serious-discussion

[modest rune]

if the idea were the proof then math would be a lot easier lol

[velvet dagger]

The discussion was literally about whether the proof, not the vague ideas, of incompleteness is harder than the halting problem

[sick burrow]

But I can see clearly that it exists

[velvet dagger]

"Clearly"

Yeah I can see clearly that the Langlands conjectures are true but

[modest rune]

lmao

[velvet dagger]

Somehow I'm not getting that fields medal yet

[sick burrow]

I mean

[tiny marten]

we begin our constructive proof of the incompleteness theorems, suppose there exists some computable function

[sick burrow]

Take the last theorem you proved Dami

[calm rampart]

is this A level work?

[modest rune]

I can clearly see that if $k>n$ then any map $S^k\to S^n$ is nullhomotopic

[fathom swallow]

MaxJ

[modest rune]

but apparently this is "trivially false"

[sick burrow]

Write it down explicitly

[modest rune]

?

[sick burrow]

Using formal logic

[modest rune]

this is cringe

[calm rampart]

heloo sexy humans

[modest rune]

touch grass

[sick burrow]

Apparently we have the same intuition Max

[modest rune]

that was a joke

[sick burrow]

Ok I'm aware this is a sorta bad take on my part

[modest rune]

its an absolutely bad take

[velvet dagger]

Look I want to give your intelligence too much credit to actually believe that you think this is a reasonable retort

[sick burrow]

I think it's like 30% reasonable

[modest rune]

you're 30% wrong

[sick burrow]

Ok 60% reasonable

[calm rampart]

tf is this

[sick burrow]

Thank you Max

[tiny marten]

you can just read the theorem and be 100% reasonable

[calm rampart]

?

[velvet dagger]

Like I think/hope we both know that the annoyingness of writing down a proof that's far away from foundations purely using foundations has literally no bearing on whether a proof that is under a page when formalized is easier than a proof that is double digit number of pages when formalized

I don't even mean formalized here in the sense of writing in terms of ZFC or some shit

[tiny marten]

??

[sick burrow]

Ok let me back up

[velvet dagger]

But writing down a correct proof rather than just giving a proof sketch

Remember the context of this whole discussion was

[sick burrow]

Halting problem is obviously much easier to prove that incompleteness

[calm rampart]

is thsi gcse stuff/

[velvet dagger]

Is the proof of Godel incompleteness noticeably more difficult than the halting problem? Not the proof outline, I can unironically give you a digestible outline of Fermat's last theorem

But when you actually spell out the details, not in the unreasonable "Oh write this proof about rep theory of Lie groups in ZFC" way but just in the "Prove that what you're saying can be done actually can be done rather than just assert it" way, that yes GIT takes some work

[sick burrow]

Ok

I feel I was just arguing the wrong thing then

[velvet dagger]

Yup

[sick burrow]

I was just a bit surprised when I saw a proof of Godel's theorem

Or at least a sketch

I remember feeling like "wait that was it?"

Because I had heard the vague idea of the proof from like GEB

[tiny marten]

oh, well

[sick burrow]

And I expected the actual thing to be much harder

And it's like yeah, I recognize the places where there is a fair bit of detail to be filled in

[tiny marten]

it is a around a hundred year old theorem at this point

[sick burrow]

And that filling in those details requires a decent amount of serious logic

But it still felt like less work that I expected

So I guess I'm just reacting to that

[sick burrow]

I'm aware it was a joke

[velvet dagger]

It's somewhat common that theorems have reasonably digestible outlines tbh

[modest rune]

The big idea behind a proof is often "obvious" in hindsight. The details rarely are.

[velvet dagger]

Depending on how much of it boils down to stuff where saying "X can be done" is much easier than the actual act of doing X

For example with Fermat's last theorem

[sick burrow]

I'm just saying my intuition struggles to see why pi_m(S^n) is so hard

Even though I know it's an open problem to compute it in the general case

Ok but also in the case of incompleteness

Me?

I've done pi_1 stuff

I know the definition of higher homotopy groups

Yeah obviously my intuition is shit here

I'm not saying it is easy to compute pi_n(S^m)

[velvet dagger]

There is a notion of "modularity" of an elliptic curve, pretty much that an elliptic curve is modular if a certain Galois rep/L-function attached to it actually comes out of a modular form

[tiny marten]

it's easy just like every other thing i dont do

too easy for me

[sick burrow]

I'm saying my intuition is shit here

And I recognize that

And I think I would almost certainly get better intuition if I studied higher homotopy groups

[modest rune]

thats where ur wrong bucko

[tiny marten]

if you put a gun to my head and told me to recreate the incompleteness theorems validly, even just one, i would be dead

[modest rune]

no one has any intuition for higher homotopy groups

[sick burrow]

By falsely generalizing things I do have intuition for

[modest rune]

this is like dunning kruger in more words

[sick burrow]

Bruh

Can you not see that I am making fun of myself perfectly fine here

[modest rune]

no

ur conduct makes it hard to believe you actually have a healthy perspective about this

[sick burrow]

Don't mistake my shitposting for actual stupidity

My Godel take was a bit bad

But I'm aware that I have zero actual intuition for higher homotopy groups

And that was the case during this entire conversation.

But I do think I have decent intuition for how the details of incompleteness work

Like obviously it would be nontrivial to fill them in

The pi_n thing was in reply to Max's joke

[cold needle]

Hurb

[sick burrow]

_ _

And I was being serious about Godel

[maiden bear]

[sick burrow]

[modest rune]

By joke I meant that that is not my actual intuition

I do think it is roughly equivalent to your Godel take

[blazing pawn]

the entire time i read k < n instead of k > n and was really confused

[modest rune]

The main difference being that the conclusion is false

[thorn brook]

lmao same

[modest rune]

But it's easy to be right when you already know the answer

[modest rune]

symbols are homotopy equivalent so the theorems must be

[blazing pawn]

Kind of woke

we need to develop the hangul of algebraic topology

send tweet

[sick burrow]

Yeah and I was saying it's not my intuition either

[blazing pawn]

basic question

[cold needle]

stated in a basic manner

[blazing pawn]

chrew

[cold needle]

?

[sick burrow]

I think that just was unclear due to the other stuff I was saying in the convo

[cold needle]

oh ic

[compact tartan]

they're definitely computable right?

like you can just

exhaustively iterate over boolean cubes or something right?

