68,440 research outputs found
Asymptotic boundary conditions for dissipative waves: General theory
An outstanding issue in the computational analysis of time dependent problems is the imposition of appropriate radiation boundary conditions at artificial boundaries. Accurate conditions are developed which are based on the asymptotic analysis of wave propagation over long ranges. Employing the method of steepest descents, dominant wave groups are identified and simple approximations to the dispersion relation are considered in order to derive local boundary operators. The existence of a small number of dominant wave groups may be expected for systems with dissipation. Estimates of the error as a function of domain size are derived under general hypotheses, leading to convergence results. Some practical aspects of the numerical construction of the asymptotic boundary operators are also discussed
Numerical Ranges of Composition Operators
Composition operators on the Hilbert Hardy space of the unit disk are considered. The shape of their numerical range is determined in the case when the symbol of the composition operator is a monomial or an inner function fixing 0. Several results on the numerical range of composition operators of arbitrary symbol are obtained. It is proved that 1 is an extreme boundary point if and only if 0 is a fixed point of the symbol. If 0 is not a fixed point of the symbol 1 is shown to be interior to the numerical range. Some composition operators whose symbol fixes 0 and has infinity norm less than 1 have closed numerical ranges in the shape of a cone-like figure, i.e. a closed convex region with a corner at 1, 0 in its interior, and no other corners. Compact composition operators induced by a univalent symbol whose fixed point is not 0 have numerical ranges without corners, except possibly a corner at 0
The Significance of the -Numerical Range and the Local -Numerical Range in Quantum Control and Quantum Information
This paper shows how C-numerical-range related new strucures may arise from
practical problems in quantum control--and vice versa, how an understanding of
these structures helps to tackle hot topics in quantum information.
We start out with an overview on the role of C-numerical ranges in current
research problems in quantum theory: the quantum mechanical task of maximising
the projection of a point on the unitary orbit of an initial state onto a
target state C relates to the C-numerical radius of A via maximising the trace
function |\tr \{C^\dagger UAU^\dagger\}|. In quantum control of n qubits one
may be interested (i) in having U\in SU(2^n) for the entire dynamics, or (ii)
in restricting the dynamics to {\em local} operations on each qubit, i.e. to
the n-fold tensor product SU(2)\otimes SU(2)\otimes >...\otimes SU(2).
Interestingly, the latter then leads to a novel entity, the {\em local}
C-numerical range W_{\rm loc}(C,A), whose intricate geometry is neither
star-shaped nor simply connected in contrast to the conventional C-numerical
range. This is shown in the accompanying paper (math-ph/0702005).
We present novel applications of the C-numerical range in quantum control
assisted by gradient flows on the local unitary group: (1) they serve as
powerful tools for deciding whether a quantum interaction can be inverted in
time (in a sense generalising Hahn's famous spin echo); (2) they allow for
optimising witnesses of quantum entanglement. We conclude by relating the
relative C-numerical range to problems of constrained quantum optimisation, for
which we also give Lagrange-type gradient flow algorithms.Comment: update relating to math-ph/070200
A boundary integral formalism for stochastic ray tracing in billiards
Determining the flow of rays or non-interacting particles driven by a force or velocity field is fundamental to modelling many physical processes. These include particle flows arising in fluid mechanics and ray flows arising in the geometrical optics limit of linear wave equations. In many practical applications, the driving field is not known exactly and the dynamics are determined only up to a degree of uncertainty. This paper presents a boundary integral framework for propagating flows including uncertainties, which is shown to systematically interpolate between a deterministic and a completely random description of the trajectory propagation. A simple but efficient discretisation approach is applied to model uncertain billiard dynamics in an integrable rectangular domain
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