Collatz-like questions

[chrome grove]

Suppose we have a polynomial g(x) and polynomials f1(x), f2(x), ..., fk(x).
If a number can be factorised into c * fi(a) where a and c are natural, then the next term is c * a.
Otherwise apply g(x) (which is non-decreasing)

I am curious if it's possible to prove that there exist such g, and fi that for every starting value the sequence does not grow without limit

@golden plover @placid ice

Now, I am aware that the problem needs to be refined to become interesting and omit trivial or ambiguous cases

[placid ice]

it'd be great if you provided an example

[chrome grove]

I just thought of it so one sec

I mean this is a generalisation of Collatz so you could take g(x) = 3x + 1 and f(x) = 2x

(and k = 1)

[golden plover]

decipher these variables rn within a minute based off of context

[chrome grove]

so obviously QM

[golden plover]

ok

[chrome grove]

one sec

hm

A is amplitude

[golden plover]

ok

[chrome grove]

e is the phase component

[golden plover]

🤨

[chrome grove]

e^ix

[golden plover]

🥴

[chrome grove]

"goes spinny"

[golden plover]

🤨

times up

[chrome grove]

do you speak

[golden plover]

you were close

phi h and v are the phase difference between the horizontal and vertical components and a rest, k=omega/c and z is your path difference

Ah and v are the horizontal and vertical amplitude

omega being angular frequency xx

now return that to i and j form

non complex

[chrome grove]

base unit vectors

^^

[golden plover]

ok...

continue

[chrome grove]

this is very exotic indeed

having a test on a course I have not seen

XD

[golden plover]

go on

no all these questions can be deduced from your knowledge of maths

and what's in front of you

[chrome grove]

honestly I am not sure why there are basis vectors if you already have complex values

[golden plover]

it's the polarisation state

has your state at any point in t

vertically and horizontally

[chrome grove]

wait

phis are real

got it (?)

[golden plover]

not quite but you got the idea

now

non complex

rn

[chrome grove]

non complex what

what do you waaaaaaaaaant

[golden plover]

the non complex formula

isn't using e^iphi

[chrome grove]

replace with cos or sin

[golden plover]

which

[chrome grove]

e^(ix) = cos(x) +i * sin(x)

I believe

[golden plover]

go on

[chrome grove]

I am actually unsure how Re(..) acts on vectors

it's just Re(z * v) = Re(z) * v ?

[golden plover]

(3+2i)i=(3)i

[opal parrot]

Well, seems to be a wavefunction some sort. Perhaps, some particles, considering the wave term?

[golden plover]

getting very close

[chrome grove]

photons

polarised

[opal parrot]

Ah, polarization...

Not something I can deal with very well 😅

[golden plover]

tryna get flr to realise that Re(cosx+isinx)=cosx

[opal parrot]

Oh, ok.

[chrome grove]

I get that

thanks

[golden plover]

so

[chrome grove]

but there is a different term

[golden plover]

now you have everything necessary

in what way??

[chrome grove]

e^(ikz-iomegat)

it's inside Re

[opal parrot]

Oh, that's the wave.

[golden plover]

the imaginary part of a+bi is closed under addition

so you can split it up

[chrome grove]

but it's multiplication

[golden plover]

try simplifying down your e's first

then distribute and simplify

[chrome grove]

honestly not sure if I can be bothered rn

=_=

[golden plover]

ok, you want the answer then?

[chrome grove]

I have a lot of enthusiasm towards new things and what people are passionate about

esp QM

but it feels like a lot of grunt work for a simple result without the theoretical meaning

which feels empty

[golden plover]

E(z,t)=iA_h(cos(kz+phih-omegat))+jA_v(cos(kz+phiv-omegat))

[chrome grove]

kewl

E is..?

Energy?

[golden plover]

it's your polarisation state vector at a point in time and in space

[chrome grove]

ah

[golden plover]

it's used in calculating polarisation patterns

so whether it's linearly, ovally, or circularly polarised

or otherwise

[chrome grove]

very cool indeed

what do you study?

[opal parrot]

What? Why do you think so? It certainly does have a lot of meaning.

[chrome grove]

for me

explained

at the moment

[opal parrot]

Oh.

