Mathematics - 2013-07-17 (page 1 of 2)

pourjour

00:29

how can I learn philosophic french?

anon

00:54

what is philosophic french?

Peter Tamaroff

01:12

@anon " Ceci n'est pas une pipe"

Gustavo Bandeira

01:23

@PeterTamaroff XD

Peter Tamaroff

@leo Missed you in chat today, sorry.

@GustavoBandeira =)

I lost so much rep due to capping yesterday! And the day before too!

amWhy

@PeterTamaroff boo-hoo... ;-)

Peter Tamaroff

@amWhy Cannot one find out how much rep one lost? I think I saw a code in SO some time ago.

amWhy

Yes, @Peter, I think so. I remember Brian M. Scott posting a link once in chat to a ranked list of users by total rep, including points lost to capping.

@PeterTamaroff So I imagine one can check it for total rep one would have had without a limit/cap, for any given day/week/month/all-time. You could then simply track down the corresponding "rep that counted" and subtract.

Peter Tamaroff

@amWhy Aha.

amWhy

01:39

@Peter Wouldn't it be a fun feature to have a "save for a rainy day" option...where, like, once day per month or so, one could cushion one's rep (that day) with accumulated rep that exceeded a daily cap, up to that (rainy) day's limit?

Now that all sounds awkward. Hopefully, you get the gist ;-)

Peter Tamaroff

@amWhy Heh, I like the ludic spirit.

robjohn

02:46

such a sleepy chat... Sleepy Hello ;-)

@amWhy I capped +60 day before yesterday, then got just the 60 yesterday :-)

amWhy

@robjohn Hello! Don't you just hate that! ;-)

Peter Tamaroff

03:11

@amWhy This is clear, yes? @robjohn

Peter Tamaroff

03:22

@amWhy Look

amWhy

+1, +1 Now spread the love ;-)

Peter Tamaroff

@amWhy ;) \meleaves

amWhy

03:51

@PeterTamaroff bye bye Peter.

robjohn

@PeterTamaroff nicely explained.

skullpatrol

06:43

@robjohn ;O...zzzZZZzzz...

Mats Granvik

11:14

I plotted this today:
$$\frac{1}{4} \left(1-\frac{1}{\exp ^{-1+(s+i r)^4}(4)}\right)$$

Reminds me of Paul Nylanders Newton Raphson fractals but I am not sure it is the same.

Mats Granvik

12:09

I notice now that this simpler expression gives a similar plot:
$$1-\frac{1}{2^{(s+i r)^4}}$$

Mats Granvik

12:59

The plots above are plots of the following expressions:
$1-\frac{1}{2^{(s+i r)^2}}$
$1-\frac{1}{2^{(s+i r)^3}}$
$1-\frac{1}{2^{(s+i r)^4}}$
$1-\frac{1}{2^{(s+i r)^5}}$
$1-\frac{1}{2^{(s+i r)^6}}$
$1-\frac{1}{2^{(s+i r)^7}}$
respectively.

Charlie

13:35

@skull hi...

:?

Peter Tamaroff

13:51

@MatsGranvik Oh! Pretty graphs!

saadtaame

14:07

Hello

skullpatrol

hi

saadtaame

I want to prove that $x+y>\sqrt{xy}/2$

oops

Peter Tamaroff

@saadtaame Start with $$(\sqrt x-\sqrt y)^2\geq 0$$

saadtaame

What should we talk about?

Peter Tamaroff

Unwrap that and see what happens.

saadtaame

14:10

@PeterTamaroff Thanks, I have a solution I just need somebody to check. I wrote: $$((x+y)-\sqrt{xy})^2 \lt (x+y)^2$$ and this gives result exactly. Right?

Peter Tamaroff

@saadtaame How so?

saadtaame

(x+y)^2 - 2(x+y)sqrt(xy) +xy < (x+y)2

simplify and you will get the required result

Peter Tamaroff

Ah, yes. That is a way.

saadtaame

@PeterTamaroff

Peter Tamaroff

Did you try what I gave?

