Topology of 2D and 3D Rational Curves

In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or use the implicit equation of the curve (in the case of planar curves) or of any projection (in the case of space curves). Moreover, these algorithms have been implemented in Maple; the examples considered and the timings obtained show good performance skills.Comment: 26 pages, 19 figure

Computing the topology of a planar or space hyperelliptic curve

We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the space. We report on a {\tt Maple} implementation of these algorithms, and present several examples. Complexity and certification issues are also discussed.Comment: 34 pages, lot of figure

Nearly free divisors and rational cuspidal curves

We define a class of plane curves which are close to the free divisors and such that conjecturally it contains the class of rational cuspidal curves. Using a recent result by U. Walther we show that any unicuspidal rational curve with a unique Puiseux pair is either free or belongs to this class.Comment: v3: title modified and the topological results strengthened. In particular, any rational cuspidal curve whose degree is either even or a prime power, or which has an abelian group, is a nearly free diviso

Logarithmic asymptotics of the genus zero Gromov-Witten invariants of the blown up plane

We study the growth of the genus zero Gromov-Witten invariants GW_{nD} of the projective plane P^2_k blown up at k points (where D is a class in the second homology group of P^2_k). We prove that, under some natural restrictions on D, the sequence log GW_{nD} is equivalent to lambda n log n, where lambda = D.c_1(P^2_k).Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper14.abs.htm

Lagrangian flow structures in 3D AC electro-osmotic microflows

This paper was presented at the 3rd Micro and Nano Flows Conference (MNF2011), which was held at the Makedonia Palace Hotel, Thessaloniki in Greece. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, Aristotle University of Thessaloniki, University of Thessaly, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute.Flow forcing by AC electro-osmosis (ACEO) is a promising technique for actuation and manipulation of microflows. Utilisation to date mainly concerns pumping and mixing. However, emerging micro-fluidics applications demand further functionalities. The present study explores first ways by which to systematically realise this in three-dimensional (3D) microflows using ACEO. This exploits the fact that continuity “organises” Lagrangian fluid trajectories into coherent structures that geometrically determine the transport properties. 3D Lagrangian flow structures typically comprise families of concentric (closed) streamtubes, acting both as transport barriers and transport conduits, embedded in chaotic regions. Representative case studies demonstrate that ACEO, possibly in combination with other forcing mechanisms, has the potential to tailor these features into multi-functional Lagrangian flow structures for various transport purposes. This may pave the way to “labs-within-a-channel” that offer the wide functionality of labs-on-a-chip yet within one microflow instead of within an integrated system