Well, explaining quantum mechanics usually takes at least a semester 😄

[golden plover]

that's a basic explanation

[chrome grove]

thanks

[golden plover]

or what it represents

icba to rewrite it for you

or anyone for that matter

[golden plover]

maths but I did particle physics and quantum physics for 2 years concurrently

[chrome grove]

like, I can imagine teh exp(ix) rotating and A going up and down

but I wanted the full picture and why there is the Re

[opal parrot]

Oh, that's what it meant! Alright.

[golden plover]

it allows for the horizontal and vertical components to have more freedom of variable

[chrome grove]

it can't be explained by "ok, but let's get real now guys, shall we"

🥁

[opal parrot]

I wonder if switching to cylindrical coordinates would be easier?

[golden plover]

fr

[golden plover]

in terms of omegat and phi? I suppose it could simplify it very slightly

[chrome grove]

I have a suspicion that it'd help in the simple casae

but then if you do anything else, it'd become unwieldy

[golden plover]

but working with cylindrical coordinates in the calculations for polarisation patterns

is a pain

just as flr said lmao

[opal parrot]

Ah, ok.

[chrome grove]

I didn't even study it

lol

just intuition

[opal parrot]

Well, as I'm a chemist, I'm sure you know which coordinates we like the most 😄

[chrome grove]

:)))

[golden plover]

also sketches of polarisation states are really simple with those

no need for 3 dimensions

just sketch them separately

[opal parrot]

True.

[chrome grove]

I sent you both requests. You are too fun to interact with

[opal parrot]

I wonder what the spectrum is for this problem.

At least as in discrete or continuous.

Or maybe mixed?

[golden plover]

which one?

[opal parrot]

Well, I assume the operators that we want to get the values of are the amplitudes. Or not?

[chrome grove]

could it be related to k?

the frequency

[opal parrot]

k is the wavenumber, not the frequency.

[chrome grove]

ah

[golden plover]

so you get the combined vector E(z,t) based on your amplitude, path difference, phase difference, etc

and then you can Parameterise it

well it already pretty much is

then you can apply the transformations enacted by a quarter, half or other wave plate

that polarises and Transforms the polarisation states in different ways

[opal parrot]

Well, yeah, I get that, but when we obtain the wavefunction, what operator is that for? Hamiltonian, as usual, or something else?

[golden plover]

this will then depict the polarisation state of the wave post transformation

[golden plover]

Hamiltonian iirc

[opal parrot]

Alright.

So yeah, that's what I'm interested in - its spectrum 😁

At least the type of its spectrum: discrete, continuous, mixed.

[golden plover]

continuous

[chrome grove]

@golden plover so you actually shamelessly stole my thread

[golden plover]

yes 100%

I can't make my own

[chrome grove]

But I love the enthusiasm etc

so

[golden plover]

make your own thread!!

[chrome grove]

I might

anyway thanks for sharing

feel free to respond to my request

and tty soon

[golden plover]

I have only accepted 1 request from someone

will you be the second?!?!?!

[chrome grove]

omg

blushing desu

[golden plover]

we'll see in the next episode

[chrome grove]

of spacetime

ok

[golden plover]

in the next loop

@chrome grove I have a joke for you

[opal parrot]

Beginner quantum chemistry students when the spectrum is continuous:

[golden plover]

when you see this in an equation: ∆
when you see this in an equation:

[opal parrot]

(live footage of them learning about the delta function)

[golden plover]

I hate del

[opal parrot]

Why? I like it.

What I don't like is Δ for ∇^2.

[golden plover]

you like the divergence/curl of a vector field surrounding a point in 3d space???

liar

[opal parrot]

Well, we call the curl rotor. But yeah! I like the notations ∇· and ∇⨯, though.

[golden plover]

I'd rather calculate a 5×5 Jacobian

[golden plover]

yes scalar and inner products are great

[opal parrot]

So you want to find a 5 by 5 determinant instead of a 3 by 3 determinant? 😄

[golden plover]

but del

[chrome grove]

:D

[golden plover]

yes

more fun

[opal parrot]

Lmao, ok...