[cold needle]

topologists, pondering they orb

[sick burrow]

Ok this is just wrong

[cold needle]

ok orbs are filled in so its whatvr

boundary of orb

[sick burrow]

I know there is a way to encode strings of symbols as numbers because computers work

[compact tartan]

I have some intuition after trying to come up with syntactic proofs

[sick burrow]

Formulas are strings of symbols

[modest rune]

the actual answer if ur curious slim is that the sphere is actually of pretty moderate difficulty

[sick burrow]

QED Godel numbering

[modest rune]

you can construct things that are sort of easy-by-design

but for example its not just spheres that have hard to compute homotopy

the stable homotopy of finite complexes is wide open

[blazing pawn]

Yeah homotopy classes of maps between manifolds in general are kind of bleak

[compact tartan]

spheres are literally iterated suspensions which is left adjoint to loops, how can loops of suspensions be complicated

[sick burrow]

Basically my intuition is that a formula can do something if a computer can

[modest rune]

virgin left adjoint loops vs stable inverse loops

[compact tartan]

stable sounds like tropo territory

[modest rune]

I dream of the day that I get to absolutely crush someone on a dating app who claims straws have more than 1 hole

4 years of topology about to get dropped on this innocent person

[blazing pawn]

Straws trivially have 1 hole

[sick burrow]

Max did you see the tumblf post about that

[maiden bear]

I wouldn't call someone who claims that innocent

[blazing pawn]

tropo does stable homotopy theory?

new developments down at the studio today

[modest rune]

I feel like this is not true

[sick burrow]

Where someone claimed that a straw is topologically equivalent to a torus

[blazing pawn]

i also feel like it is not true

[modest rune]

given tropos comments about topology earlier

[compact tartan]

it's not true

[modest rune]

okay lol

[sick burrow]

And thus has one hole

[modest rune]

you can't tease me like that

[compact tartan]

it's a pun

I'll leave it as an exercise to the reader

[modest rune]

if there was another stable homotopy theorist here

[blazing pawn]

please explain im illiterate

[modest rune]

it would make my day

[compact tartan]

stable (noun)

[modest rune]

oh

[blazing pawn]

Dont worry max once i go to ucsd ill find ur advisor and invite him here 🙂

[modest rune]

lmao he would never waste his time like this

hes better than me

[sick burrow]

Max you're a 1st year grad student right

[modest rune]

Yes

[blazing pawn]

Discord will ruin him i know it

[sick burrow]

Nice

[blazing pawn]

now its time for fallout 2

[sick burrow]

Ok but Max explain to me why my intuition on incomplentess is equivalent to thinking pi_m(S^n) is trivial

[cold needle]

[toxic schooner]

[sick burrow]

Wow ok

[maiden bear]

[modest rune]

Thinking that “seeing the intuition” for an argument is in any way similar to actually understanding the argument is bad because it only works when you already know the right answer. Being dismissive of people emphasizing the importance of details shows that you don’t really understand the practice of mathematics.

It is very easy to see a hard proof’s outline and convince yourself you understand the proof. But I could outline all sorts of reasonable sounding proofs for false results.

[sick burrow]

Bruh

Wow cool you can obfuscate that you divided by 0 and get a false result

[modest rune]

yes

thats the whole point lmfao

[cold needle]

What

[long matrix]

wtf

[bronze pelican]

Whats going on

[sick burrow]

I understand the point you're making

[modest rune]

.

[compact tartan]

Sorry please enter 1, 2 or 3

[sick burrow]

bruh

[calm rampart]

run it in python

[modest rune]

This is the type of issue

[long matrix]

its a virus

[compact tartan]

@calm rampart please don't interrupt conversations

[modest rune]

"I can clearly see that it exists" is bad practice

[sick burrow]

I'm pretty sure I've actually seen the explicit construction here

[calm rampart]

my bad

[modest rune]

Okay, explain it

[bronze pelican]

[maiden bear]

smoking that gamma pack

[sick burrow]

I mean there's a few ways to do it

[modest rune]

One is sufficient

You might see mathematicians act like they don't care about the details. I work at a high level without details most of the time on research. But I'd never tell my advisor that I've proven something just because I think I get the vibe

You can't claim a result is proven or that you understand the proof if you can't fill in the details to some sufficient level

[sick burrow]

The difference is I've actually seen the details here

[modest rune]

uh huh

[sick burrow]

For godel numbering at least

[bronze pelican]

I'm so hungry

[sick burrow]

I do think I was being too dismissive of the details earlier though

There are definitely some things I could stand to review with the proof

[modest rune]

like

the proof

[tiny marten]

blackboxing is how i do math

you just blackbox all theorems, creating a discrete system of theorems to do combinatorics over

this method can be found to be approximately correct with a large enough sample size

[sick burrow]

I blackbox too much tbh

[modest rune]

this is called "Higher Topos Theory"

[dense belfry]

Wow I missed an arithmetization of syntax pile on

[stray kite]

if i wanted to go about calculating the integer value of 2^(82589933) − 1 and dumping that into a file, is that a reasonable thing to be able to do within a sane amount of time and b) how would i do that

[calm rampart]

easy

[little vine]

How many digits is that

[calm rampart]

too many

[compact tartan]

about 20 million digits

any bignum library can do that in seconds

[calm rampart]

what

[fair mural]

reminder that 10^n is very roughly 2^(10/n)

[compact tartan]

2^(10/n)

[little vine]

Is it mersenne prime? I forget the definition

[fair mural]

[honest veldt]

is it saul on the phone

that's for me

[frail sail]

What is the meaning of blackboxing in math?

[jovial ember]

When you take a result on faith

[honest veldt]

hey you stole my question

not fair

Oh thanks

[jovial ember]

You don’t know a proof, you just assume a result which you know is true

[fair mural]

oh i thought it was ignoring something

[cold needle]

ignoring the proof yes

[fair mural]

yes it is

[modest rune]

It can also be used in similar situations, for example when you use a construction and it’s properties without having any idea how to construct the thing

[honest veldt]

does the name come from the black box you draw at the end of a proof

[modest rune]

No it comes from airplanes

[fair mural]

stare

[honest veldt]

Oh I think I do remember that

[modest rune]

They have a machine called a blackbox meant to record information in case of a crash or accident or something

[honest veldt]

ohhh yeahhh, I read that in one of my english assignments

Right right right, now I remember, thank you

[cold needle]

o o

scary

[frail sail]

Ohh so is it like just using one result of the many possible ones? Or is it like u just somehow “know” the result and u use it?

[cold needle]

like

[jovial ember]

The story goes that a mathematician was on an airplane and it was going to crash, and so he went to the black box and recorded the proof for a result he had finished on the plane, and then mathematicians took the result as being true from his vague outline of the proof on the black box

[jovial ember]

No, someone else has proved it and you don’t know a proof but you just use the result

[honest veldt]

why are stories about mathematicians always so badass (and always end in tragedy)

[fair mural]

gotta be honest i could believe this

[frail sail]

Ahh ic

[cold needle]

mathematicians are such drama queens

[jovial ember]

I made that story up

[velvet dagger]

Destroyed

[modest rune]

that was such a good story thanks for sharing chm

[frail sail]

Great u should write a book

[cold needle]

book of chmonkey

[jovial ember]

If you just “have the vibe” that a result is true and use it, that’s called “imagination”

[honest veldt]

i have lost all interest in mathematics

goodbye

mathematician d.b. cooper is not real

[fair mural]

chmonket write a book about the origin of blackboxing in math

[honest veldt]

go home

[modest rune]

(and also an important part of research)

[jovial ember]

No

I mean

Maybe

[velvet dagger]

"Cool but False reasons for etymology" would be an interesting book

[jovial ember]

But come publishing time you better have a really good reason to think it’s true

[modest rune]

for example: a professor I want to work with on a project assumed a certain construction was possible to see if it lead to interesting computational results

[frail sail]

Like the description of the book can be
Stories to read…when you have nothing to read

[jovial ember]

Theorem 3.4: I don’t know if this is true, I just think it is

[jovial ember]

Yeah this sort of thing I understand

[modest rune]

no its more like

[honest veldt]

proof by gut

[modest rune]

Lemma 4 sounds like a lot of work is it even useful

and if its useful then u prove it

[cold needle]

Lemma 4

[modest rune]

ofc you can't publish

until its done

[frail sail]

My special powers tell me that this is TRUE and the proof DONT MATTER!!!

