Mathematics - 2017-04-17 (page 5 of 6)

DHMO

17:00

that is interesting

Astyx

generate a non cyclic proper subgroup if I'm not mistaken

DHMO

how do you prove it?

Astyx

Which part ?

DHMO

that it is a proper subgroup

Astyx

Show it does not contain $a$ for instance

DHMO

17:03

how?

Astyx

In fact $ab$ and $ba$ might suffice

DHMO

not really

$(ba)(ab)(ba) = baabba = bbba = a$

Astyx

Indeed

You can show the extremities always are $ab$ or $ba$ by induction

And that's sufficient

DHMO

consider $(baba)' = abbabb$

which contradicts your theorem

Astyx

Oh yeah I frogot about inverse :p

But you can go around that

Either the two extremities are inverses, in which case you take the inverse of your element

Or one isn't and you're done

For this, let $e = abab, f = baba$, and compute $e^2, f^2, ef, fe, ef', e'f$

And then write any element of the span $e^{n_1}f^{m_1} \dots e^{n_k}f^{m_k}$

Where the $n_i, m_i\in \Bbb Z^*$

DHMO

17:15

$\begin{array}{rcl}
e^2 &=& abababab \\
f^2 &=& babababa \\
ef &=& ababbaba \\
fe &=& babbab \\
ef' &=& abababbabb \\
e'f &=& bbabbababa
\end{array}$

Astyx

Except $n_1$ and $m_k$ which can be 0

Vrouvrou

Hellô, if $y\inR $ does $(y)^(1÷3)\in R $?

DHMO

@Vrouvrou yes

$f:x \mapsto x^3$ is bijective

Vrouvrou

But I need ^(1÷3)

DHMO

Yes

$f^{-1} : x \mapsto \sqrt[3]x$

Harry Evans

17:25

Good day!

Suppose $f$ and $g$ are analytic on a domain $D \subseteq \mathbb {C}$ and that $fg \equiv 0$ on $D$. Show that either $f \equiv 0$ or $g \equiv 0$ on $D$.

Ted Shifrin

@Harry: Remember that domains are, by definition, connected.

DHMO

@TedShifrin hi

1

A: Non- linear ODE's with no independent variable

The first one: $$\begin{array}{rcl} y &=& y' \ln y' \\ W(y) &=& \ln y' \\ y' &=& e^{W(y)} \\ \dfrac{\mathrm dy}{\mathrm dx} &=& e^{W(y)} \\ \dfrac{\mathrm dx}{\mathrm dy} &=& e^{-W(y)} \\ x &=& \displaystyle \int e^{-W(y)} \ \mathrm dy \\ &=& \displaystyle \int e^{-u} \ \mathrm d(ue^u) \\ &=& \di...

Harry Evans

Hey @TedShifrin, I was thinking of a complex analysis approach

Ted Shifrin

17:40

OK, @DHMO.

Well, you need to use a fundamental result from complex analysis, yes, @Harry.

Harry Evans

I guess I can figure that one out, thanks, @TedShifrin. I do have another question, though

DHMO

Is this true? If $H$ is a subgroup of $G$, and both are finite groups, then $\#H \mid \#G$

Ted Shifrin

Yes, @DHMO, that's one of the fundamental theorems from the beginning of group theory.

DHMO

@TedShifrin that's interesting

BAYMAX

That is Lagrange's theorem!

Astyx

17:42

Salut Ted

Ted Shifrin

Salut, Astyx. Guten Tag, @DanielF.

Harry Evans

I'm asked to use the Cauchy Integral Formula on $\int_0^\pi \dfrac {d \theta} {(a + cos \theta)^2}$ where $a > 1$. I cannot seem to find a way to convert this integral to a form that can use the Cauchy Integral Formula

Ted Shifrin

@Harry: Hint: Can you make up a problem $\int_0^{2\pi} ...$ that will solve this one?

Harry Evans

@DHMO, yes, it is indeed Lagrange's Theorem. Orders of subgroups divide the order of the group

DHMO

@HarryEvans thanks

elDin0

17:44

anybody got an answer to my earlier message on hilbert spaces? or care to have a look?

Ted Shifrin

@DHMO: The key idea is cosets.

