Complete Sudoku - extra rules for more symmetry

We propose some extra rules to add to the well-known Sudoku puzzle and present an argument to justify their inclusion. The rules mean that puzzles can be created with fewer cells completed initially yet which still have only one solution. We have created a Web-based program which can be used to generate and solve both standard and extended (Complete) puzzles.

Estimating the masses of extra-solar planets

All extra-solar planet masses that have been derived spectroscopically are lower limits since the inclination of the orbit to our line-of-sight is unknown except for transiting systems. In theory, however, it is possible to determine the inclination angle, i, between the rotation axis of a star and an observer's line-of-sight from measurements of the projected equatorial velocity (v sin i), the stellar rotation period (P(rot)) and the stellar radius (R(*)). For stars which host planetary systems this allows the removal of the sin i dependency of extra-solar planet masses derived from spectroscopic observations under the assumption that the planetary orbits lie perpendicular to the stellar rotation axis.
We have carried out an extensive literature search and present a catalogue of v sin i, P(rot) and R(*) estimates for stars hosting extra-solar planets. In addition, we have used Hipparcos parallaxes and the Barnes-Evans relationship to further supplement the R(*) estimates obtained from the literature. Using this catalogue, we have obtained sin i estimates using a Markov-chain Monte Carlo analysis. This technique allows proper 1 Sigma two-tailed confidence limits to be placed on the derived sin i's along with the transit probability for each planet to be determined.
While we find that a small proportion of systems yield sin i's significantly greater than 1, most likely due to poor P(rot) estimations, the large majority are acceptable. We are further encouraged by the cases where we have data on transiting systems, as the technique indicates inclinations of similar to 90 degrees and high transit probabilities. In total, we are able to estimate the true masses of 133 extra-solar planets. Of these 133 extra-solar planets, only six have revised masses that place them above the 13M(J) deuterium burning limit; four of those six extra-solar planet candidates were already suspected to lie above the deuterium burning limit before correcting their masses for the sin i dependency. Our work reveals a population of high-mass extra-solar planets with low eccentricities, and we speculate that these extra-solar planets may represent the signature of different planetary formation mechanisms at work. Finally, we discuss future observations that should improve the robustness of this technique.

Annotated Bibliography: Grande Bretagne / Great Britain: Part Four: Administrative Law texts

(with C. N. Doe).

Annotated Bibliography: Grande Bretagne / Great Britain: Part One The Formative Writers.

(With C. N. Doe.)

Annotated Bibliography: Grande Bretagne / Great Britain: Part Two The Textbook Tradition.

(With C.N. Doe.)

Annotated Bibliography: Grande Bretagne / Great Britain: Part Three: The Problem of Rights and the Literature of Constitutional Reform

(With C.N. Doe.)