Mathematics - 2014-08-12 (page 2 of 3)

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5:00 PM

The power set is the set of all possible subsets of a set.

@Khallil Try this

Takes some thinking

also try some examples to convince yourself

its not always evident from a proof

@Alizter They're the same then.

Thanks for the link :-)

yes

by definition

Okay I got it

$$-\dfrac{10^{-n}}{10^{-n}-1}=-\dfrac{\dfrac{1}{10^{n}}}{10^{-n}-1}=-\dfrac{1}{{‌10}^n\left(10^{-n}-1\right)}=-\dfrac{1}{10^0-10^n}=\dfrac1{10^n-1}$$

@Darksonn Nice. I just multiplied by 1. $$ \dfrac{10^{-n}}{1-10^{-n}} \cdot \dfrac{10^{n}}{10^{n}} = \dfrac{1}{10^{n}-1}$$

^_^

good point

5:08 PM

@Alizter Is the empty set a subset of the empty set?

That's the argument they seem to have used in the induction.

$\mathcal{P}(\emptyset) = \{\emptyset\}$, where $\mathcal{P}(S)$ is the power set of $S$.

Yes hence $2^0=1$

The emptyset is the subset of any set

Hello @PedroTamaroff

@Khallil The emptyset is a subset of every set.

Hello.

Cool.

Can you prove it?

OOOOOOO

Its going down

5:10 PM

For some reason, I think of falling down a set of never ending stairs when I think of the empty set being a subset of the empty set.

It might involve some form of contradiction.

I think I've seen something similar before.

*paradox

What's the difference between a contradiction and a paradox?

@Khallil You're thinking $\varnothing \in\varnothing$ which is false.

@PedroTamaroff Oh, yea I was!

@Khallil A paradox is plausible but confusing. A contradiction is unattainable.

5:11 PM

Well if you prove that $A\in\mathscr P(A)$ you are done

which by def $A\subseteq A$

@PedroTamaroff Was that edit merely an aesthetic change, or is there a difference between \varnothing and \emptyset?

if you see this order of symbols $\in \emptyset$ something has gone wrong

Hello @PedroTamaroff

$$A\subseteq B\implies \forall x\{x\in A\implies x\in B\}$$

@BalarkaSen sarah is like the eye of mordor

5:13 PM

@Alizter Oh?

@Alizter That's because the empty set, by definition, has no elements.

@Khallil Exactly

@BalarkaSen Hello.

Herro, @BalarkaSen.

@Alizter If you see $\in\emptyset$ you could be making a proof by contradiction and nothing was wrong.

5:15 PM

@Darksonn But a proof by contradiction requires something to go wrong.

Then you deliberately made it go wrong, which kindof means it went right?

@Alizter Sarah's back ...

:O

Yes?

What?

Apparently, you're the eye of Mordor.

o_O

throws a pebble at @Khallil

run you fools!

3

5:18 PM

@Alizter catches the pebble and reads the message inscribed on it "Get to ze chopper!!"

I see now people!

rubs hands

Nooo $$\text{Chopper}=\emptyset$$

wtf

ftw

ftmfw

5:20 PM

bbq

I'm supposed to be proving that the empty set is a subset of every set.

@sarah it's all @Khallil 's fault/ Totally. 100%

@Khallil Done

It's trivial

$A \subset B$ means that $B$ must totally contain with the exception of at least one element that's in $B$ which isn't in $A$ right?

OK.

5:22 PM

sarah left

@Khallil You have to show that $x\in\varnothing\implies x\in A$ for any $A$:

$$A\subseteq B\implies \forall x\in A : x\in B$$

But a sentence of the form $P\implies Q$ is always true if $P$ is always false.

$\subset$ denotes a proper subset and $\subseteq$ denotes a subset, right?

yes

5:23 PM

yes

Cool. So what I said a few posts up would be correct then, right?

What was it you said?

Depends on the convention.

@Khallil Not really.

@BalarkaSen What is the galois group of a cyclotonic polynomial?

@Alizter fly you fools!

5:25 PM

@PedroTamaroff No?

@Alizter cyclotonic?

