157,316 research outputs found
A characterization of shortest geodesics on surfaces
Any finite configuration of curves with minimal intersections on a surface is
a configuration of shortest geodesics for some Riemannian metric on the
surface. The metric can be chosen to make the lengths of these geodesics equal
to the number of intersections along them.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-17.abs.htm
Circles in the Sky: Finding Topology with the Microwave Background Radiation
If the universe is finite and smaller than the distance to the surface of
last scatter, then the signature of the topology of the universe is writ large
on the microwave background sky. We show that the microwave background will be
identified at the intersections of the surface of last scattering as seen by
different ``copies'' of the observer. Since the surface of last scattering is a
two-sphere, these intersections will be circles, regardless of the background
geometry or topology. We therefore propose a statistic that is sensitive to all
small, locally homogeneous topologies. Here, small means that the distance to
the surface of last scatter is smaller than the ``topology scale'' of the
universe.Comment: 14 pages, 10 figures, IOP format. This paper is a direct descendant
of gr-qc/9602039. To appear in a special proceedings issue of Class. Quant.
Grav. covering the Cleveland Topology & Cosmology Worksho
Mapping class group orbits of curves with self-intersections
We study mapping class group orbits of homotopy and isotopy classes of curves
with self-intersections. We exhibit the asymptotics of the number of such
orbits of curves with a bounded number of self-intersections, as the complexity
of the surface tends to infinity. We also consider the minimal genus of a
subsurface that contains the curve. We determine the asymptotic number of
orbits of curves with a fixed minimal genus and a bounded self-intersection
number, as the complexity of the surface tends to infinity. As a corollary of
our methods, we obtain that most curves that are homotopic are also isotopic.
Furthermore, using a theorem by Basmajian, we get a bound on the number of
mapping class group orbits on a given a hyperbolic surface that can contain
short curves. For a fixed length, this bound is polynomial in the signature of
the surface. The arguments we use are based on counting embeddings of ribbon
graphs.Comment: 16 pages, 1 figure, generalized main resul
Magnus subgroups of one-relator surface groups
A one-relator surface group is the quotient of an orientable surface group by
the normal closure of a single relator. A Magnus subgroup is the fundamental
group of a suitable incompressible sub-surface. A number of results are proved
about the intersections of such subgroups and their conjugates, analogous to
results of Bagherzadeh, Brodskii, and Collins in classical one-relator group
theory.Comment: 15 pages, 3 figure
Complete intersection singularities of splice type as universal abelian covers
It has long been known that every quasi-homogeneous normal complex surface
singularity with Q-homology sphere link has universal abelian cover a Brieskorn
complete intersection singularity. We describe a broad generalization: First,
one has a class of complete intersection normal complex surface singularities
called "splice type singularities", which generalize Brieskorn complete
intersections. Second, these arise as universal abelian covers of a class of
normal surface singularities with Q-homology sphere links, called
"splice-quotient singularities". According to the Main Theorem,
splice-quotients realize a large portion of the possible topologies of
singularities with Q-homology sphere links. As quotients of complete
intersections, they are necessarily Q-Gorenstein, and many Q-Gorenstein
singularities with Q-homology sphere links are of this type. We conjecture that
rational singularities and minimally elliptic singularities with Q-homology
sphere links are splice-quotients. A recent preprint of T Okuma presents
confirmation of this conjecture.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper17.abs.htm
Smooth quasi-developable surfaces bounded by smooth curves
Computing a quasi-developable strip surface bounded by design curves finds
wide industrial applications. Existing methods compute discrete surfaces
composed of developable lines connecting sampling points on input curves which
are not adequate for generating smooth quasi-developable surfaces. We propose
the first method which is capable of exploring the full solution space of
continuous input curves to compute a smooth quasi-developable ruled surface
with as large developability as possible. The resulting surface is exactly
bounded by the input smooth curves and is guaranteed to have no
self-intersections. The main contribution is a variational approach to compute
a continuous mapping of parameters of input curves by minimizing a function
evaluating surface developability. Moreover, we also present an algorithm to
represent a resulting surface as a B-spline surface when input curves are
B-spline curves.Comment: 18 page
Induced Growth of Asymmetric Nanocantilever Arrays on Polar Surfaces
©2003 The American Physical Society. The electronic version of this article is the complete one and can be found online at: http://link.aps.org/doi/10.1103/PhysRevLett.91.185502DOI: 10.1103/PhysRevLett.91.185502We report that the Zn-terminated ZnO (0001) polar surface is chemically active and the oxygenterminated (0001) polar surface is inert in the growth of nanocantilever arrays. Longer and wider "comblike" nanocantilever arrays are grown from the (0001)-Zn surface, which is suggested to be a self-catalyzed process due to the enrichment of Zn at the growth front. The chemically inactive
(0001)-O surface typically does not initiate any growth, but controlling experimental conditions could lead to the growth of shorter and narrower nanocantilevers from the intersections between (0001)-O with (0110) surfaces
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