[cold needle]

where is brofib

[modest rune]

well thats not true either

[honest veldt]

4 whole lemmas?

[jovial ember]

Sometimes I even assume that some fact is true to then see where it can go

[modest rune]

theres a lot of times where people publish like

[jovial ember]

Like I think some ring has peppery P

Okay sure, but they qualify it

[modest rune]

work that depends on other unproven result

[velvet dagger]

Real topologist hours

[jovial ember]

“Contingent on…”

[modest rune]

yeah ofc

[jovial ember]

Like there’s proofs based on RH being true

[velvet dagger]

Number theory loves assuming (G)RH or Ramanujan

[jovial ember]

But I imagine they’re phrased like “if RH is true then…”

[modest rune]

ugh i want to ask my #category-theory question in the AT server

[jovial ember]

Wtf is Ramanujan?

[modest rune]

but i dont wanna look dumb

[velvet dagger]

But yeah they're always considered conditional

Ramanujan conjecture

[jovial ember]

Yeah what’s that

[frail sail]

[jovial ember]

I know who he is

[honest veldt]

it would be such a boss move, finding a counterexample to the RH and getting a multi-disprove combo

[frail sail]

Good

[surreal sapphire]

there are plenty of results that are proven assuming RH and then again assuming not RH

[honest veldt]

people love assuming huh

[jovial ember]

Okay but that’s just a proof

[surreal sapphire]

ye, i actually wanted to make another point

disproving RH does not disprove the "if RH then" results

[jovial ember]

Unless you don’t believe in LEM

Oh yeh

True true

[surreal sapphire]

i think many of those results are also like

"we think this is true and it follow from RH"

"so probably this says something about RH being true"

[fathom swallow]

Several Sloths

[velvet dagger]

Ramanujan conjecture was originally about the behavior of the tau(n)

In particular tau(ab) = tau(a)tau(b) for (a,b) = 1, also some recurrence for tau(p^k)

Finally it gives growth conditions

It's known for that function, but people ask about analogues for other automorphic forms

[honest veldt]

is this NT

[velvet dagger]

Yeah

Nowadays the statement is something like

(Almost all?) local components of certain types of automorphic representations are tempered

[sick burrow]

Blaxkboxing proofs is like using the black box in tf2

It's a perfectly reasonable thing to do in the right circumstances but people will still get mad

[clever knot]

By black boxing what exactly do you mean

Like have 0 understanding of the proof and not care

Or just you don't want to go through it all

Like I have accepted a proof after seeing some intuitive reason

[sick burrow]

Anything from "I have some sense of how these details go, I just don't want to bother filling them in" to "I'm not even going to try to understand this proof"

[clever knot]

But I didn't necessarily understand the formal proof

[sick burrow]

I do think I blackbox too much

But it also hasn't been a huge problem for me so far

[tiny marten]

sometimes you dont care about a proof

or at least i dont care at times

i usually dont blackbox something i wont figure out soon anyways

[sick burrow]

How important is it to understand a proof anyways

[tiny marten]

a professor once showed me the snake lemma and tube lemma but it's not like i really remember their proofs

[sick burrow]

The ultimate goal of math isn't being able to reconstruct proofs you've read, it's proving your own theorems

Like is there anything wrong with blackboxing proofs if you remember the general toolbox of techniques?

[tiny marten]

it depends on the method and what you are trained in i suppose

[sick burrow]

With the obvious caveat that you're never going to learn that toolbox if you skip all the details in proofs you read

[rose dock]

Well I think that might be the reason it's not common to black-box theorems

[empty stratus]

I think black-boxing is more justifiable when doing "interdisciplinary" mathematical work.

For example, using the Riesz-Thorin Interpolation Theorem for L^p spaces when working with PDEs.

Likewise, it also is justifiable when the theorems being black-boxed are being used not just in their own right, but to do something more complicated.

In that respect, virtually all people working in mathematical analysis tend to black-box the fact that the reals are complete.

And it takes a special kind of pedantic soul to want to have to go into Dedekind cuts in detail every time they bring up Cauchyness of a sequence.

[dense belfry]

This isn't what black boxing means

[tiny marten]

yeah i would agree with that, especially if you are going one direction into pure math

so like computer platforms for approximating something in algebraic geometry

[empty stratus]

I consider it more of a bad-faith example.

[tiny marten]

chances are some of those researchers will not be pure geometers

[rose dock]

Well I guess it depends what you define as black-boxing

[tiny marten]

sometimes you just arent trained in something but you probably have intellectual equipment to "deal with it"

[dense belfry]

Why are you giving bad faith examples

[empty stratus]

For example, in my dissertation, I black box a couple of facts about Berkovitch space just so I can prove two lemmas on the side about invertibility of elements of Banach algebras.

[empty stratus]

Because I'm weird.

[dense belfry]

[rose dock]

I think we were talking more about black-boxing when teaching someone something new.

[dense belfry]

[rose dock]

:dan:

This emote is actually incredibly cursed

[tiny marten]

oh well if you are learning something new and arent pretty advanced or have no choice you shouldnt black box it

[modest rune]

No liquids original comment was right

[tiny marten]

unless it's just some corollary

[modest rune]

The example of the reals is not black boxing

[dense belfry]

Citing things is not the same as blackboxing

[modest rune]

It’s not a bad faith example it’s a no example

Non*

It could be a bad faith nonexample

[empty stratus]

For someone who is doing foundational work, the completeness of the reals would be a non-trivial result.

[rose dock]

But for like an analysis course, would presenting the real numbers axiomatically be considered as black-boxing?

[dense belfry]

That's not true at all

[empty stratus]

My first real analysis course actually did this very thing. xD

[tiny marten]

if you are building a system from its axioms and that is black boxing then what isnt

[empty stratus]

Called it the "least upper bound axiom".

[empty stratus]

Many notions are relative.

[tiny marten]

i was responding to alphyte

[empty stratus]

But, anyhow, I'm just being goofy, trying to procrastinate having to do lots of matrix computations.