Daniel Fischer

Bon jour @Ted.

Harry Evans

@TedShifrin, I know that the integral I wrote is one half of the integral when taken from 0 to $2\pi$. But the denominator is not of a form $(z - a)^{n+1}$

Ted Shifrin

@Harry: What if you write $z=e^{i\theta}$?

I miss seeing you around, @DanielF.

Daniel Fischer

Too little time to hang around in the chat much these days.

Ted Shifrin

17:49

Well, I hope you are enjoying yourself some, @DanielF.

Alessandro Codenotti

Hi @Ted

DHMO

How to prove that $|G:H \cap K| \le |G:H||G:K|$ where $H$ and $K$ are subgroups of $G$?

Ted Shifrin

hi @Alessandro

Daniel Fischer

@TedShifrin I do. Not all day, but every day at least a little.

DHMO

Is it true that $|G:H \cap K| = \operatorname{lcm}(|G:H|,|G:K|)$?

Ted Shifrin

17:51

I'm glad, @DanielF, seriously.

I doubt it, @DHMO.

Daniel Fischer

@DHMO One method is to find an injection $G/(H\cap K) \hookrightarrow (G/H) \times (G/K)$.

Ted Shifrin

I doubt that, too, @DanielF. :)

Hi, Demonark.

Daminark

How's it going @Ted?

(Also hello everyone!)

Ted Shifrin

Still sick (growl), but otherwise just fine. How you?

DHMO

@TedShifrin then how would I approach it?

Alessandro Codenotti

17:55

What's $|G:H|$ for groups?

DHMO

@AlessandroCodenotti number of cosets of $H$ in $G$

Ted Shifrin

@DHMO: If you didn't know Lagrange's Theorem, you probably don't have much experience with groups. But I would suggest thinking about a reasonably small non-abelian group where you can see everything that's going on.

DHMO

alternatively, $|G/H|$

Daminark

Well, today we did the three types of integration

Ted Shifrin

At any rate, I'm confident it's false.

Daminark

17:56

So that was pretty cool

Ted Shifrin

the three types, Demonark?

Daminark

Well, "the" is not the right way to say it

But Riemann, Lebesgue, and Henstock

DHMO

@TedShifrin are all $\Bbb Z_n$ groups abelian?

Ted Shifrin

Yes, @DHMO.

DHMO

I suppose so

Ted Shifrin

17:57

What's the simplest interesting symmetry group?

DHMO

$S_3$?

Ted Shifrin

I don't recall Henstock, Demonark.

Alessandro Codenotti

What's the Henstock integral?

Ted Shifrin

OK, which is also $D_3$, the group of symmetries of an equilateral triangle. Either way, play around with that, @DHMO.

DHMO

@TedShifrin heh

well I can find 2 non-trivial subgroups

namely, $C_3$ and $\sigma$

Ted Shifrin

17:58

You can find lots more than that.

DHMO

@TedShifrin I thought there are only $6$ elements

How many subgroups are there?

Ted Shifrin

At least five.

DHMO

including the two trivial subgroups?

Ted Shifrin

Then six.

DHMO

5+2=6

Ted Shifrin

17:59

I was counting one of the trivial ones the first time; thank you very much.

DHMO

I think all subgroups are cyclic

Ted Shifrin

They are.

DHMO

so i can just $C_3 \sigma$

Daminark

The idea is that if you have a tagged partition, and a gauge $\delta: [a,b]\to (0,\infty)$, you say that the partition is $\delta$-fine if $[x_{i-1},x_i] \subset (t_i - \delta(t_i),t_i+\delta(t_i))$

Ted Shifrin

Huh?

DHMO

18:00

or $\sigma C_3$

Ted Shifrin

@DHMO. Stop talking and sit down and think.

DHMO

@TedShifrin salut

@AlessandroCodenotti arrivederci

Daminark

And then you say that the number $I$ is the Henstock integral of $f$ on $[a,b]$ if for any $\epsilon$, there exists a gauge $\delta$ and a $\delta$-fine partition $P$ such that $|\sum_P f - I| < \epsilon$

It's not much of a modification to the definition of a Riemann integral (special case where the gauge is constant) but at least on $\mathbb{R}^n$ it's more general than even the Lebesgue integral

Ted Shifrin

Does it incorporate Riemann-Lebesgue $\int f\,d\alpha$ somehow for a discontinuous $\alpha$?

