Mathematics - 2016-12-02 (page 1 of 11)

DHMO

00:01

why is it that inverse of monotone function is monotone?

@heather you only proved that it is an injection

Secret

@DHMO one to one is another name for injection. bijection is both one to one and onto (surjective)

Jasch1

00:29

Hey guys

Can you guys please help me understand something

Secret

@DHMO Let $S$ be a set. Given a map $f$ is monotone means the image $f(S)$ is a partially ordered set. The inverse map is the bijective map defined to be $f^{-1}$ such that $f^{-1} \circ f = f \circ f^{-1}=\textrm{id}$. Therefore $(f^{-1} \circ f)(S)=S$. Now for all elements $a,b\in S$, monotonicity means if $f(a)\geq f(b)$, then either $a \geq b$ or $a \leq b$. Therefore $S$ is also a partially ordered set. Thus $f^{-1}$ must also be monotone

meow-mix

what @Jasch1

Jasch1

Cantor's diagonal argument

In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began. The diagonal argument was not Cantor's first proof of the uncountability of the real numbers; it was actually published much later than...

In this

Kinda confused about this part

he constructs the sequence s by choosing the 1st digit as complementary to the 1st digit of s1 (swapping 0s for 1s and vice versa), the 2nd digit as complementary to the 2nd digit of s2, the 3rd digit as complementary to the 3rd digit of s3, and generally for every n, the nth digit as complementary to the nth digit of sn. In the example, this yields:

I'm a hs sophomore writing a paper on some stuff on infinity

Fargle

@Jasch1 What's confusing you here?

Jasch1

I'm confused about how he goes from the infinite binary series to s = (1,

wait didnt copy right

s = (1, 0, 1, 1, 1, 0, 1, ...)

I put off a 10 page reserach paper until the night before

Fargle

00:40

He's going down the diagonal, and finds (0,1,0,0,0,1,0,...)

Then he changes every one to zero, and every zero to one.

Now he has (1,0,1,1,1,0,1,...)

This can't be in the list. It can't be the first part of the list, because the first element doesn't agree.

It can't be the second part of the list, because the second element doesn't agree.

And so on.

So, he's constructed something that's outside the list, contradicting that the list had everything of that form.

Adeek

hey @Fargle

Fargle

Howdy @Adeek, and @Ted. How's it going?

Adeek

good

Ted Shifrin

hi @Fargle

Adeek

I was wondering I think there is a mistake here

Fargle

00:43

Now that I'm not qualified to answer. Going through D&F myself, as it were.

Adeek

$\sigma$ should send the root of unity to $\omega^7$ and $\tau$ should send $\omega$ to $\omega^5$

yeah I am pretty sure there is a mistake here if I am not crazy.

$\sigma(\omega) = \sigma(\theta)^4(\sigma(1/2) + \sigma(1/2 *i))$

Ted Shifrin

What is $\omega$? You mean $\zeta$?

Adeek

yeah I don't know that letter.

$\omega^4 = e^{\pi * i} = cos(\pi) + i sin(\pi) = -i$

$\theta^4 = \sqrt{2}$

so $\sigma(\theta)^4 = \omega^4 * \theta^4 = \sqrt{2}/2 - i *\sqrt{2}/2$

Ted Shifrin

They are correct, Karim.

Adeek

why ?

meow-mix

00:48

Hi @TedShifrin

Ted Shifrin

You should learn to question yourself more rather than assuming they're wrong :P

Hi @meow

Adeek

haha yeah

Ted Shifrin

$\sigma(\zeta) = \frac12\sigma((1+i)\theta^4) = \frac12(1+i)(\sigma(\theta))^4 = \frac12(1+i)(\zeta\theta)^4$

Adeek

yeah that is what I got

Ted Shifrin

$\zeta^4 = -1$, $\theta^4=\sqrt2$.

So we get $-\zeta$.

But $-\zeta = \zeta^5$.

Avoid distributing things out ... that is usually not a good thing.

Adeek

00:50

ohh haha

distributing error lool

it is been a long time since I did college algebra

Ted Shifrin

That is a totally unacceptable attitude.

Adeek

yeah

Ted Shifrin

You cannot do mathematics without doing high school algebra correctly.

