Mathematics - 2017-05-29 (page 3 of 7)

yeah right! but how do we know which function is coresponding to $u$ and $v$ since it is defined suppose f=u+iv then blah blah now i dont have an imaginary part

BAYMAX

how do we know which function is coresponding to u and v ?

we assumed that $f = u(x,y) + i v(x,y)$

Manolis Lyviakis

11:37 AM

ii mean $ |f|=u^2+v^2$ then ill take that $im|f|$=0 so $\frac{d(u^2+v^2)}{dx}=odx$

BAYMAX

why so differential operators ?

Manolis Lyviakis

i mean aprtial

partial

BAYMAX

see that $uu_{x} = vv_{y}$

right?

Manolis Lyviakis

so $2uu_x+2vv_x=0$

BAYMAX

please swim with me

that is please follow as i go

Manolis Lyviakis

11:39 AM

ok

BAYMAX

see $uu_{x} = vv_{y}$

Manolis Lyviakis

sorry for the incovinience

BAYMAX

no problem

:)

and $uu_{y} = -vv_{x}$

now apply $u_{x} = v_{y}$ and $u_{y} = -v_{x}$ to these

in the first equation that s $uu_{x} = vv_{y}$

you get $u = v$

Manolis Lyviakis

ok so far

just how did u got the first one

$uu_x=vv_y$

you applied CR eq to |f|

how ?

BAYMAX

$|f| = u^2 + v^2$

and $u,v$ are funcn of x,y

Manolis Lyviakis

11:43 AM

so real part is $u^2+v^2$ and imaginary is 0

BAYMAX

so $u_{x}$ = partial derivative

yeah

$u_{x} = (u^2)_{x} + (v^2)_{x}$

$u_{x} = 2uu_{x} + 2vv_{x}$

it is chain rule there

Manolis Lyviakis

yeap

BAYMAX

Just practice you get it!

Manolis Lyviakis

i wrote that too

BAYMAX

then

I have little time

sorry if i was wrong somewhere

Simply Beautiful Art

11:45 AM

I found something very interesting: perhaps @AkivaWeinberger @Astyx @SBM @anyone_else_who_may_find_this_interesting

53

A: How to solve $f(f(x)) = \cos(x)$?

The half-iterate of a function can be found by expressing its superfunction in a form of Newton series: $$f^{[1/2]}(x)=\sum_{m=0}^{\infty} \binom {1/2}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{[k]}(x)$$ Where $f^{[k]}(x)$ means k-th iterate of $f(x)$ This series converges if two criteria are met: 1)...

BAYMAX

may be potential users can help you

Manolis Lyviakis

so i ahve $u_x=0$

BAYMAX

bye

Manolis Lyviakis

byee

thanks alot

BAYMAX

please carry out calculations properly

my pleasure!

Secret

11:57 AM

51

Q: What is mathematical research like?

I'm planning on applying for a math research program over the summer, but I'm slightly nervous about it just because the name math research sounds strange to me. What does math research entail exactly? For other research like in economics, or biology one collects data and analyzes it and draws co...

I am mostly a theory builder in the spectrum of mathematical research

BAYMAX

i am back :)

you got that @ManolisLyviakis

Secret

I build or extend algebraic structures for various purpose:
1. Explore the implication of a set of axiomatic systems
2. Visualise the geometric properties of said algebraic structure
3. Create alien universe laws for worldbuilding purposes
4. Explore some approaches in solving mathematical problems

Having said that, my research is amateur, because I don't have as much formal training in maths compared to my physics and chemistry

BAYMAX

Interesting @Secret

Sha Vuklia

Guys, I am given the following function:
$$
\phi(k)=\sqrt{\frac{a}{\pi}}\frac{\sin(ka)}{ka}.
$$
Now in my physics book, they say that $\sin z/z$ has it's maximum at $z=0$. But I don't understand why, for I thought that $0/0$ was undefined. I'm guessing $0/0=1$ after all? A quick google search seems to give that $0/0$ is undefined. So would anyone know what the convention is?

