1,366 research outputs found
Divided Differences of Implicit Functions
Under general conditions, the equation implicitly defines
locally as a function of . In this article, we express divided differences
of in terms of bivariate divided differences of , generalizing a recent
result on divided differences of inverse functions
Reconstruction of a piecewise constant conductivity on a polygonal partition via shape optimization in EIT
In this paper, we develop a shape optimization-based algorithm for the
electrical impedance tomography (EIT) problem of determining a piecewise
constant conductivity on a polygonal partition from boundary measurements. The
key tool is to use a distributed shape derivative of a suitable cost functional
with respect to movements of the partition. Numerical simulations showing the
robustness and accuracy of the method are presented for simulated test cases in
two dimensions
A Pseudopolynomial Algorithm for Alexandrov's Theorem
Alexandrov's Theorem states that every metric with the global topology and
local geometry required of a convex polyhedron is in fact the intrinsic metric
of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a
differential equation whose solution leads to the polyhedron corresponding to a
given metric. We describe an algorithm based on this differential equation to
compute the polyhedron to arbitrary precision given the metric, and prove a
pseudopolynomial bound on its running time. Along the way, we develop
pseudopolynomial algorithms for computing shortest paths and weighted Delaunay
triangulations on a polyhedral surface, even when the surface edges are not
shortest paths.Comment: 25 pages; new Delaunay triangulation algorithm, minor other changes;
an abbreviated v2 was at WADS 200
Intersections on tropical moduli spaces
This article explores to which extent the algebro-geometric theory of
rational descendant Gromov-Witten invariants can be carried over to the
tropical world. Despite the fact that the tropical moduli-spaces we work with
are non-compact, the answer is surprisingly positive. We discuss the string,
divisor and dilaton equations, we prove a splitting lemma describing the
intersection with a "boundary" divisor and we prove general tropical versions
of the WDVV resp. topological recursion equations (under some assumptions). As
a direct application, we prove that the toric varieties ,
, and with Psi-conditions only
in combination with point conditions, the tropical and classical descendant
Gromov-Witten invariants coincide (which extends the result for
in Markwig-Rau-2008). Our approach uses tropical intersection theory and can
unify and simplify some parts of the existing tropical enumerative geometry
(for rational curves).Comment: 40 pages, 17 Postscript figures; updated to fit the published versio
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