serious-discussion

[stark charm]

if i am bothered by its presence i will handle it

[charred mortar]

I mean that’d probably send you to jail

[stark charm]

bruh no

[charred mortar]

Don’t assault other people’s pets

[errant merlin]

bro said violence is the answer

[wild lantern]

Animal abuse?

Like, don't abuse animals lol.

[stark charm]

no its not that

that isnt abusing animals

if you are threatened by animal prescene tou ahbe right to defend self

[wild lantern]

I mean, if you can leave and call animal control but choose to boot a cat in the head instead idk lmao

[stark charm]

bruh problem solved

animal controlled

[bronze pelican]

https://youtu.be/5dZKr-Z2FO4

[stark charm]

its nothing malicious

[bronze pelican]

Literally a Korean veritasium video on collatz

[wild lantern]

Yes, kill some scared cat because you lack the self control to leave lmao.

Great plan

[bronze pelican]

This is the state of the world

[stark charm]

bruh adrenaline pumping

[brave hollow]

real question is why does korean veritasium exist

[stark charm]

korean people exist

and watch veritasoyum

[bronze pelican]

It has 49k views

So there is an audience

I would never have expected it

[brave hollow]

49k is low numbers for veritasium I think

[bronze pelican]

Good on Derek

[neat lintel]

Even Chinese veritasium exist afaik

[teal lion]

hey

[azure nymph]

I wish I like the early applications of the integral more

[fervent pebble]

okay so ive done some more research

http://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/

apparently i was getting some things mixed up here

(i dont feel bad abt it tho since i guarantee ain't nobody else knew this shit either)

so, as such, my question did not encapsulate what i really wanted

my new question is as followed

Does anyone know a (mildly) elementary proof that the $\sqrt{2} \not\in \mathbb{Q}$ that is not a proof by contradiction or proof of negation, as defined in the following image:

[fathom swallow]

valley

[fervent pebble]

mildy elementary = single var calc and below

if you add like linalg or multivar calc i could probably follow but not too much

[little vine]

You mean not the classic Euclid proof?

I suppose not because that's contradiction

Folks were talking about this earlier

[fervent pebble]

yeah i asked the original question there lol

[near fox]

I feel that proving something is not in Q is somehow fundamentally a negation argument

[static loom]

I was trying to think if there's a way to prove eisenstein's criteria while side stepping that too

as applied to x^2-2 in this case ofc

[little vine]

Ohhhhh

That would do it

Because if it's irreducible over Q the root can't be rational

That's slick

[static loom]

well problem is the proof I have in mind for eisenstein is basically starting by assuming a factorization and then walking through the coefficients leading to a contradiction

so it doesn't really solve the problem

[near fox]

Yup, Eisenstein is sorta like a higher order version of the same argument

[fervent pebble]

i just think it would be ridiculous if there wasn't a way

there's gotta be one

a direct proof of something that implies sqrt(2) is irrational would also suffice

[near fox]

I guess the problem is that Q is the thing that has structure, things that aren't in Q are defined only by failing to be in Q, at least amongst elementary theory

[little vine]

This sounds hard to do

[fervent pebble]

sometimes more general statements are easier to prove

[near fox]

Perhaps a related question would be, is there a way to define or construct the set of irrationals without referring to Q

Maybe there is, but it seems unlikely to be that elementary

[static loom]

scary thought

sqrt(2) but no 2 lmao

[fervent pebble]

hmmm

yeah that seems icky

[frozen merlin]

can't we use gauss's lemma instead

[fervent pebble]

is there a direct proof of that?

[frozen merlin]

like, $x^2-2$ is primitive so it's irred in Z[X] iff it's irred in Q[X]. but 2 isn't a square so $x^2-2$ has no roots in Z

[fathom swallow]

Average J∘du=du∘j enjoyer

[frozen merlin]

honestly I never really understood how the proof of gauss goes

[fervent pebble]

oh wait i think i found a really fucking pretty working proof on the internet

wow

[errant merlin]

Do share

[near fox]

Yeah, upon googling, I do see that some exist, that is very interesting...

The proof shows that the distance between sqrt2 and any rational number is strictly greater than 0

Just purely through algebraic manipulation. I guess I stand corrected

[fervent pebble]

^

thats the one i saw

[static loom]

oh link me to it sounds cool

[fervent pebble]

https://en.wikipedia.org/wiki/Square_root_of_2#Constructive_proof

off wikipedia

[static loom]

interesting

[near fox]

I'll have to add this to my collection of interesting results

I'm curious to what extent it's possible in other cases (other than roots of integers). Because it does seem like nothing easily characterizes irrationals other than the fact that they fail to be in Q

[fervent pebble]

i think it's also really nice that you can see the motivation from the proof

[fervent pebble]

for some reason this feels like it should be very easily extendable

like very easily

to even e and pi

but i cannot fucking seem to do it

[fervent pebble]

@neat frost

[leaden crown]

Is anyone good with finance in math (loans, interest rate, etc.)? I need some help with a problem.

[sharp stump]

are all the channels occupied or is there a prob?

[fervent pebble]

@bright hill

[bright hill]

This took me an embarrassingly long time to parse

It is kinda cool tho!

I've never seen this proud of the irrationality of sqrt(2)

Constructivism as a whole sounds interesting

[fervent pebble]

yesss so facts this one is new to me

i cannot believe i never thought of it myself

and it took me a bit longer than it should have too, i think it’s just bc the concept is new

it feels so obvious but i’ve not seen it done before

[bright hill]

It seems tricky tbh

I don't think I'd come up with it alone unless I I was actively looking for it and I squander 2 days

[charred mortar]

Yeah it’s trickier than the standard way

[neat frost]

O ye I saw you posted that before, is very inchresting

[deep mango]

Weird

This reminds me of something I was learning about yesterday

about the convergents of continued fractions for irrationals

[neat frost]

Wow way to make this about you ryc

Also classic ryc doing boring math

[deep mango]

This is what shin was talking about

I guess you hate him

[neat frost]

Ryc: slurp is shin
Also ryc: slurp hates shin

Make up your mind you crazy senile boomer

[deep mango]

Or maybe it makes you sound crazy.

Anyway this is a cool proof

[neat frost]

Tbh

This is the kind of proof I don’t like

Too analysis-y

[deep mango]

this

[neat frost]

Oh

Ohh

Wow shin is really cool

You’re dumb ryc

MARKOV

Like Markov inequality???

Wowowowow

[deep mango]

@fervent pebble You should check out diophantine approximation if you think this is cool

[neat frost]

Diophantine… isn’t that a chemical?

[deep mango]

https://en.wikipedia.org/wiki/Markov_constant

[untold sapphire]

What's going on here

[neat frost]

I’m bullying ryc

[untold sapphire]

why are you talking shit?

[deep mango]

Not markov inequality

[untold sapphire]

There's turpentine

[deep mango]

I wonder that sometimes myself

But most of the time, I don't wonder anymore

[neat frost]

I can tell the difference between constant and inequality.
That’s why I said like

Stupid.

