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Frontiers in complex dynamics
Rational maps on the Riemann sphere occupy a distinguished niche in the
general theory of smooth dynamical systems. First, rational maps are
complex-analytic, so a broad spectrum of techniques can contribute to their
study (quasiconformal mappings, potential theory, algebraic geometry, etc.).
The rational maps of a given degree form a finite-dimensional manifold, so
exploration of this {\em parameter space} is especially tractable. Finally,
some of the conjectures once proposed for {\em smooth} dynamical systems (and
now known to be false) seem to have a definite chance of holding in the arena
of rational maps.
In this article we survey a small constellation of such conjectures centering
around the density of {\em hyperbolic} rational maps --- those which are
dynamically the best behaved. We discuss some of the evidence and logic
underlying these conjectures, and sketch recent progress towards their
resolution.Comment: 18 pages. Abstract added in migration
A recursively feasible and convergent Sequential Convex Programming procedure to solve non-convex problems with linear equality constraints
A computationally efficient method to solve non-convex programming problems
with linear equality constraints is presented. The proposed method is based on
a recursively feasible and descending sequential convex programming procedure
proven to converge to a locally optimal solution. Assuming that the first
convex problem in the sequence is feasible, these properties are obtained by
convexifying the non-convex cost and inequality constraints with inner-convex
approximations. Additionally, a computationally efficient method is introduced
to obtain inner-convex approximations based on Taylor series expansions. These
Taylor-based inner-convex approximations provide the overall algorithm with a
quadratic rate of convergence. The proposed method is capable of solving
problems of practical interest in real-time. This is illustrated with a
numerical simulation of an aerial vehicle trajectory optimization problem on
commercial-of-the-shelf embedded computers
Remarks on endomorphisms and rational points
Let X be a variety over a number field and let f: X --> X be an "interesting"
rational self-map with a fixed point q. We make some general remarks concerning
the possibility of using the behaviour of f near q to produce many rational
points on X. As an application, we give a simplified proof of the potential
density of rational points on the variety of lines of a cubic fourfold
(originally obtained by Claire Voisin and the first author in 2007).Comment: LaTeX, 22 pages. v2: some minor observations added, misprints
corrected, appendix modified
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