Mathematics - 2014-06-08 (page 2 of 4)

G.T.R

14:13

@Chris'ssis @robjohn is there a closed form for $\displaystyle \int_{0}^\infty \frac{cos(ux)}{x}dx$ and $\displaystyle \int_{0}^\infty \frac{sin(ux)}{x}dx$

robjohn

@G.T.R the left one does not converge ($\infty$)

The right one has one using contour integration... $\pi/2$ for all $u$

Chris's sis

@G.T.R Sure ... (for the right side one)

Dirichlet integral

In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet. One of those is : \int_0^\infty \frac{\sin \omega}{\omega}\,d\omega = \frac{\pi}{2} This integral is not absolutely convergent, and so the integral is not even defined in the sense of Lebesgue integration, but it is defined in the sense of the Riemann integral or the Henstock–Kurzweil integral. The value of the integral (in the Riemann or Henstock sense) can be derived in various ways. For example, the value can be determined from attempts to eva...

r9m

I like this problem :D ..

G.T.R

14:30

@Chris'ssis @robjohn what about $\displaystyle \lim_{M\to \infty} \int_a^b \frac{sin(Mu)}{u^2}du$ ?

90intuition

Anybody here a recommandation for a good book to study for my first course representation theory ?

Balarka Sen

@r9m Connecting stuffs is not very deep mathematics.

The real depth of mathematics comes when you develop everything from scratch, from basic observations.

r9m

@BalarkaSen I was only answering a simple question :) and found it convenient coz the OP had already proposed partial solution :)

Balarka Sen

@r9m Link?

r9m

@BalarkaSen [here](math.stackexchange.com/questions/822495/show-that-p-nx-rightrightarrows-0‌-qquad-x-in0-1/)

G.T.R

14:40

@Chris'ssis for motivation, math.stackexchange.com/questions/825869/… and let $\displaystyle I=\int ^\infty_0 \frac{b\sin{ax} - a\sin{bx}}{x}dx$, and see what $I/(ab)$ look like. Rewriting the inside as an integral, I got the computation I asked you

Balarka Sen

@r9m Interesting. That's interesting.

r9m

@BalarkaSen how does epsilon-delta stuff interest BalarkaSen ??!!.. hmm interesting :D !!

Ian

Hey @BalarkaSen I've got a question for you (more of a reference request, really)

Balarka Sen

@Ian OK.

N3buchadnezzar

I think I have 3 windows with chrome running each with 50 tabs open >.<

"Oh, this is usefull. Will just leave that open for later"

G.T.R

14:43

how much RAM do you have @N3buchadnezzar ?

N3buchadnezzar

"PING. Oh I forgot where i put facebook / this chat. Better open a new one"

@G.T.R 8 ?

G.T.R

@N3buchadnezzar good

Ian

@BalarkaSen have you ever seen the concept of "expected number of different factorizations" in a non-UFD? Like, take $\{z \in\mathbf Z[\sqrt 5]:N(z)<M\}$. Calculate the expected number of factorizations for this set. Let $M\to\infty$ ($N$ is the norm)

Balarka Sen

@Ian "expected number of factorizations"?

Ian

@BalarkaSen yes. Some numbers have two different factorizations, some four, and so on. The mean could be finite in principle

N3buchadnezzar

14:47

Still frurstrating

Balarka Sen

@Ian Ah, you mean the multiplicative partition?

N3buchadnezzar

I need to start keeping sessions, one for cubing, one for mathematics, one for misc, and one for pictures of cute kittens

G.T.R

@N3buchadnezzar cubing ?

N3buchadnezzar

Rubiks

I avreage 16 :p

G.T.R

@N3buchadnezzar speedcubing ?

N3buchadnezzar

14:48

Yeah

Balarka Sen

@N3 Have you read the algorithm?

G.T.R

I got a weird cube I need to show you

N3buchadnezzar

?

Chris's sis

@G.T.R Does the mean value theorem help?