[fervent pebble]

ooooh

will do

[neat frost]

Because if I don’t, who will?

[fervent pebble]

tyyy

[deep mango]

Dozens of other users

[neat frost]

Yeah but none as well as me

[deep mango]

You're winning the envy war

[fervent pebble]

ryc tell slurp im better

[neat frost]

Valley.

[untold sapphire]

But losing every battle

[neat frost]

Guess what.

[fervent pebble]

hmmm ur still a foot shorter than me?

[deep mango]

Better at what?

[neat frost]

Hell yeah

I saw a bunch of Americans

[deep mango]

Spending 2 hours talking about a problem for an 8th grade geometry class

[neat frost]

Why are you all so fucking BIG?

[deep mango]

Real good at that valley

[fervent pebble]

OKAY RYC

I SEE HOW IT IS

[untold sapphire]

hey ryc have you seen that there's a connection between computability and diophantine approximation

[fervent pebble]

i just hate losing arguments where ik im right okay

[neat frost]

Lmfao valley you just got rekt, even better you got rekt INSTEAD of me

[bright hill]

Ooooo

[fervent pebble]

rare i agree

[untold sapphire]

do you know what randomness is in computability theory

[deep mango]

That's the first context in which I saw diophantine approximation

[untold sapphire]

Nice.

Oh wait

you're... no nvm

[deep mango]

But I don't remember any of it

[untold sapphire]

was trying to remember where you went to school

yeah

rip

[neat frost]

You know why?

[deep mango]

Yes

[fervent pebble]

bc ur old?

[deep mango]

Because it is irrelevant to me

[neat frost]

No

It’s because you’re stupid.

[deep mango]

I remember plenty of other things that are relevant to me

[fervent pebble]

i’m not even half as mean to ryc as slurp

[neat frost]

Yeah

[fervent pebble]

and he still roasted me

[neat frost]

I’m just a mean person valley

[neat frost]

It’s because he’s scared of me

[fervent pebble]

prolly bc ur such a ryc simp tbh

[bright hill]

Einstein couldn't even remember where he lived

[neat frost]

But also because he knows that I’m doing this for his own good.

[deep mango]

I'm handling this one with kid gloves.

[neat frost]

Einstein was a fucking physicist

[deep mango]

Einstein was a dummy

[neat frost]

Yeah he had bad hair as well

And his facial hair? Don’t even get me started.

[fervent pebble]

as if urs is better

[neat frost]

I need a haircut true

[bright hill]

I'm trying to back you up lmfao

[deep mango]

Resisting urge to roast darq unprompted

[fervent pebble]

do it

[neat frost]

You’re comparing him to a fucking physicist. How is that helping him?

[deep mango]

No I can't alienate more of my allies like this

I am a physicist

Math is a waste of my gifts

[neat frost]

HAHAHAHA ADMITTED IT

[deep mango]

Maybe it's time for a new name

[deep mango]

That's better

[errant merlin]

pin

[deep mango]

Cozy mode

Lol

[neat frost]

I’m on fucking mobile ryc

[deep mango]

Ok

That's a big ass screen then

[neat frost]

I don’t have my laptop with me

[deep mango]

or your font is super small

[neat frost]

iPad

[deep mango]

omg you have an ipad

hahaha

[neat frost]

I can’t change the font size

[deep mango]

wow

👅 👢 🍎

[neat frost]

For notes…. I bet you handwrite your shit on paper you non-eco-conscious boomer fuck

[neat frost]

I am not an Apple fanboy

Stupid

[deep mango]

I do handwrite for now

[neat frost]

HAHA

[deep mango]

But I'm planning on getting a tablet

This week

[neat frost]

And you make fun of ME?

[deep mango]

Handwriting is nice

[neat frost]

Stupid stupid stupid dummy

Okay yeah true

[deep mango]

I will not be getting an ipad

[neat frost]

Samsung’s s-pen is shite tho

Apple pen is better. Not 100 dollars better but better

[deep mango]

I don't know what's good and what isn't

That's the issue

the ipad stuff is so expensive

[bright hill]

Yeah? And your a computer scientist lmao

[neat frost]

I’m good. You’re not.

[neat frost]

It is. Not really worth it imo.
My dad just really seemed to prefer me getting apple for some reason.

[deep mango]

Oh

actually i can get an ipad for like 300 bucks

wtf

[neat frost]

Then a pencil for 100 more

[deep mango]

i didn't know they were that cheap

yeah that's annoying

[neat frost]

Which iPad tho?

[neat frost]

And what are you?

[bright hill]

Not sure

[deep mango]

the keyboard is 300 dollars

what the hell

[bright hill]

I don't think I'm a mathematician tbh

[neat frost]

Lmfaoooo

Why get that tho?

[next schooner]

I have a reMarkable tablet

[neat frost]

Just use a laptop if you need a keyboard

[deep mango]

Oh ive heard of those

[neat frost]

If you have a laptop

[next schooner]

Expensive but it was worth it for me

[bright hill]

(especially after reading a mathematician's apology yesterday)

[neat frost]

I need a better note taking app tho

[neat frost]

Wassat

[neat frost]

OneNote is infuriating

[bright hill]

A book

[neat frost]

Ah

[next schooner]

I didn't want to get a tablet that I could get distracted on lmao

[neat frost]

Fucking boomer

Reading a book

[deep mango]

Huh

I can get something that's a chromebook

but is also a tablet

i wonder if there's a good notetaking app on the chromebook os

seems fine

[bright hill]

Chromebooks are still a thing?

Huh

[deep mango]

yeah

[bright hill]

I still don't like to believe people choose to be pure mathematicians coz math is the thing they're best at

[next schooner]

I don't think that's necessarily the only reason people choose to do math

Well perhaps it is necessary, but by no means sufficient

[rapid folio]

Good evening all

Is the conversation about doing mathematics or chromebooks? haha I'm trying to jump in here

[static loom]

I think that's how a lot of people go on to get higher degrees, it's the safer choice to just keep going to school

[bright hill]

Wdym?

[static loom]

I think it's pretty clear

[rapid folio]

Do you mean that people should just study independently? Because if you wanted to just "keep going to school" there would come a point where you'd be forced to seek higher degrees in order to keep going

So getting higher degrees is the same as people just keeping on going to school

[bright hill]

That doesn't explain why you'd do math instead of (god forbid) physics for example

[neat lintel]

@woven whale we're not "friends" why is it showing me your activity?

[rapid folio]

I guess I come at it from a point where my mathematics degree got me to teaching, and now I study independently because I'm interested in different facets of mathematics. It's history, its different topics, its application to every area that it reaches.

What's the gripe against physics 😅 If I can ask

[rapid folio]

I think that can just happen if they're up for showing it publicly

[neat lintel]

From any server? Is that so? New thing I learnt today, thank you!

I thought that was for people you're friends with

[woven whale]

I am just too cool

[bright hill]

It happens with people you interact with

[neat lintel]

I think he's answered by question before, I don't suppose a lil bit of typing should do that. Weird honestly

I won't be able to get used to that 🤔

[sharp stump]

Hey, do you know what happened to the channel where we were talking. It is somehow not active!

