Mathematics - 2017-10-20 (page 1 of 2)

12:00 AM

all i have inherited from my family are normie traits

it's sad

I think the (essentially completed) ethnic genocide of people like me has probably influenced my desire to connect with my heritage

Sha Vuklia

rip @balarka

Eric Silva

plus like, we make good food and shit

PVAL-inactive

pretty sure I can blame my hurting back on my ancestors.

Ted Shifrin

Well, Eric, if Drumpfolini continues, I'll be at least doubly eliminated — Jewish, gay. Probably triply — math/science.

Me too, @PVAL, and I've been in agony for over a week.

Eric Silva

12:01 AM

R.I.P. world

Sha Vuklia

^ why

Balarka Sen

i worry about getting eliminated by myself before the worldwide apocalypse comes

PVAL-inactive

Well I'm certain I'll be somewhat recovered by monday.

only to put more stress on it.

MatheiBoulomenos

My father has a diploma in math, he's an actuary. But he did really applied math, which is not the kind of math I do voluntarily, so I don't think it influenced me much

Sha Vuklia

stay for the memes, bala

PVAL-inactive

12:03 AM

My uncle has a math undergrad from Harvard where he placed above Math 55. He went on to med school though.

Eric Silva

@Balarka it already came

Balarka Sen

my father taught me a bunch of calculus when i was young. he studied economics though

Eric Silva

it was just covered in so many layers of irony that we havent been able to figure out it's all over yet

Mike Miller

i guess if i'm not motivated to write i should at least do the homework i have.

PVAL-inactive

Trump is at least an entertaining harbinger.

Balarka Sen

12:05 AM

@EricSilva Well, there's more to come in this part of the world, I feel. Mass hysteria and massive religious wars are yet to happen

Sha Vuklia

my father taught me not to travel to a different city with a plastic boat

Balarka Sen

And happen it surely would

Eric Silva

i was really being transironic @Balarka u got tooooo realllll mannnnn

Balarka Sen

hah

Ted Shifrin

interesting @Sha

PVAL-inactive

12:06 AM

I feel like there's probably some pretty good plastic boats out there by now.

Eric Silva

If I start seriously thinking about the world I get too depressed @Balarka then i feel guilty about doing something like math

Balarka Sen

well i don't because i will always be a useless person. i can't do anything about it than to observe, or die before anything bad happens

good song

Mike Miller

well that's my cue to leave

Ted Shifrin

Well, Eric, unless you want to campaign for some wonderful new people, I'm not sure what active change you can effect.

Balarka Sen

lol bye @Mike

PVAL-inactive

12:09 AM

@Ted You can show the uncultured masses logical reasoning with the dark arcane art known as "teaching", but I guess you don't know anything about that.

MatheiBoulomenos

I guess I'm useless as well, but it feels nice to be not terrible at something, that's why I keep doing maths

Ted Shifrin

@PVAL: I think I had a good effect on lots of students, but who the hell taught the idiots in power? Gross failing.

Eric Silva

@Ted i dont have any real delusions about affecting systematic changes, but I think I've experienced firsthand how important like small acts can be in peoples lives (e.g. giving food to the homeless or something)

MatheiBoulomenos

When I can explain something to others (even if its useless like the math I enjoy) I feel somewhat useful, but I when I applied as an abstract algebra tutor, I wasn't accepted, so rip my usefulness

PVAL-inactive

@TedShifrin I think at least most of the "idiots" in power know exactly what they are doing. The targets of their manipulation are the people you want to educate.

Ted Shifrin

12:12 AM

I am in a position to contribute decent money to foodbanks, things like Southern Poverty Law Center, Lambda Legal ... But I walk by homeless every day in San Diego and it breaks my heart.

@PVAL: Admittedly, at least one of my former students did vote for the idiots. I removed her as a FB friend. But she's the only one I know of.

Leaky Nun

@MatheiBoulomenos no you aren't, Sr Mathei Boulomenos

Ted Shifrin

Gerrymandering is going to make this very difficult to fix.