Daminark

I believe so, though it was created more to solve the problem of $\infty - \infty$

Like when you split up a function to positive and negative parts

Ted Shifrin

18:05

So Riemann-Lebesgue will give you as a special case a sum of values of $f$ at particular points. Can you get that with Henstock?

Daminark

I'm not totally sure, we hadn't talked about it much, it was a side note in passing. We mentioned that Riemann or Lebesgue integrable function is Henstock integrable, and that you can always integrate a derivative and get the original function

Ted Shifrin

Lebesgue integrable with respect to the standard Lebesgue measure on $\Bbb R^n$ ... but presumably not weird measures.

Daminark

We mentioned that it doesn't get you stuff like probability

Ted Shifrin

I'm guessing my hunch is right about Riemann-Stieltjes and counting measures.

Daminark

So for that reason the Lebesgue integral still has a reason for existing

:P

Adeek

18:11

@TedShifrin hi

@Daminark hi

Ted Shifrin

Hi, Karim.

Daminark

Hey @Adeek

Adeek

@TedShifrin have you heard of Qing liu algebraic geometry ?

Ted Shifrin

No, but I'm not current ...

Adeek

it is really nice intro to algebraic geometry

once exams are done I will start reading it

Daminark

18:12

But yeah, also I'm not sure if the set Henstock integrable functions gives you a nice space like the $L^p$ spaces

Ted Shifrin

You mean a Banach space, Demonark?

Daminark

Yeah

Vrouvrou

@DHMO see this quora.com/What-is-value-of-1-1-3

Daminark

shrug

We'll be using only the Lebesgue integral from now on anyway so yeah

Ted Shifrin

I'm still gonna give you some 2- and 3-dimensional integrals to evaluate :P

Daminark

18:16

th... three? That's... a lot of dimensions...

Vrouvrou

are you there @DHMO?

Daminark

shivers They won't be too hard, right? Like linear functions over rectangles, right?

Ted Shifrin

My students had to do some in 4, Demonark, and integrals over some 3-dimensional manifolds in 4 dimensions ... :P

Sid

jee advaced anyone?

Ted Shifrin

Demonark: You think you're so masterful living in infinitely many dimensions ...

Daminark

18:18

That sounds really tough

becomes an algebraist

Lol jk

I dunno, I like infinite dimensions though, they're fun

Secret

I wonder, if there are convergent examples of such expressions:
$$\lim_{R\to \infty}\int^{(R)}f(x)d^Rx$$

Well, $f(x)=e^x$ is easy, since it is just an eigenfunction of the integral operator, but are there others

Daminark

But yeah in total fairness I'm probably losing/going to lose some computational skill this quarter in particular

First quarter had some long ones, as well as last year, but starting from second quarter it was mostly just a couple small ones here and there, and this quarter likely none

So it probably would be worthwhile to practice at least some

Secret

Elaboration: $\int^{(R)}=\underbrace{\int \cdots \int}_{\textrm{R times}}$

Daminark

@Secret I was very confused for a second there

Secret

yeah, there seemed to be no nice notations for integrals beyond quadruple integrals

Therefore if $f(x)=e^x$ (and let initial condition be 0, thus the constant term will always be zero) we get the following countable sequence:

$$\begin{matrix}e^x & e^x & e^x & e^x & \cdots\end{matrix}$$

which is a "constant" sequence that obviously "converges" to $e^x$

Astyx

18:29

Hi again

Daminark

Hey @Astyx!

Astyx

How are you ?

Sha Vuklia

Random variables $X$ and $Y$ have joint density function
$$
f(x,y)=\begin{cases}
c(x^2+\frac{1}{2}xy)&\text{if }0<x<1,0<y<2,\\
0&\text{otherwise.}
\end{cases}
$$
I have to find the value of the constant $c$. I started as follows:
$$
\int_{x=0}^1\int_{y=0}^2c(x^2+\frac{1}{2}xy)dydx=\int_{x=0}^12cx^2+x\,dx=\frac{2}{3}c+\frac{1}{2}.
$$
Since this should equal 1, we have: $c=\frac{1}{2}\cdot\frac{3}{2}=\frac{3}{4}$. However, my solution set says that $c=\frac{6}{7}$. So where did I make a mistake?