[woven whale]

you deleted your original message so it got closed

[sharp stump]

oh that could be

it's open now

[simple gorge]

Could someone explain to me what contour integegration is geometrically? Like, integration of f(x) = x^2 from 0 to 1 yields the area bounded by f(x), the limits of integration, and the x-axis. I was reading about Cauchy Residue Theorem for leisure, so I apologize if this question has a really obvious answer. Thank you!

[leaden torrent]

it's integration along paths in ℂ

are you familiar with path integrals? or at least with multivariable integrals?

for example, does $\int_{D}f(x, y)$ where $D$ is some (compact) subset of $\bR^2$ and $f(x, y)\colon \bR^2 \to \bR$ make sense to you?

[fathom swallow]

Namington

[simple gorge]

I think so? It's like volume right for R^2? But what does a compact subset mean, is it like does it mean it's a closed region?

[leaden torrent]

compact = closed & bounded

im just ruling out things like D = [0, ∞) × [0, ∞)

perhaps my notation is a touch too abstract

[simple gorge]

No it's good

[leaden torrent]

https://upload.wikimedia.org/wikipedia/commons/4/42/Line_integral_of_scalar_field.gif

here's a visualization of a path integral

the coloured region represents a function from ℝ² to ℝ

so it takes a given (x, y) value and assigns it a value

which is represented by the "colour" on the (x, y) plane

then we take a path through the plane and compute the "sum" of values obtained by that path

contour integration is a method for computing path integrals for paths in ℂ, the complex plane

to be a bit more specific, when we integrate along contours, we assign a "direction" along the curve and integrate "along that direction"

this matters because the complex plane permits, well, complex multiplication

(and is therefore amenable to more integral evaluation techniques)

[ashen pike]

very nice gif

[neat lintel]

@agile wedge débloques moi stp

[neat lintel]

I'm a bit confused about this. R is one dimensional, and R^2 is 2 dimensional
So the function takes a value (x,y) and assigns it a value that is represented by the colour on the (x,y) plane

But how is the value assigned on a plane shouldn't it be assigned on a line because it is one dimensional?

[wild lantern]

f is a map from R^2 to R. The (x,y) pairs are coming from the domain (set of inputs) of f, the individual colors come from the codomain (set of possible outputs) of f. The image nami showed you gives you the color each (x,y) pair maps to under the function f.

It's similar to how if we have f from R to R given by f(x)=mx+b, each value mx+b is an element of R, but we graph f in R^2 to show what each element x in the domain maps to by the function f.

I guess nami basically said all this.

I've seen this visualization for line integrals before:

https://images.squarespace-cdn.com/content/58757ed7f5e231cc32494a1b/1507437517228-0U4F5GHDHUO4TXMAYRPQ/VIBrO.jpg?content-type=image%2Fjpeg

[cold needle]

ah stewart my beloved

[simple gorge]

@leaden torrent Thank you!

[neat lintel]

Ah so we are just representing the possible outputs of (x,y) in R^2 itself

Thanks 👍

[wild lantern]

I'm saying the colors are the outputs. At (x,y) in R^2 you draw whatever color f outputs to produce the graph nami posted.

Like, the picture is really R^3 it's just the z axis is represented by colors.

[neat lintel]

Ah I see

👍

So basically it "helps us predict" what are the same outputs f is going to produce for any (x,y)

So if a larger region is cyan in colour, all possible (x,y) values there in it will produce the same output in R

Interesting

[past stone]

is it correct to say that 1/2+1/4+1/6+1/8... is also a harmonic series, or does that title refer exclusively to 1/2+1/3+1/4...?

[wild lantern]

We're basically doing something similar when we graph something like f(x)=mx+b in the xy plane. At each x we associate a y value height corresponding to f(x).

[neat lintel]

~~The first one converges to 1 (by comparison test you can assure it converges by comparing to the latter series)

Harmonic series is specifically defined for the latter~~

And harmonic series is 1 + 1/2 + 1/3 + ....

[static loom]

wrong

it's the harmonic series multiplied by 1/2 term for term

[neat lintel]

,w \sum_{n=2}^{\infty} 1/(2n)

I see mb

[fathom swallow]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by Pencil/Idris#9129

[static loom]

comparison to the harmonic series can only be used to show divergence, not convergence

[eternal agate]

what can I do to prepare for calculus BC? I already know all of calc AB.

[stark charm]

know all of calculus BC

just spam khan academy I guess

[eternal agate]

yea but i actually wanna learn something new in that class

i feel like doing nothing would be the best option

maybe learn trig sub integrals

so they don’t kick my ass when i forget what to substitute and when

[stark charm]

they might

it’s important to know inverse function theorem

[placid coral]

hi

[placid coral]

https://tenor.com/view/calculation-math-hangover-allen-zach-galifianakis-gif-6219070

[tiny forge]

If someone in help sends two pictures one after another, is there a way to rotate them both with ,rotate

[bright hill]

The math exam was fucking torture

Jusus Christ

[deep mango]

Good job being done with it!

[bright hill]

I'm just glad I finished

Most of my classmates didn't

[neat lintel]

Btw how can one calculate the contour C here?

[rose dock]

it's given as a set and you'll have to parameterize it

[toxic gale]

Story of my life.

[neat lintel]

So a Contour C is a set of all mapped z values in this case, say for example {2,3,5,-2,6,8} from a = 2 to b = 8 (direction of the contour is from a to b)

Then we need to parameterize this, meaning we need to find a function that produces these values.
So p(t) = a, where a in {2,3,5,-2,6,8}
So by plotting these points we can deduce the function p(t) such that when t = 0, p(t) = 2; t = 1, p(t) = 3, and so on?
That's how we find the contour C?

[toxic gale]

You just need a smooth parametrization. The neat thing is that the integral doesn't actually depend on the choice of parametrization. (If I recall correctly.) Usually they'll give you relatively simple curves to parametrize like straight lines, parabolas, portions of circles, etc.

For a line segment between A and B we can just use r(t) = tA + (1 - t)B. For circles we have x = cos t, y = sin t and so on.

[neat lintel]

Ah I see so we use the already existing smooth parameterizations

But how can we frame one though by ourselves

Out of scratch

[toxic gale]

Pretty much. I highly doubt you'll run across a weird enough curve.

I mean, probably depends. If the curve in the xy-plane is a function of x you could probably fit it to a polynomial p(x). Then x = t and y = p(t) is a parametrization.

We can also just break up the curve into different pieces and find parametrizations for each piece and add up the resulting integrals.

[neat lintel]

Hmm I see

Thanks o7

[cyan goblet]

[errant ridge]

Lmao

[outer vector]

i did something cool

it makes a straight line at first glace

BUT IF YOU ZOOM IN TO LIKE 2x10^-8

HELL YEA

TOOK ME TOO LONG

[neat frost]

Why don’t you take like sin(x) and multiply it by 2x10^-8

[ancient flame]

Haha.