PVAL-inactive

538 has the generic ballot as +9 democrats.

That can change, but that should trounce any gerrymandered advantage.

Ted Shifrin

I won't remind you how badly he misled us in the big one.

Eric Silva

@Ted the thing that sucks is that homelessness is frequently criminalized in the States. It really doesn't help people escape it. It's a vicious circle that destroys your life if you're not extremely lucky. And the system basically treats the homeless like it's their fault.

Ted Shifrin

12:15 AM

(much as i like him)

PVAL-inactive

Huh

Do you mean Nate Silver?

Ted Shifrin

San Diego is trying to do positive things, Eric, but the numbers are just out of control.

Big hepatitis problems here.

PVAL-inactive

He gave trump like 35%.

Ted Shifrin

Yes, PVAL.

Eric Silva

iirc he gave trump a very nontrivial chance of winning compared to most media outlets

PVAL-inactive

12:16 AM

That is isn't misleading at all.

Ted Shifrin

Up until the end he was saying Hillary was a shoo-in.

PVAL-inactive

That was a completely reasonable odds given the data.

What no

There was maybe a point where it was 75/25

but I don't think she ever broke 80 under 538's model.

Eric Silva

He definitely got into fights with media people because he gave trump a bigger chance of winning than most others in the media, for the general

Ted Shifrin

Fair enough.

I accept the critique.

PVAL-inactive

and their models have been reasonably accurate at predicting many elections.

Eric Silva

12:18 AM

for the primary he totally flopped (but he admitted to it immediately)

Ted Shifrin

Anyway, who knows how pervasive all this Russian s*** is.

PVAL-inactive

Definitely there was a primary where Sanders one with a 1% odds or whatever.

but they have predicted lots of elections and have a good record doing it.

Ted Shifrin

Back to math(s).

Or I could go cook dinner.

Eric Silva

tbh the statistics people are doing at 538 is p interesting math

Ted Shifrin

Yes, I think most of us "pure" math people should give stat a lot more respect than we do.

I directed a masters thesis that entailed interpreting a ton of statistical methods/results in terms of geometry and linear algebra. It was fascinating.

Balarka Sen

12:21 AM

damn, this album is great

Eric Silva

i think stats is great (but i like analysis a lot so i guess i already am inclined to like a lot of the math they do)

MatheiBoulomenos

So was it titled "Statistics - A Geometric Approach?" @Ted

Ted Shifrin

It should have been @Mathei, but it wasn't by me.

Eric Silva

@Ted you should write Stats: AGA

PVAL-inactive

I've seen multiple news sources put Trumps internal numbers near election very close to 538's.

Eric Silva

12:22 AM

id read the crap out of that

Ted Shifrin

I'd have to learn an awful lot first, Eric.

PVAL-inactive

I have to give a talk in a few weeks in a topology seminar for grad students (maybe a postdoc).

Ted Shifrin

His thesis title was "The Geometry of the General Linear Model."

I had to go look it up.

PVAL-inactive

I think I'm going to talk about Nash equillibria.

Ted Shifrin

Introductory or more advanced, PVAL?

Eric Silva

12:24 AM

i think there's probably a big banach space geometry \cap stats intersection

PVAL-inactive

Well it can be whatever.

Grad students interested in topology.

Balarka Sen

i thought the LA in stat is mostly finite dimensional

Wildcard

@TedShifrin I wasn't thinking to make up my own fonts particularly but perhaps to use the Computer Modern family. :)

zigzaganimal.be/elements/the-concept-of-metafont.pdf

Ted Shifrin

Ah ... is there much topology in Nash's stuff? I have never studied it.

Meow Mix

oh @TedShifrin i have a question

PVAL-inactive

12:25 AM

There will be later ones and earlier ones. Generally I've done stuff close to my research.

Sha Vuklia

youtube.com/watch?v=Tc-wqJqUCD8 @Balarka could you make an album out of the first 12 seconds of this vid. this is what I want to listen to and look at for the rest of the evening

Ted Shifrin

@Wildcard: I'm not that fond of CMR. I have bought Lucida and New Times Roman. I like them both more.