A vacuous truth is a statement that asserts that all members of the empty set have a certain property. For example, the statement "all cell phones in the room are turned off" may be true simply because there are no cell phones in the room. In this case, the statement "all cell phones in the room are turned on" would also be true, and vacuously so, as would the conjunction of the two: "all cell phones in the room are turned on and turned off". More formally, a relatively well-defined usage refers to a conditional statement with a false antecedent. One example of such a statement is "if Ayers Rock...

@Khallil No. $\subsetneq$ denotes proper subset. $\subset = \subseteq$.

$(1-x^p)/(1-x)$

@DanielFischer I knew there was something off with the notation. That makes it much clearer!

5:26 PM

@Alizter cyclotomic.

yep failed pretty bad there +1 for ali's fail box

galois group is cyclic.

@BalarkaSen Simply correct the word.

@PedroTamaroff I was joking.

Won't happen again.

Gentlefolk observe the keyboard before you. Notice how m and n are together. The nistake can be made. Live with it.

5:28 PM

The nistake cam be nade.

@Khallil But, there are people who use the (wrong) convention $\subset = \subsetneq$, like there are people saying $0\notin \mathbb{N}$, or that compact spaces need not be Hausdorff, or $\dotsc$. Always say which convention you use, or just avoid $\subset$ and always write either $\subseteq$ or $\subsetneq$.

@BalarkaSen mo.

G'day @Ted.

@Alizter OK, can you prove that the galois group of a cyclotomic poly is cyclic?

@Alizter Here you are assuming we are using the qwerty layout

5:29 PM

@DanielFischer Herro Daniel-San.

Hello @TedShifrin

G'day, @Daniel. You and I concur on $\subset = \subseteq$.

@DanielFischer I'll do my best to make sure I stick to that notation and outline it whenever I write something down!

@TedShifrin heelo

Heya @Balarka, mr @Pedro

5:29 PM

Bom dia @Pedro.

Hi @Alizter

Just to make sure, @DanielFischer
$\subseteq$ is a subset.
$\subsetneq$ is a proper subset.

Right, @Khallil.

And $\subset$ is ambiguous

Set theoretically. It makes sense for $0\in\Bbb N$. But nobody really uses it like that

most

5:30 PM

@TedShifrin Let's see if I'm heading in the right direction with this. Let $\varphi(r)=\int_0^{\pi}e^{-r\sin t}dt$

anyway

Hope you're feeling better, mr @Pedro ... Yes, go on ...

@DanielFischer It helps to have the dashed line underneath for a proper subset because you can say that $A \neq B$ if $A \subsetneq B$.

@BalarkaSen I will have a go

Then $\varphi(r)>0$ everywhere, and $\varphi'(r)<0$ everywhere, and in fact $0\leqslant |\varphi'(r)|\leqslant \varphi(r)$.

5:31 PM

I usually use $\Bbb N$ as nonnegative integers without 0. $\Bbb N_0$ just does teh trick

@Alizter That depends. I only ever encountered people who think that $1$ is the smallest natural number when I went on the internet. Never in real life or in print.

In particular, $\varphi(r)$ admits a limit as $r\to\infty$.

@Alizter Does $\mathbb{Z}^{+}$ include $0$?

Interesting, @Pedro ... You're taking a whole new tack ... Yes ... ?

@DanielFischer Interesting.

@Khallil Is $0$ positive?

5:32 PM

@Khallil Yup. There's also $\subsetneqq$, but that goes too much out of line typographically.

@Alizter It isn't. T_T Why do I keep making that mistake‽

@Alizter $0$ is not positive

@TedShifrin I don't know what your idea is.

Who cares. Terminologies Terminologies.

Are you done with yours, @Pedro?

5:33 PM

@DanielFischer What does that mean?

Ohhhh.

It's the same as $\subsetneq$.

It just emphasises the $\neq$, right?

In meaning. Not in typography.

@TedShifrin Well, I am getting that the limit exists and is nonnegative. I need to show it is zero. I think there is some kind of theorem regarding $|f'(x)|\leqslant |f(x)|$. Cannot recall now. Or maybe it was $|f(x)|\leqslant |f'(x)|$.

