We derive a formula for the number of pre-images under non-degenerate harmonic mapping $f$, using argument principle. This reveals connection between and caustics. Our results allow to deduce $f$ geometrically every non-caustic point. approximately locate points near Moreover, we apply our prove that $k = n, n+1, \ldots, n^2$ there exists polynomial degree $n$ with $k$ zeros.

The Juno mission observed that both poles of Jupiter have polar cyclones are surrounded by a ring circumpolar (CPCs). north pole holds eight CPCs and the south possesses five, with rings positioned along latitude ~84° N/S. Here we explain location, stability number Jovian establishing primary forces act on them, which develop because vorticity gradients in background cyclone. In meridional dire...

This paper examines the action of GLN (R) on decomposable algebraic curvature tensors. The main result is that the structure group of a decomposable algebraic curvature tensor can only permute the subspaces into which the tensor decomposes: if the tensor decomposes into tensors Ri on Vi where V = ⊕k i=1 Vi, then for any matrix A in the structure group there exists a permutation σ such that A : ...