Slurp, you are a comedic individual.

[fervent pebble]

didn't desmost make that

the tiktok guy

[neat frost]

I’m so confused tho

Why

And also unnecessary parentheses

Oh

The function is just y=1

Doi

[cyan goblet]

so critical slurp

:(

[neat frost]

:D

[cyan goblet]

anyways i secured a spot in my uni’s lin alg 2 class

which is dope

which means i can drop my summer lin alg course since i wont need the credit

[neat frost]

Ayyyy

Uh

Ez 100 then tho if you know the material. Good for your average.

[cyan goblet]

it doesnt affect my gpa

[neat frost]

O idk how it works in the states

[cyan goblet]

well it’s cos im taking this thru a community college

i think that’s how ti works anyways idk tbh

[neat frost]

Ohh

Cumontity college

[cyan goblet]

:(

[neat frost]

[neat lintel]

Does a convergent sequence HAVE TO have a start? Meaning it starts somewhere. Like starts from 0 or 1.

[neat lintel]

any sequence has an initial input by definition

[azure nymph]

Just finished my calc 2 exam I either got a C or an A 💀

[neat lintel]

how did you practice for

integration?

i find it hard

[azure nymph]

Nah the integration wasn't the problem for practice I just do mock exams for me it was this torus question and some work questions that we never went over

[neat lintel]

torus?

are you doing hs calc or calc in uni

[azure nymph]

University

[neat lintel]

oh

@azure nymph did you solve a lot of integrartion problems from the texxtbook

[azure nymph]

Not really I just did ones we were assigned mainly

Is there a certain kind of integration you struggle with

[neat lintel]

yeah problems with square roots in the denominators

i find them confusing

1/(1-x^2)^.5

[azure nymph]

Well that's a standard integral so for me it's just one I have memorized when I see that I think of arcsin

[neat lintel]

yeah I think for some i just have to memorize

but i dont know which ones

[azure nymph]

Well ideally you should know all of your derivatives then use those to integrate cause if you know the derivative arcsin then you'll see $\int{\frac{1}{\sqrt{1-x^2}}$ and be like okay this is the derivative of arcsin so it's integral is arcsin

[fathom swallow]

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

[fervent pebble]

esp have arctan memorized

it came up a lot for me

[azure nymph]

Same because it is my professor's favorite

[hexed grail]

Hi, I have been studying some basic probability theory and in the process I have found out few bits about measure theory. I would like to ask: are there other applications of measure theory other than rigorous probability theory?

[deep mango]

Yes! Plenty. Ergodic theory (measure-theoretic dynamical systems) is super interesting and important, and has ties to probability but also to differential geometry, number theory, and plenty of other areas of math. Functional analysis builds off of measure theory in many ways to develop the math which underlies quantum mechanics and partial differential equations, among other types of math. Measure theory also comes up if you're studying descriptive set theory (I don't know much about that though)

[ancient flame]

Hello, RYC. That is a very interesting set of information that you provided.

[hexed grail]

Thank you, that sounds very useful indeed

[cyan goblet]

hi ricey

[hexed grail]

(I am a CS student, so I know more about discrete math and algebra than analysis)

[cyan goblet]

what are you 🤨 me for

[deep mango]

Trying to think if i'm missing important fields

[cold needle]

ricey

[deep mango]

When you study topological groups

You study haar measures on them

This is useful for Lie theory

uhhh

[cyan goblet]

aka ricky

[cold needle]

aka arr why see - ee

[cyan goblet]

richy

icci

ricc

ricci

[deep mango]

🤨

[ancient flame]

Excuse me. I would like to inquire about the individuals who attached an interesting emoji to my prior message.

[ancient flame]

This is absurd and completely unnecessary. I would like to respectfully ask you to stop doing so.

I do not have a question at the moment, Mr. RYC. However, thank you very much for the tip.

[deep mango]

You've reminded me of Toucan

[ancient flame]

I do not understand the reference.

[alpine kindle]

"hey does anyone here know functions"

[burnt dune]

what language is this Gräv ner livet, salta jorden, förtryck längd & breddgrad

[neat lintel]

swedish

[ancient flame]

Google Translate can help you detect the language.

[alpine kindle]

swedish

[burnt dune]

god

those metal swedes

[neat lintel]

snakker du svensk?

[burnt dune]

will never rest

thank you

[burnt dune]

what

[neat lintel]

nvm

[burnt dune]

check out

stort neer

stoort neer*

thall vildhjarta like metal band

[neat lintel]

dude I love viking metal

[ancient flame]

I see that other individuals have answered your inquiry. However, my prior tip could be of use to you in future situations, so please keep it in mind.

[burnt dune]

its like vilhdjarta

if u know them

[neat lintel]

I've been listening to a lot of koorpiklani

[cyan goblet]

like the one from froot loops

[burnt dune]

hwere they are like messhugah

[deep mango]

No

[neat lintel]

and finntroll

[cyan goblet]

toucan play this game

[burnt dune]

so like its like stoort neer --> vildhjarta --> messhugah

[deep mango]

toucan was a user

[cyan goblet]

caw caw

or whatever noise toucans make

[surreal sapphire]

today i learned that people listen to korpiklaani and finntroll

[neat lintel]

Y E S

[surreal sapphire]

i saw korpiklaani live twice i think and thats super cool

[neat lintel]

one of my favorite songs by korpiklaani is tervaskanto

[surreal sapphire]

but i wouldnt put them on at home

[burnt dune]

i never put metal at home

nor at the car

[neat lintel]

or maybe lempo

[neat lintel]

in finland?

[surreal sapphire]

i dont know a single song by name

no, in germany

[neat lintel]

yeah I like death metal, viking metal, power metal, and some metalcore

[surreal sapphire]

i think we have the largest medieval fair in europe

[neat lintel]

I listen to it all day long bruv

germany has some dope metal bands, like Helloween from the 80's-90's

[surreal sapphire]

i think they played the pirate ship

which is the place where the strongest alcohol is

germany has quite a good metal scene

today its better

lots of tech death

[neat lintel]

I also like listening to taiwanese folk metal too

[bronze pelican]

I'm a GRADUATE STUDENT

[surreal sapphire]

[bronze pelican]

I go to CONFERENCES

[neat lintel]

oh. my. gosh.

