A harmonic function, for the purposes of this discussion, is a function for which . A superharmonic function is one for which , and a subharmonic one one for which . Consequently, a function is harmonic if and only if it is both superharmonic and subharmonic.
Thus, a superharmonic function is (generally speaking) not harmonic.
When I explained this to the teaching-of-mathematics studying fellow the next desk over, his comment was: “Are you telling me that Superman isn’t even a man?”
(To which I should have said, “He’s from Krypton, isn’t he? I don’t even know if we should call him a he! What the hell, he might have tentacles or nothing at all down there — wait, let me check, there must be fan fiction about this. Let me google for ‘superman duck penis’.”)
To which I answered, “Ja, but if ve take der Super-Man und der Sub-Man, they together make a Man!”
A question, from the same discussion: As is well known, a topologist is a person who doesn’t know the difference between a donut and a coffee cup. This being so because in the topological sense they’re the same thing: if they were made of clay, you could morph one into the other without destroying or introducing any holes. (A donut has one, in the middle; a coffee cup has one, in the handle.)
The question now rises, how many holes should a donut have to be topologically equivalent to a human being?
Probably more than one, as the digestive pathway, mouth to fartmaker, is not the only one. But this quickly becomes a quest into the insides of the human being; it is not clear to me if even the male and the female of the species are topologically equivalent. (Either “Physiological gender as a topological concept” or “The topological equivalence of the sexes: Towards a mathematical feminism”, forthcoming once I get the funding.)
Research into this is on hiatus because the damn biologists, who surely have the requisite expertise, are far away across the frozen waste in a different building.