926 research outputs found
The C-Numerical Range in Infinite Dimensions
In infinite dimensions and on the level of trace-class operators rather
than matrices, we show that the closure of the -numerical range is
always star-shaped with respect to the set , where
denotes the essential numerical range of the bounded operator .
Moreover, the closure of is convex if either is normal with
collinear eigenvalues or if is essentially self-adjoint. In the case of
compact normal operators, the -spectrum of is a subset of the
-numerical range, which itself is a subset of the convex hull of the closure
of the -spectrum. This convex hull coincides with the closure of the
-numerical range if, in addition, the eigenvalues of or are
collinear.Comment: 31 pages, no figures; to appear in Linear and Multilinear Algebr
Pinning of interfaces in random media
For a model for the propagation of a curvature sensitive interface in a time
independent random medium, as well as for a linearized version which is
commonly referred to as Quenched Edwards-Wilkinson equation, we prove existence
of a stationary positive supersolution at non-vanishing applied load. This
leads to the emergence of a hysteresis that does not vanish for slow loading,
even though the local evolution law is viscous (in particular, the velocity of
the interface in the model is linear in the driving force).Comment: 15 Page
Non-Existence of Positive Stationary Solutions for a Class of Semi-Linear PDEs with Random Coefficients
We consider a so-called random obstacle model for the motion of a
hypersurface through a field of random obstacles, driven by a constant driving
field. The resulting semi-linear parabolic PDE with random coefficients does
not admit a global nonnegative stationary solution, which implies that an
interface that was flat originally cannot get stationary. The absence of global
stationary solutions is shown by proving lower bounds on the growth of
stationary solutions on large domains with Dirichlet boundary conditions.
Difficulties arise because the random lower order part of the equation cannot
be bounded uniformly
Stochastic homogenization for functionals with anisotropic rescaling and non-coercive Hamilton-Jacobi equations
We study the stochastic homogenization for a Cauchy problem for a first-order
Hamilton-Jacobi equation whose operator is not coercive w.r.t. the gradient
variable. We look at Hamiltonians like where
is a matrix associated to a Carnot group. The rescaling considered
is consistent with the underlying Carnot group structure, thus anisotropic. We
will prove that under suitable assumptions for the Hamiltonian, the solutions
of the -problem converge to a deterministic function which can be
characterized as the unique (viscosity) solution of a suitable deterministic
Hamilton-Jacobi problem
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
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