You get $x-2\sqrt{xy}+y\geq 0$ after expanding.

saadtaame

14:15

@pe

Peter Tamaroff

That is $$\frac{x+y}{2}\geq \sqrt{xy}$$ as desired.

saadtaame

@PeterTamaroff Yes. Much better solution

Peter Tamaroff

@saadtaame Note that if $x=y$ we have equality.

saadtaame

@PeterTamaroff Is MathJax disabled?

Peter Tamaroff

@saadtaame Guess you don't have it enabled.

Mats Granvik

14:17

@PeterTamaroff Thanks.

Peter Tamaroff

NP.

saadtaame

@PeterTamaroff It works on the forums but not here. How do I enable it?

Peter Tamaroff

@saadtaame Here

saadtaame

@PeterTamaroff Thanks. I have a slow computer...I'd better leave it as is

Do you guys have mathematician idols?

skullpatrol

robjohn is mine

saadtaame

14:28

robjohn is a user of this network I assume. So why is it that he is your idol?

skullpatrol

because he is an ideal mathematician

saadtaame

Would you care to give more details. I mean, what makes him the man?

skullpatrol

ask him a question and find out

saadtaame

@robjohn What is math for you?

awllower

Hello everyone!

skullpatrol

14:36

hi

saadtaame

@awllower Hello!

awllower

:"D

saadtaame

Any topologists over here?

@Charlie What's that palm on your profile image?

skullpatrol

@saadtaame have you ever been to India, where they paint there palms?

saadtaame

@skullpatrol No.

skullpatrol

14:49

Peter Tamaroff

@skullpatrol What is that?

Is that you?

skullpatrol

Here

saadtaame

@PeterTamaroff Henna. It's not permanent though; it vanishes in days

Peter Tamaroff

@saadtaame Ah, they use it for tattoos in the beaches here.

saadtaame

@skullpatrol It's used a lot in north Africa in wedding ceremonies and the like.

skullpatrol

14:53

hmmm...interesting

I've never really read the wikipedia article on it.

saadtaame

@skullpatrol How did you imbed that image?

skullpatrol

@saadtaame chat.stackexchange.com/faq#formatting

There should be an "upload..." rectangle beside the "send."

Insert an image

from my computer from the web

click browse to choose an image from your computer

saadtaame

Right.

Peter Tamaroff

15:25

What is the sweetest way of proving $$\sum_{(k,n)=1\;;\;1\leq k\leq n}k\equiv 0\mod n$$

@Ethan maybe?

Ah, I think I know.

saadtaame

@PeterTamaroff What does $(k,n)=1$ mean?

Peter Tamaroff

@saadtaame That $k$ and $n$ are comprime.

saadtaame

@PeterTamaroff $gcd(k,n)=1$, is this a new notation?

Peter Tamaroff

@saadtaame It is shorter, that is all.

(Also, the result is true for $n=4,5,\ldots$.

Peter Tamaroff

15:42

Ah, nope.

saadtaame

Maybe show that $$\Sum_{(k,n)\neq 1} k \equiv n(n+1)/2 \mod n$$

N3buchadnezzar

Soup ?

saadtaame

Maybe show that $$\sum_{(k,n)\neq 1} k \equiv n(n+1)/2 \mod n$$

Peter Tamaroff

Ah, we should pair numbers up with their unique inverse.

Note that we can write them like this $$\eqalign{
& \left\{ {1, - 1} \right\} \cr
& \left\{ {1, - 1} \right\} \cr
& \left\{ {1,2, - 2, - 1} \right\} \cr
& \left\{ {1, - 1} \right\} \cr
& \left\{ {1,2,3, - 3, - 2, - 1} \right\} \cr} $$ where I'm writing things $\mod n$.

And note that this works because $\phi(n)$ is even for $n\geq 3$.

Jayesh Badwaik

16:04

@PeterTamaroff How goes it?

Chat Rules | MathJAX for Chat

3

Peter Tamaroff

@JayeshBadwaik Good.

Jayesh Badwaik

Good.

Peter Tamaroff

@JayeshBadwaik =)

What are you up to?

Jayesh Badwaik

@PeterTamaroff =) I have started my graduate school, we have a refresher course in first 15 days. Kind of bring upto speed stuff in undergrad math. Solving a real analysis assignment currently.

Peter Tamaroff

@JayeshBadwaik Nice!

