We apply our model of quantum gravity to a Kerr-AdS spacetime of dimension $2 m+1$, $m\ge2$, where all rotational parameters are equal, resulting in a wave equation in a quantum spacetime which has a sequence of solutions that can be expressed as a product of stationary and temporal eigenfunctions. The stationary eigenfunctions can be interpreted as radiation and the temporal as gravitational waves. The event horizon corresponds in the quantum model to a Cauchy hypersurface that can be crossed by causal curves in both directions such that the information paradox does not occur. We also prove that the Kerr-AdS spacetime can be maximally extended by replacing in a generalized Boyer-Lindquist coordinate system the $r$ variable by $\rho=r^2$ such that the extended spacetime has a timelike curvature singularity in $\rho=-a^2$.

Solutions for many problems of interest exhibit singular behaviors at domain corners or points where boundary condition changes type. For this type of problems, direct spectral methods with usual polynomial basis functions do not lead to a satisfactory convergence rate. We develop in this paper a M\"untz-Galerkin method which is based on specially tuned M\"untz polynomials to deal with the singular behaviors of the underlying problems. By exploring the relations between Jacobi polynomials and M\"untz polynomials, we develop efficient implementation procedures for the M\"untz-Galerkin method, and provide optimal error estimates. As examples of applications, we consider the Poisson equation with mixed Dirichlet-Neumann boundary conditions, whose solution behaves like $O(r^{1/2})$ near the singular point, and demonstrate that the M\"untz-Galerkin method greatly improves the rates of convergence of the usual spectral method.

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We obtain the asymptotic symmetry algebra of Chern-Simons theory with Dirichlet boundary conditions for fixed chemical potential. These boundary conditions are obeyed by higher spin black holes. For each embedding of into , we show that the asymptotic symmetry group is independent of the chemical potential. On the one hand, starting from AdS in the principal embedding, we show that the symmetry is preserved upon turning on perturbatively spin 3 chemical potentials. On the other hand, starting from AdS in the diagonal embedding, we show that the symmetry is preserved upon turning on finite spin 3/2 chemical potentials. We also make connections between the canonical Lagrangian formalism and integrability methods based on the n = 3 KdV (Boussinesq) hierarchy.