[bronze pelican]

◉◡◉

[surreal sapphire]

best german metal bands are ahab, alkaloid and obscura

[bronze pelican]

Alkaloid

Sounds like VOCALOID

[surreal sapphire]

lol

its some chemical compound

[neat lintel]

I'll check those out

[surreal sapphire]

i wanted to go see ahab live (again) soon

but its some weird festival that is like 90% hip hop

[neat lintel]

Wow obscura is awesome

[surreal sapphire]

and then randomly ahab, which is funeral doom

so weird

[neat lintel]

loch have you listened to GOJIRA

[surreal sapphire]

ye i know gojira

[neat lintel]

literally one of my favorite bands, they're great

[surreal sapphire]

L'enfant sauvage is probably one of my favorite albums

[neat lintel]

wish I could see them live

[surreal sapphire]

i saw them in 2018 i think

[neat lintel]

Do you listen to metal while doing math loch

[surreal sapphire]

rarely

[neat lintel]

magma is my favorite, or maybe way of all flesh

really? That's interesting I almost always listen to metal while doing math

[surreal sapphire]

i need silence

also the other people in the office wouldnt appreciate

[neat lintel]

yeah lol

it just helps me tunnel focus

like I am listening to metal right now as I type lol

[surreal sapphire]

i focus more on the music

well, i am listening to music now because im done working

setting up email to be sent to my advisor in 8 hours because its 1 am and i need to keep up appearances

[neat lintel]

lol

[burnt dune]

complex analysis fucking ssuckjs

such a scam

[deep mango]

Spill

[bronze pelican]

Holy shit

People born in 2000 are now 20, 21 years old

🤯

[deep mango]

some even 22

[azure nymph]

Hi I am 22 and born in 2000

I know a person who is in med school now too if it makes you feel even older

[sleek wing]

I would never be 22 and born in 2000

[deep mango]

Wew isn't asleep

[sleek wing]

it's quarter past chicken time ofc it's not bed time yet

[next schooner]

chicken time

[deep mango]

What is chicken time anyway

[sleek wing]

1:30am

chicken time

[next schooner]

o

[ancient flame]

Excuse me Mr. Wew, why is 1:30 A.M. "chicken time?"

[next schooner]

gmod why are you chatting like a nerd today

what's got into you

[deep mango]

🤓

[ancient flame]

Please do not refer to me as a nerd.

[next schooner]

aight nerd

[ancient flame]

I am typing normally today.

[sleek wing]

I don't really understand the question

[deep mango]

I think he's been hacked

[ancient flame]

Excuse me Mr. RYC, why did you attach a duck reaction to my message?

[next schooner]

Is this like that one house episode where that man has amnesia and mimics other people he sees

but instead of imprinting on house you imprinted on CV?

[ancient flame]

Mr. RYC, I have not been hacked.

[deep mango]

I'm a wacky dude! 😵

[neat lintel]

hi guys im currently doing 2 years of homeworks rn

[sleek wing]

too quirky 4 ur own good... MODDSSSSS

[neat lintel]

@darkk

@glass mauve why u here

no

[ancient flame]

That is not appropriate for this channel, Mr. Dakkaro.

[neat lintel]

:sob

[neat lintel]

lol

#memes

nvm

[ancient flame]

Please post those videos somewhere else, such as in the #chill channel.

[sleek wing]

gmod stop talking like a moron

[neat lintel]

lmao

[ancient flame]

Please do not insult me.

[deep mango]

Please don't call respectable server users "morons"

[neat lintel]

why delete

[glass mauve]

Bruh

[deep mango]

[ancient flame]

I agree with Mr. RYC.

[sleek wing]

you bumbling fool I am beyond control... I am ZANY!!

[neat lintel]

damn are y'all teachers

[deep mango]

Please keep videos and shit to #chill (and preferably don't post unrelated nonsense)

[neat lintel]

help me with math homework 🥺

[sleek wing]

there are two PhD students and gmod in chat rn

[ancient flame]

I am not a teacher.

[neat lintel]

wtf

[next schooner]

have you read or considered #❓how-to-get-help

[neat lintel]

where

[sleek wing]

make that 3 if rat is here

[neat lintel]

im phd at gaming only

[glass mauve]

1+1=1

[next schooner]

i am not here

[sleek wing]

ok then it's just 2

[deep mango]

did you just join to post nonsense

[glass mauve]

certainly not

[deep mango]

your only posts are 2 videos and this

[sleek wing]

yeah I can't even make a funny ring joke out of that post

[ancient flame]

It would not be of comedic value, Mr. Wew.

[sleek wing]

Sir to you

[next schooner]

Lord Patrician Gmod I must politely insist that you let down this facade at once

[glass mauve]

but i need to actually learn the multiplikation like system until after summer

[empty stratus]

Eh?

[neat lintel]

🥺

[glass mauve]

I never learned it inside out

[next schooner]

oh hi CV

[empty stratus]

Hi!

[sleek wing]

Hark

[glass mauve]

any tips to do so

Over

The

Summer

[next schooner]

gmod is typing his posts with punctuation now and I can only assume that you did something to him

[ancient flame]

Good evening, Mr. ComplexVariable. How are you doing?

[dense belfry]

It's the top hat

[empty stratus]

Nicely. Today's writing went... interestingly.

[dense belfry]

gmod is a proper gentlesully

[empty stratus]

Exerted a positive influence, it sounds like! 😄

[ancient flame]

That is amazing to hear. Keep working, sir!

[next schooner]

oh it's the HAT

right of course

[glass mauve]

How do i learn the multiplication until after the summer

[empty stratus]

Tomorrow's writing is going to get quite boody.

[deep mango]

How does a hat

[ancient flame]

Hello there, Emma! How are you?

[sleek wing]

ryc it's OBVIOUS

[deep mango]

just change someone's whole personality like this

[glass mauve]

inside

[next schooner]

it's a top hat ryc

[glass mauve]

Out

[ancient flame]

I have had the hat for several days.

[empty stratus]

Join a religion, for one.

[glass mauve]

i just need a quick answer

from someone

[next schooner]

#❓how-to-get-help

[empty stratus]

What kind of multiplication?

[sleek wing]

one you shall not recieve

[glass mauve]

like the normal

[empty stratus]

For that, you should go to one of the help channels.

[ancient flame]

Please visit #❓how-to-get-help for more information, Mr. Dakkaro.

[empty stratus]

^^

[glass mauve]

any websites to learn every one of them inside out

[ancient flame]

Khan Academy is a reputable resource.

[empty stratus]

^^

[next schooner]

try quora

[sleek wing]

Inside Out (2015) is OVERRATED

[empty stratus]

I thought it was touching.

[ancient flame]

Quora is likely not the best resource for this sort of topic.

[next schooner]

actually this might help https://ncatlab.org/nlab/show/multiplication

[empty stratus]

Though I recall that I took qualm with the climactic line of dialogue. I would have phrased it slightly differently than what the screenplay ended up having the character say, and the change would have done wonders.

[sleek wing]

https://media.discordapp.net/attachments/455088502548463627/981262420012073080/D1D7.gif

[glass mauve]

Thanks

[empty stratus]

NO! IT'S A TRAP!

[Insert Admiral Ackbar meme]

[next schooner]

remember that a monoid is just a monoid in the category of monoids

[glass mauve]

I didnt

Open the link

lol

is it rly a trap tho?

[empty stratus]

YES

[empty stratus]

Doesn't this category require the axiom of choice?

[next schooner]

do i look like i care about foundations to you

[primal seal]

Lol

[fervent pebble]

he's a rat which means he plays killjoy which means yes u do

[empty stratus]

You show no indication of a stance one way or the other.

