Mathematics - 2016-10-30 (page 1 of 8)

DogAteMy!! :)

@BalarkaSen You might find this map interesting cia.gov/library/publications/the-world-factbook/graphics/…

Hi

12:37 AM

@Kaumudi: Do you vote yes or no?

How're things, DogAteMy?

user228700

We can't read that, @Kaumudi. Do you think it is an equivalence relation or is not?

user228700

^ @TedShifrin: I vote yes.

Yes we can...?

Explain to me why it's symmetric.

Well, DogAteMy, you're young.

12:40 AM

@AkivaWeinberger Heh.

I'm not.

Oh no, @Balarka is still awake? Oy.

Oh, there's a subtlety there, isn't there...

In my oldness, DogAteMy? Nah. :P

@TedShifrin Slept for like half an hour and tried to sleep for like 3

What? No, the problem.

12:41 AM

Balarka, this isn't good :(

LOL, DogAteMy :)

@Kaumudi There's a subtlety there.

user228700

Uh, that's what I wrote in that piece of paper...it's symmetric because if, for $x$ and $y$ belonging to $R$, we have $xBy$, then $x/y= \alpha$ where $\alpha$ is a rational number. I figure if I take the reciprocal, $1/ \alpha$ must be a rational number too..?

Well...

Does $x/y$ always make sense?

user228700

Ah...

12:43 AM

nods

There's a counterexample to symmetry.

Confusing terms

Nah. I just daydreamed.

Balarka's sleeplessness is making me stoooopid.

That's not my fault.

Most things in this chatroom end up being your fault.

user228700

Hm, but what if I don't take the case where $x/y$ is not defined. (At this point, I have lost hope of the relation being an equivalence relation, since it's not reflexive for 0, but what is that counterexample to symmetry?)

12:48 AM

Then you're ok, @Kaumudi. But for it to be an equivalence relation, it has to work for all ...

user228700

Yes, but it seems to be not reflexive, that's all...why isn't it symmetric?

No, it's reflexive. It fails to be symmetric.

user228700

Huh?

Think 0

It is reflexive for $0$. $(0,0)$ is in the relation, since $0=\alpha0$ for any $\alpha$ (we only need one rational $\alpha$). (Note that the definition of the relation doesn't mention $x/y$.)

user228700

12:50 AM

For 0, it's not reflexive...

$0\sim\pi$ but $\pi\sim 0$? Or maybe I have that backwards.

Sure, it's reflexive for $0$.

$0=0\cdot 1$.

You have it correct. $0$ is not $\sim$ to $\pi$.

But $\pi \sim 0$.

@Balarka Have you talked to a doctor about your insomnia? I have some form of medication that helps when I have mine. The restless leg I used to have is gone now, so it's not as bad.

In that case, @Balarka, I had it backwards :)

Yeah, Balarka, we do worry about you. Seriously.

Oh, I have it backwards too.

12:53 AM

Well, we can't both be backwards. Geez.

user228700

What does this $\sim$ notation that u're using mean? I thought it meant difference :/

They mean $(0,\pi)$ is in the relation but $(\pi,0)$ isn't

thanks DogAteMy for being the sensible interloper

user228700

'Cause u can multiply $\pi$ with 0 to get 0, but there is no rational number with which u can multiply 0 to get $\pi$, yeah..?

Yep.

12:55 AM

Bingo, @Kaumudi.

The $\sim$ thing is just a notational thing; writing it like that just makes it easier to see the analogy between equivalence relations and equality.

We didn't need to use $\pi$. Any nonzero number would work.

user228700

OK, I understand. Thank you, guys :-)

Most welcome, @Kaumudi.

@MikeMiller Not yet, but I may have to. Mine is a bit more complicated (simpler?) than insomnia, to my understanding, which is not having enough sleep. I will invariably make up my 8 hours nap time, but the whole sleep cycle will be so messed up that most of my day will end up useless, whereas it could have been constructive.

