We explicitly compute the elliptic points and isotropy groups for the action of the Picard modular group over the Gaussian integers on 2-dimensional complex hyperbolic space.

The statistical expectation values of the temperature fluctuations of cosmic microwave background (CMB) are assumed to be preserved under rotations of the sky. This assumption of statistical isotropy (SI) of the CMB anisotropy should be observationally verified since detection of violation of SI could have profound implications for cosmology. We propose a set of measures, κl (l = 1, 2, 3, . . ....

The isotropy constant of any d-dimensional polytope with n vertices is bounded by C p n/d where C > 0 is a numerical constant.

A landscape is rugged if it has many local optima, if it gives rise to short adaptive walks, and if it exhibits a rapidly decreasing pair-correlation function (and hence if it has a short correlation length). The \correlation length conjecture" allows to estimate the number of meta-stable states from the correlation length, provided the landscape is \typical". Isotropy, originally introduced as...

We consider homogeneous Abelian vector fields in an expanding universe. We find a mechanical analogy in which the system behaves as a particle moving in three dimensions under the action of a central potential. In the case of bounded and rapid evolution compared to the rate of expansion, we show—by making use of the virial theorem—that for an arbitrary potential and polarization pattern, the av...