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Caratheodory Kernel November 3, 2010

Posted by Sarah in Uncategorized.
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Last fall I had the opportunity to go to the Cryo-EM mini-symposium at the Columbia Medical Center, which was an interesting experience — a mix of biologists, computer scientists, and mathematicians. One comment from a biologist stuck with me: sometimes, given that equipment has only finite precision, microscopic images can lead to confusion as to the number of connected components. If we’re thinking about a molecule, it’s quite important to know whether we’re looking at a protrusion or a separate component. With imprecise equipment, there’s a chance of getting it wrong. But how do you know when?

The question bugged me for a while. Connected components seem like a “topological” property, but it seems that you ought to be able to talk about this more directly, without appealing to homology.

Posing this question more specifically, suppose you have a connected domain E. To make life simple, let’s suppose it’s in the plane. And suppose we approximate it at different levels of “coarseness” — perhaps we overlay a grid of squares of side length r, and let E_r be the domain that contains a square precisely when the intersection between E and the square has area at least 1/2 r^2. For some sets E, there is some scale E_r that becomes disconnected; for example, if E is dumbbell-shaped, eventually at some scale it will “split” into two components. How do you know when it’s going to do this?

I asked a professor about this, and it turns out it’s a classical question, and the relevant notion is something called the Caratheodory Kernel, developed in 1912.

Let B_n be a sequence of simply-connected domains of the z-plane containing a fixed point z_0. If there exists a disc |z-z_0| < r, belonging to all B_n, then the kernel of the sequence B_n with respect to z_0 is the largest domain B containing z_0 and such that for each compact set E belonging to B there is an N such that E belongs to all B_n, for n larger than N.

A largest domain is one which contains any other domain having the same property. If there is no such a disc, then by the kernel of the sequence B_n, one means the point z_0 (in this case one says that the sequence B_n has a degenerate kernel). A sequence of domains B_n, converges to a kernel B if any subsequence of B_n has B as its kernel.

Link.
It’s a theorem of Caratheodory that a sequence of functions f_n with positive derivatives at z_0, conformally mapping the unit disc to B_n respectively and are regular and univalent in the disc |z – z_0| < 1 then in order for the sequence f_n to converge in the disc to a finite function, it is necessary and sufficient that the kernels converge to either a point or a domain containing more than one boundary point.

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