54,322 research outputs found
Logarithmic Corrections to Extremal Black Hole Entropy in N = 2, 4 and 8 Supergravity
We compute the logarithmic correction to black hole entropy about
exponentially suppressed saddle points of the Quantum Entropy Function
corresponding to Z(N) orbifolds of the near horizon geometry of the extremal
black hole under study. By carefully accounting for zero mode contributions we
show that the logarithmic contributions for quarter--BPS black holes in N=4
supergravity and one--eighth BPS black holes in N=8 supergravity perfectly
match with the prediction from the microstate counting. We also find that the
logarithmic contribution for half--BPS black holes in N = 2 supergravity
depends non-trivially on the Z(N) orbifold. Our analysis draws heavily on the
results we had previously obtained for heat kernel coefficients on Z(N)
orbifolds of spheres and hyperboloids in arXiv:1311.6286 and we also propose a
generalization of the Plancherel formula to Z(N) orbifolds of hyperboloids to
an expression involving the Harish-Chandra character of SL(2,R), a result which
is of possible mathematical interest.Comment: 40 page
A multiscale method for heterogeneous bulk-surface coupling
In this paper, we construct and analyze a multiscale (finite element) method
for parabolic problems with heterogeneous dynamic boundary conditions. As
origin, we consider a reformulation of the system in order to decouple the
discretization of bulk and surface dynamics. This allows us to combine
multiscale methods on the boundary with standard Lagrangian schemes in the
interior. We prove convergence and quantify explicit rates for low-regularity
solutions, independent of the oscillatory behavior of the heterogeneities. As a
result, coarse discretization parameters, which do not resolve the fine scales,
can be considered. The theoretical findings are justified by a number of
numerical experiments including dynamic boundary conditions with random
diffusion coefficients
Spectral Generalized Multi-Dimensional Scaling
Multidimensional scaling (MDS) is a family of methods that embed a given set
of points into a simple, usually flat, domain. The points are assumed to be
sampled from some metric space, and the mapping attempts to preserve the
distances between each pair of points in the set. Distances in the target space
can be computed analytically in this setting. Generalized MDS is an extension
that allows mapping one metric space into another, that is, multidimensional
scaling into target spaces in which distances are evaluated numerically rather
than analytically. Here, we propose an efficient approach for computing such
mappings between surfaces based on their natural spectral decomposition, where
the surfaces are treated as sampled metric-spaces. The resulting spectral-GMDS
procedure enables efficient embedding by implicitly incorporating smoothness of
the mapping into the problem, thereby substantially reducing the complexity
involved in its solution while practically overcoming its non-convex nature.
The method is compared to existing techniques that compute dense correspondence
between shapes. Numerical experiments of the proposed method demonstrate its
efficiency and accuracy compared to state-of-the-art approaches
RG flows of Quantum Einstein Gravity on maximally symmetric spaces
We use the Wetterich-equation to study the renormalization group flow of
-gravity in a three-dimensional, conformally reduced setting. Building on
the exact heat kernel for maximally symmetric spaces, we obtain a partial
differential equation which captures the scale-dependence of for
positive and, for the first time, negative scalar curvature. The effects of
different background topologies are studied in detail and it is shown that they
affect the gravitational RG flow in a way that is not visible in
finite-dimensional truncations. Thus, while featuring local background
independence, the functional renormalization group equation is sensitive to the
topological properties of the background. The detailed analytical and numerical
analysis of the partial differential equation reveals two globally well-defined
fixed functionals with at most a finite number of relevant deformations. Their
properties are remarkably similar to two of the fixed points identified within
the -truncation of full Quantum Einstein Gravity. As a byproduct, we
obtain a nice illustration of how the functional renormalization group realizes
the "integrating out" of fluctuation modes on the three-sphere.Comment: 35 pages, 6 figure
The asymptotic behaviour of the heat equation in a twisted Dirichlet-Neumann waveguide
We consider the heat equation in a straight strip, subject to a combination
of Dirichlet and Neumann boundary conditions. We show that a switch of the
respective boundary conditions leads to an improvement of the decay rate of the
heat semigroup of the order of . The proof employs similarity
variables that lead to a non-autonomous parabolic equation in a thin strip
contracting to the real line, that can be analyzed on weighted Sobolev spaces
in which the operators under consideration have discrete spectra. A careful
analysis of its asymptotic behaviour shows that an added Dirichlet boundary
condition emerges asymptotically at the switching point, breaking the real line
in two half-lines, which leads asymptotically to the 1/2 gain on the spectral
lower bound, and the gain on the decay rate in the original physical
variables.
This result is an adaptation to the case of strips with twisted boundary
conditions of previous results by the authors on geometrically twisted
Dirichlet tubes.Comment: 15 pages, 2 figure
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