what group is that? maybe that will be easier to consider.

also if you can tex please typeset your statements. i find them hard to read.

Dan M. Katz

I would rather not say, only because, given that this is my first research problem I've ever tackled, I want to try as much as possible to reason through it on my own. I've almost got it...just trying to fill a missing piece. Hope you understand =)

Eugene

@DanMKatz so this is the part you don't want to try and reason through?

Dan M. Katz

Not quite. This is the part where I simply wished to know whether this was a theorem for general groups, since if it is, it would be a waste of time to attempt to prove the result for a specific case.

Eugene

well i think it sounds reasonable enough to be true. i can't come up with an immediate counterexample

Dan M. Katz

Haha fair enough

Hmm...So I guess I'm essentially asking whether a product of conjugates is a conjugate of a product. It somehow sounds more dubious that way. Lol

Eugene

03:27

maybe. good luck with finding the answer.

Dan M. Katz

Thanks!

Eugene

05:25

@robjohn can you see deleted questions?

robjohn

@Eugene yes

Eugene

@robjohn can you see mine?

robjohn

@Eugene If you have a link to it.

Eugene

math.stackexchange.com/questions/157250/spotting-a-crank

i was worried it would be inappropriate so i deleted it

do you think it would?

robjohn

@Eugene I don't know. It might not be on topic, but often that is quite a subjective decision

Eugene

05:29

@robjohn well if you think it's not appropo i will leave it deleted.

robjohn

@Eugene The only way to find out what otherst think is to post it and see what they think.

Eugene

@robjohn oh well i have to retype it then

robjohn

If it gets heavily downvoted, you can delete it again.

@Eugene Why?

Eugene

@robjohn because i deleted it?

robjohn

@Eugene can't you just undelete it?

Eugene

05:31

@robjohn apparently not. i can't see deleted questions i think

anon

you should be able to see your own deleted question...

robjohn

@Eugene not even your own?

Eugene

@robjohn nope!

this is what i get

"This question was voluntarily removed by the author – that's you!"

anon

doesn't it take like 3 undelete votes? robjohn, Dylan and I can do that.

robjohn

@anon it takes three to undelete someone else's question, right?

one more needed.

anon

05:33

yo @DylanMoreland

Dylan Moreland

Hey @anon.

Eugene

@robjohn don't worry about it. i don't want to stir up crap so if people might get pissed i'd rather not

anon

wait can Eugene delete his question after it's already been deleted by him and then undeleted by us?

Dylan Moreland

Can I vote to undelete? I didn't know.

robjohn

@anon I don't know. We can probably delete it again.

anon

05:34

I think it's a 10k power.

robjohn

@anon I think so

Dylan Moreland

Ah, so I can.

Not easy to find it, but it's there.

Eugene

wait before you guys do so

what do you all think?

Dylan Moreland

Right, is this something you want done?

Eugene

@DylanMoreland i'm not sure. i was wondering about all your opinions.

Dylan Moreland

05:35

I guess I probably wouldn't look at it.

Eugene

@anon how about you?

anon

(1) You might want to make it CW and (2) you might want to make it about "crankery" or "crank behavior" rather than cranks (people) to be more objective and less blunt.

Dylan Moreland

Seems kind of idle. Again, everyone's different :)

Eugene

oh well forget it then. thanks for your advice! i might just ask the guys at waterloo. maybe they have something to say about it.

Dylan Moreland

Why do you want to know?

Is this a problem in your life? Just curious, not trying to be condescending or anything.

Eugene

05:40

@DylanMoreland sometimes when i see interesting arxiv papers it is hard to tell.

also some results seem very trumpted and i'm not sure how seriously to take them.

Dylan Moreland

I guess I don't read the arXiv email so much. If there's a paper that's important to me I'll generally have heard about it through other means.

I think there are interesting things to say about evaluating unfamiliar mathematics.