Balarka Sen

@N3buchadnezzar search [rubik's cube group]

N3buchadnezzar

14:49

►

Ian

@BalarkaSen I think so, I don't know this nomenclature :-) For example, $6$ has two different factorizations: $2\cdot 3 = (1-\sqrt{-5})(1+\sqrt{-5})$ (sorry, I meant $\mathbf Z[\sqrt{-5}]$)

G.T.R

@Chris'ssis I don't know

N3buchadnezzar

@BalarkaSen Are you good at math because you know the formulas?

Are you good at driving if you know how the individual parts work?

Chris's sis

@G.T.R That is $0$.

N3buchadnezzar

Same with cubing >.< Everyone can solve it, most can do it under a minute. A few under 20. Under 15 requires real talent

G.T.R

14:59

@N3buchadnezzar image.noelshack.com/fichiers/2014/23/…

N3buchadnezzar

Thats a cool one

Chris's sis

@G.T.R Riemann-Lebesgue lemma

G.T.R

@N3buchadnezzar well I can't solve it

@Chris'ssis right, how did I omit this ...

Chris's sis

@G.T.R The challenge is to do it with high school knowledge ... (I think I have at least a proof in mind)

G.T.R

@Chris'ssis come on, here, Riemann Lebesgue is just integration by part

N3buchadnezzar

15:03

The generalized version too ?

G.T.R

no it's more troublesome

Chris's sis

I have some big issues with my comp ... hmmm

brb

N3buchadnezzar

@G.T.R I saw a proof on site, but did not fully understand it

G.T.R

@N3buchadnezzar for which version of the lemma ? for $C^1$ functions it's easy, for just continuous, you need to use density of step functions on $[a,b]$

N3buchadnezzar

$$
\lim_{n \to \infty} \int_a^b f(x)g(nx)\,\mathrm{d}x
= \left(\int_0^T g(x) \,\mathrm{d}x \right)\left( \int_a^b f(x) \,\mathrm{d}x \right)
$$

I mean this generalization

G.T.R

15:10

what are hypotheses on $f$ and $g$ ?

N3buchadnezzar

Where $g(x)$ is a periodic function with period $T$ and $f(x)$ is piecewise continuous on $[a,b]$

Heh silly me

$$\lim_{n \to \infty} \int_a^b f(x)g(nx)\,\mathrm{d}x = \frac{1}{T} \left(\int_0^T g(x) \,\mathrm{d}x \right)\left( \int_a^b f(x) \,\mathrm{d}x \right)$$

Forgot to divide by the period

Daniel Fischer

Subtract the average of $g$, and you're in the Riemann-Lebesgue department.

N3buchadnezzar

@DanielFischer Hmmm do not see it

Daniel Fischer

15:25

@N3buchadnezzar Let $\mu = \frac{1}{T}\int_0^T g(t)\,dt$ and $h(t) = g(t) - \mu$. Then $\int_0^T h(t)\,dt = 0$, and you want to show $$\lim_{n\to\infty} \int_a^b f(x) h(nx)\,dx = 0.$$ You do that like Riemann-Lebesgue, assuming $h$ is bounded [e.g. if it is continuous]; take $f\in C^1$ and integrate by parts, approximate general $f$ in $L^1$ by $C^1$ functions. If it's unbounded, a little more care to show it would be necessary.

nabla blah

15:57

Hi @DanielFischer, I don't really understand your explanation here math.stackexchange.com/questions/825958/…

If we have some closed disc centered at $x$ with points $y \leq r$ from $x$, how can $||z - y|| > r$ not belong to that disk when $y$ is inside that disc and also inside the disc centered at $z$

Daniel Fischer

@nablablah If a point is outside a closed disk (disk isn't important), then it never is "immediately adjacent" to the disk, it always has a positive distance from the disk. For an open disk, you can say that the boundary points (the points of the bounding circle) are immediately adjacent to the disk, but for a closed disk, it's either in the disk, or some distance away from it.

@nablablah Let's for simplicity look at the closed unit disk $D = \{ x : \lVert x\rVert \leqslant 1\}$. Then if $z \notin D$, that means $\lVert z\rVert > 1$, and $z$ is at a distance $\lVert z\rVert-1$ from $D$.

nabla blah

But if a point can be "immediately adjacent" to a disk, isn't that a positive distance of 1? Since the point is 1 "distance" over from the point on the disk?