Secret

$\frac{0}{0}$ is undefined but $\lim_{z\to 0}\frac{\sin z}{z}=1$, which is a well known calculus result

Simply Beautiful Art

12:08 PM

@ShaVuklia It probably meant:
$$\phi(k)=\begin{cases}\sqrt{\frac a\pi}\frac{\sin(ka)}{ka},&k\ne0\\\lim_{x\to0}\phi(x),&k=0\end{cases}$$

Alessandro Codenotti

They're extending the function by continuity

Sha Vuklia

ahh, of course

Manolis Lyviakis

im stuck @baymax here is where since $|f|=u^2+v^2$ the real part is $u^2+v^2$ and imaginary is zero . SO the cauchy rieman eq would be partialderivative of x of the real part=partial derivative of y of imaginary part. thus $2uu_x+2vv_x=0$ thus $uu_x=-vv_x$ not the one you got.!!

Sha Vuklia

right right, thanks @Secret, Alessandro, Simply

Simply Beautiful Art

:-)

Manolis Lyviakis

12:09 PM

where am i wrong?

Secret

Weird result when integrating by parts
$$\int \frac{f'(x)}{f(x)}dx=\frac{f(x)}{f(x)}-\int\frac{-f'(x)}{[f(x)]^2}f(x)dx$$

$$x\neq 0, \int \frac{f'(x)}{f(x)}dx=1+\int\frac{f'(x)}{f(x)}dx$$

$$0=1???$$

Simply Beautiful Art

@Secret Nah, constant of integration

SteamyRoot

^

It's a standard "proof of 1 = 0" by taking $f(x) = x$.

Manolis Lyviakis

always that $c_o$

Secret

Ah... , and it absorb away the numbers

[Philosophy bomb] What is not mathematics in the most abstract sense?

Background: We seemed to have a pretty good idea on whether something is part of mathematics, thus it wonders me how do we actually deduce that

(and it is also complicated by the fact that not all mathematical structures are based on sets...)

Manolis Lyviakis

12:20 PM

which mathematical structures are not base on sets?

Secret

For example:

Type theory

In mathematics, logic, and computer science, a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics. In type theory, every "term" has a "type" and operations are restricted to terms of a certain type. Type theory is closely related to (and in some cases overlaps with) type systems, which are a programming language feature used to reduce bugs. Type theory was created to avoid paradoxes in a variety of formal logics and rewrite systems. Two well-known type theories that can serve as mathematical foundations are...

They are used in computational science papers when they propose new algebraic structures that the authors often claim to simplify algorithms

But then again, I only very briefly came across these thus I cannot said much about them other than they are quite useful in formulating axiomatic systems

BAYMAX

ohk@ManolisLyviakis I think I messed it up in between

so let us try

first of all it is given that $|f(z)|$ is analytic

so let $f = u + iv$

then $|f| = u^2 + v^2$

then $(u^2 + v^2)_{x} = 0$

and also $(u^2 + v^2)_{y} = 0$

so $2uu_{x} + 2vv_{x} = 0$

and

$2uu_{y}+2vv_{y} = 0$

also i think we can conclude that $f$ is analytic!

so $u_{x} = v_{y}$ and $u_{y} = -v_{x}$

so $2uv_{y} - 2vu_{y} = 0$

$\frac{u}{v} = \frac{u_{y}}{v_{y}}$

and from the other equation

$\frac{u}{v} = -\frac{v_{y}}{u_{y}}$

implying $u^2 + v^2 = 0$

$u = 0 , v = 0$

so $f = 0$

Ok now I think we should use Liouvilles theorem only

this goes pretty mind heating :)

Manolis Lyviakis

1:10 PM

ohh nice now i fully got it @BAYMAX

we got a problem though we just proved that every f that is analytic it is constant. we didnt use the fact that |f|>1 @BAYMAX

gebra

1:27 PM

is it just me or does this question not make any sense math.stackexchange.com/questions/2300997/…

Waiting

1:56 PM

@Hippalectryon o/

Sha Vuklia

Guys, one question about the intuition behind this “physics proof” of Plancherel’s theorem. So we have a function $f(x)$ on an interval $[-a,a]$. I’m assuming that by expanding it as a Fourier series, we repeat the function infinitely often, so the function has the same shape at $[a,2a]$ for instance.