12:56 AM

Moral of the story: Be careful when you turn a statement into division.

@Balarka: Sometimes you should sacrifice constructivity for your health.

Rule 4 in Mathcamp was "No division by zero (except under staff supervision)"

In the end, the latter is much more important.

LOL, DogAteMy. I'm glad there are other people who try to inject (bad) humor.

The rules were 1) Be excellent to each other, 2) No stupid stuff, 3) No fire*, 4) No division by zero*

*except under staff supervision

I approve, DogAteMy :) ... I should have had such rules for my office.

Rule 0 was to obey the laws of the campus, country, Geneva conventions, physics, etcetera

12:59 AM

@Ted It's only a local sacrifice anyway.

@TedShifrin Thanks. I'll see a doctor about it.

How many unique solutions are there to a rubik's cube?

Thanks, @Balarka.

Infinite...?

DogAteMy: The hell with obeying the laws of physics.

1:01 AM

You add turning a face four times to get a "unique" solution... or, if that doesn't count, do RDR'D' six times (which puts the cube back where it was) at any point @idonutunderstand

I have never had restless legs, but I have a relative who's had that. That sucks.

DogAteMy: Maybe there's a notion of locally minimal :)

Or just do random twists for an hour and then solve the result (which can always be done in less than 20 moves, fun fact)

It's my mom's birthday and I solved one, then wrote h a p p y b i r t h d a y '1 6 on it, then scrambled it up again with the intention of giving it to her tonight and telling her she can either figure out the word scramble or solve the cube to find out the message. But then I realized she might solve it in a different way that would not render the intended message.

One of SamuelY's friends at UCLA had (has?) a record for speed-solving the cube while blindfolded.

1:04 AM

Cool.

LOL

? ^

@idonutunderstand you wrote message on solved cube ?

@Ramanujan yeah

my LOL was to @idontunderstand.

1:05 AM

Why would that depend on the moves used to solve it...?

:)

Then every time we solve it we get same result(same cube)

Oh, good point.

oh nice!

You can frustrate your mom, after all.

1:06 AM

Btw do your mom know how to solve it?

no lol

I have never done Rubik's cubes.

I can do 3 faces. Never managed to figure out how to do all.

@idonutunderstand then forget it,getting cube solve is much tough

I don't like that kind of math puzzles ... never have.

I've never fiddled with a Rubik's cube ... ever.

1:07 AM

@TedShifrin then which math puzzle you like?

More logic ones, geometry ones ...

Ooh, small problem, the centers might not be rotated correctly with respect to the rest of the face. Not that it would matter much; you'd still be able to guess that the center should be rotated.

Yeah it's just meant to be a gimmick. She'll just figure it out like a scramble puzzle.

I can solve a Rubik's cube

Specifically?

1:08 AM

Oy @ DogAteMy

I don't have answers, @Ramanujan. But I'm not much a fan of discrete math or number theory, either. ...

@AkivaWeinberger Not surprisingly :)

@AkivaWeinberger okay, yeah that shouldn't be a problem :)

@BalarkaSen Verily.

...OK

thanks for the good info

1:11 AM

Pretty sure I figured out some stuff I'be been puzzling on for a while. Good math day.

Good for you, @MikeM ... I hope it sits up (or stands up).

Nice!

Since it doesn't seem I'll sleep anytime soon, might as well do some math...

Seriously, Balarka, you need to fix this and get healthy.

But soon you'll be done suffering through my various texts and then you'll sleep great :P

@Ted Well, at least he hangs out in here at times convenient for us, even if he's rapidly killing himself.

Not suffering; enjoying. While I am reluctant to do the (tedious, seemingly) calculus in some contexts, intuition gained from your text+exercises and Mike's abstract exercises are doing a great job entertaining me I think.

1:22 AM

Shit, now he knows.

In all seriousness, @Balarka, I think you will be better for our influence. But health is of prime importance.

It took me way too long to realize "texts" meant "textbooks"

oops.