Ok

@DanielFischer What if we consider something like the natural numbers in $\mathbb{R}$ and we have a like a closed set $1 \leq x \leq 3$. If we consider the point $4$, isn't that a positive distance from $3$?

Daniel Fischer

16:15

@nablablah Yes, that is at a positive distance from $3$. But, in the integers, you can only move in discrete steps (of integer size), so one could then say it is immediately adjacent, since there is no integer between $4$ and the set. However, in $\mathbb{R}^n$, we can move arbitrary distances, so whenever we have a positive distance, we have lots of points in between.

nabla blah

Ok, so I think my issue is with the conceptual understanding of infinite points

Hi @JasperLoy

Jasper Loy

@nablablah Hi Bart.

I had a nightmare again.

nabla blah

About?

Jasper Loy

About being abused by some people.

nabla blah

People you don't like?

Jasper Loy

16:26

They are mostly distorted recollections of the past.

Studentmath

@Jas I have nightmares all the time. I found that looking at them as something exciting makes it easier

N3buchadnezzar

I only escape my nightmares when I sleep.

G.T.R

16:46

@chris'ssis well

N3buchadnezzar

And there chat died

Studentmath

@N3b deep

N3buchadnezzar

rollin in the deep

Studentmath

I can't be arsed to write down the code.

N3buchadnezzar

?

Studentmath

17:00

Homework assignment. Have to do this one, then five more, than have ten days to hand two more and study for the first test..

Ted Shifrin

Heya @Studentmath ;) Miss chemistry?

N3buchadnezzar

@Studentmath As a wise man told me once. Life is a poop eating contest, the one who can eat the most without giving up wins.

Studentmath

Wise words, @N3b, and I would want, Prof. @Ted, 'sadly' done with all the chemistry courses already. it's a coding course for the math major

N3buchadnezzar

Major tom to ground controll

I know too many songs and too little mathematics

Studentmath

17:18

I don't know enough of either

Ted Shifrin

I know, @Studentmath; that's why I asked if you missed (longed for) chem :)

Ugggh @N3

Studentmath

Oh, funny, I read it as 'Miss. Chemistry" as in "Lady Chemistry" - doing this course, very very much so, yeah..

N3buchadnezzar

You do not like the song @Ted ?

Jasper Loy

I just had a delicious Subway sandwich.

Ted Shifrin

It was not the song, @N3; it was the previous comment. :D

N3buchadnezzar

17:22

@TedShifrin Well it's true. You live, you die, and it's up to you to make the most out the part in between.

Ted Shifrin

They have Subway in Singapore, @Jasper? Ugh @American fast food everywhere.

Studentmath

They are the best @Jasper

They have Subway in Israel, Prof. @Ted. And we don't have Subways.

Jasper Loy

@TedShifrin Yes, although we are quite conservative, we are quite Americanised.

Ted Shifrin

Bah @Americanization = dumbing down

Jasper Loy

Yes, just look at the math, lol.

Studentmath

17:24

You are still the biggest power in the world though

N3buchadnezzar

Hebrew math all the way

@Studentmath Biggest in several ways ;)

Studentmath

@N3b Is it the Aleph or the Nebel?

N3buchadnezzar

?

Jasper Loy

I must remark again I am very surprised so few people are using Cohn's algebra books.

Studentmath

Regarding the Hebrew Math

N3buchadnezzar

17:27

Nebel

Jasper Loy

I just deleted an answer because someone posted it 6 seconds earlier and got upvoted but mine did not.

N3buchadnezzar

Wow

Jasper Loy

This voting thing is so stupid.

Studentmath

Every second counts.. I gave a very tiny hint and someone got the answer from it today, was really happy. Felt helpful in a good way.

G.T.R

Site is read-only. AGAIN ?

Jasper Loy

17:29

Offline for maintenance again?

Wow, the developers ought to be fired.

Ted Shifrin

Just as I was commenting on someone who followed the first part of my hint and pisted a separate answer ... Although he may have been typing as I posted.

G.T.R

Back again

Ted Shifrin

Salut, @Gabriel

Jasper Loy

Because I threatened to fire them.

G.T.R

@Ted salut, alors t'as regardé la finale ?