Jasch1

@Fargle just saw your response, ty

Adeek

yeah lol

Secret

00:51

Suppose I have a monotone map $f$ and an unordered set $S$. By definition of monotonicity $f(S)$ is a poset. Is this possible?

Fargle

No problemo, @Jasch1.

meow-mix

@TedShifrin how are you using the plane to describe $\mathbb{P}^2$? are you using the $\mathbb{R}^2$ part of $\mathbb{P}^2$ (i.e. the lines given by $[1,x,y]$)?

Jasch1

I suspect that I will be on a lot tonight

Ted Shifrin

@meow: I explained carefully that $\Bbb P^2 = \Bbb R^2 \cup \Bbb P^1_\infty$.

Secret

that is, is there exists a monotone map that impose a partial order to an unordered set?

Ted Shifrin

00:53

So what are you asking me now?

meow-mix

yes i understood that

what im asking is

when you use a coordinate plane to describe points in lines in $\mathbb{P}^2$

is the point $(x,y)$ on that plane corresponding to $[1,x,y]$? or something else

Jasch1

Any of you written math research papers before?

Ted Shifrin

I'm not understanding what you're saying. A line in $\Bbb P^2$ is given as the solutions of a homogeneous linear equation $ax_0+bx_1+cx_2=0$, for example.

meow-mix

yes @TedShifrin

let me get an example, one second

Ted Shifrin

OK.

Jasch1

00:56

When declaring a set in a proof

Should it be italicised?

Ted Shifrin

Not generally. You mean, defining a set while you're doing the proof?

meow-mix

what do you mean?

Ted Shifrin

LaTeX italicizes things. What do you mean with your question, @Jasch1?

Jasch1

Not using LaTeX

wondering if I should

meow-mix

you definitely should

what are you using right now?

Jasch1

00:57

Lets say I have 6 hours to do this

Ted Shifrin

But your question is ... you're babbling along in a proof. And you say "Let $A = \{x\in\Bbb Z: x\equiv 3\pmod 7\}$."

What are you wanting to italicize?

Jasch1

is the hastle of downloading and learning it worth it

Ted Shifrin

Not with only a few hours, no.

Jasch1

I have to finish this and a 5 page english paper by the morning

Ted Shifrin

But in the long term, yes.

Ah, typical college student rolls 9 of 9 eyes

Jasch1

00:58

Nope

HS Soph

Ted Shifrin

LOL, high school ... even worse

meow-mix

@TedShifrin when you smack me, do you use your 3rd or 4th arm?

Ted Shifrin

Anyhow, I asked you a question about your question?

Jasch1

But a very, very competetive high school

Ted Shifrin

LOL, @meow ... all arms that are required.

Fargle

00:59

@TedShifrin The mathematician's answer.

Ted Shifrin

@Fargle: I'm retired, not dead (yet) :D

Fargle

The slap is unique up to a permutation of arms...

Ted Shifrin

Except when you leap.

Fargle

If a Ted-smack is executed in the middle of a forest, and I leap to avoid it, am I still ashamed?

Ted Shifrin

"executed" is such a violent term and sends a mixed message.

meow-mix

01:01

so @Ted, in this figure, you use a plane to represent $\mathbb{P}^2$

Ted Shifrin

Yes. That's true. Sometimes I put the three axes in the plane, sometimes not.

meow-mix

would that plane be the "$\mathbb{R}^2$ part of $\mathbb{P^2}$?

Ted Shifrin

No, no, that's all of $\Bbb P^2$.

It's a schematic, not literal, picture.

Now I understand your question ...

meow-mix

how would we represent the $\mathbb{P}^1$ at infinity though?

Ted Shifrin

It's no different from any other line in that picture. But look at Figure 4.

The line $x_0=0$ is "the line at infinity."

G'night @MikeM.

For the purposes of Figure 8, @meow, there was no need to think about specific coordinates — just geometry as a whole.

@meow: Are we OK now (temporarily)?

meow-mix

01:06

yes

Ted Shifrin

OK.

meow-mix

today i taught a kid some basic trig stuff; the unit circle, what radians mean, and what sine and cosine really means in terms of the unit circle

Ted Shifrin

@meow, if you want, I'll send you a handout I wrote for my calc theory class years ago with a trig review (and some nonstandard derivations).

meow-mix

@TedShifrin that'd be great, thank you. you can send it via my gmail account :)

Ted Shifrin

@meow: Sent.

