Mathematical things

Graph I made:

this graph is a collatz graph. It skips even numbers (apart from 2 ("1")) and the nodes are labelled with multisets of the prime factors of the numbers (including 1 ("0") even though it's not prime), where the primes are represented by their indices rather than numerical value. it was made with edotor. all terminal nodes are prime because they are the only numbers I desired to follow the path of. This graph benefits exactly no one, and negatively affected me the most.

Fluxions

fluxions /flʌksij+on+z/ [flʌk.si.jɒnz] are inspired by isaac newton 's concept of "fluxions" /flʌks+jən+z/ [flʌk.ʃ(ə)nz]. they are the elements of a totally ordered infinite dimentional vector space over the reals, written with powers of the constant ε (and ω) differentiating its components, where ε is a number such that:

x∈ℝ, y∈ℝ, 0 ≠ x, 0 ≠ y ⇒ yεx ∉ ℝ

x∈ℝ, y∈ℝ, 0 < x, 0 < y ⇒ 0 < xε < y

ω is just ε-1

lim & ℜ

a function lim can be defined as such:

z∈ℝ<ε> ⇒ lim(z)∈ℝ<+∞,-∞>

z∈ℝ<ε>, x∈ℝ, y∈ℝ, x < z < y ⇒ x < lim(z) < y

lim is fun in that it gives nice expressions for constants and functions that are kinda nice:

π = lim((-1)ε-1/iε)

e = lim((1+ε)ω)

ln(x) = lim(xε/ε - ω) = lim((xε-1)/ε)

this expression for ln(x) does, as expected, differentiate to approximately 1/x:

d(xε/ε - ω)/dx = xε-1 ≈ x-1 = 1/x

and since ln(x) and ex are inverse functions, ex = lim((1+xε)ω) and so e = e1 = lim((1+ε)ω)

a similar function ℜ can be defined as simply taking the real part of a fluxion, similarly to ℜ in the complex numbers.

for example ℜ(7ω2+3+2ε) = 3, but lim(7ω2+3+2ε) = ∞ (ℜ is the same as lim a lot of the time, but not in this case)

Rank & norm

the characteristic function C=p(z) of a fluxion z is the function such that C(ω) = z:

I don't think "characteristic function" is a good name for this. I have only really defined this as a concept because it is useful in the definition of rank. I also feel this definition is probably not sufficient. what I really mean is "take all the ωs in a fluxion and replace them with xs"

the rank of a fluxion is the largest power of ω that occurs in it and is defined as such:

rank(z) = logω((O-1∘p(z))(ω)) = ln((O-1∘p(z))(ω))/ln(ω)

I think this definition needs explaining.

in big O notation you can say that certain programs "are" O(x) or O(log(x)) etc.

I have chosen to interpret this as the function of their time complexity *equaling* O(x) or O(log(x)) etc.

O is a bijection but not a function (just like λx.±x)

O-1 however is a function but is not an injection or surjection (just like λx.|x|).

a fluxion can be "normalised" as such:

norm(z) = z/ωrank(z)

for example:

ℜ(norm(7ω2+3+2ε)) = 7

ℜ(norm(2+7ε+5ε2)) = 2

ℜ(norm(2ε3+5ε2)) = 5

if ℜ(x) = lim(x) then ℜ(x) = ℜ(norm(x))

x∈ℝ, f(x)∈ℝ ⇒ f(n)(x) = n!ℜ(f(x+ε)/εn)

fluxions are useful when working with zero as ε is pretty much zero but you can divide by it. you can replace awkward zeros with εs and take lim when you're done and you'll probably be fine.

for any continuous function f from reals to reals, I conjecture that when f(0) is defined, f(0) = lim(f(ε)) this is trivially true if f is the identity function but I don't know if there are counter examples ( f is continuous ⇒ lim∘((Tε)f) = f ) if this is not true then the statement f∈K ⟺ (lim∘((Tε)f) = f) is true where K is the largest set of functions where it is true, functions in K would be nicely behaved in many ways.

a function f from reals to reals is continuous at a point x iff lim(f(x) - f(x+ε)) = 0 and lim(f(x) - f(x-ε)) = 0
I think? you could create a function that is continuous around every real within an infinitessimal range, but still discontinuous everywhere in the reals, but plz don't ok,,,
ℜ, lim, and norm are all idempotent.

haskelly pseudocode

(/') :: (ℝ -> ℝ) -> (ℝ -> ℝ) -> (ℝ -> ℝ)

(f /' g) x = (f x) / (g x)

d :: (ℝ -> ℝ) -> (ℝ -> ℝ<ε>)

d f x = f (x+ε) - f x

∇ :: (ℝ -> ℝ) -> (ℝ -> ℝ)

∇ f = lim . (d f) /' (d (\x -> x))

> ∇ (\x -> x2)

lim . (d (\x -> x2) /' d (\x -> x))

lim . (\x -> (x+ε)2 - x2) /' (\x -> (x+ε) - x)

lim . \x -> ((x+ε)2 - x2) / ((x+ε) - x)

lim . \x -> (x2 + 2xε + ε2 - x2) / ε

lim . \x -> (2xε + ε2) / ε

lim . \x -> 2x + ε

\x -> lim (2x + ε)

\x -> 2x

(the real deal)

--- all things thus far have been mainly about calculus in terms of fluxions, --- but there are other things that can be done with them too.

this is an interesting video that shows how fluxions are useful in measure theory. you can think of a collection of points as having rank 0, a collection of lines as having rank 1, a collection of planes as having rank 2, etc. this shows the fluxionic measure of a variety of regular topes, arbitrary over dimension and length.

this is a playlist of videos that I believe that fluxions can give you a deeper understanding of