### A Fast Transform for Spherical Harmonics.

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In this contribution, we will use the Maxwell-Cartesian spherical harmonics (introduced in [1,2]) to derive a system of partial differential equations governing transport of neutrons within an interacting medium. This system forms an alternative to the well known ${P}_{N}$ approximation, which is based on the expansion of the directional dependence into tesseral spherical harmonics ([3,p.197]). In comparison with this latter set of equations, the Maxwell-Cartesian system posesses a much more regular structure,...

Our primary goal in this preamble is to highlight the best of Vasil Popov’s mathematical achievements and ideas. V. Popov showed his extraordinary talent for mathematics in his early papers in the (typically Bulgarian) area of approximation in the Hausdorff metric. His results in this area are very well presented in the monograph of his advisor Bl. Sendov, “Hausdorff Approximation”.

Let ${\left({U}_{t}\right)}_{t\ge 0}$ be a Brownian motion valued in the complex projective space $\u2102{P}^{N-1}$. Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of $|{U}_{t}^{1}{|}^{2}$ and of $\left(\right|{U}_{t}^{1}{|}^{2},\left|{U}_{t}^{2}{|}^{2}\right)$, and express them through Jacobi polynomials in the simplices of $\mathbb{R}$ and ${\mathbb{R}}^{2}$ respectively. More generally, the distribution of $\left(\right|{U}_{t}^{1}{|}^{2},\cdots ,\left|{U}_{t}^{k}{|}^{2}\right),\phantom{\rule{0.277778em}{0ex}}2\le k\le N-1$ may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group $\mathcal{U}(N-k+1)$ yet computations become tedious. We also revisit the approach initiated in [13] and based on...