Right. But $A\subsetneqq B$

doesn't look good :(

Yea, looks a bit messy.

Differential inequalities ... Hadn't thought about that.

5:35 PM

According to WA $$\int_0^\pi \exp\left(-r\sin t\right)\,dt=\pi\left(I_0(r)-L_0(r)\right)$$

@TedShifrin But what was your idea...?!?! =D

There are two "standard" approaches. One (which is the first I think I figured out when I was your age) is to note uniform convergence on $[\delta,\pi-\delta]$.

@BalarkaSen If we define cyclotomic polys with the quotient can we just say the limits suffice as values when $x=1$?

Ah, yes. That crossed my mind. =)

Like the chicken crossing the road?

5:36 PM

@Alizter Limits? Values?

@r9m Have you seen this one? :-) $$\lim_{n\to\infty} \frac{1}{n^2} \int_0^1(x+2^2 x^2+3^2 x^3+\cdots+n^2 x^n) \log(1+x) \ dx$$

BTW, @Pedro, Daniel emailed me to say how much he enjoyed talking with you ...

@TedShifrin A trivial chicken bundle on a road?

Probably a lot of messy feathers, @Balarka.

@TedShifrin Yes, I have to reciprocate, but I was under the weather.

5:37 PM

Food poisoning, @Pedro? :D

@BalarkaSen What happens at $x=1$? Polynomial division does not suffice?

@TedShifrin Maybe! Who knows! >:)

because we have $1-1$

Well, I'm still alive @Pedro. I hope you feel better.

Take limits.

5:37 PM

@Chris'ssis nope :) Not that I can remember .. I put the Au-Yeung on my blog .. :-)

I was going to thank him, as I told you he really motivated me. =)

But what has this got to do with my problem?

@BalarkaSen So $p+1$

The other approach (which you find in most textbooks), @Pedro, is just the most obvious and standard estimate on $\sin\theta$ ...

Well, not the standard one ...

@TedShifrin $\theta-\theta^3/6\leqslant \sin \theta$?

5:38 PM

Too hard.

Oh!

You want an upper bound on $e^{-r\sin t}$.

You mean the secant line.

I do indeed.

OK.

Now I feel stoooopid.

=)

5:39 PM

Well, it's nice to be human from time to time :D

@r9m did you? Where? See above. :-)

@Chris'ssis AHHHHH !!!! :D .. AWESOME :D

@TedShifrin I have been browsing the book you gave me. Ideally I would solve all its problems in a lifetime.

Hehehhee.

@r9m Thank you! Your proof is AWESOME too! I had to learn some from there. :-)

@BalarkaSen I am being picky

5:41 PM

@Chris'ssis I have a problem for you.

I perused it a number of times over the years I've had it ... Especially stole problems a few times when I taught graduate complex analysis. @Pedro

@Chris'ssis hah .. thats a laugh .. my proof is due to H. F. Sandham and Martin Kneser :P

@TedShifrin That's something I'm planning to do. >:)

What? Steal?

Give problems from PS in midterms or final exams.

5:42 PM

@r9m Well, it doesn't matter. It's important you know it. :-)

Well, or homework.

=P

Such problems don't belong on exams ... They require time and much thought. Remember that many of us assign graded homework. That's the place for hard problems; exams should be for making sure you've learned/mastered the basics.

</soapbox>

sigh and I am sticking to my copy of Titchmarsh.

@Chris'ssis Show $$\sum_{k=1}^n (-1)^{k-1}\frac 1 k \binom nk=H_n$$

@PedroTamaroff Oh, I know I know!

5:44 PM

well, @Pedro, as long as you have the book, you'll always remember me :D

I have done that before.

@r9m I was shocked by the cleverness in that proof.

what book are you talking about?

Polya Szegö Problems in Analysis, @Darksonn

@TedShifrin Well, I don't need a book!

5:45 PM

@PedroTamaroff that one is for kids - en.wikipedia.org/wiki/Harmonic_number By the way, I have a mind-blowing proof there (I doubt you find it somewhere).

The proof I had in mind uses differencing and summing.

