Mathematics - 2014-03-21

1:10 AM

is there an algebraic proof that the least squares polynomial of an even function is even?

1:32 AM

@Alex: I never thought about it, but least squares is given by an orthogonal projection, so it should follow from linearity.

Jessy Cat

@Jasper, I was taught by Protter 's son last semester.

skullpatrol

@JessyCat he is not gonna like the ' s

Jessy Cat

@skull, why not?

Why? What I wrote was correct.

skullpatrol

Protter's not Protter 's

Jessy Cat

Oh. Well, that was a typo.

Not an act of ignorance.

skullpatrol

1:38 AM

np

Jessy Cat

Anyway, I ranted and raved about grammar earlier today because some dude posted a question on here, and the guy obviously could not write his way out of a paper bag.

Then, Karl said some things, and then deleted them before anybody could see what he wrote.

skullpatrol

Like you said, we're not the grammar police :-)

Jessy Cat

I think he does that often, though.

So, I didn't take it personally.

skullpatrol

Why anyone would take anything personally on the internet is beyond me pal.

But that's just my opinion.

say what again

Mike

3:06 AM

@AlexanderGruber is there a rule about people posting multiple homework questions in a short span of time?

Alexander Gruber

@Mike no. if he keeps posting crappy questions, though, he'll be warned by the moderators.

Jasper Loy

4:33 AM

@AlexanderGruber Warning is a strong word there, considering this is not a crime, lol.

Morning @robjohn!

robjohn

@JasperLoy morning? it is 9:35 at night ;P

@JasperLoy good morning

user4140

Do you recommend I learn integration techniques? I'm in High school.

Mike

4:56 AM

@JasperLoy it should be imo

@user4140 what kind of math do you know? what classes have you taken?

Martin Sleziak

7:15 AM

@Mike If a user posts too many questions, they will run into problems with question quota 50 q/30 days, 6 q/day.

If you wish, you can leave a comment for that user about this, there is even a comment template.

Mike

Thanks, @MartinSleziak

Martin Sleziak

However, this is for posting many questions, whatever kind they are. Not specifically for homework questions, or any particular type of questions.

But I only see 2 questions on the profile page of that user. (And I am not entirely sure how the quotas work for unregisterd users.)

Mike

@MartinSleziak At the time of posting, he'd put up 2 questions within about an hour of each other asking for help with the same thing - though it was less asking for help, and more asking for a solution.

Anyway, if it discourages a user from posting a lot of low-effort questions in a shrot period of time, I'm all for it.

r9m

7:35 AM

@robjohn Can you check math.stackexchange.com/questions/718166/… ? .. my arguments look too cheesy .. too many loop holes :'( !!

and also how to solve a Delayed DE ?

Sawarnik

8:34 AM

@skullpatrol Do you know that the T20 World Cup is going on by the way?

skullpatrol

No, what sport is that, cricket @sawarnik?

Sawarnik

@skullpatrol Yup!

skullpatrol

Cool :D

Sawarnik

8:50 AM

What does := mean? Is it the same as =?

Karl Kronenfeld

@Sawarnik Yeah, I read it as "defined to equal"

Mike

@KarlKronenfeld I feel like I'm getting bogged down in topological annoyances at the start of Lee

Karl Kronenfeld

@Mike Hm, like what?

Mike

Does the fact that we have a countable precompact basis and shit like that actuall ymatter?

Karl Kronenfeld

yeah

Mike

8:52 AM

damn

Karl Kronenfeld

partitions of unity, etc

Mike

good point

the proofs are easy, they're just a tad annoying

skullpatrol

@swarnik some things are just true "by definition" like % := 1/100.

Sawarnik

Ok.

skullpatrol

Sorry @Sawarnik :-/

Sawarnik

9:04 AM

=) [I have such a silly name.]

skullpatrol

Sa-warn-ik

Saw-ar-nik

Karl Kronenfeld

@Mike You'll be happy to hear that Lee starts his discussions in the more general topological case whenever possible. :P

skullpatrol

Some authors like to define % as /100, but I prefer 1/100 because it makes more sense @Sawarnik, right?

