6 ms·
A distribution is a function, on the space of test functions.
by practal 2y ago
A distribution is a function, on the space of test functions.
- keithalewis 2y agoTry composing two distributions.
- practal 2y agoTry composing f : A -> B with g : A -> B, for A ≠ B. Still, f and g are functions. So, what exactly is your point?
- keithalewis 2y agoWhat is a delta function at a composed with a delta function at b <> a?
- gunnihinn 2y agoA 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 2y agoA functional is a function.
- ogogmad 2y agoThe 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 2y agoDon’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 2y agoThe 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.
- skhunted 2y agoBy 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.
- ogogmad 2y ago> What changes from subject to subject is what the underlying spaces of interest are. I'm not sure I understand what you mean here. I need some clarification. How does this have any bearing on whether functionals count as functions or not? What is the "underlying spaces of interest" in this example? In some trivial way, every mathematical object can be seen as a function. You can replace sets in axiomatic set theory with functions.
- skhunted 2y ago
- 082349872349872 2y agoOK, 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 ?
- deleted 2y ago[deleted]
- gradschoolfail 2y agoWait 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 https://archive.is/eaYRs Note how the independent diagonals are what i consider interesting]
- 082349872349872 2y agoif 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/ https://mmozgovoy.dev/posts/solar-matter/
- gradschoolfail 2y ago[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 2y agoIs 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/002240499390169T 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...