PVAL-inactive

Well the existence theorem is an application of Brouwer fixed point theorem.

Meow Mix

if i want to have another bound can i add it to the $\min$?

Eric Silva

@Balarka probably, there's definitely also function spaces tho

Ted Shifrin

12:25 AM

Ah, fair enough, PVAL.

Absolutely, @Meow.

Balarka Sen

@EricSilva Ah yeah I guess

Ted Shifrin

you mean adding the requirement to achieve that bound, @Meow

Wildcard

@TedShifrin Fair enough. You answered my question, though, in that I can use CMR in LaTeX. Yay!

Ted Shifrin

@Wildcard: That's the default.

Eric Silva

i had a friend who used statistics to analyze taylor swifts musical trends over time and it was p quality

Balarka Sen

12:26 AM

@ShaVuklia :P

@Eric lmao

Sha Vuklia

i'm not joking:l

MatheiBoulomenos

@Balarka my stats professor said that sometimes you want to estimates functions, which brings banach spaces into play

Wildcard

@TedShifrin Ah, cool. And, where do you recommend starting? I could just use start with the man page....

Eric Silva

@Balarka patternsofideas.wordpress.com/2017/09/19/…

Ted Shifrin

I learned by referring to books, @Wildcard. It was way before the internet era.

Kasmir Khaan

12:27 AM

Hello guys :D

Wildcard

Oooooh.

Ted Shifrin

@Meow: I'm curious. For which problem do you need 3 constraints?

Kasmir Khaan

@MatheiBoulomenos I got few last questions about orbit stab theorem then I wont ask about it no more :D

Meow Mix

none, i was just wondering :)

MatheiBoulomenos

@Kasmir sure, go ahead

Kasmir Khaan

12:28 AM

:)

Ted Shifrin

In more advanced stuff, it will certainly show up, @Meow.

Balarka Sen

@EricSilva Pretty well researched.

Eric Silva

quality content

Kasmir Khaan

So first question , G/ stab (x) , we have elements in G, some of them dont move x at all, those are in stab x , and others do move x so some other element in X , is it the case that the cardinality od stab (x) , and the elements that move x to y , is the same ?

Eric Silva

oh god i need to do algebraic topology

Kasmir Khaan

12:30 AM

ie, if we had 5 elements in stab(x) , do we have to have 5 elements that move x to y , x to z ect ? @MatheiBoulomenos

Balarka Sen

Uh a case study on Taylor Swift is um er ah interesting i guess

Meow Mix

ok so

Balarka Sen

I am just mostly impressed by the amount of research put into it :)

Ted Shifrin

@Meow: If you want more practice, try 7, 8, 9, 10, 11

MatheiBoulomenos

@Kasmir, yes, assuming that x,y and z are in the same orbit of course

Meow Mix

12:30 AM

for $\lim_{x\to 0}x^2 \sin \frac{1}{x}$, $\delta = 1$ works right?

Ted Shifrin

Nooooo.

Better do #7 first.

Meow Mix

???? let me look through my algebra again

Eric Silva

@Balarka i think it's hilarious and quality because of the fact that it goes so far. I super dont care about her music

Ted Shifrin

I can't think of any function (other than a constant) where $\epsilon$ isn't involved in the formula for $\delta$ @Meow

Kasmir Khaan

@MatheiBoulomenos can we chat in a room alone ? i get distracted with many ppl writing

MatheiBoulomenos

12:32 AM

If you prefer that, okay

thanks man :)

True true

Bye Kasmir!

I didn't know you could do that

Kasmir Khaan

ill try to invite you , i never do thsi before

Meow Mix

12:32 AM

actually

okay so i realized my mistke

so here's what i was doing

$|x^2 \sin \frac{1}{x}| = |x|^2|\sin\frac{1}{x}| \leq |x|^2 < \varepsilon$

idk where i got 1 lmao

Eric Silva

I get shook when people don't tell me what my $\varepsilon$s and $\delta$s depend on

anakhronizein

Any representation theorists around?