Sawarnik

What is /100?

skullpatrol

"divided by 100"

That^ is the tradition way of teaching/learning it :(

Utkarsh

9:15 AM

_____

skullpatrol

*traditional

...and it is still taught that way, to this day.

Alex Youcis

9:38 AM

@Mike You on, brohamine?

Complex analysis

10:30 AM

@DanielFischer Hi :)

Daniel Fischer

Hi, @Complexanalysis.

Complex analysis

@DanielFischer for what $a$ does the limit of $\lim_{n\to \infty} \frac{(2n)!}{2^{n+1} (n!)^2}. n^a$ exist ? i can't get it.

Daniel Fischer

@Complexanalysis If you count $+\infty$, for all real $a$.

$$\binom{2n}{n} \sim \frac{4^n}{\sqrt{\pi n}}$$

Complex analysis

@DanielFischer I would like to exclude $+\infty$ .

Daniel Fischer

Then, for no $a$, @Complexanalysis. It's asymptotically equal to $$\frac{2^{n-1}}{\sqrt{\pi}}\cdot n^{a-1/2}.$$

Jasper Loy

10:43 AM

@robjohn I miscalculated.

Complex analysis

@DanielFischer Stirling gives me the approximation . ok ok. =)

Vrouvrou

11:24 AM

@robjohn see this please

Complex analysis

11:50 AM

what does tangent space to a matrix $M$ mean ?

robjohn

12:55 PM

@Vrouvrou so how does this cause a problem with my answer?

Vrouvrou

i dont understand what happen at infinity ?

robjohn

@Vrouvrou what do you mean? If the book says the things vanish, they vanish. That doesn't conflict with what I said.

@Vrouvrou I don't know what $H_k$ and $\beta_k$ are, so I only know what you tell me the book says.

Vrouvrou

1:14 PM

ah ok so they want to say that at infiniti $\sum M_k=\sum \beta_0=$

H_k is the the k-th singular homology

and $\beta_k$ is the beti number

user4140

1:35 PM

@Mike I took some classes in olympiad math because I was picked to represent my city. Other than that I self-learn random stuff all the time. But I haven't taken anything equivalent to a full university course as such.

skullpatrol

2:53 PM

@user4140 instead of choosing random stuff, try starting from the beginning of any good first year calculus textbook and systematically working your way through it.

Mats Granvik

@skullpatrol You just gave me inspiration to open my calculus book.

skullpatrol

:D

Américo Tavares

3:38 PM

Hi, everybody! I've just created the following Chat Room: Conversas sobre Matemática/Chatting about Mathematics (in Portuguese)
Chatting about Mathematics for Portuguese speaking users or users interested in learning Portuguese for Mathematics
http://chat.stackexchange.com/rooms/13749/conversas-sobre-matematica-chatting-about-mathematics-in-portuguese

skullpatrol

Thanks for sharing :-)

Américo Tavares

@skullpatrol You are welcome! Is "Chatting about Mathematics (in Portuguese) " grammatically correct?

Mike

yep

Américo Tavares

@Mike Thanks!

Boni Tea

3:54 PM

What is this "rarely if ever expressible as a ratio of integers" in the chatroom description referring to?

Mike

the joke is that we're usually irrational

Boni Tea

Ah. My first thought was that it is the ratio of general : maths questions mentioned just before that and that is obviously false. So it has to be something else.

skullpatrol

4:13 PM

Some of the questions are irrational too :-)

Balarka Sen

Is there any complex analytic proof of Mellin inversion? I wanted to ask this for a long time as I feel quite uncomfortable with real analysis, especially Fourier. CA is much more natural to me.