And...?

Jayesh Badwaik

16:09

Courses start from 1st aug.

And? ?

skullpatrol

good luck :-)

Jayesh Badwaik

thanks.

Peter Tamaroff

@JayeshBadwaik What is the assignment?

Jayesh Badwaik

@PeterTamaroff A host of problems from all over analysis, sequence, integrals, uniform convergence. some 15 problems in total.

Peter Tamaroff

@JayeshBadwaik Ah! Any trouble?

Jayesh Badwaik

16:11

Not really too hard. Basic ones.

@PeterTamaroff Not really. :-)

Peter Tamaroff

@JayeshBadwaik Heh, not surprised.

Jayesh Badwaik

We are also doing the proof of the fact the there exists a ordered field with least upper bound property, and that it is unique upto isomorphism and dedekind cut is one construction of it. I think the proof is very basic, but we are doing everything from set theory. That part is kind of a refresher in mathematical communication. So, it is kind of long.

saadtaame

@PeterTamaroff I did not understand your solution

Peter Tamaroff

@saadtaame Well, the sum is actually $$\frac{n\phi(n)}2$$

The proof can go as follows.

saadtaame

What is $\phi (n)$?

Peter Tamaroff

16:18

As I said, $\phi(n)=2m$ when $n\geq 3$.

@saadtaame Euler's Totient.

N3buchadnezzar

Do anyone know the problem with measure the amount of liquid in a barrel?

Eg drop a stick into semi full barrel, to measure its volume.

It is quite a well known problem, but I can not seem to find the solution to the problem.

Peter Tamaroff

16:36

@N3buchadnezzar You see how much the water level rises, brahw.

N3buchadnezzar

You can not see into the barrel

robjohn

@skullpatrol but I am an analyst, I don't do ideals ;-)

Peter Tamaroff

@N3buchadnezzar Why not?

Are you blind?

robjohn

@saadtaame to me, math is fun. It is the opportunity to push myself and help others to do the same.

2

N3buchadnezzar

@PeterTamaroff The barrel is closed..

skullpatrol

16:47

@robjohn when you teach first year calculus students that infinity is not a number, how would you logically present to them -infinity < x < +infinity?

robjohn

@skullpatrol $x$ is a real number.

skullpatrol

@robjohn Yes.

N3buchadnezzar

Antonio Vargas

I'll show you a real number...!

robjohn

@skullpatrol As I said, $+\infty$ is a point in the topological two point compactification of $\mathbb{R}$. The reason we use a two point compactification for $\mathbb{R}$ and a one point compactification for $\mathbb{C}$, is that we can use $\lt$ and $\gt$ to describe subsets of the two point compactification of $\mathbb{R}$. Since there is no order on $\mathbb{C}$, we use the one point compactification.

N3buchadnezzar

16:53

@AntonioVargas Platon would turn in his grave

robjohn

@skullpatrol Therefore, $-\infty\lt x\lt+\infty$ describes the standard reals as a subset of the two point compactification of $\mathbb{R}$.

skullpatrol

@robjohn Thank you, I did not know that the reason was the use of < and > to describe subsets.

N3buchadnezzar

@robjohn Any first year highschool student would totally understand and accept this answer ;)

robjohn

@skullpatrol well, that is my take on it, at least. Others may have differing opinions.

@N3buchadnezzar Yeah, but they listen to proof by intimidation, so it's cool...

skullpatrol

lol

N3buchadnezzar

17:02

"Why I study mathematics? Well at highschool, my teacher robjohn threatened to beat me if I did not accept his proofs. That was when I really started appreciating abstract mathematics"

3

:p

skullpatrol

Proof by intimidation

Proof by intimidation (or argumentum verbosium) is a jocular phrase used mainly in mathematics to refer to a style of presenting a purported mathematical proof by giving an argument loaded with jargon and appeal to obscure results, so that the audience is simply obliged to accept it, lest they have to admit their ignorance and lack of understanding. The phrase is also used when the author is an authority in his field presenting his proof to people who respect a priori his insistence that the proof is valid or when the author claims that his statement is true because it is trivial or becau...

Little Child

Anyone familiar with Fourier Integral Representation ?

Peter Tamaroff

@LittleChild Explain, please.