Meow Mix

anyways

i assume you dont want me to take $\sqrt{\varepsilon}$ cuz you said not to in the other problem

Ted Shifrin

Right, @Meow, which takes us back to #7.

Meow Mix

let me do 7 first

i didnt realize this would boil down to 7

Balarka Sen

12:36 AM

I should get some sleep now

Meow Mix

hmm whats a function less than or equal to $\sqrt{\epsilon}$

Ted Shifrin

Night, i.e., morning, Balarka.

Hint, @Meow: You need to stipulate.

Meow Mix

12:50 AM

i have an idea

let me check first

yes i think this is right

$\min(\epsilon, 1)$ maybe

then $|x|^2 < \varepsilon \cdot 1 < \varepsilon$

look good?

Ted Shifrin

@Meow: You didn't ping, so I didn't know you'd answered.

Meow Mix

sorry

Ted Shifrin

I was working on my AoPS challenging homework problem :)

mercio

homework o..õ

Ted Shifrin

So if $\epsilon\le 1$, $|x|<\epsilon\implies |x^2|<\epsilon$, and if $\epsilon > 1$, then $|x^2|<1^2<\epsilon$. OK.

Meow Mix

12:57 AM

or you could apply both conditions

Ted Shifrin

Right. Well done.

Meow Mix

$|x| < \epsilon$, $|x| < 1$, so $|x|^2 < \epsilon$

Ted Shifrin

$<$ not $\le$

nope, $<$ everywhere

Meow Mix

but what if $\epsilon = 1$?

Ted Shifrin

Yup.

Then both conditions are the same.

Meow Mix

12:59 AM

ohhh nevermind

because its $|x| \mathbf{<} \min{\epsilon, 1}$

Ted Shifrin

This is sorta a fun game once you catch on. (Not as fun as projective geometry, but ... :P)

Right.

Do a few more if you want. I'm going to cook dinner. :)

Meow Mix

have fun

Ted Shifrin

OK

Meow Mix

i also gotta do my real math homework

Ted Shifrin

Right, good idea.

Fun chatting with you.

Meow Mix

1:00 AM

blegh, real life quadratic systems

you as well

Simply Beautiful Art

15

Q: Golf a number bigger than TREE(3)

The function TREE(k) gives the length of the longest sequence of trees T1, T2, ... where each vertex is labelled with one of k colours, the tree Ti has at most i vertices, and no tree is a minor of any tree following it in the sequence. TREE(1) = 1, with e.g. T1 = (1). TREE(2) = 3: e.g. T1 = (1...

I struggling a little :(

anakhronizein

1:18 AM

So I finally figured out the solution to the determinant tic tac toe game.

It's pretty surprising, actually.

(Recap: player 1 and player 0 take turns filling in either a 1 or 0 (corresponding to their player #) in a tictactoe grid. Player 1 always begins. After it is completely filled, the determinant of the grid is taken and then winning is decided as follows: if det is 0, then player 0 wins. Otherwise player 1 wins. Question: is there a winning strategy? What is it?)

Leaky Nun

I wonder if $|(fg)'(0)| \le |(f^2)'(0)| |(g^2)'(0)|$ where $f^2(x) = f(x)^2$ not $f(f(x))$

Leaky Nun

1:47 AM

one problem is that this inner product doesn't have positive-definiteness

ah, that's why this fails

Daminark

2:17 AM

@EricSilva any progress?

Eric Silva

lol

rip me

i have no clue how to do problem 1.3: 13 in hatcher

i dont even really know what it's asking

Daminark

Huh, strange, I'll check it out

I don't even know covering spaces so maybe not :P

Meow Mix

how are you daminarlk

Daminark

Doing aight, how about you?

Meow Mix

tired

Daminark

2:47 AM

Yeah same

Almost done with my algebra pset, and then I'm gonna just do AT/Smash until I drop

Daminark

3:00 AM

Hey @nitsua60!

nitsua60

@MeowMix What's delta_0? It crops up somewhere never having been defined. Perhaps mixing up delta_1 and delta_0?

