Algorithm 1018: FaVeST—Fast Vector Spherical Harmonic Transforms

Vector spherical harmonics on the unit sphere of ℝ 3 have broad applications in geophysics, quantum mechanics, and astrophysics. In the representation of a tangent vector field, one needs to evaluate the expansion and the Fourier coefficients of vector spherical harmonics. In this article, we develop fast algorithms (FaVeST) for vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has a computational cost proportional to N log √ N for N number of evaluation points. The adjoint FaVeST, which evaluates a linear combination of vector spherical harmonics with a degree up to ⊡ M for M evaluation points, has cost proportional to M log √ M . Numerical examples of simulated tangent fields illustrate the accuracy, efficiency, and stability of FaVeST.

On Copula-Itô processes

We study the dynamics of the family of copulas {Ct}t≥0 of a pair of stochastic processes given by stochastic differential equations (SDE). We associate to it a parabolic partial differential equation (PDE). Having embedded the set of bivariate copulas in a dual of a Sobolev Hilbert space H1 (ℝ2)* we calculate the derivative with respect to t and the *weak topology i.e. the tangent vector field to the image of the curve t → Ct. Furthermore we show that the family {Ct}t≥0 is an orbit of a strongly continuous semigroup of transformations and provide the infinitesimal generator of this semigroup.

Branches of Forced Oscillations Induced by a Delayed Periodic Force

We study global continuation properties of the set of T-periodic solutions of parameterized second order delay differential equations with constant time lag on smooth manifolds. We apply our results to get multiplicity of T-periodic solutions. Our topological approach is mainly based on the notion of degree of a tangent vector field.

On some notions of tangent space to a measure

We consider some definitions of tangent space to a Radon measure μ on ℝn that have been given in the literature. In particular, we focus our attention on a recent distributional notion of tangent vector field to a measure and we compare it to other definitions coming from ‘geometric measure theory’, based on the idea of blow-up. After showing some classes of examples, we prove an estimate from above for the dimension of the tangent spaces and a rectifiability theorem which also includes the case of measures supported on sets of variable dimension.