Little Child

I am attempting to solve a question and I am confused with the notation

Peter Tamaroff

@LittleChild OK, keep it coming.

Little Child

17:14

0

Q: Learning Fourier Integral

I am learning Fourier Integral for real numbers. I downloaded a presentation of some university. I am summing up what I know and hopefully someone will correct me where I am wrong. Fourier integral can be calculated only for functions that are decaying (what ever that means) and non-periodic ...

$\lambda$ is ?? 0,1,2,3.... right ?

So, here is the problem I am solving:

$f(x)$ is $x$ when -1 < x < 2 and 0 elsewhere. Calculate Fourier Integral Representation @PeterTamaroff

Peter Tamaroff

What is $A,B$ in that question?

Little Child

-1 and 2

oh sorry that

$A= \frac{1}{\pi}\int_{-\infty}^{\infty}f(x)Cos(\lambda x)dx$

B is the same. Cos becomes Sin @PeterTamaroff

I am unable to evaluate the integral after calculating it

brb

robjohn

what is the difference between evaluating and calculating?

Victor

@robjohn - Are you good in Calculus?

Had you seen that variance appear in taylor's expansion?

robjohn

@PeterTamaroff: I have added a Motivation section to my answer that might help explain why I chose the approach I used.

@Victor pardon? what variance, where?

Victor

17:27

equation (5.5) in i.sstatic.net/j0dyh.png

robjohn

@Victor Is that some actuarial text? The terminology is hard for me to understand.

Victor

@robjohn - Yes, however, how come variance could appear in some taylor's expansion?

Little Child

@robjohn by that I mean .. substituting in the values :)

@PeterTamaroff were you able to get the answer ?

robjohn

@Victor well, given a probability distribution $\phi(x)$, the variance is $$\int (x-x_0)^2\phi(x)\,\mathrm{d}x$$

@Victor where $x_0=\int x\phi(x)\,\mathrm{d}x$

Peter Tamaroff

@LittleChild Sorry, I don't know about Fourier.

robjohn

17:37

@Victor If you take the fourier transform of $f$, the variance becomes the second derivative of the fourier transform of $f$

Peter Tamaroff

@robjohn Ah, Laplace's method, aye?

Little Child

@PeterTamaroff ok :)

robjohn

@Victor and the second derivative is part of the Taylor expansion. This is actually the basic idea behind some proofs of the Central Limit Theorem

Little Child

@PeterTamaroff $\frac{1}{\pi}\int_{-1}^{2}x.Cos(nx)dx$

robjohn

@LittleChild integrate by parts

Little Child

17:42

@robjohn Did.

Peter Tamaroff

@LittleChild I think you want $\cos(\pi nx)$.

Little Child

I dont know if the formula is right (ref. the question I linked to)

The formula just says Cos(nx)

robjohn

@PeterTamaroff Then they wouldn't need the $\frac1\pi$... it is weird

Victor

@robjohn - Thank you, May you answer more completely on my put on hold question

-1

Q: How to derive the variance premium formula

How to derive formula (5.5) with Taylor's expansion in the following link? https://i.sstatic.net/j0dyh.png

calculus actuarial-science

Little Child

@PeterTamaroff people.ucsc.edu/~lewis/Math140/Fourier_transf.pdf

Peter Tamaroff

17:45

@robjohn But, what period does the function have?

robjohn

@Victor As I said, I don't know most of the terminology used there. I will look and see if I can say anything intelligent.

Little Child

Fourier Integral Representation is for non-periodic function, I guess

robjohn

@PeterTamaroff as I said, it is weird.

Little Child

The formula is wrong? @robjohn

Jayesh Badwaik

@robjohn Please pin chat rules. :-) Thanks

robjohn

17:47

@LittleChild well, in general, as Peter says, you want to integrate over a whole period of the sin and cos...

Jayesh Badwaik

@PeterTamaroff How do you type the little o notation in $\LaTeX$?

robjohn

Etiquette Guidelines | $\LaTeX$ support for chat

6

Peter Tamaroff

@JayeshBadwaik Just write o(x)

Jayesh Badwaik

I see.

Peter Tamaroff

Alternatively, use \omicron.