Astyx

What is a primitive of $x\mapsto x$ ?

Sha Vuklia

$1/2x^2$?

Daminark

18:32

I'm alright, how about you?

Astyx

@ShaVuklia Re read yourself :)

I'm fine thanks :)

Sha Vuklia

which part?@Astyx

Daminark

How's the exam situation?

Sha Vuklia

I've reread it countable times :P

Astyx

Second line of equations

Ted Shifrin

18:34

You made the standard mistake for which I always yelled at my students.

Astyx

You did not integrate $x$ with $x^2/2$

Ted Shifrin

Do not distribute $c$. Factor it out once and for all.

Sha Vuklia

ohhhh

right

haha :P thanks

Ted Shifrin

Break these bad habits!

Astyx

@Daminark Exams start in 36 hours or so

Sha Vuklia

18:35

but, I thought I made things easier by distributing?

why is it better not to distribute then?

Astyx

And here I am sitting on my chair solving rubik's cubes

Ted Shifrin

NOOOO ... you made it easier to screw up !! Always happens.

@Astyx: Bonne chance :)

Sha Vuklia

I'm not even sure if I can do the integral without distribution:P I'm gonna see if I know how it works

Astyx

Merci :)

He means get it out of the integral @Sha

Sha Vuklia

OHhhh!

righttt

that is indeed better

Ted Shifrin

18:37

Funny how people never trust me ... :D

Astyx

Poor Ted :p

Ted Shifrin

Exactement.

Sha Vuklia

hahaha :P no, I thought you meant I had to integrate $c(x^2+\frac{1}{2}xy)$. But it is true that there have been instances when I didn't show trust where due :P

Daminark

Good luck @Astyx!

Astyx

Funny story also. For Centrale-Supelec I mistakenly registered myself to take the exam in Poitiers instead of Paris

So I'll be doing them in Poitiers

Ted Shifrin

18:42

How far an excursion is that?

Astyx

300 km or so

But there is a direct TGV connection so it's okay

Daminark

Oh damn

Convenient

Astyx

Is TGV an international slang btw ?

Daminark

No, I think it's specific to France

Astyx

It's probably quicker for me to get to Poitiers than to get to the place the exams would have been in Paris

Daminark

18:44

Or at least to French-speaking countries

Ted Shifrin

Je m'en doute.

Astyx

But would a random person still understand what it is ?

Ted Shifrin

No

Daminark

Nope

Ted Shifrin

But, as I've taken it several times and will be taking it again in June, I'm fully aware :P

Pretty crazy error, Astyx.

Astyx

18:46

Yeah, I'm not 100% sure it's my fault really

Ted Shifrin

My favorite word for this is ... ditzy.

Astyx

ditzy ?

Ted Shifrin

Oh, how is it not your fault? Sort of the way I ended up with a train ticket in Italy where we switch trains and end up in second class, even though I never saw any way to avoid that? The "help" people were worse than unhelpful.

Astyx

I'm pretty sure I remember taking Poitiers and not having any other choice on the registration website. This being said, my memory might be failing me

Ted Shifrin

ditzy = out of it, dumb ... :P

Astyx

18:48

In fact taking it in Poitier might even be an improvement after all, since it's very calm and quiet there (compared to Paris)

Lozansky

Italy is conservative, no?

Ted Shifrin

But you have to allow all sorts of time for travel and getting lost.

Astyx

I'll be staying there for 4 nights, so I have a whole day to find my way, and once I'm there I've been told it's only 5 minutes away, so that's settled

Ted Shifrin

Oh, not to mention hotel :P

But it should be a fun adventure.

Astyx

Centrale is not really a school I want in any case

Maybe I'll go to the Futuroscope instead ...?

:p

Adeek

18:51

@TedShifrin I want to make sure I understand something correctly

here it should be finitely generated as a ring over $\phi(A_0)$ right ?

Ted Shifrin

Karim: I have told you multiple times not to ask me algebra stuff. I haven't thought about this in 40 years.

