7,938 research outputs found
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
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
Extended matter coupled to BF theory
Recently, a topological field theory of membrane-matter coupled to BF theory
in arbitrary spacetime dimensions was proposed [1]. In this paper, we discuss
various aspects of the four-dimensional theory. Firstly, we study classical
solutions leading to an interpretation of the theory in terms of strings
propagating on a flat spacetime. We also show that the general classical
solutions of the theory are in one-to-one correspondence with solutions of
Einstein's equations in the presence of distributional matter (cosmic strings).
Secondly, we quantize the theory and present, in particular, a prescription to
regularize the physical inner product of the canonical theory. We show how the
resulting transition amplitudes are dual to evaluations of Feynman diagrams
coupled to three-dimensional quantum gravity. Finally, we remove the regulator
by proving the topological invariance of the transition amplitudes.Comment: 27 pages, 7 figure
Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory
This paper is a concise introduction to virtual knot theory, coupled with a
list of research problems in this field.Comment: 65 pages, 24 figures. arXiv admin note: text overlap with
arXiv:math/040542
Learning Redundant Motor Tasks With and Without Overlapping Dimensions: Facilitation and Interference Effects
Prior learning of a motor skill creates motor memories that can facilitate or interfere with learning of new, but related, motor skills. One hypothesis of motor learning posits that for a sensorimotor task with redundant degrees of freedom, the nervous system learns the geometric structure of the task and improves performance by selectively operating within that task space. We tested this hypothesis by examining if transfer of learning between two tasks depends on shared dimensionality between their respective task spaces. Human participants wore a data glove and learned to manipulate a computer cursor by moving their fingers. Separate groups of participants learned two tasks: a prior task that was unique to each group and a criterion task that was common to all groups. We manipulated the mapping between finger motions and cursor positions in the prior task to define task spaces that either shared or did not share the task space dimensions (x-y axes) of the criterion task. We found that if the prior task shared task dimensions with the criterion task, there was an initial facilitation in criterion task performance. However, if the prior task did not share task dimensions with the criterion task, there was prolonged interference in learning the criterion task due to participants finding inefficient task solutions. These results show that the nervous system learns the task space through practice, and that the degree of shared task space dimensionality influences the extent to which prior experience transfers to subsequent learning of related motor skills
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