Time to get to work. Toodles. 🙂

[tiny marten]

though i bet some people working in computer stuff skip over completeness in a practical way

as in not using it

still, most people in analysis?

it's not what i would consider black boxing

[modest rune]

You completely miss the point. Black boxing is about whether one understands why a result is true or why a construction works. It has nothing to do with whether any particular reference to that fact is explained in detail

[tiny marten]

most people just know all that stuff

[rose dock]

as opposed to constructing the reals

[tiny marten]

oh so like constructing the reals from the rationals

this would not be black boxing, no

not unless you purposefully dont learn another definition/construction of the reals

[dense belfry]

It's hard not to learn the construction of the reals if you study math

[tiny marten]

like there isnt something more fundamental about constructing the reals as the completion of Q than tarski's axioms

yeah

the completion method is just more immediately useful to analysis and topology for reasons that become more clear

[mellow moon]

I did a very long Homework on Dedekind cuts which introduce everything by the mean of big problem with interdemiate questions during my 4th year uni. I did it. And know I forgot almost everything about it, and I never mind about using real numbers like people that doesnot know about it.

[mellow moon]

Indeed, and it only use archimedean property

[tiny marten]

yeah it's actually pretty beautiful

minimal construction

but you know, you only use the reals so much

[mellow moon]

But again every the huge part of Analyst don't mind about R, and even generally of non complete space, just considering completion.

But there is some reasons so that in some cases, it's better to leave the space uncomplete.

[tiny marten]

does the completion end up requiring more laborious methods or something?

[mellow moon]

Generally completion is just about quotient of Cauchy sequences, the problem is the meaning you want to give to the "constructed limits"

When you deal with spaces of a.e. defined function, you want its completion to be a set of a.e. defined function and also distributions.

Some times it fails to be distributions

[tiny marten]

oh, interesting

yeah it makes sense that there has to be some down side to distributions

you never get a free lunch

[mellow moon]

That's the point

So you leave the space uncomplete to be sure to deal with distributions

But this makes it more difficult to work with since you lose density argument to extend functionals etc...

[tiny marten]

yeah it sounds like a headache some of the nicest theorems of functional analysis start breaking down

big brain stuff

[mellow moon]

In some explicit cases you can change the point of view to recover some results

but those are special and explicit cases

[uneven hound]

In the zeta function, how does -2 equal zero? Wouldnt it be equal to 1 + 4 + 9 ...
and it would go to infinity?

[mellow moon]

No

[uneven hound]

Why

[mellow moon]

Again no

[uneven hound]

Uhh

How

[mellow moon]

Because the Zeta function is first only defined for Re(z)>0

But with some formula you can extend it on larger subset of C

[empty stratus]

Can I ask a question about hypothetical notation?

[mellow moon]

And this extension vaanish at -2k, k in IN, greater than 1

[tiny marten]

sure

[empty stratus]

Okay, so here's the set-up.

In my dissertation, I'm doing loads of computations involving Fourier analysis over the p-adics.

[tiny marten]

sounds fancy

[empty stratus]

It is.

All of this is done with respect to scalar-valued functions of a single variable.

[tiny marten]

okay i at least know the words and p-adics

[empty stratus]

However, I'm now generalizing to vector-valued functions of vector variables, and I want to keep things at a happy medium between "not horrible" but also "not confusing".

Consider an integer $r\geq2$, and prime numbers $p_{1},\ldots,p_{r}, with p_{1}\mid p_{2}, p_{2}\mid p_{3}, etc.$

[uneven hound]

Does this k have a limit

[fathom swallow]

ComplexVariable

[mellow moon]

what ?

[uneven hound]

Limit

[tiny marten]

it is unbounded

[mellow moon]

k is an integer

[tiny marten]

so for all k natural numbers, with no bound

[empty stratus]

The setting is then:

$\mathbb{Z}{P}\overset{\textrm{def}}{=}\mathbb{Z}{p_{1}}\times\cdots\times\mathbb{Z}{p{r}}$

[fathom swallow]

ComplexVariable

[empty stratus]

where P is the collection of primes p_1, ... ,p_r

[uneven hound]

Okay thanky all

[tiny marten]

okay got some primes and a decomposition

[empty stratus]

Letting $\hat{\mathbb{Z}}{p}=\mathbb{Z}\left[1/p\right]/\mathbb{Z}$, we have: $\hat{\mathbb{Z}}{P}\overset{\textrm{def}}{=}\hat{\mathbb{Z}}{p{1}}\times\cdots\times\hat{\mathbb{Z}}{p{r}}$

[fathom swallow]

ComplexVariable

[empty stratus]

That's the decomposition of the dual of Z_P

I then write $\bar{\mathfrak{z}}=\left(\mathfrak{z}{1},\ldots,\mathfrak{z}{r}\right)\in\mathbb{Z}{P}$ and $\bar{t}=\left(t{1},\ldots,t_{r}\right)\in\hat{\mathbb{Z}}_{P}$

[fathom swallow]

ComplexVariable

[empty stratus]

I let $p=\prod_{m=1}^{r}p_{m}$

[fathom swallow]

ComplexVariable

[empty stratus]

And I write $\bar{p}\bar{\mathfrak{z}}=\left(p_{1}\mathfrak{z}{1},\ldots,p{r}\mathfrak{z}_{r}\right)$ and likewise for $\bar{p}\bar{t}$.

[fathom swallow]

ComplexVariable

[empty stratus]

So far, is this sensible?

[tiny marten]

yeah it seems fine

pretty straightforward

[empty stratus]

The next issue regards denoting limits of summation.

[stray kite]

this was so pointless but i spent this time getting linux on a VM, compiling and installing GNU MP, writing a program to do this and getting it

but hey at least now i have the integer value of the largest known prime number

[empty stratus]

Let $\mathbf{n}\in\mathbb{N}_{0}^{r}$ be an $r$-tuple of non-negative integers.

[fathom swallow]

ComplexVariable

[empty stratus]

I then write: $\sum_{\mathbf{n}=\mathbf{0}}^{\bar{p}^{N}-1}=\sum_{n_{1}=0}^{p_{1}^{N}-1}\cdots\sum_{n_{r}=0}^{p_{r}^{N}-1}$

[fathom swallow]

ComplexVariable

[empty stratus]

When working in a single variable, it's easy to write sums over subsets of $\hat{\mathbb{Z}}{p}$ using the p-adic absolute value. For instance:$\sum{\left|t\right|_{p}\leq p^{N}}$ means sum over all ts with denominators at most $p^{N}$

[unborn trellis]

I like that notation

[fathom swallow]

ComplexVariable

[empty stratus]

Thanks

[mellow moon]

Use Integrals instead, and product measure over finite sets

this will give you a consistent notation

[empty stratus]

Integrals are for continuous quantities, though.

Anyhow... the problem emerges when I extend this to $\hat{\mathbb{Z}}_{P}$ because I end up in a conflict with the "norm" on that space.

[mellow moon]

No using counting measure

[fathom swallow]

ComplexVariable

[empty stratus]

Viewing $\mathbb{Q}{P}$ as the direct sum of $\hat{\mathbb{Z}}{P}$ and $\mathbb{Z}_{P}$

[fathom swallow]

ComplexVariable

[empty stratus]

we can get a "norm" that induces the product topology by defining:
$\left\Vert \bar{\mathfrak{z}}\right\Vert {P}\overset{\textrm{def}}{=}\max\left{ \left|\mathfrak{z}{1}\right|{p{1}},\ldots,\left|\mathfrak{z}{r}\right|{p_{r}}\right}$

[fathom swallow]

ComplexVariable

[empty stratus]

Compared to this, however, that leads to a conflict.

