20,701 research outputs found
Geodesic systems of tunnels in hyperbolic 3-manifolds
It is unknown whether an unknotting tunnel is always isotopic to a geodesic
in a finite volume hyperbolic 3-manifold. In this paper, we address the
generalization of this problem to hyperbolic 3-manifolds admitting tunnel
systems. We show that there exist finite volume hyperbolic 3-manifolds with a
single cusp, with a system of at least two tunnels, such that all but one of
the tunnels come arbitrarily close to self-intersecting. This gives evidence
that systems of unknotting tunnels may not be isotopic to geodesics in tunnel
number n manifolds. In order to show this result, we prove there is a
geometrically finite hyperbolic structure on a (1;n)-compression body with a
system of core tunnels such that all but one of the core tunnels
self-intersect.Comment: 19 pages, 4 figures. V2 contains minor updates to references and
exposition. To appear in Algebr. Geom. Topo
A correction to the enhanced bottom drag parameterisation of tidal turbines
Hydrodynamic modelling is an important tool for the development of tidal
stream energy projects. Many hydrodynamic models incorporate the effect of
tidal turbines through an enhanced bottom drag. In this paper we show that
although for coarse grid resolutions (kilometre scale) the resulting force
exerted on the flow agrees well with the theoretical value, the force starts
decreasing with decreasing grid sizes when these become smaller than the length
scale of the wake recovery. This is because the assumption that the upstream
velocity can be approximated by the local model velocity, is no longer valid.
Using linear momentum actuator disc theory however, we derive a relationship
between these two velocities and formulate a correction to the enhanced bottom
drag formulation that consistently applies a force that remains closed to the
theoretical value, for all grid sizes down to the turbine scale. In addition, a
better understanding of the relation between the model, upstream, and actual
turbine velocity, as predicted by actuator disc theory, leads to an improved
estimate of the usefully extractable energy. We show how the corrections can be
applied (demonstrated here for the models MIKE 21 and Fluidity) by a simple
modification of the drag coefficient
Distributed Exact Shortest Paths in Sublinear Time
The distributed single-source shortest paths problem is one of the most
fundamental and central problems in the message-passing distributed computing.
Classical Bellman-Ford algorithm solves it in time, where is the
number of vertices in the input graph . Peleg and Rubinovich (FOCS'99)
showed a lower bound of for this problem, where
is the hop-diameter of .
Whether or not this problem can be solved in time when is
relatively small is a major notorious open question. Despite intensive research
\cite{LP13,N14,HKN15,EN16,BKKL16} that yielded near-optimal algorithms for the
approximate variant of this problem, no progress was reported for the original
problem.
In this paper we answer this question in the affirmative. We devise an
algorithm that requires time, for , and time, for larger . This
running time is sublinear in in almost the entire range of parameters,
specifically, for . For the all-pairs shortest paths
problem, our algorithm requires time, regardless of
the value of .
We also devise the first algorithm with non-trivial complexity guarantees for
computing exact shortest paths in the multipass semi-streaming model of
computation.
From the technical viewpoint, our algorithm computes a hopset of a
skeleton graph of without first computing itself. We then conduct
a Bellman-Ford exploration in , while computing the required edges
of on the fly. As a result, our algorithm computes exactly those edges of
that it really needs, rather than computing approximately the entire
Theory of small charge solitons in one-dimensional arrays of Josephson junctions
We identify and investigate the new parameter regime of small charge solitons
in one-dimensional arrays of Josephson junctions. We obtain the dispersion
relation of the soliton and show that it unexpectedly flattens in the outer
region of the Brillouin zone. We demonstrate Lorentz contraction of the soliton
in the middle of the Brillouin zone as well as broadening of the soliton in the
flat band regime.Comment: 4 pages, 6 figures, final published versio
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