Jasch1

01:17

quick question

just about how stuff is named

Cantor's diagonal argument

In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began. The diagonal argument was not Cantor's first proof of the uncountability of the real numbers; it was actually published much later than...

s is a list right?

Not a set

Ted Shifrin

it's a sequence

meow-mix

well

Jasch1

Because I know a set generally uses {}

k

meow-mix

what ted said

Jasch1

ty

meow-mix

01:18

a sequence is like an ordered set

Jasch1

how old are you guys?

meow-mix

13

Ted Shifrin

that's confusing, @meow, because you index by the positive integers, not by the order of the real numbers

@Jasch: Add 50 for me :P

Jasch1

50 years the wiser

meow-mix

how true. :D

Ted Shifrin

01:19

40+ years of torturing math students :D

Jasch1

teacher?

Ted Shifrin

I used to be a professor ... retired.

Jasch1

k

meow-mix

@Ted likes to make a smooth connection between "tutor" and "torture"

Jasch1

In that same example

of all infinite binary sequences

Ted Shifrin

01:20

@meow: I don't throw the word "smooth" around lightly.

No, no, @Jasch1, only a very few binary sequences.

The point is that you cannot list them all. It's an uncountable set.

Jasch1

Yes I understand that

Ted Shifrin

That's what Cantor's argument shows.

OK, so don't say all :D

Jasch1

k sorry

ty tho

Ted Shifrin

No need to apologize. Just trying to make sure you say stuff that's right :)

Adeek

hey @TedShifrin I was wondering why is determine the lattice of subgroup of this group G is straightforward ?

Fargle

01:21

@TedShifrin There's a "smooth operator" joke in here somewhere.

Jasch1

I have to say, listening to Chance the Rapper helps so much while writing

off a nice speaker

so calming

Adeek

how do they get the lattice so fast ?

Ted Shifrin

The subgroups are pretty obvious (because you know the symmetries of the square), Karim. The intermediate fields are not all. That's what I was telling you to do.

Fargle

@Jasch1 I can't listen to rap while I write, though I love Chance. I listen mostly to instrumental stuff.

meow-mix

im going to upload a new picture

my hair is wet in this picture

Jasch1

01:22

Nah none of Chance's kinda hard stuff

Ted Shifrin

@Fargle: But a singular operator is so much better.

Adeek

oh I see @TedShifrin

Fargle

@TedShifrin :I

Jasch1

@Fargle Just a repeat playlist of Juice, Cocoa Butter Kisses, Favorite Song, Angels, and No Problem

Harder rap is harder to write wit

Fargle

@Jasch1 I just get distracted by words in general, is what I mean. Can't listen to Cocoa Butter Kisses without rapping the whole thing.

Jasch1

01:23

lmao it is fucking amazing I see what your mean

Ted Shifrin

@Karim: In particular, you know about conjugate subgroups and so it's pretty easy to write everything down correctly.

Jasch1

What time zone you guys in?

Fargle

Plini, Sigur Rós and Explosions in the Sky are my go-tos.

Central, @Jasch.

Jasch1

Thats 2 hours behind EST right?

nah 1

Ted Shifrin

Mountain is 2

meow-mix

01:24

i am in EST right now

Jasch1

All I know is between here(NYC) and LA is three hours

Ted Shifrin

Then we good guys in CA are 3 hrs behind. (Although I spent 44 of my years in EST.)

Jasch1

boooooo

NYC is the best

amazing public high schools

meow-mix

@TedShifrin have you proved any neat theorems in your life time? :P

Fargle

My sleep schedule is somewhere in the Pacific, however.

Not the time-zone, mind--the ocean.

Ted Shifrin

01:25

I think so, @meow, yes.

I think you, Brody, and Balarka all need to move to the moon, @Fargle.

Fargle

What'd I do? :O

Ted Shifrin

time zone issues

Jasch1

I've gotten 3 hours of sleep each day this week

probably none today

Ted Shifrin

You'll end up very sick, @Jasch.

Stop chatting and work.

Jasch1

I am

Ted Shifrin

01:27

@Jasch: I went to college with many, many kids from Bronx High School of Science.