Eugene

@DylanMoreland i sometimes browse through it to read find papers i think are interesting i guess.

Dylan Moreland

Like, "is this worth time investigating".

Any such discussion is going to be really broad, probably too broad.

Rajesh D

Hi

Eugene

hm... so i should wait for for specific instances and ask about those?

Dylan Moreland

05:42

No, that might be bad too :)

Eugene

@anon i was quite concerned about pissing people off.

@DylanMoreland yeah. i saw an instance recently where this guy asked about the riemann hypothesis

Dylan Moreland

But you're right, people make claims all the time. It's hard to know what's important.

Eugene

yes. like de branges for instance. when i was writing my thesis i saw "proof of the riemann hypothesis" and i was confused.

Dylan Moreland

People have plans for proving RH and BSD and all that and there's generally more smoke than fire in them, it seems to me.

Noncommutative geometry ugh.

Eugene

and he's a serious mathematician so i was curious about it

then i saw conrey completely disproved his approach

Dylan Moreland

05:45

Was, it seems to me.

Eugene

@DylanMoreland that was also another question i had. how do you know if someone is becoming a crank

Rajesh D

the derivative of $f:\mathbb{R}^N \to \mathbb{R}^M$ is of the form $f':\mathbb{R}^N \to \mathbb{R}^{M \times N}$. How would the double derivative be, i.e, how would $f''$ be ?

Eugene

oh well i defer to you, @robjohn, and @anon. if you guys think it's off topic and broad i think i shouldn't ask it

Rajesh D

"defer with you" no?

Eugene

sorry guys for the interruption

anon

05:48

no, defer to you @Rajesh - meaning handing over the decision to us

Rajesh D

ok

oh ok

i thought "i don't agree with you on this"

Dylan Moreland

I think you can develop yourself such that if it's unclear what's going on in a piece of mathematics then there is probably an actual problem in the mathematics. It's not just you being slow or whatever.

And until you hit that point you hopefully have someone telling you what to do.

Rajesh D

so any one knows my question?

Eugene

@DylanMoreland yeah you're probably right.

anon

@RajeshD You might find this helpful

basically $f\,'':\Bbb R^n\to\Bbb R^{m\times n\times n}$

I mean you might as well set $g=f\,'$ and use the same formula to find $g\,'$

Eugene

05:53

@DylanMoreland cool. ok then. thanks for the time!

@robjohn sorry.

Dylan Moreland

But again, there is like zero harm in asking a question here or on the main site.

Eugene

@DylanMoreland hahah. that is true!

anon

but more abstractly we're working with linear maps on tangent spaces of tangent spaces or whatever, though in Euclidean space these objects are all more Euclidean spaces and can be viewed as matrices by setting bases and whatnot

Rajesh D

posted it on main

@anon : this must be having some name right? it must have been delt with in some books?

anon

in the more abstract sense it's dealt with in differential geometry. with coordinates set you also have tensor calculus (same thing basically).

but seriously just apply the formula to itself to get the double derivative...

Rajesh D

06:02

yes i know what you are saying. I need a reference book so that i can sail through

Eugene

@DylanMoreland oh well. thanks again.

@robjohn sorry to have bothered you in the end.

@anon thanks for your help

Benjamin Lim

06:22

@DylanMoreland I edited my answer

Gigili

08:44

Ohoi, @Ilya.

Old John

Hi folks

Ilya

Ave @Gigili

@OldJohn Hi, I prefer rock :)

Old John

classical here :)

Gigili

Only pop.

Ilya

Lonely pop

Lolly pop

@Gig :p

@OldJohn which composer is your favorite one? Maybe you also like contemporary classics (like Saint-Preux)

Old John

08:50

Mostly I listen to stuff like Bach - and lute music :-)

Gigili

@Ilya Very creative.

Holy pop.