No need for complicated integral manipulations.

Why does my pdf viewer treat F5 as fullscreen instead of reload pdf...

@Chris'ssis You have seen $300$(II) .. you know what Athenian Shock Combat is all about .. :P LOL

@r9m hahahaha :-))))))

@Chris'ssis I had to ask one of my professors to check the old volumes of AMM .. he has them all :-) .. the proof was essentially the same as the one given by G. Klambauer in his book :-) ..

5:52 PM

@r9m That series along with the proof should be taught even in high school. It's amazing.

@Chris'ssis Take $s,n$ natural numbers. Then the cardinality of the set $\{(x,y,z)\in [-n,n]^3\cap\Bbb Z^3:|x+y+z|\leqslant s\}$ is $$\frac{1}{2\pi}\int_{-\pi}^{\pi}D_n(t)^3D_s(t)dt$$

Who knew.

What is D_n?

@TedShifrin You got a cat?

Hahaha, suspected that.

@Chris'ssis Ya .. the most subtle plan of making the High-Schoolers go crazy and run haywire into mayhem and complete chaos :P

5:54 PM

+1 cats

No, @Pedro. I'm at school. :)

@r9m lollllll :-)

@Pedro Please explain the $D_n$s.

@BalarkaSen Dirichlet Kernel.

@PedroTamaroff Should I google?

@TedShifrin Again?

5:56 PM

smacks @Balarka

@Chris'ssis Another good exercise that is not that hard is to show that $$\lVert D_n\rVert_1-\frac 4{\pi^2}\log n=O(1)$$

@Chris'ssis but seriously .. the Romanian High School curriculum is amazingly advanced compared to ours :O

@PedroTamaroff I like that one :-)

@PedroTamaroff When I'm going to reach your very high level during this life (if this is possible), then I'll try to answer. :-)

@PedroTamaroff Wait, I have some for you ...

@Chris'ssis What you want to do is break up $\lVert D_n\rVert _1$ into the pieces where $\sin $ keeps its sign, and then make rough estimates.

@PedroTamaroff $$ \space\lim_{n\to\infty} n\left(n\int_0^{n} \frac{\displaystyle \arcsin\left(\operatorname{arcsinh}\left( \arctan \left( \operatorname{arctanh}\left(\frac{x}{n}\right)\right)\right)\right)}{x(x^2+1)} \ dx - \frac{\pi}{2}\right)$$

5:58 PM

How do I put larger vertical distance between two elements in a latex array

@PedroTamaroff It's pretty easy, it's not a joke.

@Chris'ssis That looks adhoc-ish.

@Chris'ssis -_- that evil expression there .. I think I know what may be going in that integral .. although I'm not sure :|

@r9m Is it? As fas as I know, Ramanujan comes from India, isn't it? :D

@Chris'ssis It is.

Ramanujan knew very less mathematics before getting hold of that book, BTW.

6:01 PM

@BalarkaSen I see.

He was extremely talented, not a know-it-all.

@Chris'ssis wait .. thats not what I wanted to say ... I mean is Ramanujan the only name you could come up with ?! :P

yeah.

@Darksonn: you could use \renewcommand{\arraystretch} ... I dunno what you're really trying to do.

@r9m lol, no. I also remember some guys that generalized some of the integrals and series of Ramanujan. I don't remember their names. :D

6:04 PM

@Chris'ssis Did you try any of the two problems I posed?

@Chris'ssis Note that Ramanujan knew very less mathematics. (the one to blame is our educational system). Ramanujan was a problem solver -- the only thing he knew was how to manipulate through sums and integrals.

For the first one, I'll give you a hint. $$\int_{-\pi}^{\pi}e^{it(k_1+k_2+k_3+j)}dt=\begin{cases}1,k_1+k_2+k_3+j=0\\0,\rm else\end{cases}$$

Now sum through $|k_i|\leqslant n,|j|\leqslant s$.

@Chris'ssis Hehe .. I saw what you wrote the other day .. I can see that nothing below the caliber of Ramanujan pleases you :P ..

