34,835 research outputs found
Bohr-Sommerfeld Quantization of Space
We introduce semiclassical methods into the study of the volume spectrum in
loop gravity. The classical system behind a 4-valent spinnetwork node is a
Euclidean tetrahedron. We investigate the tetrahedral volume dynamics on phase
space and apply Bohr-Sommerfeld quantization to find the volume spectrum. The
analysis shows a remarkable quantitative agreement with the volume spectrum
computed in loop gravity. Moreover, it provides new geometrical insights into
the degeneracy of this spectrum and the maximum and minimum eigenvalues of the
volume on intertwiner space.Comment: 32 pages, 10 figure
Analyticity and criticality results for the eigenvalues of the biharmonic operator
We consider the eigenvalues of the biharmonic operator subject to several
homogeneous boundary conditions (Dirichlet, Neumann, Navier, Steklov). We show
that simple eigenvalues and elementary symmetric functions of multiple
eigenvalues are real analytic, and provide Hadamard-type formulas for the
corresponding shape derivatives. After recalling the known results in shape
optimization, we prove that balls are always critical domains under volume
constraint.Comment: To appear on the proceedings of the conference "Geometric Properties
for Parabolic and Elliptic PDE's - 4th Italian-Japanese Workshop" held in
Palinuro (Italy), May 25-29, 201
Convergence and Optimality of Adaptive Mixed Finite Element Methods
The convergence and optimality of adaptive mixed finite element methods for
the Poisson equation are established in this paper. The main difficulty for
mixed finite element methods is the lack of minimization principle and thus the
failure of orthogonality. A quasi-orthogonality property is proved using the
fact that the error is orthogonal to the divergence free subspace, while the
part of the error that is not divergence free can be bounded by the data
oscillation using a discrete stability result. This discrete stability result
is also used to get a localized discrete upper bound which is crucial for the
proof of the optimality of the adaptive approximation
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