If that’s correct, then I’m confused what happens when we take $a\to\infty$. Does that mean we stretch the function infinitely much? That would leave of with a nearly flat function, I would think. So I’m guessing that instead of repeating the function infinitely often, we actually just increase its interval, but we just define it to be zero outside of $[-a,a]$. Is that the right interpretation? How about the Fourier series extension then?

(also, hi @Waiting !)

Manolis Lyviakis

2:21 PM

math.stackexchange.com/questions/1902231/… is that correct where can i find the formula for |z|>1

Secret

The complex geometry of Islamic design - Eric Broug

►

Simply Beautiful Art

@Semiclassical chat.stackexchange.com/transcript/message/37722744#37722744

Semiclassical

2:39 PM

@ShaVuklia not going to be able to answer right now, but remind me again later

Kane

hey guys, anybody know about footrule distance between two vectors?

im trying to find minimum footrule for two lists of rankings

Simply Beautiful Art

@TheGreatDuck Have you ever dealt with half iteration functions?

TheGreatDuck

@user243301 use the actual chat where this supposed comment exists instead of randomly going on here and spouting nonsense.

@SimplyBeautifulArt no and I'm just hopping on here to check notifications. In 5 minutes we're leaving

Simply Beautiful Art

Like $f(f(x))=e^x$

Astyx

@Sha I think you're looking at the mean value of $f$ on $[-a,a]$, no ?

barrycarter

2:58 PM

@SimplyBeautifulArt I've played with them, but nothing concrete.

Simply Beautiful Art

@barrycarter chat.stackexchange.com/transcript/message/37722744#37722744

barrycarter

@SimplyBeautifulArt Interesting, though a bit (a lot) above my level.

Simply Beautiful Art

Yeah

barrycarter

My from scratch approach would be power series, but that's probably wrong.

Simply Beautiful Art

@barrycarter You can prove $f(x)$ is not analytic from $f(f(x))=e^x$

barrycarter

3:02 PM

@SimplyBeautifulArt Well, so THAT won't work.

@SimplyBeautifulArt My "goto" example is f(f(x)) = x^2 + 1

Simply Beautiful Art

xD

but the link works for any iteration on a function

assuming you follow the conditions

barrycarter

Yes, the question seems to have been generalized

Simply Beautiful Art

quite beautifully too

SteamyRoot

@SimplyBeautifulArt This is contradiction with the answers here

barrycarter

I think it's been around for a while. Ages ago, I was looking for a functor with the property: B(f(g)) = B(f)*B(g) or something similar. A composition to simple function functor. Never found one

Simply Beautiful Art

3:07 PM

@SteamyRoot It's actually an approximation to an $f(x)$

@SteamyRoot where is the contradiction though?

barrycarter

I did consider: f0(x) = x, and then iterating with Sqrt(fn(x)*e^x) or something-- some sort psuedoaveraging.

SteamyRoot

The fact that there is an analytic function $f(x)$ such that $f(f(x)) = \exp(x)$ ?

barrycarter

The approximations are analytic, but there's no guarantee the limit exists or is analytic.

Simply Beautiful Art

@SteamyRoot the given link is not analytic I think

SteamyRoot

The second answer refers to a paper that is exactly that...

Waiting

3:12 PM

@ShaVuklia Hi awesome user @ShaVuklia! And sorry for the delay. I was in the middle of a meeting with an important project of mine.

@ShaVuklia how are you doing lately? :P

Simply Beautiful Art

:D

Sha Vuklia

@Waiting haha, well I got exams this week, so not sure what to say!:P but I think overall I'm doing well, how are you?

@Astyx I'm not sure?

I didn't really think of it in terms of mean value to be honest

@Semi and thanks, I will!

SBM

analytic?

Waiting

@ShaVuklia Glad for you!:P You wouldn't like to know the whole pressure here!:P I've been preparing the last steps for a very important project of mine. It's a crazy atmosphere here, pressure all over the place. :P Is everything just fine?

SBM

All the best for exams @ShaVuklia

Sha Vuklia

3:17 PM

haha yea I'm allergic to pressure @Waiting :P but good luck to you!!

thanks @SBM I will need it:(

Astyx

@Sha Don't listen to me, I'm a pile of garbage today

Sha Vuklia

right @Astyx :P

SBM

Don't get into self depression @Astyx

It does no good.

