814 research outputs found
AG Codes from Polyhedral Divisors
A description of complete normal varieties with lower dimensional torus
action has been given by Altmann, Hausen, and Suess, generalizing the theory of
toric varieties. Considering the case where the acting torus T has codimension
one, we describe T-invariant Weil and Cartier divisors and provide formulae for
calculating global sections, intersection numbers, and Euler characteristics.
As an application, we use divisors on these so-called T-varieties to define new
evaluation codes called T-codes. We find estimates on their minimum distance
using intersection theory. This generalizes the theory of toric codes and
combines it with AG codes on curves. As the simplest application of our general
techniques we look at codes on ruled surfaces coming from decomposable vector
bundles. Already this construction gives codes that are better than the related
product code. Further examples show that we can improve these codes by
constructing more sophisticated T-varieties. These results suggest to look
further for good codes on T-varieties.Comment: 30 pages, 9 figures; v2: replaced fansy cycles with fansy divisor
Smooth Parametrizations in Dynamics, Analysis, Diophantine and Computational Geometry
Smooth parametrization consists in a subdivision of the mathematical objects
under consideration into simple pieces, and then parametric representation of
each piece, while keeping control of high order derivatives. The main goal of
the present paper is to provide a short overview of some results and open
problems on smooth parametrization and its applications in several apparently
rather separated domains: Smooth Dynamics, Diophantine Geometry, Approximation
Theory, and Computational Geometry.
The structure of the results, open problems, and conjectures in each of these
domains shows in many cases a remarkable similarity, which we try to stress.
Sometimes this similarity can be easily explained, sometimes the reasons remain
somewhat obscure, and it motivates some natural questions discussed in the
paper. We present also some new results, stressing interconnection between
various types and various applications of smooth parametrization
High-Order AFEM for the Laplace-Beltrami Operator: Convergence Rates
We present a new AFEM for the Laplace-Beltrami operator with arbitrary
polynomial degree on parametric surfaces, which are globally and
piecewise in a suitable Besov class embedded in with . The idea is to have the surface sufficiently well resolved in
relative to the current resolution of the PDE in . This gives
rise to a conditional contraction property of the PDE module. We present a
suitable approximation class and discuss its relation to Besov regularity of
the surface, solution, and forcing. We prove optimal convergence rates for AFEM
which are dictated by the worst decay rate of the surface error in
and PDE error in .Comment: 51 pages, the published version contains an additional glossar
Decomposition into pairs-of-pants for complex algebraic hypersurfaces
It is well-known that a Riemann surface can be decomposed into the so-called
pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3
points. We show that a smooth complex projective hypersurface of arbitrary
dimension admits a similar decomposition. The n-dimensional pair-of-pants is
diffeomorphic to the complex projective n-space minus n+2 hyperplanes.
Alternatively, these decompositions can be treated as certain fibrations on
the hypersurfaces. We show that there exists a singular fibration on the
hypersurface with an n-dimensional polyhedral complex as its base and a real
n-torus as its fiber. The base accomodates the geometric genus of a
hypersurface V. Its homotopy type is a wedge of h^{n,0}(V) spheres S^n.Comment: 35 pages, 9 figures, final version to appear in Topolog
Analysis of Csupra(k)-subdivision surfaces at extraordinary points
This paper gives an analysis of surfaces generated by
subdividing control nets of arbitrary topology. We assume
that the underlying subdivision algorithm is stationary on
the regular parts of the control nets and described by a
matrix iteration around an extraordinary point. For these
subdivision schemes we derive conditions on the spectrum of
the matrix and its generalized eigenvectors such that
surfaces are produced which are regular and k-times
differentiable at their extraordinary points
A narrow-band unfitted finite element method for elliptic PDEs posed on surfaces
The paper studies a method for solving elliptic partial differential
equations posed on hypersurfaces in , . The method allows
a surface to be given implicitly as a zero level of a level set function. A
surface equation is extended to a narrow-band neighborhood of the surface. The
resulting extended equation is a non-degenerate PDE and it is solved on a bulk
mesh that is unaligned to the surface. An unfitted finite element method is
used to discretize extended equations. Error estimates are proved for finite
element solutions in the bulk domain and restricted to the surface. The
analysis admits finite elements of a higher order and gives sufficient
conditions for archiving the optimal convergence order in the energy norm.
Several numerical examples illustrate the properties of the method.Comment: arXiv admin note: text overlap with arXiv:1301.470
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