[glass mauve]

Goodnight sry for sending memes in #serious-discussion

[deep mango]

@sleek wing I made it to the grey world in Glyph

😭

[next schooner]

i think that any and all adults are welcome to choose whichever axiomatic formalism they like

[deep mango]

I'm on a screen with a bunch of chained wavedashes and falling conveyor belts and it's truly horrid

woog watched me playing it for a while

[next schooner]

minors however should not be allowed to play with choice

[fervent pebble]

what consenting adults do in their math papers is none of other ppls business 👍

[next schooner]

arguably they should be restricted to ultrafinitism but this is controversial

[fervent pebble]

as it should be

personally im pro-choice

[glass mauve]

i have no idea about half the stuff you guys are saying

[next schooner]

i'm pro-choice-choice

[dense belfry]

People should realize how much modern constructive math is entwined with category theory

[glass mauve]

Are you guys going to be math proffesors when you get older or

[dense belfry]

Yeah

Hopefully

[empty stratus]

I'm applying for several positions at the moment, yes.

Though my current priority is finishing my novel.

[next schooner]

I'm leaving academia after my phd

smell you later

[glass mauve]

So you earn money by mastering math

[empty stratus]

Dunno.

[dense belfry]

I think I may dip my toes into more general logic than just mathematical logic

At some point

[bronze pelican]

I'm gonna be a MATH PRPFESSOR

[dense belfry]

idk

[next schooner]

i'm gonna teach high school

[dense belfry]

Nice

I can see you there Josh

[next schooner]

Yeah I like teaching

Plus my wife and I want a child soon, so postdoc hell is probably not a good choice

[dense belfry]

Makes sense

[neat lintel]

What does that look like in practice? Philosophy?

[dense belfry]

Yes

[teal acorn]

What is explicit description and implicit description of the solution sets of linear system? (Linear algebra)

[bronze pelican]

@dense belfry do u know about p-automatic sequences

[dense belfry]

No I don't

[empty stratus]

I've heard of them.

[dense belfry]

Is that an automata thing?

[empty stratus]

If it's what I'm thinking of, then yes.

[dense belfry]

I don't do automata theory

My roommate does dynamics and actually she works with automata a lot more than me

[empty stratus]

https://oeis.org/A003842/a003842.pdf

Though my interest in them was due to their connection to transcendental number theory, and from there to Collatz.

[dense belfry]

Of course

[empty stratus]

And Mahler Theory.

(Transcendental number theory using holomorphic functions satisfying certain functional equations)

[bronze pelican]

Ah

[dense belfry]

Why do people have such bad foundations takes

Especially about choice

Smh

[bronze pelican]

Yeah the Thue Morse sequence is a 2-automaruc sequence

[empty stratus]

My original approach to Collatz-type problems was to reformulate them in terms of fixed points of linear operators of spaces of functions on the open unit disk.

[dense belfry]

Yeah she likes that stuff pt

[empty stratus]

And automatic sequences can be reformulated in terms of analytic generating functions using the so-called Cartier operators.

[neat lintel]

ComplexVariable ELI5 your research on the collatz conjecture

[bronze pelican]

Collatz research can be safely disregarded as crank work so don't worry about it 🙃

[dense belfry]

People not realizing you can literally just study choice principles in general

And look at all of them at once

[bronze pelican]

Lol

[empty stratus]

1. Given a Collatz-type map H : Z^d —> Z^d, I found a function from the p-adics to the q-adics (for distinct primes p & q) whose image completely characterizes the periodic points of H and, conjecturally, completely characterizes the divergent points of H.

2. You can reformulate Collatz-type conjectures as eigenvalue problems in terms of the aforementioned function.

Specifically, you want to study the values x for which the (p,q)-adic function f(z) - x has a non-singular reciprocal.

[bronze pelican]

I met someone who said they are hesitant about the axiom of choice and so didn't want to work with profinite groups like Gal(Q-bar/Q)

[neat lintel]

So how does that tie back to the conjecture

[empty stratus]

It reformulates it.

[dense belfry]

Giving up LEM gives you so much more leeway

[empty stratus]

Have you heard of Wiener's Tauberian Theorem?

[neat lintel]

No

[empty stratus]

It's a chimerical result.

[dense belfry]

But you can still prove things using choice

[empty stratus]

Do you know what a Banach algebra is?

[dense belfry]

You just need to state that you are using it as an assumption

[neat lintel]

No

[dense belfry]

It is the algebra that captures continuous logic

Yes

[neat lintel]

What does this mean?

[dense belfry]

Lol

[empty stratus]

It has many different forms.

[bronze pelican]

[empty stratus]

Do you know of the Fourier transform?

[neat lintel]

So you’re proving it in a different form?

[neat lintel]

Yes I know that one

[bronze pelican]

Fourier analysis on number fields

[empty stratus]

Great.

Do you know what L^1 space is?

[neat lintel]

No but go ahead

[empty stratus]

It's functions f:R—>C such that |f(x)| is integrable over R.

[neat lintel]

Ok

[empty stratus]

This forms a metric space using the metric:

(f,g) —> integral of |f(x) - g(x)|dx

Let f:R—>C be integrable.

A translate of f is a function of the form x —> f(x + a), where a is a real constant.

[neat lintel]

So what does this do in regards to the problem

[empty stratus]

I'm getting to it.

[near temple]

[empty stratus]

I'm very pedantic, and explain all details.

So, just be patient. 😄

Do you know what linear combinations are?

[neat lintel]

Yes

Very well

[empty stratus]

Do you know what it means for a set to be dense in a metric space?

[neat lintel]

No I’m currently reading about them tho

[empty stratus]

It means you can approximate elements of the space to arbitrary accuracy using the elements of the dense set.

The quintessential example is the set of rational numbers, which is dense in the reals.

[neat lintel]

Oh yeah

Ok

[empty stratus]

Let f:R—>C be integrable.

[neat lintel]

I know about sets being dense just not in metric spaces

[empty stratus]

And let T be the set of all translates of f (so, every element of T is of the form x —> f(x+a) for some real number a).

The L^1 version of Wiener's Tauberian Theorem describes necessary and sufficient conditions for T to be dense in L^1, meaning that, for any integrable function g:R—>C, you can choose a function h which is a linear combination of elements of T so that the integral of |g - h| can be made arbitrarily small.

Specifically, the WTT says that T is dense in L^1 if and only if there are no real numbers at which the Fourier transform of f vanishes.

Do you know what the Fourier transform does to convolutions of functions?

[neat lintel]

I have a vague idea

[empty stratus]

It turns convolution into point-wise multiplication.

So, the Fourier transform of f * g is the product of the Fourier transform of f and the Fourier transform of g.

[neat lintel]

Ok

[empty stratus]

Likewise, the Fourier transform of the product of f and g is the convolution of the Fourier transforms of f and g.

[teal acorn]

What is explicit description and implicit description of the solution sets of linear system? (Linear algebra)

[modest rune]

#❓how-to-get-help

[empty stratus]

A very important inequality (Young's Convolution Inequality) tells us that given two integrable functions f and g, their convolution f * g will also be integrable.