[compact tartan]

Box topology

[empty stratus]

It makes the "norm" non-archimedean.

[compact tartan]

Who cares if its just for the topology?

[empty stratus]

I need to work with limits and the like.

So I need to have a notion of norm in order to talk meaningfully about limits and convergence.

[mellow moon]

I should write $|\mathfrak{z}|_P$ instead

[fathom swallow]

Anatole

[empty stratus]

You're getting ahead of me. 😉

[fathom swallow]

ComplexVariable
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[empty stratus]

The issue is I want to sum over the set:

$\left{ \bar{t}\in\mathbb{Z}{P}:\left|t{m}\right|{p{m}}\leq p_{m}^{N},\textrm{ }\forall m\in\left{ 1,\ldots,r\right} \right}$

[fathom swallow]

ComplexVariable

[mellow moon]

I get the troubles

[empty stratus]

The compromise I've been leaning toward is this:

$\left\Vert \bar{t}\right\Vert {P}\leq$ x means all $\bar{t}\in\mathbb{Z}{P}$ with norm $\leq x$

[fathom swallow]

ComplexVariable

[mellow moon]

hmmmm

[empty stratus]

and:

$\left{ \left|\bar{t}\right|{\bar{p}}\leq\bar{p}^{N}\right} \overset{\textrm{def}}{=}\left{ \bar{t}\in\mathbb{Z}{P}:\left|t_{m}\right|{p{m}}\leq p_{m}^{N},\textrm{ }\forall m\in\left{ 1,\ldots,r\right} \right}$

[fathom swallow]

ComplexVariable

[empty stratus]

so as to mimic the notation for the sum over n.

But I'm worried that the proliferation of overbars will make things unreadable.

Example:

$\frac{\mathbf{1}{\mathbf{0}}\left(\bar{\varrho}^{N}\bar{t}\right)}{\varrho^{N}}\sum{\mathbf{n}=\mathbf{0}}^{\bar{\varrho}^{N}-1}\chi_{H}\left(\mathbf{n}\right)e^{-2\pi i\mathbf{n}\cdot\bar{t}}$

[fathom swallow]

ComplexVariable

[mint patio]

[mellow moon]

use mathbf for vectors

[empty stratus]

The 1_0 means all ts whose mth entires have p_m-adic magnitude ≤ p_m to the N

[empty stratus]

I am.

[mellow moon]

why not on rho

[empty stratus]

Because rho is the product of the rho_ms.

[mellow moon]

overlined rho is a product ?

[empty stratus]

whereas rho-bar represents the tuple of rhos.

rho1, rho2, .... rho r

[mellow moon]

write it with mathbf

consitent with tuples

[molten egret]

Can someone help me out with math workplace grade 11?

[mellow moon]

for n

[empty stratus]

The big N, you mean?

[mellow moon]

the little one

n is a vector

[empty stratus]

Here:

$\bar{\varrho}^{N}\bar{t}=\left(\varrho_{1}^{N}t_{1},\ldots,\varrho_{r}^{N}t_{r}\right)$

[fathom swallow]

ComplexVariable

[empty stratus]

Little n is already bold.

[mellow moon]

yes

[molten egret]

Somone pls help with part b

[empty stratus]

Hmmm... I suppose I can use bold letters for the p-adic quantities.

[mellow moon]

$\mathbf{\varrho}^N\cdot \mathbf{t}$

instead of

$\bar{\varrho}^{N}\bar{t}=\left(\varrho_{1}^{N}t{1},\ldots,\varrho_{r}^{N}t{r}\right)$

[empty stratus]

That would then reduce the overbar notation to being used to distinguish the tuple and product.

[fathom swallow]

Anatole

Anatole

[empty stratus]

That's better... but... I have actual dot products in my exponentials.

[mellow moon]

ast ?

[empty stratus]

n dot t-bar.

[mellow moon]

you need the point wise product for product of Groups

[empty stratus]

But I'm going to make it n dot t

[empty stratus]

I'm showing that by juxtaposition.

Okay, I think I've got it.

Let me go back to work and see if things can be cleaner.

Thanks.

[mellow moon]

np

[sick burrow]

what reason might someone write $|E\cap (E-v)|$ instead of $|E| - 1$

[fathom swallow]

Worst Geometer NA

[sick burrow]

with $v \in E$

[fathom swallow]

Worst Geometer NA

[sick burrow]

because this paper I'm reading is

[empty stratus]

They don't want you to forget which v was removed. Its legacy must be remembered.

[sick burrow]

I guess?

the wording here is just very strange

like I get that they use this v later in the paper, but also idk

if this holds for some v with $v\in E$ then it holds for all v

[fathom swallow]

Worst Geometer NA

[empty stratus]

No, it's a difference set.

It's not E take away v.

It's {e - v : e is in E}

[sick burrow]

ohh

I guess that would make more sense

I'm just used to authors using - instead of $\setminus$

[fathom swallow]

Worst Geometer NA

[sick burrow]

to denote set subtraction

[empty stratus]

I use \

[sick burrow]

oh I guess yeah

but I feel like I've seen the notation S-x used to denote S\{x} before

[open aspen]

There are lots of notations for setminus

Those two. There's also ~, which Royden uses, for example.

[tiny marten]

i use \ but then i mix it up with / a lot

or start writing it as /

and then i have to go by context clues lol

which isnt that bad i suppose, for myself

[sick burrow]

wow this paper I'm reading is a clusterfuck

this final section seems to be pulling a magic number out of its ass

except the magic number is literally 100

so I'm not sure if any arbitrary number works and they just chose 100 because "wowie round number" or if 100 is somehow special

I think the authors figured the best way to make sure people don't blackbox their results is to explain them so poorly the reader is forced to rederive them

"By Cauchy Schwartz"

not a single inner product in sight

the entire paper reads like the author's poorly written notes to himself

describing a sketch of the results in terms only he can understand

[frail sail]

lol😂

[sick burrow]

no

how are you so good at finding papers what

I swear

one day once I've done some actual research I'm going to vaguely refer to some result I proved here and you're going to find the paper and dox me

what did you even search

"cauchy schwarz lemma 100 awful wording"?

wait where did I say this

oh wait

here

[sick burrow]

_ _

the content of the paper is actually really cool

the wording not so much

mathematician actually explain your proofs challenge (impossible)

yeah

this is what I've heard

is like academia puts a lot of emphasis on proving good results but not as much emphasis on communicating them effecitvely

unless you do such a bad job that no one can read your paper at all

has anyone made any progress on understanding IUTT?

or is it just considered a meme at this point

[leaden torrent]

reset the counter

days since iutt was last mentioned:
0

[sick burrow]

what is our record

wait we should have a bot

but instead of IUTT it should be Godel

hours since Godel was mentioned 0️⃣

minutes since Godel was mentioned 0️⃣

[empty stratus]

It's a shitfest.