Astyx

I'm not, thanks for caring though :)

Waiting

@ShaVuklia haha, well, after a while you become a friend with it which is not that bad, rather than thinking all the time how to get rid of it. :P

@ShaVuklia In the evening I'll go jogging which is amazing to me!

Simply Beautiful Art

3:20 PM

@barrycarter xD Don't think that would work

@SteamyRoot Hm

Astyx

Jogging in the evening is amazing, almost as much as jogging at dawn

Waiting

:P

@Astyx True!

Secret

In maths stuff, simpleart has verified that my recent attempt at understanding the ordinal collapse function is fine so far

As for my PhD project, there's one stubborn calculation that refuses to work. Since no one in my group had ideas on how to solve it (and CSE people are also suggesting there is no way around it) currently its just me and the problem. I am still trying to figure out ways to punch through it

Simply Beautiful Art

@Secret I have been mentioned!

@Secret Oh?

Anonymous

3:41 PM

@BalarkaSen If we have a system of linear equations of the form $Ax=b$ ($A$ is a $m\times n$ matrix) then it is said that we can obtain a diagonal system of equations using the transformations $x=Py$ and $b=Pz$. My question is: How do we know what $P$ is?

Anonymous

Also what is $y$ and $z$? :P

Balarka Sen

@blue Remind me what a diagonal system is? $x = Py$ and $b = Pz$ means you're changing coordinates, this is like linear substitution replacing $x_i$ with variables $y_i$.

Anonymous

@BalarkaSen Diagonal system means the equations are like $2x_1=3,4x_2=5,x_3=3,...$. All the variables are decoupled...

Balarka Sen

I see, so simply when $A$ is diagonal.

Anonymous

Yup, so how do we find $P$ ?

Anonymous

3:46 PM

So as to decouple the system

Balarka Sen

Er, it suffices to construct a $P$ such that it sends the column vectors of $A$ to $a_i \mathbf{e}_i$, right?

Anonymous

@BalarkaSen Ah, what does $a_ie_i$ mean? (noob alert :P)

Balarka Sen

Sorry, that gives a $P$ such that $PA$ is diagonal. You want $AP$ to be diagonal. $a_ie_i$ means a multiple of the $i$-th coordinate vector.

Hi @Ted (<-- this is the person you should be asking stuff like this @blue)

Ted Shifrin

Hi a Balarka

Anonymous

I think I get it now. Suppose $Ax=b$ then we let $y=Kx$. Now $K^{-1}=P$. I see. $P$ may or may not always exist such that $(P^{-1}AP)y=z$

Anonymous

3:56 PM

$K$ is a $n\times n$ invertible matrix

Balarka Sen

Ok, yeah, I guess this is a diagonalizability question. You want a $P$ such that $P^{-1}AP$ is diagonal.

Anonymous

@BalarkaSen Right right :) Gotcha!

Anonymous

@BalarkaSen Hehe. I was just watching @Ted's linear algebra lecture sometime back :) I am also using this lecture series by IISC (youtube.com/watch?v=NAAa_eQOh2s)

user193319

I recently proved the following theorem: Let $G$ be a permutation group on the set $A$ (i.e., $G \le S_A$), and let $\sigma \in G$ and let $a \in A$. If $G$ acts transitively on $A$, then $$\bigcap_{\sigma \in G} \sigma G_a \sigma^{-1} = \{e\}$$, where $G_a$ denotes the stabilizer of $a$.

Balarka Sen

@blue Nice, nice. I should relearn some linear algebra.

user193319

4:01 PM

My question is, does this mean that the transitive action of a permutation group of $A$ is always faithful?

Is that the interpretation?

arctic tern

if $G\le S_A$ then the action is automatically faithful

@BalarkaSen missed a fun discussion with danu about Aut(CP^1-{z1,...,zn})

Balarka Sen

@arctictern Ah, did I? let me look in the transcript

user193319

@arctictern I also forgot to mention that I proved $\sigma G \sigma^{-1} = G_{\sigma(a)}$. So, what is the interpretation of these results, if not that it is faithful (since you said that this always holds)?

arctic tern

It is not always true that the normal core of a stabilizer (your intersection) is trivial; the normal core of Stab(a) is the kernel of the restricted action G->Orb(a), so the intersection being trivial may indeed be interpreted as G acting faithfully on Orb(a)

Adam Warlock

4:38 PM

Hi there, can someone take a look at my question about Fourier series?

