13,942 research outputs found
Gravity and compactified branes in matrix models
A mechanism for emergent gravity on brane solutions in Yang-Mills matrix
models is exhibited. Newtonian gravity and a partial relation between the
Einstein tensor and the energy-momentum tensor can arise from the basic matrix
model action, without invoking an Einstein-Hilbert-type term. The key
requirements are compactified extra dimensions with extrinsic curvature M^4 x K
\subset R^D and split noncommutativity, with a Poisson tensor \theta^{ab}
linking the compact with the noncompact directions. The moduli of the
compactification provide the dominant degrees of freedom for gravity, which are
transmitted to the 4 noncompact directions via the Poisson tensor. The
effective Newton constant is determined by the scale of noncommutativity and
the compactification. This gravity theory is well suited for quantization, and
argued to be perturbatively finite for the IKKT model. Since no
compactification of the target space is needed, it might provide a way to avoid
the landscape problem in string theory.Comment: 35 pages. V2: substantially revised and improved, conclusion
weakened. V3: some clarifications, published version. V4: minor correctio
Galois theory of fuchsian q-difference equations
We propose an analytical approach to the Galois theory of singular regular
linear q-difference systems. We use Tannaka duality along with Birkhoff's
classification scheme with the connection matrix to define and describe their
Galois groups. Then we describe \emph{fundamental subgroups} that give rise to
a Riemann-Hilbert correspondence and to a density theorem of Schlesinger's
type.Comment: Prepublication du Laboratoire Emile Picard n.246. See also
http://picard.ups-tlse.f
Comparison and Rigidity Theorems in Semi-Riemannian Geometry
The comparison theory for the Riccati equation satisfied by the shape
operator of parallel hypersurfaces is generalized to semi-Riemannian manifolds
of arbitrary index, using one-sided bounds on the Riemann tensor which in the
Riemannian case correspond to one-sided bounds on the sectional curvatures.
Starting from 2-dimensional rigidity results and using an inductive technique,
a new class of gap-type rigidity theorems is proved for semi-Riemannian
manifolds of arbitrary index, generalizing those first given by Gromov and
Greene-Wu. As applications we prove rigidity results for semi-Riemannian
manifolds with simply connected ends of constant curvature.Comment: 46 pages, amsart, to appear in Comm. Anal. Geo
Distributed Hierarchical SVD in the Hierarchical Tucker Format
We consider tensors in the Hierarchical Tucker format and suppose the tensor
data to be distributed among several compute nodes. We assume the compute nodes
to be in a one-to-one correspondence with the nodes of the Hierarchical Tucker
format such that connected nodes can communicate with each other. An
appropriate tree structure in the Hierarchical Tucker format then allows for
the parallelization of basic arithmetic operations between tensors with a
parallel runtime which grows like , where is the tensor dimension.
We introduce parallel algorithms for several tensor operations, some of which
can be applied to solve linear equations directly in the
Hierarchical Tucker format using iterative methods like conjugate gradients or
multigrid. We present weak scaling studies, which provide evidence that the
runtime of our algorithms indeed grows like . Furthermore, we present
numerical experiments in which we apply our algorithms to solve a
parameter-dependent diffusion equation in the Hierarchical Tucker format by
means of a multigrid algorithm
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