[teal acorn]

I asked bro like 20 mins ago and ping helpers 2 times too but no one came! 😞

[empty stratus]

In the algebraic sense, convolution defines a multiplication operation on L^1.

The Fourier transform turns convolution into point-wise multiplication.

What's important is that the Fourier transform is injective.

[empty stratus]

Do you know what the Dirac Delta function is?

[neat lintel]

YES

I love that thing so much

[empty stratus]

The dirac delta is the identity element of the convolution operation.

[neat lintel]

Ohhhhhh

[empty stratus]

∂ * f = f • ∂ = f

The Fourier transform of ∂, meanwhile, is the constant function 1.

The reverse is also true.

[neat lintel]

I use the DDF for computational neuroscience stuff

[empty stratus]

1's Fourier transform is ∂ (in the sense of distributions)

Given an integrable function f, I say that f has a convolution inverse if there is a function g so that f * g = ∂.

[modest rune]

That is unfortunate but does not excuse asking the question here. Sometimes questions do not get answered. There is a conversation going on here and I would rather not keep interrupting it.

[empty stratus]

Because the Fourier transform turns convolution into multiplication, observe that the Fourier transform of f will be equal to the reciprocal of the Fourier transform of g whenever f * g = ∂.

In other words, a function has a nice convolution inverse if and only if its Fourier transform has a nice reciprocal.

This fact is equivalent to the Wiener Tauberian Theorem.

[dense belfry]

talks about being pedantic
talks about two theorems being equivalent despite the fact that every two theorems are equivalent

[empty stratus]

...

[dense belfry]

Lol

[empty stratus]

One of the topics analysts like to study is value distribution theory.

[neat lintel]

Like prime gaps?

[empty stratus]

Not quite.

[neat lintel]

Sorry I didn’t see what you quoted

[empty stratus]

It's the study of the images of functions.

In subjects like complex analysis, you can say a surprising amount of stuff about a function if you know that a certain number of complex number are not in the function's range.

[neat lintel]

Yeah I’m computational neuroscience we use the DDF to mimic a neural impulse

[empty stratus]

One of the most powerful toolkits for studying value distributions is Nevanlinna Theory.

This exploits the properties of integration in the complex plane (such as Jensen's Formula) to count the number of zeroes that a function has.

[neat lintel]

Ok

[empty stratus]

In my research, as I said, I showed that given a Collatz-type map H, there is a certain function you can construct—I call it Chi_H—such that the range of Chi_H completely determines the periodic points of H.

In other words, an integer n gets iterated back to itself by H if and only if there is some input z so that Chi_H(z) = n.

[neat lintel]

Whoa

But what’s a collatz map?

Like a functional mapping

[empty stratus]

Do you know what the Collatz Conjecture is?

[neat lintel]

I know the elementary phrasing

[empty stratus]

(It's also known as the 3x+1 Problem)

The kinds of maps I study are of the flavor: fix an integer p ≥ 2. For each integer j in {0,1,2,...,p-1}, define an affine linear map H_j(x) = (a_j x + b_j) / d_j

Then, define a map H:Z—>Z by H(n) = H_j(n), where j is the value of n mod p.

I call this a Hydra map.

They are also known as residue-class-wise affine maps, but that's a horrible, horrible name, so I don't use it.

xD

You can also study generalizations of these maps on Z^d (or, equivalently, on a d-dimensional ring extension of Z).

ANYHOW...

We say n is a periodic point of H if the sequence n,H(n), H(H(n)), ... is periodic.

(That is, there is an integer k ≥ 1 so that if you apply H to n exactly k times, the output will be n)

We say n is a divergent point of H if the sequence n,H(n), H(H(n)),... tends to positive or negative ∞.

By "Collatz-type conjectures"/"Collatz-type problem"/"the dynamics of H", I mean, "determining the periodic points and divergent points of a given H".

[neat lintel]

So you are trying to show that H is always convergent?

[empty stratus]

My methods show that given H, the dynamics of H are determined by the whole numbers (if any) which are outputted by the function Chi_H.

[empty stratus]

Nope. I'm just trying to find out what the periodic points of H are and what the divergent points of H are.

[neat lintel]

Ohhhhhh

[empty stratus]

In this terminology, the Collatz Conjecture is the assertion that the 3x+1 map has no divergent points in the positive integers, and that 1,2,4 are the only periodic points of the 3x+1 map in the positive integers.

[neat lintel]

OK

[empty stratus]

So, my methods show that given H, the dynamics of H are determined by the whole numbers (if any) which are outputted by the function Chi_H.

Thus, instead of studying H directly, to determine H's dynamics, we just need to understand Chi_H.

[neat lintel]

What’s Chi_H?

[empty stratus]

A function I can construct.

[shrewd bobcat]

damn man you type really quickly

[empty stratus]

I have a certain recipe which, given any H, tells us how to construct Chi_H.

[empty stratus]

Thank you. 🙂

[neat lintel]

So is that a big result?

[empty stratus]

It's just part of the whole.

A big result is the fact that Chi_H's output values determine the dynamics of H.

I call this the Correspondence Principle (CP).

The CP tells us, for example, that the problems:

1. Determine the periodic points of H

2. Determine the whole number values that Chi_H(z) takes as z varies over the set of rational numbers

are equivalent.

The problem is that Chi_H is a rather strange function. For any suitably well-behaved H, there are distinct primes p and q (which depend on H) so that Chi_H is a function from the p-adic integers to the q-adic integers.

(or, as I like to call it, a (p,q)-adic function)

Because (p,q)-adic functions behave very strangely, standard tools for studying their value distributions don't apply.

However, there is one method that we can use.

Consider a function f. If we want to show that a number w lies in the range of f, we need only show that the function f(z) - w vanishes for some input z.

Right?

[neat lintel]

Right

[empty stratus]

On the other hand, if f(z) - w is always non-zero, then f(z) ≠ w for any z.

Now, consider the reciprocals.

w is in the range of f if and only if function 1/(f(z) - w) blows up to ∞ at some z.

If f is, say, continuous, then 1/(f(z) - w) is continuous if and only if w is not in the range of f.

For example, because x^2 + 1 never vanishes for real x, its reciprocal 1/(x^2 + 1) is also continuous.

On the other hand, x - 1 vanishes for x = 1, so its reciprocal 1/(x-1) is discontinuous, having a singularity at x = 1.

Here's where Wiener's Tauberian Theorem comes into play.

Let f be a continuous (p,q)-adic function, and let g(z) = f(z) - w, where w is fixed.

Then, letting G be the Fourier transform of g, the following are equivalent:

1. 1/g is continuous
2. G has a convolution inverse

Because the Fourier transform turns multiplication of functions into convolution (and vice-versa), to study whether or not a function has a well-behaved reciprocal, it suffices to study whether that function's Fourier transform has a convolution inverse.

[hybrid quest]

Wiener

[empty stratus]

In this context, the periodic points of H are precisely those whole numbers x so that function Chi_H(z) - x vanishes.