Lies, nonsense, and shamelessness.

[jade crane]

yo ryc or hatcherstan, unrelated to the current discussion but are the survivor seasons similar in setting? Like are they always on islands with similar premises? Or are there twists in the rules between different seasons

[deep mango]

Umm

So the premise is always similar but the rules do change a little over time

E.g. the introduction of immunity idols in season 11

But they are always in a remote place (a few seasons are not on an island, sometimes these are the coolest locations - like season 18 tocantins is in the Brazilian highlands)

[jade crane]

ah yea I had to look what those were when watching season 37 lol

the immunity idols I mean

[fair mural]

season 37

bruh

[toxic schooner]

is there a single season a year?

[deep mango]

And the premise is always the same: it starts as 2-4 tribes, they compete and the losing tribe has to vote someone out each episode. they merge about halfway in (exact merge time is inconsistent) and then everyone competes for individual immunity instead of team immunity

The end has changed over the years

It started as a final 2 where the people eliminated after the merge vote for a winner

Then it became a final 3 (because the best player kept getting cut at 3 lol)

[jade crane]

ah yea makes sense

pat was a tragedy though I assume that doesn't happen often?

i.e. an actual injury?

[deep mango]

More recently, now there's no vote at final 4 - someone wins immunity, they choose someone else to go to the final 3 with them, and the remaining two have to compete to build a fire which burns a rope.

The firemaking winner takes the third spot

Lots of people prefer the final 4 vote but the final 4 firemaking is really exciting in my opinion haha

[jade crane]

when you say wins immunity, do you mean they still have a hidden immunity idol?

or do you mean by they win a game

[deep mango]

This is pretty rare. Especially so close to the start

Pats injury was also one of the worst

And it sucks cause it was on the boat

Its not like it was something he did

[jade crane]

yea the boat of all things

[deep mango]

So bad

Yeah thats the trouble with 37. The first episode is so sad

[jade crane]

I honestly thought he would get injured from the way he was standing on the tree at the start

with the machete

[deep mango]

Yeah they set it up haha

[jade crane]

😂

[deep mango]

Also like

Idk

Pat seemed to be bothering people so he might have been voted out anyway

Its hard to tell after the fact

[jade crane]

yea it was probably gonna happen since they had the tribal council that night anyways but an injury sucks

[deep mango]

After the merge starts, you can win an individual immunity in challenges every cycle (not hidden, but also doesnt need to be played after the votes - you just cant be voted for)

[jade crane]

oh ok makes sense, just like what happens after a team win

[deep mango]

Exactly

But there's usually only one immunity per vote

So it usually doesnt matter

But when it does matter it's really fun

Like, the target wins immunity and everyone has to scramble to not be the collateral damage

[jade crane]

do you recommend just going chronologically from 37 then?

seems like the early seasons would be too far removed in terms of strategy and the general setting

[deep mango]

Uh, I dont recommend 38-40 since they all have big spoilery returnees

[jade crane]

oh

alright then 41 it is

[deep mango]

41 is the newest one and is super interesting, but is also very very twisty and confusing

So if you like that then it's great

[jade crane]

yea I'll give it a shot 🙂

[deep mango]

If you want another season with modern strategy and really fun gameplay which is less twisty, season 33 millenials vs gen x is good (stupid theme but excellent season).
But yeah while I adored 41, its very polarizing

[jade crane]

lmao ok that actually sounds pretty good

for the millenials vs gen x

[deep mango]

37 is really good episode to episode, while 33 and 41 form a few more broad, cohesive storylines. It's sort of different feeling, but still very fun.

[jade crane]

yea sure, thanks for the recs. Will probably be seeing 33 first then

just out of curiousity, have you seen nearly all of the seasons?

[deep mango]

Ive actually only seen half of them haha

[jade crane]

wouldn't blame you, 43 seasons is way too many

[deep mango]

I know basically everything about way more than half of the seasons though

So

Its hard to avoid that if you want to discuss it online

But i will still slowly get through them

[jade crane]

yea I can imagine, big fanbase online?

[deep mango]

Yeah

And I wasnt so careful about waiting to watch returnee seasons

I just watched them anyway

[jade crane]

Are those just

[deep mango]

But then you find out how those people did beforehand

[jade crane]

all winners

[deep mango]

It depends

The only All Winners season is 40

But they do bring back winners sometimes in other ones

Its usually people who were very memorable / good players but lost late in the game

[jade crane]

yea makes sense, ppl probably more attached to them

[deep mango]

And there are returnee seasons with just a few returnees and a bunch of new people, as well as returnee seasons which are purely returnee

Those are interesting, because they're either the best or the worst

Season 20 (heroes vs villains) is regarded to be the absolute best season

I havent seen it

I want to see a bunch more of the pre-20 seasons first

[tender musk]

Hi @deep mango! @deep mango for Moderator!

[toxic schooner]

wha..

[tender musk]

?

[toxic schooner]

is this a thing now, people naming themselves this way for people to be promoted to certain positions?

[fair mural]

yes

[tender musk]

Other people are doing this?

[toxic schooner]

No, but i have a feeling other people will start

[tender musk]

Oh, ryc is really helpful and generally is online so that is why I think he should be Moderator

[toxic schooner]

[tender musk]

Same with @fair mural. Hi @fair mural!

[toxic schooner]

i am generally online, so make me honorable

[tender musk]

You have a point

[tender musk]

Oooo, another Hamilton fan!

[toxic schooner]

😌

yessir

[fair mural]

[tender musk]

?

[tender musk]

Do you have a Windows 64-bit computer?

[toxic schooner]

yeah i do

[tender musk]

Oooo, finally....

[toxic schooner]

im not into linux rn

[sleek wing]

literally 98% of all computers running right now are windows 64 bit

probably higher

[tender musk]

But no one responds that they have one

They always respond with, "I have a Mac"

[toxic schooner]

[tender musk]

https://drive.google.com/drive/folders/1aakWrKivzu8u_5JZWsVEzuuYYbHMVdtS?usp=sharing

Do you run Processing?

[toxic schooner]

what?

[tender musk]

I am assuming that is a no.

[tender musk]

Would you be willing to download this file?

I perfectly understand if you don't want to

I mean I am a stranger

[toxic schooner]

[tender musk]

And this https://java.com/en/download/

[toxic schooner]

what is that file

[tender musk]

It is a game I programmed over the weekend

I want to see if people without Processing can open it

But everyone I know that has a Windows computer has Processing

This is the area with the most people I know

Therefore I am seeing if anyone here doesn't have Processing

[toxic schooner]

What does it mean to have processing?

[tender musk]

It is an application for Programming with Java... It combines oracle with its own library...

[toxic schooner]

i see

[tender musk]

https://www.google.com/search?q=processing&oq=processing&aqs=chrome..69i60j69i57j0i131i433i512l3j69i60l3.2720j1j7&sourceid=chrome&ie=UTF-8

https://processing.org/

[toxic schooner]

ok i dont have this

[tender musk]

So will you be willing to try the file?

