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Register a new userExplicit vector spherical harmonics on the 3-sphere
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We build a family of explicit one-forms on $S^3$ which are shown to form a complete set of eigenmodes for the Laplace-de Rahm operator.
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Many applications in science call for the numerical simulation of systems on manifolds with spherical topology. Through use of integer spin weighted spherical harmonics we present a method which allows for the implementation of arbitrary tensorial evolution equations. Our method combines two numerical techniques that were originally developed with different applications in mind. The first is Huffenberger and Wandelts spectral decomposition algorithm to perform the mapping from physical to spectral space. The second is the application of Luscombe and Lubans method, to convert numerically divergent linear recursions into stable nonlinear recursions, to the calculation of reduced Wigner d-functions. We give a detailed discussion of the theory and numerical implementation of our algorithm. The properties of our method are investigated by solving the scalar and vectorial advection equation on the sphere, as well as the 2+1 Maxwell equations on a deformed sphere.
Spherical Harmonics, $Y_ell^m(theta,phi)$, are derived and presented (in a Table) for half-odd-integer values of $ell$ and $m$. These functions are eigenfunctions of $L^2$ and $L_z$ written as differential operators in the spherical-polar angles, $theta$ and $phi$. The Fermion Spherical Harmonics are a new, scalar and angular-coordinate-dependent representation of fermion spin angular momentum. They have $4pi$ symmetry in the angle $phi$, and hence are not single-valued functions on the Euclidean unit sphere; they are double-valued functions on the sphere, or alternatively are interpreted as having a double-sphere as their domain.
This paper is devoted to study discrete and continuous bases for spaces supporting representations of SO(3) and SO(3,2) where the spherical harmonics are involved. We show how discrete and continuous bases coexist on appropriate choices of rigged Hilbert spaces. We prove the continuity of relevant operators and the operators in the algebras spanned by them using appropriate topologies on our spaces. Finally, we discuss the properties of the functionals that form the continuous basis.
We analyze the local field of stellar tangential velocities for a sample of $42 339$ non-binary Hipparcos stars with accurate parallaxes, using a vector spherical harmonic formalism. We derive simple relations between the parameters of the classical linear model (Ogorodnikov-Milne) of the local systemic field and low-degree terms of the general vector harmonic decomposition. Taking advantage of these relationships we determine the solar velocity with respect to the local stars of $(V_X,V_Y,V_Z)=(10.5, 18.5, 7.3)pm 0.1$ kms. The Oorts parameters determined by a straightforward least-squares adjustment in vector spherical harmonics, are $A=14.0pm 1.4$, $B=-13.1pm 1.2$, $K=1.1pm 1.8$, and $C=-2.9pm 1.4$ kmspc. We find a few statistically significant higher degree harmonic terms, which do not correspond to any parameters in the classical linear model. One of them, a third-degree electric harmonic, is tentatively explained as the response to a negative linear gradient of rotation velocity with distance from the Galactic plane, which we estimate at $sim -20$ kmspc. The most unexpected and unexplained term within the Ogorodnikov-Milne model is the first-degree magnetic harmonic representing a rigid rotation of the stellar field about the axis $-Y$ pointing opposite to the direction of rotation. This harmonic comes out with a statistically robust coefficient $6.2 pm 0.9$ kmspc, and is also present in the velocity field of more distant stars. The ensuing upward vertical motion of stars in the general direction of the Galactic center and the downward motion in the anticenter direction are opposite to the vector field expected from the stationary Galactic warp model.
A fast and exact algorithm is developed for the spin +-2 spherical harmonics transforms on equi-angular pixelizations on the sphere. It is based on the Driscoll and Healy fast scalar spherical harmonics transform. The theoretical exactness of the transform relies on a sampling theorem. The associated asymptotic complexity is of order O(L^2 log^2_2(L)), where 2L stands for the square-root of the number of sampling points on the sphere, also setting a band limit L for the spin +-2 functions considered. The algorithm is presented as an alternative to existing fast algorithms with an asymptotic complexity of order O(L^3) on other pixelizations. We also illustrate these generic developments through their application in cosmology, for the analysis of the cosmic microwave background (CMB) polarization data.
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