Adeek

oh okay

Ted Shifrin

Seriously. I'm getting annoyed.

Adeek

ok sorry

Daminark

@Astyx Come to Chicago so you never have to compute things ever again!

(@Ted Must think I'm a great influence right about now)

Astyx

18:57

Do people in Chicago never compute anything ?

Daminark

I mean computation isn't like, nonexistent

Astyx

Reverso translates "ditzy" as "écervelé", didn't realise it was that bad :p

Daminark

But say, as opposed to the typical track of students doing mostly computational calculus, linear algebra, and ODEs before reaching proofs

Here it's more, Spivak first year, which is proofs plus a couple examples here and there to make sure you don't totally forget how it works

Sha Vuklia

Random variables $X$ and $Y$ have joint density function
$$
f(x,y)=\begin{cases}
e^{-x-y}&\text{if }x,y>0,\\
0&\text{otherwise.}
\end{cases}
$$
Find $\mathbb P(X+Y\leq 1)$. I started as follows:
$$
\mathbb P(X+Y\leq 1)=\int_{x=0}^1\int_{y=1}^{1-x}e^{-x-y}dydx=\int_{x=0}^1-e^{-1}+e^{-x-1}dx=-e^{-1}-e^{-2}+e^{-1}=-e^{-2}.
$$
This can't be correct. The answer should be $1-2e^{-1}$. Could someone tell me where I made the mistake?

Daminark

Then second year, you do one of the 3 analysis classes. First is more calculation-based, has the feel of rigorous multi plus a side of metric spaces. Second is Rudin throughout the year, third kinda fast tracks through Rudin in a couple weeks and focuses on functional analysis, measure theory, and whatever the professor feels like doing

Astyx

19:01

@Sha $y=0$ as a bound, not $1$

@Daminark I really don't understand how you can do maths without doing proofs ...

Sha Vuklia

@Astyx oh yea, thanks

Daminark

@Astyx A lot of schools in America have very small math departments and get a good amount of funding from having service courses to engineering departments and all that, and don't have the resources to have a side track that just jumps into proofs for actual majors

A lot of schools are somewhere in the middle

UGA, for example, has an honors multivariable calc class that Ted used to teach, which is taken right after first year calculus and is proof-based (though computation does actually exist in a non-trivial way)

From what I've gathered, the main places where undergrad math looks like how it's done in France would be Harvard, Princeton, Chicago, and so forth

Those math departments have more room to have math classes outside the standard funnel of "math for science/engineering"

Astyx

@Sha it's not $-x-1$ but $-x$ once you've integrated once

@Daminark I believe that works better for producing mathematicians, but I guess engineers don't care

Daminark

Yeah, that's why I really cannot appreciate more having come here

I'm now in the fast track class and we're covering a lot of very fun content

The speed makes it such that you might get through material only vaguely, or at a cost of some computational fluency

But it's a whole lot of fun

Astyx

Are you doing only maths ? Or do you also take physics, chemistry etc ?

Daminark

19:13

So, we have a core curriculum

So everyone does some amount of humanities, social science, civilization studies, physical science, biology, and math

Astyx

We're doing approximately the same for preparation of the exams, I can't wait to be in school so I can learn more deeply about stuff

Oh all that

Daminark

That takes about a third of the time

Then a third is dedicated to major requirements (give or take)

Then a third is just whatever you want to do

Second major, minor, going deeper into your field to prepare for grad school, random classes for fun, whatever you feel like

I got through most of them, I just have to finish one more quarter in biology along with 2 in civilization

Astyx

Is civilization like history ?

And what are humanities and social science ?

Daminark

Yeah, and humanities is stuff like philosophy, art, linguistics, that kind of thing

Social science usually entails stuff like psychology, politics, economics, and so forth

The core classes have mostly the feel of a discussion-based class on political philosophy

(core in social science)

Astyx

Sounds cool

Daminark

19:19

Yeah, most of the core classes have been very enjoyable

The math department requires you to take a few classes outside math as part of the major, in either computational neuroscience, physics, chemistry, statistics, or computer science

(Which is not a decision I agree with, like they should decrease this requirement and add point-set topology and complex analysis at least...)