Again, feel no pressure from me...

[toxic schooner]

so i just install it?

if its not too complicated i can try i guess

[tender musk]

Yes and then run it, after getting Java 8

[deep mango]

Processing is great

it's so fun

[tender musk]

I know

[toxic schooner]

ic

i really dont know shit about computers

[deep mango]

Coding Train

[tender musk]

It is amazing

[tender musk]

That is how I learned

[toxic schooner]

even tho i have been using them for quite literally 16 years at this point 😭

[deep mango]

I learned so much cool stuff from Coding Train

[tender musk]

Same

This project is the first one I have made without the Coding Train

[tender musk]

I am sure you already have Processing, but if you want to play test my game...

https://discordapp.com/channels/268882317391429632/849776666161971240/941679825683374090

https://discordapp.com/channels/268882317391429632/849776666161971240/941680370523463762

[deep mango]

this one is really cool
https://www.youtube.com/watch?v=7gNzMtYo9n4

oh i am not willing to download files from discord haha, sorry

[tender musk]

I haven't seen any of the remotely new ones, I watched him a long time ago.

[deep mango]

yeah same

well

this one is slightly new

it depends on what you mean

[tender musk]

2 years

[deep mango]

my friend did a big research project about the Chaos Game

inspired by Dan's video

so good

[tender musk]

Okay, no pressure

[tender musk]

I will watch it

[deep mango]

i did a project about this one

https://www.youtube.com/watch?v=diGjw5tghYU

sandpiles fractal

in high school

omg i remember when the coding train intro was posted LMAO

when he switched from coding rainbow due to copyright stuff

[tender musk]

Did you install it, and is it running? (Again, if you haven't don't feel like this message is pressuring you...)

[toxic schooner]

i am installing

at least i think so

[tender musk]

Yeah you are... that is interesting, why is it zipping?

[toxic schooner]

google drive is zipping it all into one file

and then i will install it

[tender musk]

Hmm

Thanks for helping!

[toxic schooner]

its very slow in zipping it

[tender musk]

😦

[leaden torrent]

with all due respect, your behaviour is indistinguishable from a "test my game" phishing bot

[tender musk]

What is that?

[leaden torrent]

and the fact that you changed your name so recently doesnt help your credibility here

[tender musk]

Sorry, I will change it back

[leaden torrent]

its a common type of discord phishing scam

send a .rar or .jar or .exe with malicious code that steals account info when ran

[tender musk]

Oh, sorry, I was trying to make sure they weren't being pressured

[leaden torrent]

say "hey can you test my game"

i dont think thats whats going on here but its very suspicious looking

i would, at the very least, prefer if you sent the decompiled code

[tender musk]

?

How would I go about doing that?

[leaden torrent]

huh

how do you make a java applet without knowing what i mean by that

[tender musk]

This is literally what I see

[leaden torrent]

ah.

[leaden torrent]

okay, thanls for the cooperation

still wont guarantee that its safe but i feel a little less peeved

[tender musk]

Sorry again...

[tender musk]

If you want a text file of the code, I can send you a text file.... (Can text files be dangerous, and could I send a text file?)

[leaden torrent]

it's fine, don't worry about it

[leaden skiff]

hey can you download my game? www.fake-nitro-link.gov

[tender musk]

I am not sure that is a site...

[leaden torrent]

though to answer your question, yes discord supports uploading text files and no theyre typically not dangerous (though files that use other software, like .py files using a python interpreter, can be)

unless theres some new text file vulnerability i dont know about, but that vulnerability would probably be used on the government of a major power, not a random discord user

soooo

[tender musk]

?

[toxic schooner]

ok i cant even install it even if i wanted to

[toxic schooner]

its stuck here

i stopped it 2 times

still stuck there

sadge

[tender musk]

Huh?

[prisma swallow]

Given three players A,B,C. Suppose player A wins against player B, and player B wins against player C, but player "A does not necessarily win against player C. What is this called mathematically?

[surreal sapphire]

a non transitive relation?

hard to tell what you are looking for

[prisma swallow]

I thought there might be some way to describe this kind of relation

But yeah, probably non transitive is the correct phrasing

[charred mortar]

Sounds like rock/paper/scissors

Maybe antitransitive if the relation is never transitive

[prisma swallow]

Oh yeah sorry, I was not specific enough

It should be "A does not necessarily win against C"

Alright thanks

[lilac agate]

How many forms linear equations have? I know three of them but I think there is a lot of ways to write them differently but idk if there are infinite forms

[steep shadow]

@stray kite can you help

[stray kite]

with what lol

[steep shadow]

Math idk if helpers do that

[pale orchid]

read #rules and #❓how-to-get-help

[honest veldt]

fellas, is it fine to say "E is a subset of the metric space X" or should I be saying "... of the set X"

regardless of whether X is a metric space, top. space, etc.

[honest veldt]

So using "subset" when I'm talking about a metric space (X, d), rather than the set X, doesn't make sense, right?

I understand that, but can I still use "subset" if I only wish to reference a set of points in the topological space, without caring about any structure?

or should I specify that it's a subset of the set X rather than the topological space (X, T)?

[humble solar]

It doesn't really matter. Remember that you're communicating with humans and not machines.

[deep mango]

coq moment

[humble solar]

They can figure out that when you say E is a subset of (X,d) you mean that E is a subset of the underlying set X. They're not going to puzzle through the Kuratowski definition of pairs.

[tiny marten]

it depends a lot on what you are doing, too

like if you are constructing a space out of a subset you should be referencing topologies and stuff

but if your thing just happens to have a topology in the background of your work it's not necessary to designate it as a topo space

[modest rune]

The only time when this is like, something to worry about

is if like

I have spaces X and Y

and Y is a subset of X

but Y has a topology other than the subspace topology

Outside of that you can play fast and loose

[tiny marten]

yeah you sort of learn when you should unload a bunch of names, hopefully

this is more so people dont get a headache by reading your work

[honest veldt]

Alright, got it, thanks folks. Just not gonna worry about it for now then

[tiny marten]

oh yeah i think i see what you were asking too

in topology a subspace is especially weak, it only needs to be a subset and it automatically inherits a subspace topology

it's not like algebra where they have to have closure under an obvious operation

the trick with subspaces is figuring out which to use at all

since they are so easy to define

a linear subspace has much more explicit structure than a topological subspace (at first blush)

[honest veldt]

Right, I see. So for a metric space, where the subspace is obvious, it's not that significant, right? Subspace, subset, they're all nice

[tiny marten]

yeah you might say subspace if you are treating it like one

otherwise you would just refer to it as a subset

[honest veldt]

Perfect, that's exactly what I needed, thanks a bunch

[tiny marten]

a subspace can be created by any subset and its subsets are all subsets of the original space intersected with the subset you mean to make into the subspace

[honest veldt]

I hadn't thought of making the distinction when the structure doesn't allow for subspaces to be so easily chosen though, thanks for bringing my attention to that