Jasch1

In low-level research papers would it be acceptable to use diagrams from wikipedia

Ooooo

Stuy life

Ted Shifrin

NO, it would not. But in a high school paper ...

Jasch1

Its a draft too

meow-mix

@TedShifrin I've been considering dual enrollment, but i dont think it would work because
1. may be too much stress
2. costly

Jasch1

I know in a real paper, it would not be, but solely for portraying what I am saying and not for the intent of publishing

Ted Shifrin

01:28

@meow: I keep telling you it's ok to slow down. Seriously.

Adeek

@TedShifrin what do you mean by conjugate subgroups ?

Ted Shifrin

Only worth joint-enrolling if you go to a school with good classes for you.

That's a standard term, Karim.

meow-mix

of course, @Ted

Ted Shifrin

$gHg^{-1}$ for fixed $g$.

Jasch1

@TedShifrin My school has a big rivalry with BxSci

Adeek

01:29

oh ok I see

Aksel'sRose

the adjoint of $\frac {1}{2i}$ is just $\frac {1}{2i}$ correct?

Ted Shifrin

No, the conjugate of $-i$ is $+i$.

Aksel'sRose

oh yeah that it is...I really need to brush up on my basic linear...

Jasch1

the context of classical mathematics, this is impossible, and the diagonal argument establishes that, although both sets are infinite, there are actually more infinite sequences of ones and zeros than there are natural numbers.

From the wikipedia article

Ted Shifrin

This isn't linear. It's complex numbers :D

Jasch1

01:31

How can this be derived from the fact that T is uncountable

Aksel'sRose

true! Im more a statistics person

Ted Shifrin

"Uncountable" means "more than there are natural numbers" :) @Jasch1

Jasch1

just realized that

Algebra and Trig are my bread and butter

Not much experience with sets and sequences

Prepare for large text paste

Ted Shifrin

Ugh.

Jasch1

wait il post imgur link

imgur.com/a/4suYt anybody has the time to review this paragraph I would be greatful

meow-mix

01:37

ok updated picture

@Jasch1 umm the first sentence is kind of incomprehensible

Ted Shifrin

It still looks like a grim mug shot, @meow :D

Jasch1

whops meow-mix

i can do the math, just me no grammars

Ted Shifrin

@Jasch: I'll be your English teacher and say grateful.

Nor spelling.

Jasch1

lmao I also have to do that

A 5 page paper on macbeth

Ted Shifrin

I'll leave @meow to be the editor. I've done that my whole life. :)

meow-mix

01:39

i feel like if i edit this im going to end up telling you "say this instead" and i don't want to write your whole paper for you, so im only going to be giving little nudges like that one :P

Ted Shifrin

LOL, @meow sounds like me. (Sorry to insult.) :D

Jasch1

K thats what I need

I dont want you to tell me what to write instead

I need to understand what is wrong, I figure out how to correct it

Ted Shifrin

Good boy :)

Jasch1

Better for me to do that

Because the rest is gonna be a pain in the ass if I don't understand this part

meow-mix

sees Ted wink with his third and seventh eyes

Ted Shifrin

01:41

The extra eyes only roll. Do they wink too?

meow-mix

not sure. maybe you can train them to?

@Jasch1 umm not sure what you mean by "infinite binary set"

you mean a sequence? or what do you mean

Jasch1

yes sequence

Ted Shifrin

infinite set of binary decimals?

Jasch1

i forgot to change them back from when you corrected me on the terminology

Ted Shifrin

Oh, sequence of binary decimals

Jasch1

01:43

As an inexperienced writer of research papers, would it be expected of me to include a subsection on the history of this discover?

meow-mix

its not necessary, but you could include it in say the abstract of the paper

Ted Shifrin

Abstracts are short.

meow-mix

or you could add a footnote

Ted Shifrin

This is actually some complicated history, I think.

Jasch1

My intro goes over the history of infinity

Started by mentioning Zeno's Achilles vs. Turtle paradox

Ted Shifrin

01:44

tortoise :P

oh, no, it's the tortoise and the hare. Never mind.

Jasch1

lmao uk what I mean

yea

The Achilles one is with a Turtle

PVAL-inactive

@Ted What do you think about the Mattis appointment? That has to be really positive news for our foreign policy.