Jessy Cat

0

Q: Showing something is a smooth solution of a Laplace-like equation

Let $(M, g_{ij}$ be a compact Riemannian manifold. Let $f \in C^{\infty}(M)$ (I.e., $f$ is smooth on the manifold) be a given function. Consider the family of equations $\Delta u-4u^{3}+tf=0$ (*) for each $t \in [0,1]$. And let $I\equiv \{ t \in [0,1]; \text{ (*)}_{t} \text{ admits a smooth solu...

Jasper Loy

Hi @JessyCat!

Jasper Loy

4:35 PM

Brian Scott is getting so much rep even when not around, lol.

This chat is dead. All the kids are out partying this weekend.

Alizter

@robjohn Hi! Can you link me that triple sum with the i!j!k! I cannot find it on the main.

Jessy Cat

5:03 PM

Daniel Fischer might be able to help.

Enjoys Math

t('-' t)

skullpatrol

-_-

Mats Granvik

5:25 PM

I think I have an elementary proof of the Prime Number Theorem, given the version of the PNT involving the average order of the von Mangoldt function. Should I post it?

It is not exactly rocket science.

Wikipedia:
The prime number theorem is equivalent to the statement that the von Mangoldt function Λ(n) has average order 1

$$\displaystyle T = \begin{bmatrix} +1&+1&+1&+1&+1&+1&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&-2&+1&+1&-2&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&+1&+1&-4&+1&+1 \\ +1&-1&-2&-1&+1&+2&+1 \\ +1&+1&+1&+1&+1&+1&-6 \end{bmatrix}$$

You see the average of the rows is $1$ for the first row, and zero for the rest. The von Mangoldt function is in turn the sum: $$\Lambda(k)=\sum\limits_{n=1}^{\infty} \frac{T(n,k)}{n}$$

@skullpatrol What do you think?

skullpatrol

Sure @Mats posting it sounds like a good idea to me.

Mats Granvik

@skullpatrol Yes, but am I correct or not?

skullpatrol

Dunno much about number theory, sorry.

Mats Granvik

@skullpatrol Ok, but you do see that average of 1,1,1,1,1,1,... is 1, while average of 1,-1,1,-1,1,-1,1,-1,1,-1,... is zero?

skullpatrol

5:40 PM

Is that an infinity long string?

Mats Granvik

@skullpatrol Yes it is.

Américo Tavares

I renamed the room "Conversas sobre Matemática/Chatting about Mathematics (in Portuguese)" to the shorter title "Da Matemática/Chatting about Mathematics (in Portuguese)" so that "Da Matemática"(On Mathematics) appears in full. Link chat.stackexchange.com/rooms/13749/…

@robjohn I've created the Portuguese room about Mathematics chat.stackexchange.com/rooms/13749/….

skullpatrol

@Mats does the idea of "average" apply to such a long string?

robjohn

@AméricoTavares Good luck :-)

Américo Tavares

@robjohn Thanks!

Mats Granvik

5:46 PM

@skullpatrol By taking periods of the rows I believe it does.

skullpatrol

Hmm... interesting. The -1/12 argument on numberphile is similar.

Américo Tavares

@IanMateus I've created the room "Da Matemática/Chatting about Mathematics (in Portuguese)". Chatting about Mathematics for Portuguese speaking users or users interested in learning Portuguese for Mathematics here chat.stackexchange.com/rooms/13749/…

Mats Granvik

-1/12 is for the partial sums of the sequence, and some zeta function thing.

Partial averages of the sequence:
1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1,...
are:
1, 0, 1/3, 0, 1/5, 0, 1/7, 0, 1/9, 0, 1/11, 0,...
which tends to zero.

@skullpatrol

skullpatrol

I'm just saying one can look at an infinite sequence in an infinite number of ways.

Mats Granvik

Yes, infinte ways maybe. But the fundamental method of counting is the most analytic.

Fundamental method of counting: one, two, three, four,...

skullpatrol

6:02 PM

How about 1, 1/2, 1/3, 1/4,...

