8,589 research outputs found
Minimum Polygonal Separation
Coordinated Science Laboratory was formerly known as Control Systems LaboratoryNational Science Foundation / ECS 84-10902Amoco Fnd. Fac. Dev. Comput. Sci
Majorana states in prismatic core-shell nanowires
We consider core-shell nanowires with conductive shell and insulating core,
and with polygonal cross section. We investigate the implications of this
geometry on Majorana states expected in the presence of proximity-induced
superconductivity and an external magnetic field. A typical prismatic nanowire
has a hexagonal profile, but square and triangular shapes can also be obtained.
The low-energy states are localized at the corners of the cross section, i.e.
along the prism edges, and are separated by a gap from higher energy states
localized on the sides. The corner localization depends on the details of the
shell geometry, i.e. thickness, diameter, and sharpness of the corners. We
study systematically the low-energy spectrum of prismatic shells using
numerical methods and derive the topological phase diagram as a function of
magnetic field and chemical potential for triangular, square, and hexagonal
geometries. A strong corner localization enhances the stability of Majorana
modes to various perturbations, including the orbital effect of the magnetic
field, whereas a weaker localization favorizes orbital effects and reduces the
critical magnetic field. The prismatic geometry allows the Majorana zero-energy
modes to be accompanied by low-energy states, which we call pseudo Majorana,
and which converge to real Majoranas in the limit of small shell thickness. We
include the Rashba spin-orbit coupling in a phenomenological manner, assuming a
radial electric field across the shell.Comment: 14 pages, 16 figures, accepted for publication in Phys. Rev.
Interacting electrons in polygonal quantum dots
The low-lying eigenstates of a system of two electrons confined within a
two-dimensional quantum dot with a hard polygonal boundary are obtained by
means of exact diagonalization. The transition from a weakly correlated charge
distribution for small dots to a strongly correlated "Wigner molecule" for
large dots is studied, and the behaviour at the crossover is determined. In
sufficiently large dots, a recently proposed mapping to an effective
charge-spin model is investigated, and is found to produce the correct ordering
of the energy levels and to give a good first approximation to the size of the
level spacings. We conclude that this approach is a valuable method to obtain
the low energy spectrum of few-electron quantum dots
Two-Dimensional Pursuit-Evasion in a Compact Domain with Piecewise Analytic Boundary
In a pursuit-evasion game, a team of pursuers attempt to capture an evader.
The players alternate turns, move with equal speed, and have full information
about the state of the game. We consider the most restictive capture condition:
a pursuer must become colocated with the evader to win the game. We prove two
general results about pursuit-evasion games in topological spaces. First, we
show that one pursuer has a winning strategy in any CAT(0) space under this
restrictive capture criterion. This complements a result of Alexander, Bishop
and Ghrist, who provide a winning strategy for a game with positive capture
radius. Second, we consider the game played in a compact domain in Euclidean
two-space with piecewise analytic boundary and arbitrary Euler characteristic.
We show that three pursuers always have a winning strategy by extending recent
work of Bhadauria, Klein, Isler and Suri from polygonal environments to our
more general setting.Comment: 21 pages, 6 figure
Lipschitz spaces and M-ideals
For a metric space the Banach space \Lip(K) consists of all
scalar-valued bounded Lipschitz functions on with the norm
, where is the Lipschitz constant
of . The closed subspace \lip(K) of \Lip(K) contains all elements of
\Lip(K) satisfying the \lip-condition . For , , we
prove that \lip(K) is a proper -ideal in a certain subspace of \Lip(K)
containing a copy of .Comment: Includes 4 figure
Efficient Multi-Robot Motion Planning for Unlabeled Discs in Simple Polygons
We consider the following motion-planning problem: we are given unit
discs in a simple polygon with vertices, each at their own start position,
and we want to move the discs to a given set of target positions. Contrary
to the standard (labeled) version of the problem, each disc is allowed to be
moved to any target position, as long as in the end every target position is
occupied. We show that this unlabeled version of the problem can be solved in
time, assuming that the start and target positions are at
least some minimal distance from each other. This is in sharp contrast to the
standard (labeled) and more general multi-robot motion-planning problem for
discs moving in a simple polygon, which is known to be strongly NP-hard
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