On the cohomology groups of certain quotients of products of upper half planes and upper half spaces
Abstract A theorem of Matsushima–Shimura shows that the space of harmonic differential forms on the quotient of products of upper half planes under the action of certain groups, when the quotient is compact, is the direct sum of two subspaces called the universal and cuspidal subspaces. We generalize this result to compact quotients of products of upper half planes and upper half spaces (hyperbolic three spaces) under the action of certain groups to obtain a similar decomposition.
Hilbert Modular Forms
A discrete subgroup Γ ⊂ SL, (2ℝ) acts discontinuously on the upper half-plane H. The parabolic elements of Γ give rise to a natural extension of H/Γ by the so-called cusp classes. We are mainly interested in the case where this extension is compact. Our basic example is Γ = SL(2, Z). The method of construction is such that it can easily be generalized to the case of several variables, i.e. discrete subgroups of SL(2, ℝ) n acting on the product of n upper half-planes. This will be done in the next section (§2).
Reflection coefficient of inhomogeneous anisotropic impedance plane
Solved the problem of constructive synthesis of an inhomogeneous anisotropic impedance plane reflecting a homogeneous plane wave in two different preset directions on a matched and cross polarization. Obtained the reflection coefficients and expressions for the scattering diagrams. The impedance is implemented by the frequent array of orthogonal strips. The law of distribution of the reactance was obtained in an explicit form. It is shown that for arbitrarily given (unequal) directions of reflections on matched and cross polarizations, the orientation of the reactance strips turns out to be variable along the plane. Formulated restrictions on class of implemented scattering diagrams. Shown the results of numerical studies.