375,319 research outputs found
Dirac-Foldy term and the electromagnetic polarizability of the neutron
We reconsider the Dirac-Foldy contribution to the neutron electric
polarizability. Using a Dirac equation approach to neutron-nucleus scattering,
we review the definitions of Compton continuum (), classical
static (), and Schr\"{o}dinger () polarizabilities
and discuss in some detail their relationship. The latter is the
value of the neutron electric polarizability as obtained from an analysis using
the Schr\"{o}dinger equation. We find in particular , where is the magnitude of the magnetic moment
of a neutron of mass . However, we argue that the static polarizability
is correctly defined in the rest frame of the particle, leading to
the conclusion that twice the Dirac-Foldy contribution should be added to
to obtain the static polarizability .Comment: 11 pages, RevTeX, to appear in Physical Review
On the expected diameter, width, and complexity of a stochastic convex-hull
We investigate several computational problems related to the stochastic
convex hull (SCH). Given a stochastic dataset consisting of points in
each of which has an existence probability, a SCH refers to the
convex hull of a realization of the dataset, i.e., a random sample including
each point with its existence probability. We are interested in computing
certain expected statistics of a SCH, including diameter, width, and
combinatorial complexity. For diameter, we establish the first deterministic
1.633-approximation algorithm with a time complexity polynomial in both and
. For width, two approximation algorithms are provided: a deterministic
-approximation running in time, and a fully
polynomial-time randomized approximation scheme (FPRAS). For combinatorial
complexity, we propose an exact -time algorithm. Our solutions exploit
many geometric insights in Euclidean space, some of which might be of
independent interest
Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties
In the sl\_n case, A. Berenstein and A. Zelevinsky studied the
Sch\"{u}tzenberger involution in terms of Lusztig's canonical basis, [3]. We
generalize their construction and formulas for any semisimple Lie algebra. We
use for this the geometric lifting of the canonical basis, on which an analogue
of the Sch\"{u}tzenberger involution can be given. As an application, we
construct semitoric degenerations of Richardson varieties, following a method
of P. Caldero, [6]Comment: 22 pages, 3 figure
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