Jessy Cat

1

Q: Showing something is a smooth solution of a Laplace-like equation

Let $(M, g_{ij}$ be a compact Riemannian manifold. Let $f \in C^{\infty}(M)$ (I.e., $f$ is smooth on the manifold) be a given function. Consider the family of equations $\Delta u-4u^{3}+tf=0$ (*) for each $t \in [0,1]$. And let $I\equiv \{ t \in [0,1]; \text{ (*)}_{t} \text{ admits a smooth solu...

pde manifolds

@DanielFischer, perhaps you might be able to provide some insight?

Or @Tomas

robjohn

@Alizter are you talking about chat.stackexchange.com/transcript/message/12778725#12778725 ?

Alizter

@robjohn Yes I swear you have a post

I spent an hour trying to get it through a query :P

robjohn

@Alizter there is also chat.stackexchange.com/transcript/36?m=12777499#12777499

They would be in chat, but I don't exactly know where. I can post them to chat later, unless you want to ask on main.

Alizter

6:18 PM

@robjohn I remember there being a post that you wrote and you showed the solution

robjohn

I have to go proctor an exam now...

OK thanks!

Have fun.

@Alizter I will look

@robjohn It would be good to have a reference on the main

Jessy Cat

6:19 PM

I need somebody who is good with PDEs. Especially on manifolds.

Mats Granvik

@skullpatrol I was away to the shop. Yes that is similar: 1, 1/2, 1/3, 1/4,...

Daniel Fischer

@JessyCat Would $u_0 \equiv 0$ do the job? If not, I guess there are some theorems about $\Delta$ on a Riemannian manifold that say there is a smooth solution, but PDEs are not my cup of tea, I know very little about them, sorry.

Jessy Cat

@Daniel, I think it would b/c that's how we did an example in class. But how do I know that $u_{0}=0$ here?

Daniel Fischer

By inspection. $\Delta 0 = 0$ also on manifolds, I expect, so then you see it is a (the?) solution.

Jessy Cat

Right. That makes sense. Thank you @Daniel, as always, you are awesome! :)

skullpatrol

6:37 PM

@Daniel have you seen OldJohn around?

Daniel Fischer

Not today, @skullpatrol. But yesterday.

skullpatrol

I see, I see; thanks

I'm sure he'd be interested in the new math educators.SE site opening up soon.

Chris's sis

Greetings

Jasper Loy

@JessyCat The best books I have seen on PDE on manifolds is Michael Taylor's PDE (Vol 1, 2, 3).

@skullpatrol Hmm, I will take a look too, though I probably won't join.

Ethan

6:56 PM

@JasperLoy you there?

Jasper Loy

@Ethan Yes!

Ethan

@JasperLoy admissions.ucla.edu/applicant/notify.htm

I will get it sometime in the next 8 hours I think

Jasper Loy

@Ethan I see.

Mike

@Ethan Let me know your decision

Your result too

Jasper Loy

@Ethan OK, but UCLA and UChicago are both great.

Mike

6:58 PM

of course it does you no harm to wait until the April 15 deadline

@Daniel Can I ask how you learned all your math? A curiosity?

Jasper Loy

@Ethan No point doing such things, just wait.

@Ethan I wish I can redo my undergrad, lol.

Ethan

@JasperLoy why

Jasper Loy

@Ethan Well, I kind of didn't take the courses I should.

Ethan

why can't you teach yourself

Jasper Loy

@Ethan I can. So it's not a big deal, lol.

Jessy Cat

7:02 PM

@Jasper, take them now as a non-matriculated student at the best university near you. That will look good on grad school apps.

That's what I'm doing.

Jasper Loy

@JessyCat I see. I upvoted some of your posts just now. =)

Jessy Cat

@Jasper, thanks, guy!

Jasper Loy

@Ethan Actually, I find it best to learn myself from books. Anyway, take all the math courses you can as undergrad.

Daniel Fischer

@Mike A long time ago, I studied mathematics. Since then, I've occasionally read this or that book on topics I didn't study, some of it I have not yet forgotten.

Jessy Cat

@Ethan, where do you go now, just out of curiosity.