(considering possible alternatives)

Ted Shifrin

I haven't seen the nightly update yet, @PVAL. I need more and more martinis to read the news.

meow-mix

it is quite chilly here

Ted Shifrin

@PVAL: Every appointment is about as regressive, inexperienced, and right-wing as you can imagine. This one doesn't seem all that much better.

@meow: You got the email?

meow-mix

01:49

@TedShifrin yes, i did.

Ted Shifrin

@PVAL: I do see a few of the "better" points in the NYT, though.

PVAL-inactive

@TedShifrin Well he certainly isn't inexperienced, and seems to have been vocal against "questionable interrogation techniques", and relatively non-partisan.

Ted Shifrin

yeah, and skeptical about getting into bed with Russia.

But after you've picked Petraeus and Palin and all the rest homophobic, misogynist, racist, and xenophobic, it's not hard to look "better."

Jasch1

I found something rare in America!

a moderate!

Ted Shifrin

@PVAL: Did you pursue that discussion of the integral with function limits with your student?

where did you find that, @Jasch1?

Akiva Weinberger

01:52

@meow-mix So your question was to find a bunch of four-cycles, no two of which leave the same numbers fixed, which satisfy $\sigma_1\dotsb\sigma_n(1234)=\rm id$?

Jasch1

you

Akiva Weinberger

(I don't have a solution, I just want a clarification)

Ted Shifrin

no, I'm pretty far left, @Jasch1.

PVAL-inactive

@Ted No they were completely satisfied with the geometric interpretation.

Jasch1

Same

but at least you have hope that he will do well

Ted Shifrin

01:52

You made me work that hard for nothing, @PVAL? :D

meow-mix

@AkivaWeinberger well $\sigma_n \sigma_{n-1} \dots \sigma_1(1234) = \mathrm{id}$

Ted Shifrin

@Jasch1: Maybe you missed all my sarcasm.

Jasch1

Probably

PVAL-inactive

Well it was a case of a student unintentionally asking a much harder question.

meow-mix

opposite direction

Akiva Weinberger

01:53

Same thing but in reverse, then

meow-mix

either way it shouldnt matter since you can reverse it

Jasch1

pretty numb to sarcasm now

meow-mix

whats your solution :D

Akiva Weinberger

I'll think about it, sounds interesting

Ted Shifrin

I understand, @Jasch1.

Akiva Weinberger

01:53

As I said, I don't have a solution

Ted Shifrin

@meow, DogAteMy: what are we solving?

Akiva Weinberger

I posted it two minutes ago

PVAL-inactive

Most of them have difficulty understanding what the notation means, and don't want to spend "even more" energy on math thinking about possible applications.

meow-mix

(2345)(3451)(4512)(5123) id imagine it would be like something cyclic like that

that might even be a solution, idk

Akiva Weinberger

2 mins ago, by Akiva Weinberger

@meow-mix So your question was to find a bunch of four-cycles, no two of which leave the same numbers fixed, which satisfy $\sigma_1\dotsb\sigma_n(1234)=\rm id$?

meow-mix

01:54

@TedShifrin i posed a little permutation question :)

Ted Shifrin

@PVAL: No one had ever asked me the question before, so I was pleased.

meow-mix

@AkivaWeinberger you forgot a restriction; none of them may be "congruent" (as i described) to (1234)

Ted Shifrin

Is there something special about 4?

meow-mix

yes, that its not prime :D

and not 1

Ted Shifrin

Gotcha.

Akiva Weinberger

01:56

"No two of which leave the same numbers fixed"

meow-mix

im pretty sure this "theorem" has been proved for prime cycles

Akiva Weinberger

Got to go

PVAL-inactive

I think its a really good question, because while I can believe students understand that being able to calculus with products, exponentials, trigonometric, sums etc is reasonable because these are such natural operations.

Ted Shifrin

Bye, DogAteMy.

PVAL-inactive

But that operation isnt really natural.

Ted Shifrin

01:57

@PVAL: Well, moving targets aren't so unnatural, especially in applications.

Oh, @PVAL: Good point.

In fact, the problem I gave you that's true.

My fault.

PVAL-inactive

I didn;t mean to delete that whoops

Transcript for

$Mathematics$ Mathematics