I'll be satisfying those more by theoretical compsci classes, like algorithms and combinatorics

Astyx

Yeah I also have computer science, and I enjoy it tbh

Daminark

I'll possibly take, like, general relativity or something, plus some philosophy classes and one on Italian Renaissance

But from here on it'll mostly be math and theoretical compsci

Yeah same

Though I'm more into the theory side, along with functional programming

C just annoyed me to no end

Astyx

Oh we mostly do theorical stuff, with graphs, trees, languages

And the language we use is CamL, so lots of functionnal programming :)

Daminark

Nice

The first class I took in compsci used Typed Racket

I'm also hoping to get into Haskell

Astyx

I've barely ever heard of those

Daminark

19:24

Alright, well, I'm gonna have to head out in order to finish my sosc stuff now, I've got class in about half an hour

Oh they're pretty fun, I probably would say Haskell is better

Astyx

See ya

Larry

any way i can see the chat when i last logged on

Vrouvrou

19:43

hello please if $y$ is real then $(y)^(1/3)$ is real ?

Simply Beautiful Art

@Vrouvrou Usually yes

Use y^{1/3}

Vrouvrou

why ? please because i found this and i don't undertand :quora.com/What-is-value-of-1-1-3

Simply Beautiful Art

That's why I said usually

Vrouvrou

so not alwayse ?

Simply Beautiful Art

So here's the thing. $x^2=1$ does not mean $x=\sqrt1$, but rather $x=\pm\sqrt1$.

Vrouvrou

19:45

it can be real it cn be not

?

Simply Beautiful Art

Well note the above

Meow Mix

@SimplyBeautifulArt Depends if you mean principal root or not

Simply Beautiful Art

There are two different solutions to $x^2=1$

@MeowMix Yes, I was getting there :P

@Vrouvrou Likewise, there are three solutions (not necessarily unique) to any third degree polynomial.

Meow Mix

Also, there is always a real root of $x^3-y = 0$

(thus implying that $y^{1/3}$ always has a real solution)

Simply Beautiful Art

One of the solutions to $x^3=1$ is $x=1$. But the other two solutions are not real.

Kosta Butbaia

19:48

@Vrouvrou The function y=x^(1/3) is defined for whole real number line, No matter what is x if it is a real number the output will be real too.

Meow Mix

It's also not differentiable on its domain :]

Simply Beautiful Art

@Vrouvrou So to sum it up, our point is that the equations $x^2=1$, $x^3=1$, and $x^3=-1$ have multiple solutions. But one of them is always a real solution.

Vrouvrou

my problem is that i want to prove that $(R,d)$ where $d(x,y)=|x^3-y^3|$ is complete, and i found that any Cauchy sequence converge to $(l)^{1\3}$ where $l$ is real to say that $(R,d)$ is complete i need to have that $(l)^{1\3}$ is real

Sha Vuklia

If $X$ and $Y$ have joint density function
$$
f(x,y)=\begin{cases}
\frac{1}{2}(x+y)e^{-x-y}&\text{if }x,y>0,\\
0&\text{otherwise,}
\end{cases}
$$
find the density function of $X$ and $Y$. I was considering two things:
1) I could check whether $X$ and $Y$ are independent, and if so, use the convolution formula $f_Z(z)=\int_{-\infty}^\infty f_X(x)f_Y(z-x)dx$,
2) I could differentiate $\mathbb P(X+Z\leq z)=\int_{-\infty}^\infty\int_{-\infty}^{z-x}\frac{1}{2}(x+y)e^{-x-y}dydx$.
I'm not sure if 2) is reasonably doable. Are these my two only options? Or is there a better way?

LeGrandDODOM

@ShaVuklia yes, and it's over

Sha Vuklia

19:56

What do you mean, @LeGrandDODOM ?

LeGrandDODOM

@ShaVuklia there's nothing more to write

Sha Vuklia

ohh, you're referring to that other question

yea, that's solved, thank you

LeGrandDODOM

for your last question, you should have seen in class that you get marginal densities by integrating the joint density with respect to one variable, letting the other one be fixed

If you haven't, it's a consequence of Fubini's theorem

Sha Vuklia

yes I've seen that

but that's only useful if they are independent, right?

Transcript for

$Mathematics$ Mathematics