Jayesh Badwaik

17:49

Now, that's what I needed. :-)

Little Child

@robjohn if I made the PDF I am using as reference available, will you explain to me what is going on ? :)

robjohn

$o(x)$ vs $\omicron(x)$ hmmm

Peter Tamaroff

@robjohn There must be a difference!

Arkamis

They look identical in STIX fonts

Peter Tamaroff

@anon $$\prod_{\zeta \in \Bbb G_n^{\times}}\zeta=(-1)^{\varphi(n)}$$ yes?

Little Child

17:52

@robjohn ??

robjohn

@LittleChild is this a book or class notes?

Little Child

Looks more like someone made a slide show, converted it into PDF and uploaded it

There r examples n graphs too

robjohn

@PeterTamaroff That is $-1$ only for powers of two, right?

anon

@PeterTamaroff if that means over prim roots of unity then yes

robjohn

I think $\varphi(n)$ is odd only for powers of two

Peter Tamaroff

17:55

@anon Yes.

@robjohn Nope, just for $2$.

$\varphi(n)$ is even for $n\geq 3$.

robjohn

@PeterTamaroff ah, of course

Peter Tamaroff

@anon What is the good notation?

robjohn

there is the $(p-1)p^{e-1}$

anon

I'd use $\langle\zeta\rangle=\mu_n$ probably.

Peter Tamaroff

@anon What is that?

anon

17:57

$\mu_n$ is notation for $n$th roots of unity

robjohn

@LittleChild where is it uploaded?

Little Child

Putting it on mediafire. Wait

Do you guys use cloud storage ??

skullpatrol

You can store clouds? :D

Arkamis

Just put it somewhere easily and commonly accessible

Peter Tamaroff

@anon What do you think?

Little Child

18:00

@Arkamis oh hello there, astronaut

Arkamis

Why am I an astronaut?

Little Child

just saying. You said you were into aeronautics.

Arkamis

Most of my interests stay below LEO.

skullpatrol

@Arkamis How about putting it in the patriot's super bowl trophy case, that is easily accessible to the giants :D

robjohn

@skullpatrol yes, but it's quite nebulous how to access it...

Little Child

18:02

mediafire.com/view/34ftzl03c12czy4/Fourier_integral_intro.pdf

@Arkamis

Well... use copy.com for cloud storage.

you start with 15gb. You refer people and get 5gb more

:)

So if you get all your friends n family to use it, you can get a lot if free space

@robjohn See the link

Arkamis

@skullpatrol >:(

anon

@PeterTamaroff I am not a fan of all the letters

robjohn

@PeterTamaroff the integral is over all $\mathbb{R}$ but the function is $0$ off that interval $[-1,2]$

Little Child

@skullpatrol Of course you can store the clouds. I keep them in a jar. Their molecular formula is $H_2O$

Atomic mass is like.... (2X1 + 1x16) = 18

Angstrom!

skullpatrol

@LittleChild But how do you count the number of clouds?

Arkamis

18:08

@anon I am going to use that exact phrase whenever I come across any article or opinion I don't like.

robjohn

@LittleChild see what I said to Peter

Little Child

The weight of water is 1gm / 1cm^3. You can estimate it. Get the weight in grams.

robjohn

Darn, the links take you to the transcript even if the reference is still on the screen. >8(

Little Child

@Arkamis u here ?

skullpatrol

@robjohn It is best to use the "drag-down" option then.

Arkamis

18:11

Yes

Little Child

so .. I am attempting to solve the first example..

and I cant

Arkamis

What are you having trouble with?

Little Child

The final answer .. of $A(\lambda)$

Arkamis

Well, that's not the final answer, but an intermediate step

Little Child

wait.. let me calculate.

I will show you where I am stuck.

Arkamis

18:14

Just integrate by parts

Little Child

doing

Arkamis

Remember, in computing $A(\lambda)$ you're integrating over $x$, so $\lambda$ is constant

Little Child

ok, here is what I get.. without plugging the upper and lower limit

$\frac{1}{\pi}[\frac{xSin(\lambda x)}{\lambda} + \frac{1}{\lambda^2}Cos(\lambda x)]$

@Arkamis

Arkamis

That's not right.

