DOUBLE STRUCTURES AND DOUBLE SYMMETRIES FOR THE EINSTEIN–KALB–RAMOND THEORY

We further study the two-dimensional reduced Einstein–Kalb–Ramond (EKR) theory in the axisymmetric case by using the so-called double-complex function method. We find a doubleness symmetry of this theory and exploit it so that some double-complex d×d matrix Ernst-like potential can be constructed, and the associated equations of motion can be extended into a double-complex matrix Ernst-like form. Then we give a double symmetry group [Formula: see text] for the EKR theory and verify that its action can be realized concisely by a double-complex matrix, form generalization of the fractional linear transformation on the Ernst potential. These results demonstrate that the theory under consideration possesses more and richer symmetry structures. Moreover, as an application, we obtain an infinite chain of double-solutions of the EKR theory showing that the double-complex method is more effective. Some of the results in this paper cannot be obtained by the usual (nondouble) scheme.

Optimal bounds correlating electric, magnetic and thermal properties of two-phase, two-dimensional composites

The effective conductivity function of a two-phase two-dimensional composite is known to be a tensor valued analytic function of the component conductivities, assumed here to be isotropic. Optimal bounds correlating the values this function can take are derived. These values may correspond to the measured effective magnetic permeabilities, thermal conductivities, or any other transport coefficient mathematically equivalent to the effective conductivity. The main tool in this derivation is a new fractional linear transformation which maps the appropriate class of conductivity functions passing through a given point to a similar class of functions which are not subject to the restriction of passing through a known point. Crude bounds on this class of functions give rise to sharp bounds on the original class of effective conductivity functions. These bounds are the best possible, being attained by sequentially layered laminate microgeometries.

Notes on projective structures on complex manifolds

Consider a Riemann surface X equipped with a projective structure, that is, a covering of X with coordinate neighborhoods U and corresponding (holomorphic) local coordinates {t} such that in the intersection U ∩ U′ of any two such coordinate neighborhoods U and U′ change of local coordinates is mediated by a fractional linear transformation