Meow Mix

how does my proof look

nitsua60

(well, that was unexpected)

Mambo

6:54 AM

@TedShifrin Look at this sequence construction. Suppose $\sum_n c_n < \infty$. Choose $n_1 < n_2 < \ldots$ such that $\sum_{n > n_k} c_n < 1/2^{-k}$. Now let $b_1 = b_2 = \cdots = b_{n_1} = 1$ and for $ i \geq 1, b_{n_i + 1} = b_{n_i + 2} = \cdots = b_{n_{i+1}} = i+1$. Then $\sum_n b_n c_n$ converges as $\sum_k k/2^{-k}$ converges.

Leaky Nun

9:00 AM

$$Q_8 \star Q_8 \cong D_8 \star D_8$$

basicaly I sat in a lecture about group theory and they talked about this amazing fact about central products

Tobias Kildetoft

@LeakyNun Hmm, usually I would indicate which central product it was by having the shared subgroup be a subscript

I mean, the direct product is also a central product

Secret

NB: Not understanding what f(x)==(x<y) means shows how terrible I am at boolean algebra

f : N -> Bool

boolean strings for some reason, I am not very intuitive to them. Perhaps I need revision on classical logic

Leaky Nun

@TobiasKildetoft the center

Tobias Kildetoft

@LeakyNun I know, since that is the necessary one for the isomorphism in this case

Alessandro Codenotti

10:09 AM

When is the intersection of principal ideals principal? That should be true in UFDs but I think it fails in general

Secret

does that statement need axiom of choice to hold in general, I vaguely recall?

Bysshed

10:33 AM

Does anyone know what happened to the website crazyproject.wordpress.com ? It used to contain solutions to Abstract Algebra by Dummit and Foote, but has now disappeared. I found the site really helpful.

Silent

Are 'Dedekind completeness' and 'Least-upper-bound property' synonyms?

@Bysshed My god! its no more? I feel so sad.

I was planning using it for next semester.

@Bysshed I have screenshots of almost all answers.

Bysshed

@Silent They are both equivalent statement of the completeness axiom. This axiom comes in many forms, see the wikipedia page en.m.wikipedia.org/wiki/Completeness_of_the_real_numbers.

If you have some time I recommend this article of the subject. Their equivalence is shown on page 9.@Silent

google.co.uk/url?sa=t&source=web&rct=j&url=http://…

Silent

@Bysshed Thanks!

Tobias Kildetoft

10:49 AM

@Silent I would say that they are only synonyms when applied to a specific case, namely the reals, where they are both just true statements.

Silent

@TobiasKildetoft So, for what sets they differ?

Tobias Kildetoft

@Silent For one thing, there are sets where only one of them even makes sense

I am not sure I know an example where both make sense and they differ

Silent

@TobiasKildetoft Will you please give such example?

Tobias Kildetoft

@Silent Ohh, never mind actually, it seems that Dedekind completeness can also be formulated just using an ordering

They might actually be equivalent in general in that setup then, sorry.

Silent

@TobiasKildetoft, ok, no problem. Thanks.

FuzzyPixelz

10:54 AM

If you don't mind taking a look at this, please explain to me the "There are only finitely many rational numbers p/qp/q in that interval with 1/q≥ϵ1/q≥ϵ" part in the first definition, I understand it well, but I fail to see its place within the proof.

quora.com/…

Secret

::Talking on phone:: No, that room will remain star-free no matter what!

Leaky Nun

11:40 AM

somehow under Z16, the multiplicative subgroup <5> is equal to the additive coset 1+<4> and is even isomorphic

Tobias Kildetoft

@LeakyNun What do you mean by isomorphic?