Periodical solutions of differential equations, by bad. The fourier series one is coming up :P

0

Q: Find the periodic solutions of the following differential equations:

I am not sure how to continue with the following question: Find the periodic solutions of the differential equations (a)$\frac{dy}{dx}+ky=f(x)$, (b) $\frac{d^3y}{d^3x}+ky=f(x)$ where k is a constant and f(x) is a $2\pi$-periodic function.Consider a Fourier series expansion for $f(x)$ using...

calculus fourier-series

Zophikel

$$\int_{-\infty}^{\infty}\frac{sin^{2}(z)}{z}$$

$$\int_{\Gamma}\frac{sin^{2}(z)}{z}$$

$$\int_{\Gamma}\frac{e^{2}(z)}{z}$$

$$\int_{\Gamma}\frac{e^{2}(re^{i\theta}}{re^{i\theta}}$$

@Simply my work so far ^

I used the regular semicular contour $\Gamma$

after prematerzing the integral I observed that

$$\int_{\Gamma}\frac{e^{2}(re^{i\theta)}}{re^{i\theta}}$$

$$|\int_{\Gamma}f(z)dz| \leq \max_{z \in \partial{D} \frac{e^{2}(re^{i\theta}{re^{i\theta}| \leq |\int_{0}^{\pi}\frac{e^{2}(re^{i\theta}){re^{i\theta}}$$

^ Is my initial steps valid

arctic tern

you did not need to take up that much space

Zophikel

sorry @arctictern compied this over from another chat room

^ I integrated over that contour

Balarka Sen

4:58 PM

Hi @Krijn

BAYMAX

I think to use Liouville theorem would be a nice idea, also it appears a bit tedious to do through CR eqns @ManolisLyviakis

hey any1 up for differential equations ?

Mats Granvik

Do you think anything useful could come out of this triangle?
https://oeis.org/A255195

Krijn

@BalarkaSen Hey!

How's the weather over there?

Balarka Sen

It was extremely hot until today, when it finally rained a little. More to come, I think

Krijn

I'm hoping for rain over here

Astyx

5:00 PM

Hi chat

Krijn

Very warm, and smothery

Balarka Sen

semiconscious insertion of lines from Eliot/ "Ganga was sunken, and the limp leaves waited for rain" /end of semiconsciousness

Abdullah UYU

@Secret The video you've mentioned above is so impressed me. I live in Konya which is mentioned in video. And you can see the pattern almost everywhere. Actually the pattern speaker says is the offical logo of the municipality.

Balarka Sen

@Krijn we had 40C (maximum temp) on average so

Krijn

@BalarkaSen ...

Damn

Balarka Sen

5:02 PM

Hi @Astyx

Astyx

I'm already melting at 29

By the way, I just got my copy of Guillemin&Pollack

Balarka Sen

ooh nice

I really like AMS's hardcover edition

Abdullah UYU

@Secret ekohaber.com.tr/haber_o/15215.jpg

Astyx

Nice indeed :)

Ted Shifrin

Greetings, a Balarka, @Astyx, @Krijn

Balarka Sen

5:05 PM

rehi, the Ted

Astyx

That's the one I got @Balarka

Hullo Ted

Ted Shifrin

I wonder if they corrected any of the list of three pages of typos I gave Guillemin.

Krijn

Hi @Ted, long time no see

Astyx

@TedShifrin I can check if you want ?

Balarka Sen

I got your list of typos pinned to my front page

Ted Shifrin

5:06 PM

The typos are linked on my webpage (linked in the profile).

Balarka Sen

There are at least one other typo you are missing from your list, as I mentioned to you before, by the way

Ted Shifrin

I don't remember. I usually add to the list.

Balarka Sen

Let me check.

Krijn

When an author says "It is known that [statement I need]" without any source, what is the usual course of action?

