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246C notes 3: Univalent functions, the Loewner equation, and the Bieberbach conjecture

We now approach conformal maps from yet another perspective. Given an open subset of the complex numbers , define a univalent function on to be a holomorphic function that is also injective. 6,044 more words


Embedding the Boussinesq equations in the incompressible Euler equations on a manifold

The Boussinesq equations for inviscid, incompressible two-dimensional fluid flow in the presence of gravity are given by

where is the velocity field, is the pressure field, and is the density field (or, in some physical interpretations, the temperature field). 1,694 more words


Heat flow and zeroes of polynomials

Let be a monic polynomial of degree with complex coefficients. Then by the fundamental theorem of algebra, we can factor as

for some complex zeroes (possibly with repetition). 761 more words