Leaky Nun

12:14 PM

5x5 = 9, 9x5 = 13, 13x5=1

5+4=9, 9+4=13, 13+4=1

@TobiasKildetoft

Tobias Kildetoft

@LeakyNun But how can two things with different types of structure be isomorphic?

one is a group, the other is not

Secret

12:37 PM

does "isolabel" exist?

user84215

The second week of the Abstract Algebra Course will start at 9:30 GMT on Saturday, October 21, 2017 in this room.

Secret

[Random]

The desire to build a Metropolis algebra

Let $C$ be the class of all algebraic structures. Then the metropolis algebra $M$ is an infinite sized algebra whoose subalgebra are all elements in $C$

[Super random]

Let $S$ be a sunset

Then a finite sunset is one that occured a finite number of times

An infinite sunset occurs indefinite number of time

Now

Bysshed

@Secret Have you heard of category theory, it sounds similar to your 'metropolis algebra'

Secret

@Bysshed I don't know if category theory can represent nonassociative algebras since functors and categories are associative under composition

Tobias Kildetoft

@Secret Well, that starts to get into the territory of operads

Though as far as I recall, not all algebras are algebras over some operad (possibly if those algebras form a variety they are, but I never really did enough of that stuff to know this)

Secret

12:53 PM

hmm, a very brief read of operads looks very general and will cover a lot of associative and lie bracket kinds of algebra

Tobias Kildetoft

right, there is an operad for "associative" and one for "Lie"

Secret

It seems that the associative law of the objects determines what kind of associative or nonassociative algebra we will get

but that's as far I can comment about it so far, anything more in depth I need to read up the category theory book

Secret

On an unrelated note, I really wish I can formally define a dedekind finite sunset so I can use it as a valid running gag

Balarka Sen

1:08 PM

@EricSilva I think you should draw the universal cover, and quotient that by all 3-length paths

Well, I don't mean 3-length paths. But it's not clear what it looks like even if I do the construction I have in mind

Hi @Balarka

Hi @Alessandro

how is it going?

Not much, just got some food

Alessandro Codenotti

1:28 PM

So I want to describe the ideals in $S^{-1}A$ where $S$ is some multiplicative subset of $A$

Let $J_s:A\to S^{-1}A$ be the inclusion $a\mapsto\frac a1$, I want to show that all ideals of $S^{-1}A$ can be constructed as $(J_s(I))$ for some ideal $I$ of $A$

So if I have $J\subseteq S^{-1}A$ an ideal I guess that $(J_s(J_s^{-1}(J)))=J$, but probably I should first work out what does $(J_s(I))$ look like for a given $I$

Balarka Sen

@Alessandro Right, so if for some $a \in A$ and $s \in S$, $a/s \in J$, you must have $a \in J_s^{-1}(J)$

Alessandro Codenotti

So I think $(J_s(I))=\{\frac is :i\in I,s\in S\}$

Balarka Sen

1:43 PM

That's correct, yes

Which would tell you, as $(J_s(a)) \subset (J_s(J_s^{-1}(J))$, $a/s \in (J_s(J_s^{-1}(J))$

So the establishes $J \subset (J_s(J_s^{-1}(J))$ I guess

Secret

I am thinking about how to write $\Bbb{Q}$ as a tree, but then I kinda lost count on how many nodes each branch have

Alessandro Codenotti

@BalarkaSen Right, and the other inclusion is the easy one

Balarka Sen

True

Secret

One thing about $\Bbb{Q}$ that caught my attention is how the repetends are actually finite strings, meaning that whatever that branch in the tree has only finite depth

Balarka Sen

@Secret If you could enumerate $\Bbb Q$ as a tree, I'm pretty sure you'd need $\Bbb Q$ to be uncountable.

Secret

1:48 PM

hmm...

but is such uncountable tree have a simple kind of structure e.g. a line with n nodes and each nodes having finite number of branches?

Balarka Sen

Well, no, I guess I am thinking about the boundary of the tree

@Secret Sorry, it wasn't clear to me what you were doing.

Secret

yeah, the boundary will be uncountable since it will correspond to the irrationals I think

@BalarkaSen So back in months ago the chat was discussing about finding uncountable chains in the rationals and reals

Balarka Sen

Boundary of any tree which is not valence 2 is uncountable.

