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).
APPLICATION OF THE FINITE ELEMENT METHOD TO ROTARY-WING AEROELASTICITY.
Recent research in rotary-wing aeroelasticity has indicated that all fundamental problems in this area are inherently nonlinear. The non-linearities in this problem are due to the inclusion of finite slopes, due to moderate deflections, in the structural, inertia and aerodynamic operators associated with this aeroelastic problem. In this paper the equations of motion, which are both time and space dependent, for the aeroelastic problem are first formulated in P.D.E. form. Next the equations are linearized about a suitable equilibrium position. The spatial dependence in these equations is discretized using a local Galerkin method of weighted residuals resulting in a finite element formulation of the aeroelastic problem. As an illustration the method is applied to the coupled flap-lag problem of a helicopter rotor blade in hover. Comparison of the solutions with previously published solutions establishes the convergence properties of the method. It is concluded that this formulation is a practical tool for solving rotary-wing aeroelastic stability or response problems.