Alright... but if we project x onto n... then we get a value of a*b right ?
That should fit... in order to get the distance to the plane we used en.wikipedia.org/wiki/Hesse_normal_form in school... which tells us : E: ((x-a)*n)/|n| so we project our x onto n and divide it by |n|... so we would get b as a result, right ?
I feel like a technologically impaired old guy, but how can I get a list of all the papers published on the ArXiV in a field (let's say logic) in a specified time interval?
Oh I figured it out, the category option was well hidden in the search form
@Astyx set $N = V, L = K = k$ for a field $k$ and you already have the double dual operation, which preserves finite-dimensional vector spaces but sends $k^\kappa$ to $k^{|k|^{|k|^\kappa}}$ for an infinite cardinal $\kappa$
ok so from the implicit function theorem we know that the principal branch of log satisfies $\mathrm{Log}'(e^w) = 1/e^w$, from the ${f^{-1}}^\prime(f(x)) = 1/f'(x)$ thing
how do we go from there to $\mathrm{Log}'(z) = 1/z$
The other direction (given a harmonic function, come up with a holomorphic function with that as real part) is a little tricky
But essentially
All of these conditions are at it's core the "mean value property" - the value of the function at a point is the average of the nearby values. It's possible to prove that a continuous function satisfying this property is harmonic and vice versa.
I need a Brownian motion starting at every point on the disk
$\tau < \infty$ with probability 1, as for every other bounded domain, so we can forget about that measure zero event. Define $F : D^2 \to \Bbb R$, $F(x) := \Bbb E f(B^x_\tau)$
This is harmonic because of the expectation analogy you said @Soham, plus the fact that Brownian motions are Markov processes
And this construction gives an immediate solution to Dirichlet boundary value problem
So you start a Brownian motion at $x$, and look at the first time this particle escapes the disk. The point on the boundary circle through which it escapes is $B_\tau^x$
Your influence definitely got to me, @TedShifrin. I ended up studying the Atiyah-Singer Index Theorem via psuedodifferential operators and heat trace asymptotics for my dissertation.
Well, I wouldn't say that was my doing, @Kari, but neat. That stuff was really big back when I was in grad school. I'm glad people are still thinking about it.
interestingly enough, many years ago i went to a little class on solving rubik's cubes and asked the guy who was teaching about solving cubes in the fewest moves possible
