Topics on Solid Angles

Brownian Motion on a Sphere: Distribution of Solid Angles

We study the diffusion of Brownian particles on the surface of a sphere and compute the distribution of solid angles enclosed by the diffusing particles. This function describes the distribution of geometric phases in two state quantum systems (or polarised light) undergoing random evolution. Our results are also relevant to recent experiments which observe the Brownian motion of molecules on c...

Measuring Solid Angles Beyond Dimension Three

The dot product formula allows one to measure an angle determined by two vectors, and a formula known to Euler and Lagrange outputs the measure of a solid angle in R3 given its three spanning vectors. However, there appears to be no closed form expression for the measure of an n-dimensional solid angle for n > 3. We derive a multivariable (infinite) Taylor series expansion to measure a simplici...

Computing Solid Angles by Barycentric Subdivision ∗

Bounds for solid angles of lattices of rank three

We find sharp absolute constants C1 and C2 with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval [C1, C2]. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical c...

Three New Topics in Solid-Liquid Flow (TSE03P, TSE03Q)

The projects proposed here are combined computational and experimental studies of three new problems of solid-liquid flows. The first problem is to obtain explicit formulas for the lift on particles in shear flows of pure fluids and slurries by correlating data from direct numerical simulation (DNS). The numerical studies to be undertaken look to the development of methods for interrogating the...