152,283 research outputs found
Regularity of Kobayashi metric
We review some recent results on existence and regularity of Monge-Amp\`ere
exhaustions on the smoothly bounded strongly pseudoconvex domains, which admit
at least one such exhaustion of sufficiently high regularity. A main
consequence of our results is the fact that the Kobayashi pseudo-metric k on an
appropriare open subset of each of the above domains is actually a smooth
Finsler metric. The class of domains to which our result apply is very large.
It includes for instance all smoothly bounded strongly pseudoconvex complete
circular domains and all their sufficiently small deformations.Comment: 14 pages, 8 figures - The previously announced main result had a gap.
In this new version the corrected statement is given. To appear on the volume
"Geometric Complex Analysis - Proceedings of KSCV 12 Symposium
Two-dimensional Stokes flow driven by elliptical paddles
A fast and accurate numerical technique is developed for solving the biharmonic equation in a multiply connected domain, in two dimensions. We apply the technique to the computation of slow viscous flow (Stokes flow) driven by multiple stirring rods. Previously, the technique has been
restricted to stirring rods of circular cross section; we show here how the prior method fails for noncircular rods and how it may be adapted to accommodate general rod cross sections, provided only that for each there exists a conformal mapping to a circle. Corresponding simulations of the flow are described, and their stirring properties and energy requirements are discussed briefly. In particular the method allows an accurate calculation of the flow when flat paddles are used to stir a fluid chaotically
Numerical range for random matrices
We analyze the numerical range of high-dimensional random matrices, obtaining
limit results and corresponding quantitative estimates in the non-limit case.
For a large class of random matrices their numerical range is shown to converge
to a disc. In particular, numerical range of complex Ginibre matrix almost
surely converges to the disk of radius . Since the spectrum of
non-hermitian random matrices from the Ginibre ensemble lives asymptotically in
a neighborhood of the unit disk, it follows that the outer belt of width
containing no eigenvalues can be seen as a quantification the
non-normality of the complex Ginibre random matrix. We also show that the
numerical range of upper triangular Gaussian matrices converges to the same
disk of radius , while all eigenvalues are equal to zero and we prove
that the operator norm of such matrices converges to .Comment: 23 pages, 4 figure
Relative "-Numerical Ranges for Applications in Quantum Control and Quantum Information
Motivated by applications in quantum information and quantum control, a new
type of "-numerical range, the relative "-numerical range denoted
, is introduced. It arises upon replacing the unitary group U(N) in
the definition of the classical "-numerical range by any of its compact and
connected subgroups .
The geometric properties of the relative "-numerical range are analysed in
detail. Counterexamples prove its geometry is more intricate than in the
classical case: e.g. is neither star-shaped nor simply-connected.
Yet, a well-known result on the rotational symmetry of the classical
"-numerical range extends to , as shown by a new approach based on
Lie theory. Furthermore, we concentrate on the subgroup , i.e. the -fold tensor product of SU(2),
which is of particular interest in applications. In this case, sufficient
conditions are derived for being a circular disc centered at
origin of the complex plane. Finally, the previous results are illustrated in
detail for .Comment: accompanying paper to math-ph/070103
Second Order Freeness and Fluctuations of Random Matrices: I. Gaussian and Wishart matrices and Cyclic Fock spaces
We extend the relation between random matrices and free probability theory
from the level of expectations to the level of fluctuations. We introduce the
concept of "second order freeness" and derive the global fluctuations of
Gaussian and Wishart random matrices by a general limit theorem for second
order freeness. By introducing cyclic Fock space, we also give an operator
algebraic model for the fluctuations of our random matrices in terms of the
usual creation, annihilation, and preservation operators. We show that
orthogonal families of Gaussian and Wishart random matrices are asymptotically
free of second order.Comment: 46 pages, 13 figures, second revision adds explanations, figures, and
reference
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