@PedroTamaroff You give me hints? You're very kind today. :-)))

@r9m What I wrote? :-)

@Pedro: You should look up the Hardy-Littlewood circle method sometime ...

6:08 PM

@TedShifrin Hem hem.

Then you can start to use Fourier analysis for -- gasp -- number theory. :)

=P

OK, enough chat for today. I've to got to go.

@TedShifrin OK. I'm reading about it. Looks interesting.

@Chris'ssis the short conversation you had with Caleb the other day .. imo you sure are exceptionally extremal !! :D

Then you can look at my colleague Neil Lyall's webpage for some stuff along those lines, @Pedro. Some expository papers are there.

6:12 PM

@r9m Heya. Got that paper about the frac integrals? I remember you posting it a few weeks back.

hi @N3b

@r9m and what does that mean here? "exceptionally extremal"? :D

@TedShifrin Yarr

@N3buchadnezzar which one ? :-) I forgot .. :) There are some crazy frac integrals in Furdui's book :D

@r9m Some crazy generalization of $\{ 1/x \}^n \{1/(1-x\}^m$

6:14 PM

@Chris'ssis just sayin ya know .. meant no harm in ayway .. :D

@N3buchadnezzar omg !! :-) okay .. link plz :)

@r9m No pb with that, but I wasn't sure about the meaning. :-)

@r9m You linked me that paper :p 27 pages by some chinese dude

@N3buchadnezzar oh !! ^^ I forgot about that too :p irks ..

:17103190

This is the first time I do this. What do you think?

Out for some jogging. brb

6:21 PM

Ha ha ha @Pedro. Some might call it downright obnoxious :)

@TedShifrin Oh, noes.

Really?

I had a colleague who taught calculus with fill-in-the-blank tests.

@PedroTamaroff Perhaps you should say that $(m,n)$ denotes gcd. (Some people might be used to $(m,n)$ for ordered pair and $\gcd(m,n)$ for g.c.d.)

Yes, indeed, when I teach Algebra, I always write gcd ...

Unless we've talked about ideals generated by ...

@TedShifrin Does that mean the greatest common divisor? What kinds of applications does it have in math?

6:23 PM

@MartinSleziak $(m,n)=1$ makes no sense if $(m,n)$ is an ordered pair.

IIRC I have seen in some discussion on MO that someone mentioned they were giving students homeworks of the type "fill missing part of this proof". The intention was to get them to used to lingo.

People know how to interpret context.

Don't worry.

Well, it's not universally known that $(m,n) = \gcd(m,n)$ and $[m,n]=\text{lcm}(m,n)$.

For example, LaTeX doesn't even know lcm :D

(I think it was either in a comment on the main or on meta - now tea.)

@TedShifrin \defineMathOperator

6:24 PM

it's easier just to insert \text{}.

@N3buchadnezzar \DeclareMathOperator

Since I can't write macros to use, this chat gets annoying.

@PedroTamaroff It would take some time to check that, if we use Kuratowski's definition of ordered pair and von Neumann's definition of natural nubmers.

We should try to make some CSi smarty write down a script for us to have macros here @TedShifrin

I guessed :p

6:26 PM

@MartinSleziak Now you're being pedantic!

@PedroTamaroff We had that earlier, it was removed for a reason.

Isn't he usually? @Pedro :D

\definenewcommand{pedro}{fill in the blanks}

That doesn't work, @N3b.

I could not help myself.

6:28 PM

@TedShifrin Thats what I told pedro, but it worked back in the old days.

hmm, define "old days" ?

Google says: "a period in the past, typically regarded as significantly better or worse than the present."

Gee, thanks, @DanielF.

omicron is the coolest name for a character, and then it's symbol is just a boring ο

Is there a \var version for omicron?

It appears not.

Interestingly enough, this is the only result you get in Google if you type in \varomicron. limg.sakura.ne.jp/LimgWiki/…

6:41 PM

@Darksonn Well, the name means "small o", so what did you expect?

I don't know greek, I just see characters with cool names

I also really like kappa but kappa is $\kappa$

almost a k

$\kappa k$

so no exactly the same

$\varkappa\kappa$

hmm

what does the var in varkappa, varphi and so on mean?

