Rotation matrices (or Wigner D functions) are the matrix representations of the rotation operators in the basis of spherical harmonics. They are the key entities in the generation of symmetry-adapted functions by means of projection operators. Although their expression in terms of ordinary (complex) spherical harmonics and Euler rotation angles is well known, an alternative representation using...

Univariate symmetry has interesting and diverse forms of generalization to the multivariate case. Here several leading concepts of multivariate symmetry — spherical, elliptical, central and angular — are examined and various closely related notions discussed. Methods for testing the hypothesis of symmetry, and approaches for measuring the direction and magnitude of skewness, are reviewed.

For a Lagrangian scheme defined in the cylindrical coordinates, two important issues are whether the scheme can maintain spherical symmetry (symmetry-preserving) and whether the scheme can maintain positivity of density and internal energy (positivity-preserving). While there were previous results in the literature either for symmetry-preserving in the cylindrical coordinates or for positivity-...

The calculation of LEED intensities in a spherical-wave representation can be substantially simplified by symmetry relations. The wave field around each atom is expanded in symmetry-adapted functions where the local point symmetry of the atomic site applies. For overlayer systems with more than one atom per unit cell symmetry-adapted functions can be used when the division of the crystal into m...

The electronic structure of C60 has been in the focus of interest of physicists during the last decade. A variety of different theoretical approaches from purely phenomenological to ab initio calculations has been applied to find the electron energy structure and explain main experimental features. Among the experimental data, the high frequency peak (∼ 25 eV) in the response of the cluster cal...

Although the crystallographic notation for two-dimensional Eu-clidean and spherical symmetry groups is widely used, Conway's orbifold notation is both simpler and more transparent, and provides a uniied naming scheme for symmetry groups of all three two-dimensional geometries. Based on it, this paper dicusses the concept of symmetry mutation, an operation that maps a two-dimensional symmetry gr...

Approximately calculating integrals over spherical surfaces in R3 can be done by simple extensions of one dimensional quadrature rules. This, however, does not make use of the symmetry or structure of the integration domain and potentially better schemes can be devised by directly using the integration surface in R3. We investigate several quadrature schemes for integration over a spherical sur...

Geodesics are studied in one of the Weyl metrics, referred to as the M–Q solution. First, arguments are provided, supporting our belief that this space–time is the more suitable (among the known solutions of the Weyl family) for discussing the properties of strong quasi–spherical gravitational fields. Then, the behaviour of geodesics is compared with the spherically symmetric situation, bringin...

The full classical Dirac-Maxwell equations are considered in a somewhat novel form and under various simplifying assumptions. A reduction of the equations is performed in the case when the Dirac field is static. A further reduction of the equations is made under the assumption of spherical symmetry. These static spherically symmetric equations are examined in some detail and a numerical solutio...

The issue of the possible presence or absence of jets in GRBs is here re-examined for GRB991216, which has a claimed half-opening beaming angle of θ◦ = 3 ◦. We compare and contrast our theoretically predicted afterglow luminosity in the 2–10 keV band for spherically symmetric versus jetted emission. Jetted emission can be excluded and data analysis confirms spherical symmetry. These theoretical...