Ted Shifrin

17:33

Bien sûr. J'ai même posté VAMOS, RAFA sur mon Facebook :)

Jasper Loy

@TedShifrin I would like to ask your opinion. Do you think it is important to teach Krull Schmidt theorem and Maschke theorem in a grad algebra course?

Ted Shifrin

Don't ask me. The only grad alg I ever took was a comm alg course when I was an undergrad.

Then I took a bunch of alg geo, and a year-long course on Lie groups/algebras in grad.

Studentmath

Gonna force myself to work. Good night @Ted, @N3b!

N3buchadnezzar

Nighty

Jasper Loy

@Studentmath See you in your dreams!

Hello @ant, why this username?

Ted Shifrin

17:52

night, @Studentmath

Jasper Loy

@TedShifrin May I ask which differential geometry text you use to teach a grad differential geometry course?

Ted Shifrin

18:20

@Jasper: My taste is not typical. I have never followed any single text, and much of what I do is more differential-forms based than any of the standard texts. These days, doCarmo and Petersen are the two most-used texts, but neither fits my personal taste. This last time I proved Chern-Gauss-Bonnet, which is not standard; the previous time, I taught some complex geometry because the students (all undergrads) wanted to see that.

If I get totally bored when I've retired, I may get inspired to TeX up my various grad course notes, but I doubt it.

Ant

@JasperLoy I'm Italian, and my full name is Antonio. Hence Ant ;-)

I didn't really realize the connection with and, the insect :)

*with ant, the insect

Ted Shifrin

Ciao, @Ant. :)

Ant

Saluti a tutti da Milano ! I have to leave now ;)

G.T.R

@Ted is there an intuitive explanation of why the Fourier coefficients get smaller as a function is differentiated ?

Ted Shifrin

Ciao, @Ant :)

Well, it's more like the smaller they are the more differentiable the function is. @GTR

American Luke

18:31

@TedShifrin hello?

Balarka Sen

@N3buchadnezzar Am I even good at math? But that wasn't the point : I simply asked if you know the theory. It's interesting, but not of much use.

Ted Shifrin

Hello @AmericanLuke

American Luke

@AmericanLuke self-ping :P

Balarka Sen

1 hour ago, by Ted Shifrin

They have Subway in Singapore, @Jasper? Ugh @American fast food everywhere.

Hahahaha

Ha..oof...heh... ahem

Hullo @TedShifrin

Ted Shifrin

@GTR, if $f(x)\sim \sum c_ke^{ikx}$, then $f'(x)\sim \sum ikc_ke^{ikx}$, so they're not getting smaller when you differentiate. But if $f$ is $d$ times differentiable, that tells you that you need the $c_k$ to decay sufficiently rapidly for $\sum k^dc_k$ to converge appropriatley @GTR

hello @Balarka

Balarka Sen

18:34

@TedShifrin I got through that problem.

Ted Shifrin

Oh, which problem, @Balarka?

American Luke

Did you mean to ping me?

Balarka Sen

@TedShifrin $L/\Bbb C(z)$ and $L'/\Bbb C(z)$ be galois extensions. If the galois groups are isomorphic then is $L = L'$?

Ted Shifrin

@AmericanLuke, to the best of my knowledge, I only pinged you after you'd said hello? to me ...

ohhh, that one @Balarka.

American Luke

that one... :P

Daniel Fischer

18:36

@TedShifrin You wrote @American fast food in a message to Jasper. That pinged Luke.

Ted Shifrin

Oh, crap. Thanks, @DanielF. I should have left a space after the @ ... My apologies, @AmericanLuke.

Balarka Sen

Yeah, and that's why I laughed.

Ted Shifrin

Well, you're a big help, @Balarka.

American Luke

np :P

Ted Shifrin

Luke probably is a majority stockholder in McDonalds or something ...

how many letters does it take to earn a ping, @DanielF? (Note that I now always put the F in your name so as not to incure DanielRust's ire.)

That was NOT star-worthy.

American Luke

18:39

@TedShifrin I wish :P

@ted

@te

@t

three it seems

Ted Shifrin

so it takes three, huh?

user3222184

Hi all

Ted Shifrin

hi @use

user3222184

does someone here know how to use mathematica?

