N\'eron-Tate heights of cycles on jacobians

We develop a method to calculate the N\'eron-Tate height of tautological integral cycles on jacobians of curves defined over number fields. As examples we obtain closed expressions for the N\'eron-Tate height of the difference surface, the Abel-Jacobi images of the square of the curve, and of any symmetric theta divisor. As applications we obtain a new effective positive lower bound for the essential minimum of any Abel-Jacobi image of the curve and a proof, in the case of jacobians, of a formula proposed by Autissier relating the Faltings height of a principally polarized abelian variety with the N\'eron-Tate height of a symmetric theta divisor.Comment: 35 pages, SAGE file written by David Holmes is available as an ancillary file, v2: minor revision

Constraint-based adaptation for complex space configuration in building services

In this paper an object-based CAD programming is used to take advantage of standardization to handle the schematic design, sizing and layout planning for ceiling mounted fan coil system in a building ceiling void. In order to deal with more complex geometry and real building size, we have used a hybrid approach combining case-based reasoning and constraint programming techniques. Very often, building services engineers use previous solutions and adapt them to new problems. Case-based reasoning mirrors this practical approach and did help us deal effectively with increasingly complex geometry. Our approach combines automation and interactivity. From the specification of the building 3D BIM model, our software prototype proceeds through four steps. First, the user divides the building into zones, each zone being defined by a geometrical primitive (i.e. rectangle zone, triangle zone, curved zone, etc.). Next, for each zone a similar case is retrieved from the case library. The retrieval process will generate a first incomplete 3D solution containing some inconsistencies. Next, the incomplete solution is adapted, using constraint programming techniques, to provide a consistent solution. Finally, distribution routes (i.e. ducts and pipes) are generated using constraint programming techniques. The 3D fan coil solution can be modified or improved by the designer, while providing further contribution by concentrating on interactivity. The project has been funded by the Engineering and Physical Sciences Research Council (EPSRC) in the UK

Chabauty-Coleman experiments for genus 3 hyperelliptic curves

We describe a computation of rational points on genus 3 hyperelliptic curves $C$ defined over $\mathbb{Q}$ whose Jacobians have Mordell-Weil rank 1. Using the method of Chabauty and Coleman, we present and implement an algorithm in Sage to compute the zero locus of two Coleman integrals and analyze the finite set of points cut out by the vanishing of these integrals. We run the algorithm on approximately 17,000 curves from a forthcoming database of genus 3 hyperelliptic curves and discuss some interesting examples where the zero set includes global points not found in $C(\mathbb{Q})$.Comment: 18 page