Little Child

err... ?

Arkamis

18:22

Or, maybe it is, hang on

Little Child

woohoo !! got the answer !!

It is right !!

I iz happy.

Arkamis

Yeah, I missed a sign

Yeah that's right

Now plug in the limits

Little Child

did. Got the answers

robjohn

$$
\begin{align}
\frac1\pi\int_{-1}^2x\cos(x)\,\mathrm{d}x
&=\left.\frac1\pi x\sin(x)\right]_{-1}^2-\frac1\pi\int_{-1}^2\sin(x)\,\mathrm{d}x\\
&=\left.\frac1\pi x\sin(x)+\frac1\pi\cos(x)\right]_{-1}^2\\
&=\frac1\pi(2\sin(2)-\sin(1)+\cos(2)-\cos(1))
\end{align}
$$

Arkamis

After 15 years, I still miss the damn negative sign when integrating by parts.

Little Child

18:28

@Arkamis chill, bro. Math is wicked

I calculated both. A and B

now just substitute the values in integral formula ?

Arkamis

Yeah

then do it all over again

Little Child

DO. IT. ALL. OVER. AGAIN. ???

Arkamis

You know, lots of fun integrals

Peter Tamaroff

@LittleChild Yay! Computations be tough. Bitches be crazy.

Arkamis

We all love integrals

Little Child

18:30

@PeterTamaroff tough*

:p

Peter Tamaroff

@LittleChild I salute you.

Arkamis

Naw, though. Be was an active verb!

It's deep philosophical chat here today.

2

"Computations be."

woah

Little Child

Let computations be. Leave them alone.

Arkamis

You can't not be on a boat.

Little Child

►

Arkamis

18:33

►

Peter Tamaroff

@LittleChild HAHAHAHAHA

Little Child

I apply rigorous standards at home. I see no reason why they should not be applied at workplace

►

Peter Tamaroff

@LittleChild I prefer this

Note how I used [TEXT](link)

Little Child

oh ok :)

Inglorious Basterds

@Arkamis Still here ?

Arkamis

Nope

Little Child

18:44

then who is ?

Arkamis

Someone must be.

Little Child

$\frac{Sin\lambda}{\pi\lambda^2}[\lambda-1] + \frac{Cos 2 \lambda}{\pi \lambda^2}[1-2\lambda]-\frac{Cos\lambda}{\pi \lambda^2}[\lambda + 1] + \frac{Sin 2 \lambda}{\lambda^2 \pi}[2\pi + 1]$

@Arkamis Now, I must integrate that

Arkamis

Looks like a chore.

Little Child

These are all constants so all I need to do is add an $x$ right ?

Arkamis

No

Little Child

18:47

why ?

Arkamis

Now you're integrating by $\lambda$

Little Child

awwwh ok.

Maybe I am not required to integrate that in exam :)

@Arkamis How old r u ?? : )

Arkamis

Integrating it should not be a problem.

Actually, it's basically the same result as what you've found previously

Little Child

@Arkamis what do u mean ?

Arkamis

$\int y \cos ky\ dy$

Or $\int y\sin ky\ dy$

Little Child

18:49

so I must leave it at that ?

Arkamis

In the first case, you integrated with respect to $x$

Now you're integrating with respect to $\lambda$

So in the first case, you had $y = x$, now you have $y = \lambda$. Basically, you'll be doing the same basic integral a few different times.

Try it and see.

$y$ is just a dummy variable I made up to illustrate the point

Little Child

ok

Arkamis

Just do the integrals out and see what happens

It's easy, just tedious.

Little Child

The $\frac{1}{\pi}$ comes out

$\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{Sin\lambda}{\lambda^2}[\lambda - 1]$

@Arkamis correct ?

Arkamis

Yeah

Actually, I forgot about that $1/\lambda$

So it's not the same integral, but it's similar enough.

Little Child

18:54

It is going to be 8 different integrals

@Arkamis $\int\frac{Sin\lambda}{\lambda}d\lambda$ How do I proceed ? Integrate By Parts ?

Arkamis

I don't know. Maybe. Look it up in an integral table. I find very little actual value in memorizing the forms of antiderivatives.

Transcript for

$Mathematics$ Mathematics