@Darksonn It's just an alternative way of writing it down.

"Variant", another variant of writing the letter. Note that $\varphi$ and $\vartheta$ are actually the (old) standard variants.

6:46 PM

They're quite different! Atleast $\vartheta$ and $\theta$

Think of it as printed letters versus cursive ?

I'm too young to know what cursive is. What is it, @Ted?

Also, I was just doing an integral and got something of the form $\log |x|$.

Handwriting @Khallil

heya @blue

heya

Would it be true that, in the case of the integral, $\log |x| = \log \sqrt{x^2} = \frac{1}{2} \ \log (x^2) = \frac{2}{2} \log (x)$?

6:56 PM

Only if $x>0$.

Taking the obvious case of $x=-1$, it's clear that $\log |-1| \neq \log (-1)$.

Yea, so if I had an indefinite integral, I really ought to leave it as $\log |x|$ right?

We're working with real numbers here. Complex $\log$ is a multivalued function.

Yeah, you should, although I don't usually make my students do that unless they're doing an applied problem where the absolute value becomes relevant.

Oh, wait! The original integral had a condition that $0 < x < 1$.

All's well that ends well.

@BalarkaSen I compute $\operatorname{Gal}(\Phi_3)\cong\Bbb Z_2^2$

Therefore counter example to cyclicity

7:15 PM

@Alizter Isn't $\Phi_3(X) = 1 + X + X^2$?

Hi @DanielFischer if you have a second will you look at my post maybe you have an insight into an approach of some kind for the problem...

@DanielFischer A hint if possible...if you have chance at some stage.

7:29 PM

@DanielFischer Yes

@JohnDoe I know practically nothing about nonlinear. I'd think you need the hemicontinuity. Maybe looking at some $w \notin \bigcap_{z\in M} T_1(z)$ helps. If $w \notin T_1(z_0)$, you may be able to derive a contradiction assuming that $w\in T_2(z)$ for all $z$ close enough to $z_0$ using the hemicontinuity. That's what I'd try first.

@Alizter So the splitting field is $\mathbb{Q}[e^{2\pi i/3}]$.

@DanielFischer I thought $\Bbb Q(\sqrt{3}, i)$?

Is it correct to try and decompose a rational function, say $\frac{p(x)}{q(x)}$ where $q(x) \neq 0$, if the degrees of the $p$ and $q$ are equal?

because then I get the klein four group

@Alizter $\mathbb{Q}[e^{2\pi i/3}] = \mathbb{Q}[\sqrt{-3}]$ is normal.

7:33 PM

@DanielFischer Okay thanks, I will try that. Are you familiar at all with this definition of hemicontinuity? I famaliar with another definition. One that looks more like:

@JohnDoe No, not at all familiar with hemicontinuity.

@DanielFischer $T: K \rightarrow X^{*}$ is hemicontinuous if and only if the real function $t \mapsto \langle T(x + ty), z \rangle$ is continuous for all $x,y,z \in K$.

@DanielFischer oh so I messed up with the splitting field there

Okay thanks for the idea, will have a look.

Good luck, @JohnDoe.

7:35 PM

@DanielFischer So it should be $\Bbb Z_3$

@Alizter $\mathbb{Z}_2$, the degree is $2$.

wait let me think

Conjugation is the only nontrivial automorphism.

@DanielFischer I had a what the four-letter-word am I thinking about moment sorry. Of course it is $\Bbb Z_2$

Don't worry. It won't be your last.

7:47 PM

It's part of the growing pains :-)

Out of interest. Why are capital variables used in aa?

@r9m youtube.com/watch?v=xMzDTaq9XW4 :-)))))

@Alizter ? What's aa?

sorry Abstract Algebra

What was the motivation behind writing $P[X]$ rather than $P[x]$?

Ah, for indeterminates.

7:51 PM

yes

@Chris'ssis :-)

Lower case is also used, but upper case is used to hammer home the point that it's not a point of the ring one inserts, but an abstract thingy. $P(X)$: polynomial; $P(x)$: the polynomial evaluated at the point $x$.

That makes sense

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