[082349872349872]

Do I understand properly that in a different universe distributions could have been called prefunctions?

[practal]

A distribution is a function, on the space of test functions.

[082349872349872]

OK, so if we have a distribution D (less nice than the average function) and a test function T (nicer than the average function), we have ⟨D,T⟩ = c: ℂ, so ⟨D,—⟩: test fn→ℂ and ⟨—,T⟩: distribution→ℂ ?

[gradschoolfail]

Wait i thought functions are predistributions..

[My bad, it was Matvei, not Manuel, no idea how i mixed that up..

Checkout his childrens books, as well as

https://archive.is/eaYRs

Note how the independent diagonals are what i consider interesting]

[082349872349872]

if there are no interiors (maybe edges but no faces nor volumes) then the vertices on the diagonals are truly independent: eg QM on small scales, GR on large ones.

[I'm currently pondering how the "main diagonal" of a transition matrix provides objects, while all the off-diagonal elements are the arrows. This implies that by rotating into an eigenframe (diagonalising), we're reducing the diversion to -∞ (generalised eigenvectors have nothing to lose but their Jordan chains) and hence back in the world of classical boolean logic?]

EDIT: https://mmozgovoy.dev/posts/solar-matter/

[gradschoolfail]

[Righhht, maybe you can excite me even more by relating this to quantales?? Or maybe expand on fns vs distributions a bit more?]

L: quantal (quasiparticles)

[082349872349872]

Is this sufficient relation: rel'ns (matrices which are particularly "irrreducible"/"simple" in that they've forgotten their weights to the point where these are either identity or zero) are concrete models of abstract quantales?

Lagniappe: https://www.sciencedirect.com/science/article/pii/0022404993...

EDIT: I'm afraid I'm just learning fns vs distributions (curried fns?) myself.

I wonder how quasiparticles might relate to ideals (nuclei in quantale-speak I believe)? Note that something very much like quasiparticles is how regexen turn exponential searches into polynomial...

[gunnihinn]

A distribution is not a function. It is a continuous linear functional on a space of functions.

Functions define distributions, but not all distributions are defined that way, like the Dirac delta or integration over a subset.

[skhunted]

A functional is a function.

[ogogmad]

The term "function" sadly means different things in different contexts. I feel like this whole thread is evidence of a need for reform in maths education from calculus up. I wouldn't be surprised if you understood all of this, but I'm worried about students encountering this for the first time.

[skhunted]

Don’t know if you are a mathematician or not but mathematically speaking “function” has a definition that is valid in all mathematical contexts. Functional clearly meets the criteria to be a function since being a function is part of the definition of being a functional.

[ogogmad]

The situation is worse than I thought. The term "function", as used in foundations of mathematics, includes functionals as a special case. By contrast, the term "function", as used in mathematical analysis, explicitly excludes functionals. The two definitions of the word "function" are both common, and directly contradict one another.

[keithalewis]

Try composing two distributions.

[practal]

Try composing f : A -> B with g : A -> B, for A ≠ B. Still, f and g are functions. So, what exactly is your point?

[keithalewis]

What is a delta function at a composed with a delta function at b <> a?