Ted Shifrin

(Now all people with name @user..... will get pinged. GOOD.)

I hate to admit knowing anything in here, but, yes, @user322222222184.

Daniel Fischer

18:40

@TedShifrin I'm not sure about chat. In comments, you need at least 3, and if there are several matches, the least recent contributor is pinged.

user3222184

sorry for interrupting your conversion, because I got a quick question. If you have x+y=x^2, how do you solve in terms of y in mathematica?

Ted Shifrin

A college kid bitched at me when I commented on his comment that made absolutely no sense. :) He told me he wasn't done editing. Geez ... @DanielF

@user322...: Solve[x+y==x^2,y]

Oh, wait, in terms of y? You want to solve for x. Then put x at the end, not y. Sorry.

Daniel Fischer

@TedShifrin I wonder what became of the custom to finish editing before posting.

Ted Shifrin

Especially in comments, where one only has milliseconds @DanielF

Daniel Fischer

@TedShifrin You have more time for editing there than in chat.

user3222184

18:43

@TedShifrin, thank you

Ted Shifrin

Of course, the OP had garbage in his post, which I kindly corrected in my comment :) $(\operatorname{Col}(A))^\perp = (\operatorname{Nul} A)^\top$, eh?

Daniel Fischer

? No comprende.

Balarka Sen

@TedShifrin I noted that $\Bbb C(z)$ is the function field of $\Bbb P^1$ and $G = \text{Gal}(L/\Bbb C(z))$ is the deck transformation group of the (branched) galois covering $X \to \Bbb P^1$ for some Riemannsurface $X$. Another covering is $X_0 \to \Bbb P^1$ corresponding to $L'$. One has $\Bbb P^1 \cong X/G$ and $\Bbb P^1 \cong X_0/G$ so $X \cong X_0$.

Ted Shifrin

That was the garbage. But then the kid told the OP to do that in his "comment." Sigh. After I said it was nonsense, he said he hadn't finished editing. @DanielF

Whoa, in general, that can be subtle, @Balarka.

I don't think $X/G = Y/G \implies X=Y$ without something.

But I don't want to think about it now. I have to go back to the kitchen and finish decalcifying my espresso machine :D

Balarka Sen

@TedShifrin Hmm. Counterexamples?

@TedShifrin Okey-dokey.

Ted Shifrin

18:46

I don't have one off the top of my head, @Balarka. But I think I've fallen in that trap ... years ago.

I'll ponder later.

Balarka Sen

:15976286 So?

Ted Shifrin

You're assuming $G$ acts freely and properly or something? Otherwise, I can take $G$ acting trivially in one case and not in the other, for a stupid counterexample @Balarka.

Daniel Fischer

Yeah, not the same group.

Balarka Sen

@TedShifrin $G$ acts freely on the fibers of the cover, right?

So free action.

Daniel Fischer

@BalarkaSen Problem at branch points.

Ted Shifrin

18:48

Yes ... Just saying we need hypotheses for starters. But I still think it might be wrong. What if we have an example with $X/G \cong X$?

Balarka Sen

@DanielFischer What problem?

Daniel Fischer

@BalarkaSen Not all points above $p(\text{branch point})$ need to be branch points.

(In general, I don't know about the specific situation)

Balarka Sen

@DanielFischer Hmmm.

@TedShifrin Perhaps start by the fact that $X \to X/G$ is always a galois covering whenever $G$ acts on $X$ properly discontinuously.

Ted Shifrin

Yeah, you're doing covering spaces (well, branched). I'm thinking more generally, just in terms of homogeneous spaces. I am pretty sure it's wrong.

OK, outta here for now.

Balarka Sen

Bye @TedShifrin

Jasper Loy

18:58

There are not enough lhf for me.

G.T.R

@JasperLoy you missed one of mine earlier today

Jasper Loy

@G.T.R Nah, your questions are too hard for me, I have forgotten all the math I learnt, except high school stuff, lol.

@IlanAizelmanWS Nice haircut, lol.

Ilan Aizelman WS

@JasperLoy Oi!