Well, in some contexts, yes, DogAteMy :)

the sacred texts

1:27 AM

not to mention the sacrosanct ones

donaldson, donaldson and kronheimer, kronheimer and mrowka, mrowka ruberman and saveliev, saveliev, morgan

let's throw in besse for good measure

I know too many of these people. List some godly texts farther removed from me.

Gromov, "Hyperbolic groups"

That's mostly just the books I don't travel without.

Well, I'll just leave. The air is getting too lofty for me. Balarka, get some sleep!

1:32 AM

Starring that so I can look them up later

@TedShifrin Thinking about Clairaut now.

lol I just saw a video about Trump mimicking our prime minister.

1:50 AM

Interesting. That the angular momentum of an object along a path is preserved on a surface in $\Bbb R^3$ if that path is a geodesic is a nice and neat way to check if something is a geodesic or not.

How am I supposed to prove this beautiful theorem? $$\dfrac{\sin x}x \equiv \cos\left(\dfrac x2\right) \cos\left(\dfrac x4\right) \cos\left(\dfrac x8\right) \cos\left(\dfrac x{16}\right) \cdots$$

I'd call that sinc(x), but that's just my engineering background.

Hint: $\sin x=2\sin\frac x2\cos\frac x2$

and $N\sin\frac xN\to x$ as $N\to\infty$

Wasn't that Euler who figured that out?

Wikipedia says yes

2:01 AM

@AkivaWeinberger thanks

https://en.wikipedia.org/wiki/Viète's_formula

I have no idea how to reference you though

@Edward Does that have a name?

:) Euler is a good guess for most mathematical things...

If it has a name, I don't know it.

> a result discovered in 1593 by Francois Viète (Kac 1959, Morrison 1995) and sometimes known as Euler's formula (Prudnikov et al. 1986, p. 757; Gearhart and Shulz 1990)

so who discovered it?

1593 predates Euler by some centuries, so apparently Viète.

2:06 AM

but I can't write "Viete's formula" because that's for polynomials

Also, Viete's formula is specifically for pi

Viète's formula

In mathematics, Viète's formula is the following infinite product of nested radicals representing the mathematical constant π: 2 π = 2 2 ⋅ 2 + 2 2 ⋅ 2 + ...

For $x=\pi/2$, specifically

How on earth should I name it then?

@AkivaWeinberger that's cool

It's the same as $(\sin x)/x\to1$ as $x\to0$

the line above, I meant

2:12 AM

Ah.

I know that fact; it's in every calculus course I ever taught

the trick is real cute

@MikeMiller but how on earth am I supposed to name it?

"An infinite product for sinc"

"cute formula"

@MikeMiller genius

2:18 AM

"Michael"

Unless it's a girl

what does the "c" in "sinc" mean?

Or maybe even if it is a girl idk

$\operatorname{sinc}(x)=\dfrac{\sin x}x$

oh, "sine cardinal"

@Balarka: Be careful. Only forces normal to the surface to constrain the particle to the path doesn't mean angular momentum is conserved in general!

?

2:19 AM

but what on earth does "cardinal" mean?

mathworld.wolfram.com/SincFunction.html

> The full name of the function is "sine cardinal," but it is commonly referred to by its abbreviation, "sinc."

meh

> The term "sinc" /ˈsɪŋk/ is a contraction of the function's full Latin name, the sinus cardinalis (cardinal sine).[2] It was introduced by Phillip M. Woodward in his 1952 paper "Information theory and inverse probability in telecommunication", in which he said the function "occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own",[3] and his 1953 book "Probability and Information Theory, with Applications to Radar".[2][4]

Wikipedia

but what on earth does "cardinal" mean?

I never encountered the name "sine cardinal" but have known the term "sinc" for many years. I don't know that it stands for anything.

cardinal adjective. Of the greatest importance; fundamental.

2:20 AM

@AkivaWeinberger thanks

kind of defeats the point to abbreviate "cardinal" if it means "of the greatest importance"

You'd prefer a gorgeous red bird?

