30,715 research outputs found
Dirichlet sigma models and mean curvature flow
The mean curvature flow describes the parabolic deformation of embedded
branes in Riemannian geometry driven by their extrinsic mean curvature vector,
which is typically associated to surface tension forces. It is the gradient
flow of the area functional, and, as such, it is naturally identified with the
boundary renormalization group equation of Dirichlet sigma models away from
conformality, to lowest order in perturbation theory. D-branes appear as fixed
points of this flow having conformally invariant boundary conditions. Simple
running solutions include the paper-clip and the hair-pin (or grim-reaper)
models on the plane, as well as scaling solutions associated to rational (p, q)
closed curves and the decay of two intersecting lines. Stability analysis is
performed in several cases while searching for transitions among different
brane configurations. The combination of Ricci with the mean curvature flow is
examined in detail together with several explicit examples of deforming curves
on curved backgrounds. Some general aspects of the mean curvature flow in
higher dimensional ambient spaces are also discussed and obtain consistent
truncations to lower dimensional systems. Selected physical applications are
mentioned in the text, including tachyon condensation in open string theory and
the resistive diffusion of force-free fields in magneto-hydrodynamics.Comment: 77 pages, 21 figure
Witness (Delaunay) Graphs
Proximity graphs are used in several areas in which a neighborliness
relationship for input data sets is a useful tool in their analysis, and have
also received substantial attention from the graph drawing community, as they
are a natural way of implicitly representing graphs. However, as a tool for
graph representation, proximity graphs have some limitations that may be
overcome with suitable generalizations. We introduce a generalization, witness
graphs, that encompasses both the goal of more power and flexibility for graph
drawing issues and a wider spectrum for neighborhood analysis. We study in
detail two concrete examples, both related to Delaunay graphs, and consider as
well some problems on stabbing geometric objects and point set discrimination,
that can be naturally described in terms of witness graphs.Comment: 27 pages. JCCGG 200
Integrability and action operators in quantum Hamiltonian systems
For a (classically) integrable quantum mechanical system with two degrees of
freedom, the functional dependence of the
Hamiltonian operator on the action operators is analyzed and compared with the
corresponding functional relationship in
the classical limit of that system. The former is shown to converge toward the
latter in some asymptotic regime associated with the classical limit, but the
convergence is, in general, non-uniform. The existence of the function
in the integrable regime of a parametric
quantum system explains empirical results for the dimensionality of manifolds
in parameter space on which at least two levels are degenerate. The comparative
analysis is carried out for an integrable one-parameter two-spin model.
Additional results presented for the (integrable) circular billiard model
illuminate the same conclusions from a different angle.Comment: 9 page
Ordered Level Planarity, Geodesic Planarity and Bi-Monotonicity
We introduce and study the problem Ordered Level Planarity which asks for a
planar drawing of a graph such that vertices are placed at prescribed positions
in the plane and such that every edge is realized as a y-monotone curve. This
can be interpreted as a variant of Level Planarity in which the vertices on
each level appear in a prescribed total order. We establish a complexity
dichotomy with respect to both the maximum degree and the level-width, that is,
the maximum number of vertices that share a level. Our study of Ordered Level
Planarity is motivated by connections to several other graph drawing problems.
Geodesic Planarity asks for a planar drawing of a graph such that vertices
are placed at prescribed positions in the plane and such that every edge is
realized as a polygonal path composed of line segments with two adjacent
directions from a given set of directions symmetric with respect to the
origin. Our results on Ordered Level Planarity imply -hardness for any
with even if the given graph is a matching. Katz, Krug, Rutter and
Wolff claimed that for matchings Manhattan Geodesic Planarity, the case where
contains precisely the horizontal and vertical directions, can be solved in
polynomial time [GD'09]. Our results imply that this is incorrect unless
. Our reduction extends to settle the complexity of the Bi-Monotonicity
problem, which was proposed by Fulek, Pelsmajer, Schaefer and
\v{S}tefankovi\v{c}.
Ordered Level Planarity turns out to be a special case of T-Level Planarity,
Clustered Level Planarity and Constrained Level Planarity. Thus, our results
strengthen previous hardness results. In particular, our reduction to Clustered
Level Planarity generates instances with only two non-trivial clusters. This
answers a question posed by Angelini, Da Lozzo, Di Battista, Frati and Roselli.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Convective stabilization of a Laplacian moving boundary problem with kinetic undercooling
We study the shape stability of disks moving in an external Laplacian field
in two dimensions. The problem is motivated by the motion of ionization fronts
in streamer-type electric breakdown. It is mathematically equivalent to the
motion of a small bubble in a Hele-Shaw cell with a regularization of kinetic
undercooling type, namely a mixed Dirichlet-Neumann boundary condition for the
Laplacian field on the moving boundary. Using conformal mapping techniques,
linear stability analysis of the uniformly translating disk is recast into a
single PDE which is exactly solvable for certain values of the regularization
parameter. We concentrate on the physically most interesting exactly solvable
and non-trivial case. We show that the circular solutions are linearly stable
against smooth initial perturbations. In the transformation of the PDE to its
normal hyperbolic form, a semigroup of automorphisms of the unit disk plays a
central role. It mediates the convection of perturbations to the back of the
circle where they decay. Exponential convergence to the unperturbed circle
occurs along a unique slow manifold as time . Smooth temporal
eigenfunctions cannot be constructed, but excluding the far back part of the
circle, a discrete set of eigenfunctions does span the function space of
perturbations. We believe that the observed behaviour of a convectively
stabilized circle for a certain value of the regularization parameter is
generic for other shapes and parameter values. Our analytical results are
illustrated by figures of some typical solutions.Comment: 19 pages, 7 figures, accepted for SIAM J. Appl. Mat
Bragg spectroscopy of an accelerating condensate with solitary-wave behaviour
We present a theoretical treatment of Bragg spectroscopy of an accelerating
condensate in a solitary-wave state. Our treatment is based on the
Gross-Pitaevskii equation with an optical potential representing the Bragg
pulse and an additional external time-dependent potential generating the
solitary-wave behaviour. By transforming to a frame translating with the
condensate, we derive an approximate set of equations that can be readily
solved to generate approximate Bragg spectra. Our analytic method is accurate
within a well defined parameter regime and provides physical insight into the
structure of the spectra. We illustrate our formalism using the example of
Bragg spectroscopy of a condensate in a time-averaged orbiting potential trap.Comment: 9 pages, 3 figure
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