12,964 research outputs found
Computing largest circles separating two sets of segments
A circle separates two planar sets if it encloses one of the sets and its
open interior disk does not meet the other set. A separating circle is a
largest one if it cannot be locally increased while still separating the two
given sets. An Theta(n log n) optimal algorithm is proposed to find all largest
circles separating two given sets of line segments when line segments are
allowed to meet only at their endpoints. In the general case, when line
segments may intersect times, our algorithm can be adapted to
work in O(n alpha(n) log n) time and O(n \alpha(n)) space, where alpha(n)
represents the extremely slowly growing inverse of the Ackermann function.Comment: 14 pages, 3 figures, abstract presented at 8th Canadian Conference on
Computational Geometry, 199
Quantum Separability and Entanglement Detection via Entanglement-Witness Search and Global Optimization
We focus on determining the separability of an unknown bipartite quantum
state by invoking a sufficiently large subset of all possible
entanglement witnesses given the expected value of each element of a set of
mutually orthogonal observables. We review the concept of an entanglement
witness from the geometrical point of view and use this geometry to show that
the set of separable states is not a polytope and to characterize the class of
entanglement witnesses (observables) that detect entangled states on opposite
sides of the set of separable states. All this serves to motivate a classical
algorithm which, given the expected values of a subset of an orthogonal basis
of observables of an otherwise unknown quantum state, searches for an
entanglement witness in the span of the subset of observables. The idea of such
an algorithm, which is an efficient reduction of the quantum separability
problem to a global optimization problem, was introduced in PRA 70 060303(R),
where it was shown to be an improvement on the naive approach for the quantum
separability problem (exhaustive search for a decomposition of the given state
into a convex combination of separable states). The last section of the paper
discusses in more generality such algorithms, which, in our case, assume a
subroutine that computes the global maximum of a real function of several
variables. Despite this, we anticipate that such algorithms will perform
sufficiently well on small instances that they will render a feasible test for
separability in some cases of interest (e.g. in 3-by-3 dimensional systems)
Nonlinear Dirac Equations
We construct nonlinear extensions of Dirac's relativistic electron equation
that preserve its other desirable properties such as locality, separability,
conservation of probability and Poincar\'e invariance. We determine the
constraints that the nonlinear term must obey and classify the resultant
non-polynomial nonlinearities in a double expansion in the degree of
nonlinearity and number of derivatives. We give explicit examples of such
nonlinear equations, studying their discrete symmetries and other properties.
Motivated by some previously suggested applications we then consider nonlinear
terms that simultaneously violate Lorentz covariance and again study various
explicit examples. We contrast our equations and construction procedure with
others in the literature and also show that our equations are not gauge
equivalent to the linear Dirac equation. Finally we outline various physical
applications for these equations
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