Mike

7:04 PM

@Daniel You certainly have better retention than most I know.

@Jessy He's a high school student looking at undergrad schools

Jessy Cat

@Ethan, oh you're in HS.

Jasper Loy

@Ethan I was in a good high school, but went to a bad university, lol.

@Ethan However, many people think that the university I went to is good, lol.

Jessy Cat

You're from the inner city; don't discount that. That gives you a better perspective than a lot of people.

Jasper Loy

Just take care of your mental health and soon you will be a great mathematician @ethan.

Jessy Cat

For one thing, I think you appreciate your opportunities more than a lot of the overprivileged undergrads who have had everything handed to them, and waste their time in uni partying and not going to class.

Ethan

7:07 PM

@JessyCat jasper or me haha

Jessy Cat

And then wondering why they're not doing well.

You, @Ethan.

Jasper Loy

I applied to Cambridge but did not get a place. I will try to go there next life.

Jessy Cat

@Jasper, any reason why you can't try again?

Jasper Loy

@JessyCat Well, I was talking about undergrad Cambridge. Now I will just apply to grad schools, if I do.

Jessy Cat

There's a guy in one of my classes who has a masters from Oxford. I'm not that impressed with him.

Ethan

7:10 PM

@JasperLoy at cambridge they give you an interview and ask you about what your majoring in

if they consider you I think, so thats good

Jasper Loy

@Ethan US and Canada would be my choices for grad school. Easier to get funding.

Jessy Cat

@Jasper, I think you should! You should apply to places in the states!

Jasper Loy

@JessyCat Yup, I will. And I hope to migrate there too, lol.

@JessyCat A Masters from Oxford is pretty impressive.

Jessy Cat

@Jasper, who knows? Maybe we'll end up in the same program.

Jasper Loy

@JessyCat Hehe, I won't be going there too soon, need to get better first.

Jessy Cat

7:13 PM

@Jasper, you think it would be, but I have to tell you sometimes things aren't as impressive as you think they should be.

Jasper Loy

@JessyCat My email is s8124939c@gmail.com, if you wanna email me.

Jessy Cat

:)

Jasper Loy

@ethan So do you go to university later this year?

Chris's sis

@JasperLoy the characters before @ look like the code of a prisoner. :-)

Jasper Loy

@Ethan You take SAT verbal and math right?

@Chris'ssis Hehe.

Ethan

7:14 PM

yea they don't count the new writing section

@JasperLoy I sent them a bunch of my work though lol

Jasper Loy

@Ethan They must be impressed!

@Ethan My only "work" is the paper I sent you. I am quite happy with it.

Ethan

not too impressed lol their are some really amazing people out there

Jasper Loy

@Ethan But I think you are amazing too.

@Ethan I got 680/800 for verbal and 790/800 for math, IIRC.

@Ethan Oh yeah I remember.

@Ethan The reason for my not doing too well in verbal is my extremely small vocabulary. I know very few words.

Chris's sis

I'd like to meet such amazing people, but in the real life, not on internet. I'd be curious to see how they would cope with my last 3 multiple integrals in terms of fractional part. (I mean those amazing people from MIT, Princeton,Harvard and so on)

Jasper Loy

@ethan LOL at your (removed).

@Ethan You can learn to abandon your fears slowly...

@Ethan Gradually do more and more things you cannot do, until you are completely free from OCD.

@Ethan It's alright as long as you are not committing a crime, lol.

@Ethan Well, if you did you would not even apply to X, so it does not matter.

:14417416 The university I went to was ranked 9th in the world by subject mathematics ranking last year on topuniversities.com, but to me the syllabus was crap.

Ethan

7:24 PM

I wouldn't necessarily trust any particular ranking system

Jasper Loy

But the Cambridge 3 year undergrad course is to me one of the best in the world because I am really impressed by the curriculum.

No where else in the world can you learn so many topics in 3 years.

Ethan

i don't understand

Jasper Loy

Don't understand what? I meant math topics, lol.