Ilya

oh, Bach is a cool dude

@Gigili the kindest

Old John

one of the most mathematical of composers IMO

Ilya

@OldJohn I didn't finish even the first chapter of Esher, Gegel, Bach. But yeah, WTK is something quite theoretic.

Old John

I read it more than 20 years ago - but lost my copy :(

Ilya

08:52

I mean the idea to write 24 pieces in all possible tones, and to do this 2 times!

Gigili

I have a strong sense of deja vu that I saw Old-John somewhere before.

Ilya

in some of them alteration signs were driving me crazy

Old John

really?

Ilya

@Gigili :(

@OldJohn she also told me that we met 10 years ago (not something that I am aware of)

Old John

LOL

which part of the world are you in Gigili?

Ilya

08:53

@OldJohn emmmm. Would you put a comma there?

Gigili

Umm.

Good question, good question.

@Ilya Haha.

Ilya

@Gigili :-x (say ``West'' to him) since there is not West pole, it's a meaningless piece of the information. In the meantime you can hide

Old John

I'm baffled by the comma comment :(

Ilya

@OldJohn: this reads like there are several parts of the world in Gigili

Gigili

says West to him and hides

Old John

08:56

AH! OK :)

Gigili

@Ilya There are. A part of Germany is in Gigili, for instance.

Ilya

@Gigili which one? Warsteiner and a sausage?

Gigili

@Ilya The West part of Germany.

Ilya

@Gigili Oh! Oh! that's close to me

Old John

UK here

Ilya

08:58

rains here

morning here

Ilya here

table here

laptop here

Gigili

Nothing here, nothing.

Ilya

that would be a long list

I wonder if the West pole would be in LA, where would be the correspondent East pole

Old John

probably somewhere near India?

Very rough guess from time-zones

Ilya

@OldJohn -34.04697,61.75827 if I'm not mistaken

it's within a time zone of India, right - but it's a piece of Indian Ocean, South-East from Madagascar

Let me try NY as a West pole

then the East pole is 40.7, 105 - in the desert on the Chinese-Mongolian border

fair enough

100 years I would take London or Paris as a benchmark, but time makes it all change...

@Gigili: what would you say?

Gigili

@Ilya I'm speechless right at the moment, you should have told me earlier that you're Indian.

Ilya

09:09

..a Jones

Gigili

Isn't $lim_{x \to 0} \frac{sinx}{x}=1$?

0

A: Limit finding of an indeterminate form

$$\lim_{x \to 0} \frac{x^3}{\tan (2x)^3}=\frac{x^3}{(2x)^3}=\frac18$$

Ilya

sorry, I was afraid Harrison Ford is around. He played me very bad the last time :(

@Gigili it is

Old John

yes - that limit is 1.

Gigili

@Ilya I murdered him, don't you worry. Just tell me next time that you're Indian.

Old John

So, for small $x$, $\sin x$ and $\tan x$ are both approximately $x$.

Ilya

09:12

@Gigili why do you think I am?

@Gigili: I like you answer, +1

Gigili

Huh, it got downvoted.

What in the name of Tarzan is this Alex thingy saying? I don't get it.

@Ilya Hey, I am the kindest, not you!

@Ilya Hugh!

Ilya

still thinking I'm Indian?

Gigili

Was that you?

Ö

Old John

Time to go - bye for now

Ilya

bye

Gigili

09:20

Does it look good now @Ilya?

Ilya

@Gigili just put a $\lim$ for the middle function in your first version, don't use $\approx$

maybe also mention that $\tan x \sim x$ for $x\to 0$ in the beginning

I don't think it's necessary - but certain people may have need this explanation

have to go, bye

Gigili

Bye!

robjohn

09:56

@Eugene Sorry about what?

@JM good day! I see that you entered the final undelete vote for Eugene's question

J. M.

@robjohn Hi! Yeah, I think it's a nice question...

robjohn

@JM He was debating it quite a bit earlier.

Transcript for

$Mathematics$ Mathematics