Rank-preserving geometric means of positive semi-definite matrices

The generalization of the geometric mean of positive scalars to positive definite matrices has attracted considerable attention since the seminal work of Ando. The paper generalizes this framework of matrix means by proposing the definition of a rank-preserving mean for two or an arbitrary number of positive semi-definite matrices of fixed rank. The proposed mean is shown to be geometric in that it satisfies all the expected properties of a rank-preserving geometric mean. The work is motivated by operations on low-rank approximations of positive definite matrices in high-dimensional spaces.Comment: To appear in Linear Algebra and its Application

Scale Invariant Cosmology

An attempt is made here to extend to the microscopic domain the scale invariant character of gravitation - which amounts to consider expansion as applying to any physical scale. Surprisingly, this hypothesis does not prevent the redshift from being obtained. It leads to strong restrictions concerning the choice between the presently available cosmological models and to new considerations about the notion of time. Moreover, there is no horizon problem and resorting to inflation is not necessary.Comment: TeX, 20 page

Spectral Geometry of Heterotic Compactifications

The structure of heterotic string target space compactifications is studied using the formalism of the noncommutative geometry associated with lattice vertex operator algebras. The spectral triples of the noncommutative spacetimes are constructed and used to show that the intrinsic gauge field degrees of freedom disappear in the low-energy sectors of these spacetimes. The quantum geometry is thereby determined in much the same way as for ordinary superstring target spaces. In this setting, non-abelian gauge theories on the classical spacetimes arise from the K-theory of the effective target spaces.Comment: 14 pages LaTe

Casimir effect in hemisphere capped tubes

In this paper we investigate the vacuum densities for a massive scalar field with general curvature coupling in background of a (2+1)-dimensional spacetime corresponding to a cylindrical tube with a hemispherical cap. A complete set of mode functions is constructed and the positive-frequency Wightman function is evaluated for both the cylindrical and hemispherical subspaces. On the base of this, the vacuum expectation values of the field squared and energy-momentum tensor are investigated. The mean field squared and the normal stress are finite on the boundary separating two subspaces, whereas the energy density and the parallel stress diverge as the inverse power of the distance from the boundary. For a conformally coupled field, the vacuum energy density is negative on the cylindrical part of the space. On the hemisphere, it is negative near the top and positive close to the boundary. In the case of minimal coupling the energy density on the cup is negative. On the tube it is positive near the boundary and negative at large distances. Though the geometries of the subspaces are different, the Casimir pressures on the separate sides of the boundary are equal and the net Casimir force vanishes. The results obtained may be applied to capped carbon nanotubes described by an effective field theory in the long-wavelength approximation.Comment: 24 pages, 5 figure