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 ?
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?]
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?
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...
I ought to get overly emotional (in a bittersweet way) about all this, and i almost did, but Teddy reminded me to stay ataraxic (i.e. keeping his role in formulating key management policies purely in the cortex )
thank you for that blogpost about MPB (its one small step for fuzzablekind!)
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.
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.
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.
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 each other.
This is incorrect. In mathematics there is a single definition of function. There is no conflict or contradiction. In all cases a function is a subset of the cross product of two spaces that satisfies a certain condition.
What changes from subject to subject is what the underlying spaces of interest are.