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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.

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.


Vector Spherical Harmonics May 26, 2010

Posted by Sarah in Uncategorized.
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Spherical harmonics are the angular part of the solution to Laplace’s equation on the sphere: they have the property that
\Delta Y_{lm} = \frac{l (l+1)}{r^2} Y_{lm}

They form an orthonormal basis in Hilbert space and have many nice properties. In particular, the space of spherical harmonics is an irreducible representation of the group of rotations SO(3).

Vector spherical harmonics are an extension of the concept for use with vector fields. We define three vector spherical harmonics,
\mathbf{Y_{lm}} = Y_{lm} \hat{\mathbf{r}}
\mathbf{\Psi_{lm}} = r \nabla Y_{lm}
\mathbf{\Phi_{lm}} = \vec{r} \times \nabla Y_{lm}
These are orthogonal just like the usual spherical harmonics, and they allow every vector field to be expanded in vector spherical spherical harmonics.

They also turn out to be useful in the study of magnetostatics (that is, the study of static magnetic fields) as in this paper by Barrerra, Estevez, and Giraldo (Here.)

The gist: expanding functions on the sphere in spherical harmonics makes it possible to replace the Poisson equation with an ordinary differential equation. This means that if we define a charge distribution and require it to be bounded in extent, we can determine the potential outside the charge distribution. The key is canceling out the angular component to simplify the partial differential equation.

Extending this notion to vector fields allows us to do the same thing with differential equations involving the gradient (and Laplacian and curl etc), expanding in vector spherical harmonics and letting the angular components cancel. This will allow us to calculate magnetic multipoles analogous to electrical multipoles. When there is no charge, the magnetic induction field is the gradient of some function, the magnetic scalar potential.
\mathbf{B} = \nabla \mathbf{\Phi}.
The potential can be expanded in spherical harmonics; the coefficients are called the magnetostatic multipole moments.