Defective and Clustered Graph Colouring

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in the Electronic Journal of Combinatoric

Colouring the Square of the Cartesian Product of Trees

We prove upper and lower bounds on the chromatic number of the square of the cartesian product of trees. The bounds are equal if each tree has even maximum degree

The Physics Case for Extended Tevatron Running

Run II of the Tevatron collider at Fermilab is currently scheduled to end late in 2011. Given the current performance of the collider and of the CDF and D0 detectors, it is estimated that the current data set could be approximately doubled with a run extended into 2014. A few examples are presented of the physics potential of these additional statistics. These are discussed in the context of the expected reach of the LHC 7 TeV data and the existing Tevatron data. In particular, an extraordinary opportunity is described which could probe the existence of a standard model Higgs boson with mass in the currently preferred region between 115 GeV and 150 GeV.Comment: contribution to proceedings of HCP2010 - Toront

The pulsation modes, masses and evolution of luminous red giants

The period-luminosity sequences and the multiple periods of luminous red giant stars are examined using the OGLE III catalogue of long-period variables in the Large Magellanic Cloud. It is shown that the period ratios in individual multimode stars are systematically different from the ratios of the periods at a given luminosity of different period-luminosity sequences. This leads to the conclusion that the masses of stars at the same luminosity on the different period-luminosity sequences are different. An evolutionary scenario is used to show that the masses of stars on adjacent sequences differ by about 16-26% at a given luminosity, with the shorter period sequence being more massive. The mass is also shown to vary across each sequence by a similar percentage, with the mass increasing to shorter periods. On one sequence, sequence B, the mass distribution is shown to be bimodal. It is shown that the small amplitude variables on sequences A', A and B pulsate in radial and nonradial modes of angular degree l=0, 1 and 2, with the l=1 mode being the most common. The stars on sequences C' and C are predominantly radial pulsators (l=0). Matching period ratios to pulsation models shows that the radial pulsation modes associated with sequences A', A, B, C' and C are the 4th, 3rd, 2nd and 1st overtones and the fundamental mode, respectively.Comment: 16 pages, 10 figures, 1 tabl

The controllability of the aeroassist flight experiment atmospheric skip trajectory

The Aeroassist Flight Experiment (AFE) will be the first vehicle to simulate a return from geosynchronous orbit, deplete energy during an aerobraking maneuver, and navigate back out of the atmosphere to a low earth orbit It will gather scientific data necessary for future Aeroasisted Orbitl Transfer Vehicles (AOTV's). Critical to mission success is the ability of the atmospheric guidance to accurately attain a targeted post-aeropass orbital apogee while nulling inclination errors and compensating for dispersions in state, aerodynamic, and atmospheric parameters. In typing to satisfy mission constraints, atmospheric entry-interface (EI) conditions, guidance gains, and trajectory. The results of the investigation are presented; emphasizing the adverse effects of dispersed atmospheres on trajectory controllability

The aerospace developments concept

The viability of using airships for the transport of natural gas, and the initial design of such a system, the airship and its associated subsystems together with a continuing economic analysis of the project were investigated. Investigations, on a funded basis, were also carried out into the application of the airship for A.S.W. and A.E.W. uses, and a further investigation into the transport of mineral concentrates for an Australasian mining concern was completed