Secret

and then we dug up an MO link telling us how to find dense linear order for set of any cardinality by basically putting two branches for each element in the set as if they are nodes

This caused me to wonder whether the infinitely decreasing sequence in the rationals can be captured as a tree somehow

13

Q: Uncountable chains

$P(\mathbb N)$ = power set of $\mathbb N$. $A \subset P(\mathbb N)$ is a chain if $a,b \in A \implies$ either $a \subseteq b$ or $ b \subseteq a$ That is we have something like this: $$\ldots a \subseteq b \subseteq c \subseteq\ldots$$ where $a,b,c \in A$ are distinct. We can show easy enough...

set-theory cardinals

^ 1st ans of above are the details

Stern–Brocot tree

In number theory, the Stern–Brocot tree is an infinite complete binary tree in which the vertices correspond one-for-one to the positive rational numbers, whose values are ordered from the left to the right as in a search tree. The Stern–Brocot tree was discovered independently by Moritz Stern (1858) and Achille Brocot (1861). Stern was a German number theorist; Brocot was a French clockmaker who used the Stern–Brocot tree to design systems of gears with a gear ratio close to some desired value by finding a ratio of smooth numbers near that value. The root of the Stern–Brocot tree corresponds to...

hmm found something

So the rationals can be represented as a tree where all branches have finite length (because all rationals have terminating repeated fractions)

Therefore, if we insert one branch of $\omega$ length, then we will get to an irrational

Secret

2:45 PM

A well ordering of the rationals can also be established as follows:

Much better than the usual zig zag mapping, there are no repeats

This is reasonable because each level of this tree there are only finite number of nodes

Secret

3:02 PM

Actually screw it, infinitely decreasing chains...

Eric Silva

@Balarka I ended up drawing it without any trouble

My problems were mostly me trying to resist doing any work tbh

Kasmir Khaan

4:02 PM

@LeakyNun Sup leaky =p

@LeakyNun hope you did not forget about the examples i sent you :D

Leaky Nun

I left it open in a tab

Balarka Sen

@Eric I guess the picture looks like 6 triangles joined vertex-by-vertex in a circular fashion so you have a hexagon in the middle

Or at least locally so

Leaky Nun

doing it tonight @KasmirKhaan

Kasmir Khaan

@LeakyNun okay :D

Balarka Sen

The 6 triangles are alternately the aaa and bbb triangles

Kasmir Khaan

4:12 PM

Am almost done with orbit stab theorem and sylow =p

Leaky Nun

do we have any number theorist here

I am wondering why for Z16 for the powers of 5, multiplying by 5 is the same as adding by 4

I guess it trivially follows from binomial theorem [(4+1)^n = 4n+1] and i wonder how to generalize it

it’s really intriguing

Kasmir Khaan

any odd number can be written as 4n+1 or 4n+3

that might be usefull for what you are doing :D

Leaky Nun

i don’t think it is

Eric Silva

4:27 PM

@Balarka yeah I just drew a square with lines going zip zap zoom and it was done

Balarka Sen

coolio

Astyx

Holidays !

I can finally do maths

Balarka Sen

4:44 PM

Yay

Akiva Weinberger

4:56 PM

@Secret Look at this one as well https://en.wikipedia.org/wiki/Calkin–Wilf_tree

Enumerate the rationals the same way, going across the rows

Notice: The denominator of each one is equal to the numerator of the one after it

Look at the section on Stern's diatomic sequence

0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, …

To enumerate the rationals (the same enumeration gotten from the Calkin–Wilf tree), take each one divided by the one after it:

0/1, 1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, …

ÍgjøgnumMeg

@LeakyNun Likely something to do with the subgroup $\left\lbrace 1, 5, 9, 13\right\rbrace = (5) \subset \Bbb Z/(16)$. These are all elements congruent to $1 \bmod 4$ and so $4k \equiv 4 \bmod 16$ when $k \in (5)$, just an observation