As such, we have the equivalences:

1. x is a periodic point of H;

2. Chi_H(z) - x vanishes for some z;

3. 1/(Chi_H(z) - x) has a singularity at some z;

4. The Fourier transform of Chi_H(z) - x does NOT have a convolution inverse.

[hybrid quest]

Oh shiit this is related to collatz

[empty stratus]

Yes.

It reformulates Collatz and Collatz-type conjectures on finite dimension lattices in terms of non-archimedean spectral theory.

(i.e., eigenvalue problems)

The point of all this is that, instead of studying the Collatz-type map H, you can study the Fourier transform of Chi_H.

[neat lintel]

Gotcha

[empty stratus]

Not only does this give a reformulation of Collatz-type conjectures...

It also allows us to study these conjectures systematically in a unified way.

For example, if you consider the 5x+1 map (which is Collatz, but with the 3 replaced by a 5), it can be shown that almost every positive integer should be a divergent point.

Moreover, this result holds for the qx+1 map, where q is any odd integer ≥ 5.

If you consider the Fourier transforms of their Chi_Hs, you end up with the following result:

Chi_3 (the one associate to Collatz) is different from Chi_q (the one associated to qx+1) for all q ≥ 5.

For q ≥ 5, Chi_q's Fourier transform has a q-adic magnitude of 1 at every input.

For q = 3, though, Chi_q gets arbitrarily close to 0 in q-adic absolute value infinitely often.

This is a "smoking gun": it's a concrete, non-heuristic, non-probabilistic distinction between 3x+1 and the other qx+1 maps.

Moreover, there is a simple map due to K.R. Matthews which can be easily demonstrated to have divergent points.

It's Chi_H's Fourier transform is bounded away from zero.

This suggests that "Chi_H's Fourier transform is bounded away from zero" is needed in order for H to have divergent points.

[neat lintel]

So what is going on with q=3? Why does it have this different behavior from all other q?

are you still working on this problem CV, or have you moved on since your thesis?

[devout nacelle]

What can you say when Chi_H's Fourier transform is not bounded away from zero? You said for the case q=3, Chi_q gets close to 0 infinitely often in the q-adic norm, is this necessary (or even sufficient) to guarantee the existence of periodic points?

[empty stratus]

I'm still working on it, though not at the moment.

[tawdry smelt]

2 + 2 = 11.5

[empty stratus]

I'm currently trying to publish the various pieces of my dissertation.

[devout nacelle]

@tawdry smelt It wasn't funny the first time you said it. No need to go around spamming it.

[empty stratus]

That remains to be seen.

[tawdry smelt]

I'm not being funny.

I'm being Legit lol

[devout nacelle]

No need to post this across channels without any context.

[empty stratus]

Ultimately, what it boils down to is studying the density properties of the translates of Chi_H-hat - x1, where 1 is the indicator function for zero.

[neat lintel]

So that's essentially the next step?

[empty stratus]

Yes.

Establishing the existence or non-existence of divergent points and periodic points boils down to establishing bounds on the q-adic absolute value of linear combinations of translates of Chi_H-hat - x1 in terms of x.

[neat lintel]

Do you think the collatz conjecture will ever be solved CV?

[empty stratus]

Yes.

[neat lintel]

interesting take

why?

[empty stratus]

If human civilization gets extinguished before we solve it, I'm pretty confident aliens will eventually discover it and puzzle over it, themselves. xD

More seriously...

The (p,q)-adic analysis I discovered/invented in the course of doing this research just works too well for Collatz-type problems.

I'd be very surprised if this was an accident.

Also, my methods work for Collatz-type maps on Z^d.

For arbitrary integers d ≥ 1.

It works for ALL OF THEM!

[devout nacelle]

Do you have an informal heuristic for why (p,q)-adic function setup is relevant in problems concerning iteration maps on integers? I'm only aware this is the kind of stuff people within arithmetic dynamics study, but don't know anything about it.

[empty stratus]

It's not about maps on the integers per-se.

Rather, it's due to the affine linear structure of the maps.

Long story short, you construct Chi_H by considering the monoid generated by composition of the branches of a given Collatz-type map H.

[devout nacelle]

[empty stratus]

For example, for the shortened qx+1 map (sends even n to n/2 and odd n to (qn+1)/2)...

[neat lintel]

what would an example of Chi_H look like?

[empty stratus]

Chi_q is completely characterized by the following functional equations:

f(2n) = f(n)/2

f(2n+1) = (qf(n) + 1)/2

Subject to the condition that as n tends to a 2-adic limit z, where z has infinitely many 2-adic digits, f(n) converges to f(z) in the q-adic topology.

[neat lintel]

how many people are working on this problem at the moment at a professional level CV?

[empty stratus]

On Collatz? A couple.

On my particular approach? 1.

[neat lintel]

what are the other approaches?

are the others aware of this method now?

[empty stratus]

No, but that's why I'm trying to get myself published.

[neat lintel]

what does that process entail?

Like do you have to submit it to a journal?

[empty stratus]

Writing up papers and submitting them for publication.

Attending conferences.

[empty stratus]

Yep.

[empty stratus]

Very depressing.
Most people ONLY study the 3x+1 map.

[neat lintel]

Do you think it is going to get published?

[empty stratus]

I hope so!

[empty stratus]

And, for those that DO study the maps in greater generality, the methods used are almost always probabilistic in nature.

[neat lintel]

huh that seems inefficient after hearing your explanation

[empty stratus]

EXACTLY!

All of the most important studies in 3x+1 have been probabilistic in nature.

My driving concern was to find an approach that allows us to study a large family of Collatz-type maps SIMULTANEOUSLY.

One of the reasons why Collatz studies have lingered in shame in the mathematical ghetto is that literally anyone with at least an elementary school education can investigate the problem and write up patterns that they notice in it.

The problem is: how do you know which patterns actually matter?

[neat lintel]

so it's kind of frowned upon to research it?

[empty stratus]

Answer: by comparing them across many different maps

[empty stratus]

Yes, and for three reasons:

1. The fact that it draws so many cranks and amateurs.
2. Its extraordinary difficulty.
3. Its apparent isolation from other areas of mathematics.

[neat lintel]

Did your advisor try to steer you away from the problem because of those reasons?

[empty stratus]

Yes, and also because my advisors weren't knowledgeable in the tools and subjects I drew from for my research.

These included: analytic number theory, p-adic analysis, non-archimedean analysis, and harmonic analysis.

No one at my university knew any of these things at an expert level.

[neat lintel]

So how did you learn all that stuff?

[empty stratus]

On my own.

[neat lintel]

Books? Conferences?

[empty stratus]

Books.

[azure nymph]

Wow no one knew about harmonic analysis?

[empty stratus]

Yep.

Our analysis department was all PDEs and SDEs.

[neat lintel]

I thought you were at a top school though

[empty stratus]

No.

University of Southern California.

[azure nymph]

I think there's at least 3 professors at my school who do harmonic analysis, we don't have a phd program though