I was thinking of exactly such a bird building its nest in the trough of an ordinary sin function.

I'm getting bogged down in gradings.

But then we'd have to ask what the n in sin means...

And I can only take so much recursion.

> Etymologically, the word sine derives from the Sanskrit word for chord, jiva*(jya being its more popular synonym). This was transliterated in Arabic as jiba جــيــب, abbreviated jb جــــب . Since Arabic is written without short vowels, "jb" was interpreted as the word jaib جــيــب, which means "bosom", when the Arabic text was translated in the 12th century into Latin by Gerard of Cremona.

> The translator used the Latin equivalent for "bosom", sinus (which means "bosom" or "bay" or "fold").[8][9] The English form sine was introduced in the 1590s.

@Edward three recursions saved for you

2:26 AM

I have to confess that now that I have heard that that "sine" and "bosom" are related, I will never again consider any period shorter than $4\pi$

@Edward ...

Apparently the Sanskrit word jiva is (very very) distantly related to the word vital in English

As well as the prefix bio-

@TedShifrin I am not applying any external forces.

@AkivaWeinberger source?

They have more or less the same meaning

user228700

2:29 AM

I've a quick question. When given two intervals, how to find their intersection?

A bunch of Wiktionary pages. Apparently they keep track of Proto-Indo-European roots

@Kaumudi You mean, like, "$(0,3]\cap[2,4)$"?

@AkivaWeinberger I've gone there, can you elaborate?

user228700

Eg: $[2n\pi, (2n+1)\pi]$ and $(-4,-4)$

user228700

Where $n$ is an integer.

I am hungry and there's literally nothing to eat. Darn.

2:31 AM

@AkivaWeinberger the PIE root for "jiva" is *gʷiH- while the PIE root for "vital" and "bio-" is *gʷeyh₃-

en.wiktionary.org/wiki/%E0%A4%9C%E0%A5%80%E0%A4%B5 -> en.wiktionary.org/wiki/Reconstruction:Proto-Indo-European/… -> en.wiktionary.org/wiki/…

I've spent all day doing algebra and I am so bad at it

@AkivaWeinberger No, I'm talking about en.wiktionary.org/wiki/…

while "i" can be converted to "ey", the exact quality of "H" is not determined, so you can't say that they are the same word

user228700

Anybody?

@MikeM I sympathize, although I suppose that's harder than any algebra I have ever done

2:33 AM

@Kaumudi Try drawing it out

I said jiva, not jya

@DHMO

user228700

The answer is supposedly $[-4,-\pi]$ U $[0,\pi]$.

@AkivaWeinberger alright

user228700

Is that correct?

Sounds right, although probably should be $(-4$ and not $[-4$

because $-4\notin(-4,4)$

user228700

Yeah, OK. How tho? I don't get how to obtain the answer by drawing :/ If I put $n=0$, I see that it should lie b/w 0 and $\pi$.

2:38 AM

so, you meant (-4,4) not (-4,-4) right?

the intersection of two intervals will always be either empty or a single interval, not a disjoint union of two of them

Right. And if you put $n=-1$?

user228700

@arctictern Yes, typo :|

Also I assume you mean $\bigcup_{n\in\Bbb Z}[2n\pi,(2n+1)\pi]$ — that is, the union of all intervals of that form.

So, the intersection of that and $(-4,4)$.

user228700

@AkivaWeinberger Uh, yes, that is precisely what I meant.

user228700

So if I take one of those sets and find the intersection b/w that and $[-4,4]$ (Yes, [ ]. Typo again, sorry :|), then I get my answer, right?

2:42 AM

"$\displaystyle\bigcup_{n\in\Bbb Z}$" (or "$\displaystyle\bigcup_{n=-\infty}^\infty$") just means we're letting $n$ be every integer and unioning them together

Well, no, you have to do every $n$ individually — but, for all $n$ other than $0$ and $-1$, they don't intersect each other at all, so they don't affect the final result.

