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).
L1 Adaptive Controller for Attitude Control of Multirotors
In this work we show an application of L1 Adaptive control theory for attitude control of UAVs. We implement the flight control system on a multirotor to show robustness and precise attitude tracking in the presence of modeling uncertainties and environmental disturbances. We choose backstepping control architecture, since the kinematics and dynamics of multirotors in most cases can be written in strict feedback form. We further exploit the fact that the kinematics of the plant, while free of uncertainties, is nonlinear, which makes it highly suitable for dynamic inversion control at each level of backstepping. On the other hand, plant dynamics is uncertain and is affected by environmental disturbances such as wind gusts, unmodeled dynamics etc. Therefore, we consider 3 variants of the control architecture. The first method uses backstepping to determine the moment demand and augments it with the L1 adaptive controller to account for uncertainties and provide robustness with guaranteed transient performance. The second architecture apply the concept of L1 adaptive backstepping to the same problem; and the third architecture uses L1 backstepping for quaternion representation of the system dynamics, which helps to avoid the singularities associated with Euler angles.