Thursday, July 22, 2010

why can't everything be smooth ..?

so i was wrong earlier, at least about the first part.

ye gods, sometimes i hate dealing with issues of regularity. i blame the notation from calculus -- neither newton or leibniz is innocent here -- because integration and differentiation are operations so seemingly common that to use them is almost instinctual.

specifically, for a while i had written a differential equation in a certain form, including such terms as $|\nabla f|^2$.
the thing is, the ODE should be interpreted in the sense of distributiοns; a priori there's a way for it to make sense classically, but only in an almοst everywhεre sense.

so strangely enough, $|\nabla f|$ made sense in that setting, but not $|\nabla f|^2$. [0]

i think i have a proof now, though. let's see if i say the same tomorrow.

when i read this the other day, i was struck how much it resembled my own life .. though not completely. i don't smoke, and coffee is more my poison than tea.
In a cold but stuffy bed-sitting room littered with cigarette ends and half-empty cups of tea, a man in a moth-eaten dressing gown sits at a rickety table, trying to find room for his typewriter among the piles of dusty papers that surround it.

He cannot throw the papers away because the wastepaper basket is already overflowing, and besides, somewhere among the unanswered letters [1] and unpaid bills it is possible that there is a cheque for two guineas which he is nearly certain he forgot to pay into the bank. [2]

There are also letters with addresses which ought to be entered in his address book. He has lost his address book, and the thought of looking for it, or indeed of looking for anything, afflicts him with acute suicidal impulses.

~ from "confessions of a book reviewer" by george orwell.
[0] for those in the know, $f \in BV_{loc}(\mathbb{R})$, so $|\nabla f| \in L^1_{loc}(\mathbb{R})$ yet $|\nabla f| \notin L^2_{loc}(\mathbb{R})$.

[1] this reminds me too much of emails to which i meant to reply, but still haven't .. \-:

[2] speaking of which, i still haven't deposited a few conference reimbursement checks into my bank account ..