Jasper Loy

@IlanAizelmanWS After my basic military training, I was a clerk in an office most of the time. So military service was not too bad for me.

Ilan Aizelman WS

@JasperLoy Awesome.

user3222184

19:05

Who is from Canda!?

Ilan Aizelman WS

@JasperLoy What do you tell the girls when they ask you what you did in the military?

user3222184

Anyone here from CANDA?

Ilan Aizelman WS

@JasperLoy Oh wait, nvm.

user3222184

NO CANADIAN>

Jasper Loy

@IlanAizelmanWS I tell them the truth, lol.

user3222184

19:05

?

Jasper Loy

Not many Canadians in this chat.

Ilan Aizelman WS

@JasperLoy The truth they want to hear?

G.T.R

@user3222184 The user JessicaK is

Jasper Loy

@IlanAizelmanWS Just the truth and nothing but the truth. I try not to lie.

user3222184

►

Jasper Loy

19:06

@user3222184 What is the video about?

@G.T.R I have searched in vain for a brand new copy in English of all the volumes of Dieudonne's Treatise On Analysis.

The last time I saw it was in my university library, six big books.

The problem with SE questions is that many of them have redundant commas. Commas trip up even native speakers.

There seems to be a slight problem with chat.

Daniel Fischer

@JasperLoy Which?

G.T.R

@JasperLoy lhf math.stackexchange.com/questions/827145/…

Jasper Loy

@DanielFischer Oh I keep having to refresh my browser. Maybe it is just me.

@G.T.R What book do you use in France for analysis course?

G.T.R

we don't have iconic recent books like they have in the US, but this one has a good rep amazon.fr/Analyse-MPSI-Jean-Marie-Monier/dp/2100498371 @JasperLoy

Jasper Loy

19:22

I just answered a lhf.

Jasper Loy

19:41

I finally got 1200 rep.

@mike I still think you are going to UCLA, lol.

nabla blah

Test

Jasper Loy

Wow, the user I hate most has cast over 1500 downvotes...

A @BalarkaSen

Balarka Sen

A

G.T.R

@JasperLoy Mr.hungry /angry ?

Balarka Sen

@JasperLoy How'd you know?

Jasper Loy

19:52

@G.T.R No idea who that is.

@BalarkaSen From his user profile.

Ted Shifrin

@Jasper: You'll respect me more when I downvote more :D

Jasper Loy

@TedShifrin Nah, I have reached the conclusion, somehow, that downvotes should be abolished from SE.

Mike Miller

Stupid conclusion.

Jasper Loy

@MikeMiller Flag, lol.

G.T.R

@JasperLoy This math.stackexchange.com/users/147263/words-that-end-in-gry

Balarka Sen

19:54

Did you find any counterexamples, Professor @Ted?

Ted Shifrin

Yeah, I disagree, but I use them sparingly. There are too many people posting wrong stuff, including me from time to time.

Mike Miller

Need to discourage bad questions and bad answers somehow.

Jasper Loy

@G.T.R OMG, only downvotes, no upvotes!

Ted Shifrin

I haven't been thinking, @Balarka.

Balarka Sen

I never downvote. I am barely active in MSE. Not enough questions of my type.

G.T.R

19:55

@JasperLoy I think he favorites things instead of upvoting

Jasper Loy

@G.T.R I think he is a nutcase, lol.

Ted Shifrin

Hmm, 1/4 of my votes are $\downarrow$

Jasper Loy

I have cast 0 downvotes.

Balarka Sen

@TedShifrin How'd you get that?

G.T.R

Look at the last edit here... math.stackexchange.com/questions/827184/…

Ted Shifrin

19:57

Basic arithmetic @Balarka

@Jasper: Have you ever seen Wedding Banquet?

Mike Miller

I'm surprised, mine are about 1/6 downvotes.

Jasper Loy

@TedShifrin No, if that is a show.

Ted Shifrin

Movie ... Early Ang Lee ... One of my favorites

Kevin Driscoll

I know I need to be downvoted sometimes. I told someone on Reddit the other day that if $U = e^{i x}$ then $U U^* = I$ for any $x$ in $\mathbb{C}$ facepalm

Transcript for

$Mathematics$ Mathematics