[golden plover]

the 5 stages to insanity

calculate 5 partial differentials

repeat 5 times

[opal parrot]

Tell your idea of fun to the computer trying to calculate the Slater determinant of the most insane stuff you can make up 😄

[chrome grove]

it's not based at all

[golden plover]

inverting a matrix except you're solving it in exponential time

P vs NP would be proud

[chrome grove]

yikes

[opal parrot]

Ok but y tho?

Oh, here's a better one.

[golden plover]

Inverting a 2×2 matrix 🥱
inverting a 3×3 matrix 🥴
inverting a 4×4 matrix
inverting a 5×5 matrix
inverting a 6×6 matrix
inverting a 7×7 matrix
inverting a 7×6 matrix 🗣️(it doesn't exist)

[chrome grove]

I mean this is technically fooni but

[opal parrot]

Me when solving a pretty difficult DE VS me when encountering partial fractions:

[chrome grove]

partial fractions XD

[golden plover]

me when encountering a 2nd order non homogeneous partial differential equation

[opal parrot]

Oh yeah, and several quasipolynomials on the right...

[chrome grove]

ok what are quasipolynomials

[golden plover]

partial fractions are so easy tbh

[opal parrot]

Easy, but takes about 3877 years (so, about the whole class time).

[golden plover]

I could do them in my sleep

used them so much when I was doing ODEs when I was 13/14

they're good fun

[chrome grove]

takes me back

[opal parrot]

Just a generalized class of function for which linear DEs with constant coefficients are solvable without integration if they are on the right side.
They are of the form e^(ax) (P(x)cos(bx) + Q(x)sin(bx)), where P and Q are polynomials.

[golden plover]

partial fractions of 2/x²+1

[chrome grove]

ah cool

thanks

[opal parrot]

Well, that doesn't really need partial fractions 😄

[chrome grove]

unless C

[golden plover]

yes yes remove the constant and arctan

however

A/(x+i) + B/(x-i)

[chrome grove]

.

[opal parrot]

Well, yeah.

But there's rarely any need for that.

Unless in complex analysis, maybe? Haven't learned it, but sounds like a good place to maybe do that.

[chrome grove]

I can hardly see the use for that

[golden plover]

A(x-i)+B(x+i)=2
Ax+Bx=0
A+B=0
Bi-Ai=2

B-A=-2i

[golden plover]

nah not really

doing it for the sake of doing it

hardly any partial fractions are too too much harder than any other

[chrome grove]

Stubborn factorisation enthusiasts

[golden plover]

quit complaining and memorise the quartic formula

[opal parrot]

Oh, those are probably number theorists, rather.

[golden plover]

get a grip

[chrome grove]

what even is that

[golden plover]

I'm building a steering wheel for a project/driving simulator or mine

[chrome grove]

cool

[golden plover]

gonna 3d print a set of grips

that's the grip model

[chrome grove]

looks like it's compliant with checks list Somalia's driving laws

no other country

[opal parrot]

Oh, fantastic, now I'm getting Vietnam flashbacks from last year when we had engineering graphics. Thanks!

[golden plover]

it's a track car

[golden plover]

I'm just better than you then

[chrome grove]

ah

[golden plover]

cad so easy

[opal parrot]

You can probably imagine how ""fun"" that was for us, pure chemists 😄

[golden plover]

so much fun

physics made me dedicate to pure maths

I needed a challenge

full offense to any physicists

[opal parrot]

Well, anyway, it's 1 AM for me now, so I'm going to sleep.

Good night!

[golden plover]

gn

[chrome grove]

gn

[opal parrot]

Well, seeing as how you like to spend your time: derive the hydrogen-like atom wavefunction all the way from the beginning to the end, fully 😁

[chrome grove]

the burn is so 🔥 that their bodies became quark plasma

[chrome grove]

then go for Helium

pencil and paper only

[opal parrot]

The specific atom doesn't matter, only thing needed is there only needs to be one electron, otherwise there's no analytical solution.

[golden plover]

that's a warm up

[chrome grove]

that's why Helium

2 electrons

[golden plover]

nah

oganesson

[chrome grove]

frustratingly close to 1

[golden plover]

go all the way

it's a noble gas

so basically helium

but 118 electrons

[chrome grove]

ffs

[golden plover]

and then choose the greatest of methods of solution

... guess and check

[chrome grove]

right

I think Imma leave too for today