Dynamic Programming for Graphs on Surfaces

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2^{O(k log k)} n steps. Our approach combines tools from topological graph theory and analytic combinatorics. In particular, we introduce a new type of branch decomposition called "surface cut decomposition", generalizing sphere cut decompositions of planar graphs introduced by Seymour and Thomas, which has nice combinatorial properties. Namely, the number of partial solutions that can be arranged on a surface cut decomposition can be upper-bounded by the number of non-crossing partitions on surfaces with boundary. It follows that partial solutions can be represented by a single-exponential (in the branchwidth k) number of configurations. This proves that, when applied on surface cut decompositions, dynamic programming runs in 2^{O(k)} n steps. That way, we considerably extend the class of problems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve most previous results in this direction.Comment: 28 pages, 3 figure

Dynamic programming for graphs on surfaces

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2O(k·log k). Our approach combines tools from topological graph theory and analytic combinatorics.Postprint (updated version

Development of a carbon fibre composite active mirror: Design and testing

Carbon fibre composite technology for lightweight mirrors is gaining increasing interest in the space- and ground-based astronomical communities for its low weight, ease of manufacturing, excellent thermal qualities and robustness. We present here first results of a project to design and produce a 27 cm diameter deformable carbon fibre composite mirror. The aim was to produce a high surface form accuracy as well as low surface roughness. As part of this programme, a passive mirror was developed to investigate stability and coating issues. Results from the manufacturing and polishing process are reported here. We also present results of a mechanical and thermal finite element analysis, as well as early experimental findings of the deformable mirror. Possible applications and future work are discussed.Comment: Accepted by Optical Engineering. Figures 1-7 on http://www.star.ucl.ac.uk/~sk/OEpaper_files

From stellar to planetary composition: Galactic chemical evolution of Mg/Si mineralogical ratio

The main goal of this work is to study element ratios that are important for the formation of planets of different masses. We study potential correlations between the existence of planetary companions and the relative elemental abundances of their host stars. We use a large sample of FGK-type dwarf stars for which precise Mg, Si, and Fe abundances have been derived using HARPS high-resolution and high-quality data. A first analysis of the data suggests that low-mass planet host stars show higher [Mg/Si] ratios, while giant planet hosts present [Mg/Si] that is lower than field stars. However, we found that the [Mg/Si] ratio significantly depends on metallicity through Galactic chemical evolution. After removing the Galactic evolution trend only the difference in the [Mg/Si] elemental ratio between low-mass planet hosts and non-hosts was present in a significant way. These results suggests that low-mass planets are more prevalent around stars with high [Mg/Si]. Our results demonstrate the importance of Galactic chemical evolution and indicate that it may play an important role in the planetary internal structure and composition.Comment: Accepted by A&A (Letter to the Editor

Constraining planet structure from stellar chemistry: the cases of CoRoT-7, Kepler-10, and Kepler-93

We explore the possibility that the stellar relative abundances of different species can be used to constrain the bulk abundances of known transiting rocky planets. We use high resolution spectra to derive stellar parameters and chemical abundances for Fe, Si, Mg, O, and C in three stars hosting low mass, rocky planets: CoRoT-7, Kepler-10, and Kepler-93. These planets follow the same line along the mass-radius diagram, pointing toward a similar composition. The derived abundance ratios are compared with the solar values. With a simple stoichiometric model, we estimate the iron mass fraction in each planet, assuming stellar composition. We show that in all cases, the iron mass fraction inferred from the mass-radius relationship seems to be in good agreement with the iron abundance derived from the host star's photospheric composition. The results suggest that stellar abundances can be used to add constraints on the composition of orbiting rocky planets.Comment: A&A Letters, in pres

Halocarbon ozone depletion and global warming potentials

Concern over the global environmental consequences of fully halogenated chlorofluorocarbons (CFCs) has created a need to determine the potential impacts of other halogenated organic compounds on stratospheric ozone and climate. The CFCs, which do not contain an H atom, are not oxidized or photolyzed in the troposphere. These compounds are transported into the stratosphere where they decompose and can lead to chlorine catalyzed ozone depletion. The hydrochlorofluorocarbons (HCFCs or HFCs), in particular those proposed as substitutes for CFCs, contain at least one hydrogen atom in the molecule, which confers on these compounds a much greater sensitivity toward oxidation by hydroxyl radicals in the troposphere, resulting in much shorter atmospheric lifetimes than CFCs, and consequently lower potential for depleting ozone. The available information is reviewed which relates to the lifetime of these compounds (HCFCs and HFCs) in the troposphere, and up-to-date assessments are reported of the potential relative effects of CFCs, HCFCs, HFCs, and halons on stratospheric ozone and global climate (through 'greenhouse' global warming)

Properties of branes in curved spacetimes

A generic property of curved manifolds is the existence of focal points. We show that branes located at focal points of the geometry satisfy special properties. Examples of backgrounds to which our discussion applies are AdS_m x S^n and plane wave backgrounds. As an example, we show that a pair of AdS_2 branes located at the north and south pole of the S^5 in AdS_5 x S^5 are half supersymmetric and that they are dual to a two-monopole solution of N=4 SU(N) SYM theory. Our second example involves spacelike branes in the (Lorentzian) plane wave. We develop a modified lightcone gauge for the open string channel, analyze in detail the cylinder diagram and establish open-closed duality. When the branes are located at focal points of the geometry the amplitude acquires most of the characteristics of flat space amplitudes. In the open string channel the special properties are due to stringy modes that become massless.Comment: 41 pages; v2:typos corrected, ref adde