So you end up just intersecting $[-2\pi,-\pi]\cup[0,\pi]$ with $[-4, 4]$.

seriously, draw things on a number line. the intersection should be obvious by inspection then

^

user228700

@arctictern I didn't know what to draw, since $n$ is involved.

@Kaumudi pick values for n, get intervals, draw the intervals

user228700

Which values to pick?

2:45 AM

start picking them

do things

don't sit on your hands

can you draw all the intervals [2n,2n+1] for instance? it's just all intervals from even numbers to the next odd one

user228700

Well, I picked $n=1$, found the intersection and it was completely wrong so I figured something must be going terribly wrong, the way I did it.

user228700

That's when I asked here.

I said values. you can't just draw one interval, you have to draw enough of them.

@Kaumudi How about $n=-2,-1,0,1,2$ or something

Do more than one

Oh, I have a packet of biscuit in my drawer for emergency purposes

great

2:48 AM

what

@Balarka: Then highly unlikely the particle doesn't fly off the surface :P

@DHMO hi

@Ramanujan hi

Hi, tern :)

@MikeMiller chat.stackexchange.com/transcript/message/33206494#33206494

@TedShifrin yikes true

2:50 AM

How to solve this?

I wish people would learn to type in MathJax and not expect us to read photographs.

to be fair, that'd be a lot of stuff to type

user228700

I pick values, I plot them and then what..? Essentially, I'm supposed to find the interval b/w $[-4,4]$ and all of the other intervals involving $\pi$?

LOL ... you apologist, you.

with the einstein summation convention, it won't

2:52 AM

@Ramanujan: Have you considered formulas for $\tan\alpha+\tan\beta$?

@Kaumudi I drew the thing

@arctictern

@TedShifrin it is sum of n not 2 angles

Start with 2.

Ok

On the top the things surrounded by big brackets are the first set and the thing surrounded by small brackets is the second set. The bottom is the intersection

2:53 AM

There is tan inverse :-(

Yes, and if you take $\tan(\tan^{-1}x)$, what do you get?

@TedShifrin it's inverse tan no?

Then apply tan to it.

Thank goodness @Balarka is here to defend my sanity.

Don't you need the formula for $\tan(\alpha+\beta)$ here

2:55 AM

OH, I mistyped. My apologies.

Yes, that's what Ted's suggesting

I meant $\tan(\alpha+\beta)$. I mistyped.

to derive the formula for $\tan^{-1}(a)+\tan^{-1}(b)$

Kudos to DogAteMy.

I automatically corrected that in my head

2:55 AM

Well, I was stooopid.

I apologize.

user228700

@AkivaWeinberger OK. I'll do this on my own again. I just didn't know that I as supposed to find the intersection for literally all of the $n$s.

@Kaumudi Only two $n$s end up mattering

leaves DogAteMy to run the room

@Kaumudi $(\bigcup I_i)\cap J$ means all things that are in $J$ and at least one of the $I_i$

Um

user228700

2:56 AM

I was confused as to what I was supposed to do, that's all... ._.

Is @DHMO here? Looks you are typing answer for me

@arctictern Confusing wording there but OK

I just realized - is @DHMO water?

Like, chemically

Dihydrogen monoxide

H2O

user228700

Alright, thank you :-)

user228700

@AkivaWeinberger Yeah, it's this conspiracy thing. Has a Wikipedia page.

@Ramanujan $\displaystyle \tan^{-1}\left(\frac{x}{1\cdot2+x^2}\right) = \tan^{-1}\left(\frac{1/x}{(1/x)\cdot(2/x)+1}\right) = \tan^{-1}\left(\frac{(2/x)-(1/x)}{1+(1/x)\cdot(2/x)}\right) = \tan^{-1}\frac2x - \tan^{-1}\frac1x$

2:59 AM

@Kaumudi Did you know! An excess of DHMO can kill you!

As I was meaning to say, $\tan(\alpha+\beta) = ...$.