Ethan

oh

Jasper Loy

Math, math, and more math, and nothing else, lol.

I think forcing you to take other courses is pretty stupid.

Oh, Jessy has left, lol.

Bye @ethan, email me when you get news!

Ethan

7:35 PM

alright later

Mike

@Jasper Make sure to email me when Erhan gets news, lol

Chris's sis

(in this life I mean)

Ethan

I think I've read a paper which shows how to evaluate similar looking integrals lol

Chris's sis

@Ethan I really doubt ... (especially for the last one)

brb, I have some some things to do ...

Anthony

7:59 PM

How do you taylor expand at infinity?

Jasper Loy

OMG, so many (removed), lol.

sea turtles

8:33 PM

@Anthony taylor expand f(1/x) at x=0

for instance say f(x)=x/(x-1). then f(1/x)=1/(1-x) has maclaurin expansion 1+x+x^2+x^3+...
and therefore the original function has the taylor expansion f(x)=1+1/x+1/x^2+... around x=inf

Chris's sis

9:25 PM

@Ethan here is another cute limit you'd probably like :-) $$\lim_{n\to\infty} \frac{1}{n\log(n)}\sum_{k=1}^{n-1} \csc\left(\frac{k\pi}{n}\right)$$ (newly created)

Américo Tavares

9:37 PM

Da Matemática/On Mathematics (in Portuguese) chat.stackexchange.com/rooms/13749/…

robjohn

@Alizter I don't think I answered that on main.

Complex analysis

@Chris'ssis I would strongly suggest you to write a book on limit and integral problems =D

it would be super cool

Chris's sis

9:53 PM

@Complexanalysis :-))) Thank you. I really wanna write a book since I have lots of such problems, but ... who would read the book of someone with no background in mathematics :-(?

Complex analysis

I would @Chris'ssis

@Chris'ssis you will be a good mathematician :D

Chris's sis

@Complexanalysis OK, one reader! Thanks! :-)

Complex analysis

although don't make it too expensive :P @Chris'ssis

Chris's sis

@Complexanalysis Some copies can be given for free I suppose. Let's say one to you if this ever comes true. :-)

Complex analysis

In that case i would definitely want one , I would definitely buy one even if i don't get it for free @Chris'ssis

@Chris'ssis It will definitely come true i guess but hopefully soon enough :D

Chris's sis

10:01 PM

@Complexanalysis :D

Complex analysis

i guess you already have a soft copy of lots of problems and solutions

Chris's sis

I have some. Maybe you like this one I created yesterday. Prove that
$$\int_0^1\int_0^1\cdots\int_0^1 \log\left(\log\left(\frac{1}{x_1 x_2\cdots x_n}\right)\right) \log\left(\frac{1}{x_1 x_2\cdots x_n}\right)^{s} \ dx_1 \ dx_2\cdots dx_n=\frac{\Gamma'(s+n)}{\Gamma(n)} $$

where $s>-n, \space n\ge 1$

Complex analysis

@Chris'ssis looks dangerous , i feel scared by seeing such ones :P

Chris's sis

@Complexanalysis No, no danger around. This is meant to be aestetically nice. :-)

user4140

@JasperLoy have you seen the curriculum of UNAM?

Complex analysis

10:10 PM

@Chris'ssis I can't think of an idea on how to start .

Chris's sis

10:29 PM

@Complexanalysis Everything can be done in a very brilliant way (it's possible a dream solution). You don't have to do it now, I only showed you its beauty.

Complex analysis

@Chris'ssis yes :)

robjohn

@Alizter here is my post: chat.stackexchange.com/transcript/message/12844136#12844136

@Chris'ssis It would look nicer with $\Gamma(n)$ in the denominator

Chris's sis

@robjohn Yeah, far nice. Could you change that please?

@robjohn thanks!

user4140

How can I prove if in a binary relation a(ba)=b always then (ab)a=b always??

Chris's sis

10:48 PM