Ted Shifrin

Oh, @Daminark, @Astyx: In honor of American Memorial Day, "We can never truly compensate those who fought for our country, but we can ..." (3/7) EPYCSTAPER

2

@Krijn: Now you have google.

Lozansky

5:09 PM

Oh I got it

Ted Shifrin

Or you talk to others more knowledgeable.

Yeah, it's easy, @Lozansky.

Astyx

Me too

(for once)

user379685

X is the random variable with the normal distribution with parameters m, s. Knowing that P(X<5,5)=0,72 and P(X>=3,2)=0,98 calculate P(4,1<X<5,2). Could someone help me with this? I have no clue :c

Ted Shifrin

It's annoying to answer questions on main and get no response from the OP.

Lozansky

Not for me. My english is subpar

Balarka Sen

5:10 PM

@TedShifrin page 136, in the equation divided by the norm a few lines down the top.

It should be |v(x) + t r(t, x)|, whereas they put a -

I don't think this is in your list

Ted Shifrin

Oh, right, @Balarka, maybe you did mention this. I never covered that material (as you know, I taught it in a totally different way).

Mike Miller

@TedShifrin I will conscientiously object from this jumble.

Astyx

it's not corrected in my book

Lozansky

Hmm $\nabla^2 (\nabla^2 u) = 0$ where $u = u(\sqrt{x^2+y^2+z^2})$. This should be juicy

Ted Shifrin

You have far more important things to which to object conscientiously, @MikeM.

OK, @Balarka, thanks. I'll retypeset the list.

Fargle

5:11 PM

That jumble was easier than the amount of time I put into it.

Good morning, chat.

Balarka Sen

@TedShifrin No problem, thanks a lot for updating. :) Always helpful for us mortals to know where the typos are.

Astyx

What's an easy amount of time ? And hi

Fargle

@Astyx Bah, you know what I mean, lol

Mike Miller

scontent-lax3-2.xx.fbcdn.net/v/t1.0-9/fr/cp0/e15/q65/…

Sha Vuklia

Guys, is it possible (without explicit calculations) to see from the Fourier transformation that if $\psi(x)$ has a narrow range, $\phi(k)$ has a large range, and vice versa? Because apparently this is the case, and my book states is quite carelessly, as if it’s very obvious.

oh I think I know how I should do it

in the case I'm working with, $\phi(k)$ is given as a constant function on an interval, and then it's (relatively) easy to plot $\phi(x)$

I guess that's what they wanted anyways

Mike Miller

5:18 PM

@Ted Want to give me advice?

Ted Shifrin

Hang on: I'm trying to retypeset the G&P errata.

Fargle

That might be the actual nerdiest thing I've ever read, @Ted.

And I used to write really bad sci-fi short stories.

...so actually, never mind.

Balarka Sen

I want to write a bad sci-fi screenplay as my school project.

Fargle

Just steal a novella and screenplay-ify it.

Mike Miller

I have an hour to kill before going somewhere to write (for like an hour). Do I read Dieudonne's history, the latest Jacobin, or bring myself to write more?

Fargle

5:22 PM

I've always wanted to see Ethan Frome in Space.

Balarka Sen

Meh, that's not original though. And most novellas I read aren't that bad.

Fargle

Oh, you want bad AND original?

I can't help you there.

I'm merely the former.

Ted Shifrin

OK, @Balarka, @Astyx: Updated.

Yes, @MikeM?

Mike Miller

It's posted above.

Ted Shifrin

Oh, bring yourself.

Balarka Sen

5:25 PM

@Fargle Well, if I find multiple bad scripts I can cut-up them to create an original bad one.

Fargle

Like a ransom note, where you're holding hostage the sanctity of the stage.

Mike Miller

aw

Balarka Sen

That's what I did in my last project; I had to write an autobiography so I chose the title "An Autobiography of The Ill", wrote pages of unrelated horrid things and scrambled them up in random order.

Ted Shifrin

@Sha: What do you mean when you say narrow/broad "range"? You actually mean the set of values or do you mean the support (in the domain)?

Balarka Sen

And stapled them.

I got an A+ so

Mike Miller

5:28 PM

They probably think you're fucked in the head.

Balarka Sen

Oh that's an established fact

Krijn

5:54 PM