[tiny marten]

so for X a TS and A a subset the set A becomes a (sub)space by looking at its subspace topology defined by O_A is open in A iff there exists O open in X with O_A=O\cap A

[honest veldt]

that to my attention*

[tiny marten]

this allows for incredibly fine structures

[honest veldt]

I actually just hit this subspace topology part, so very convenient stuff

[tiny marten]

yeah this stuff feels like nothing for a bit

then it becomes clear you must have topology

[honest veldt]

Yeah, Hatcher's notes got me a little worried here, he says this part is a pretty big deal

[tiny marten]

i would say understanding the significance of subspaces and how the subspace-set say A as above interacts with the topology of X is one of the most important parts of point set topology

you gotta spiritually do it all again with measure theory so you had might as well learn it well

[frozen merlin]

what do you mean by 'how it interacts'

[honest veldt]

Ah man, I was just struggling with this part a day ago too, this is lovely

[tiny marten]

so like, if you take R and Q the rationals as a subset

when you look at the subspace topology of Q as a subspace of the reals R, you have to be careful with stuff

you often need a bunch of examples

until you really get it

upon observation as sets Q and R are totally different topologically

so you can define Q to have some topology, maybe a discrete topology because it is countable so whatever

but as a subspace pairing Q and R must be thought of together, and you must consider this topology as yet another that can be defined on Q

the "natural" topology of R is the metric topology of the euclidean metric and this works out nicely because R is complete, where Q is clearly not (and countable)

but Q inherits this in a not too bad way as a subspace

this isnt even a bad example, just the essential one for a dense set

as Q is dense in R, but not nearly as numerous overall

people have come up with some real bizarre subspaces and subalgebras (of measure theory)

topology and measure theory have a very free form quality

[tender tulip]

Insane mathematical insanity:
Let P(z) be an elliptic function, and has an order n pole at every point on the lattice.
Let K(z) = f(P’(z)) - f(P(z))
What is the lowest orders of f and g such that periodic function K has an order n pole at every point at the lattice

1. Due to the definition of an elliptic function, P at minimum must have an order 2 or above pole at every point on the lattice
2. It’s derivative must have a minimum of an order 3 pole at every point in the lattice due to the definition of a derivative
3. Due to the definition of a polynomial, g(P(z)) must have a pole equal to the order of g multiplied by n, this also applies to f but instead n + 1
4. For the poles to cancel, they must have equivalent order. Therefore, the order of both f(P’(z)) and g(P(z)) must be the same, aka the LCM of n and n +1, which is always n(n+1) due to coprimality. This means you must make the polynomial of P have the order of P’s derivative’s Pole, and vice versa.
5. The lowest n can be is 2, thus the minimum order of f is 2, and the minimum order of g is 3.

Thus f(z) at minimum is a quadratic, and g(z) a cubic

That’s why elliptic functions correlate to cubics

[simple raven]

Basically that if X is a topological space and A a subset of X, then every open set of A for the induced topology is the intersection of an open set lf X with A
Same thing with the closed sets

[frozen merlin]

yeah I know the definition, just the way it was phrased made me think there was some cooler stuff going on

[sick burrow]

Ok you ready for an extremely cold take

A good professor makes such a massive difference

[charred mortar]

Wow what a take

[sick burrow]

ikr

I do feel like I'm finding myself increasingly picking classes based on the prof

[fair mural]

big no it’s cursed

[jovial ember]

Blushysully gang

[little vine]

Que horror

[wooden flax]

thats nsfw

too sexy

[steep mountain]

no.

[sick burrow]

See right now blushysully is special since only Emma has it

But I also am always in favor of more cursed emotes

[limber thunder]

es horrible

[stray kite]

do you think its a good idea to take a few CS courses while im in undergrad? will that do something for future job prospects when i decide to not go into academia

[jaunty marlin]

wtf is a cs course im still in highschool 🤣

[leaden torrent]

computer science.

this can be googled btw

[stray kite]

i mean i do programming anyway so im building up a portfolio and whatnot, but does having it on your degree as taken courses make a difference?

[leaden torrent]

wasnt talking to you, i dunno how much the mark on the transcript matters

probably not at all iff you have a good github

and ideally an internship

[compact tartan]

this is not a democracy

[ripe wasp]

has anyone taken an aops class online

im thinking in enrolling in their calculus one over the summer

they seem cool but im worried like its all gonna be like directed for comp math

im really bad at comp math

[toxic schooner]

please no

[dawn bridge]

do your cs courses teach the skill of 'fluent microsoft excel user' 😂

being able to program is a rly rly good skill tho

so just go for it

[stray kite]

i mean i do know how to program due to wasted time during the past few years , i just want to know if its actually useful to have those things on your degree listed as courses taken

[dawn bridge]

intro to programming is useless if u already have projects

but maybe like some more specific courses like blockchain might be useful

[stray kite]

hmm i see

[uncut socket]

@spring wyvern I honestly wouldn't have helped that person, if one they had 15 minutes and 2,they didn't look back at the material they missed

Makes them responsible

[vast surge]

[neat lintel]

wtf

[bronze pelican]

If mods dont add blushy sully now even when there is MASSIVE PUBLIC SUPPORT

twenty fucking seven sadcatthumbsup

[deep mango]

2 more 's than 's...

"massive public support"

[bronze pelican]

the notthis'ers are haters

[sleek wing]

bandwagoning nay-sayers... you hate to see it in such a thriving democracy

[bronze pelican]

you know what we do to haters?

[neat lintel]

1 more hehe

[bronze pelican]

we BLUSHYSULLY on the haters!

[deep mango]

it's almost even

then you'll be sorry

[bronze pelican]

get BLUSHYSULLIED

[sleek wing]

https://cdn.discordapp.com/attachments/849776666161971240/942089207080435772/zsullyhappy-2.png

[neat lintel]

we need a different emote

wait a min

[deep mango]

is already the best blushy emote

idk why we need more

[sleek wing]

anti-blushy cope

[deep mango]

lol

[neat lintel]

we need this

[deep mango]

energy

[neat lintel]

my pfp is the overall best emote tho

[bronze pelican]

rycoping right now

[fast ivy]

[deep mango]

[sleek wing]

the zased blushysullyists take the lead

[vast surge]

If we're adding new emojis can we also add the commutative algebra emoji

[vast surge]

It's just an absolutely amazing image

[toxic schooner]

yes i am a big hater

there will be no blushy sully

[neat lintel]

blushy sully is absolutely cursed and should be banned

[toxic schooner]

based

[deep mango]

DOWNVOTES HERE

[bronze pelican]

You can't stop thr blushysully train

[toxic schooner]

i will downvote with my alt

😈

[deep mango]

we're almost even again

[charred mortar]

I will upvote with my two alts

[steep mountain]

this gonna be a good list to know who to never be friends with

[deep mango]

we can win without voter fraud

[toxic schooner]

i didnt even add my alt here yet

[deep mango]

IT'S EVEN

FUCK YOU BLUSHIES

[sleek wing]

even stevens...

but not over dover...

[deep mango]

it will be soon

you sniveling swine