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This paper concerns improper λ-colourings of graphs and focuses on the sizes of the monochromatic components (i.e., components of the subgraphs induced by the colour classes). Consider the following three simple operations, which should, heuristically, help reduce monochromatic component size: (a) assign to a vertex the colour that is least popular among its neighbours; (b) change the colours o...

Although Bayes’s theorem demands a prior that is a probability distribution on the parameter space, the calculus associated with Bayes’s theorem sometimes generates sensible procedures from improper priors, Pitman’s estimator being a good example. However, improper priors may also lead to Bayes procedures that are paradoxical or otherwise unsatisfactory, prompting some authors to insist that al...

We first give a geometrical description of the action of the parity operator (P̂ ) on non relativistic spin 12 Pauli spinors in terms of bundle theory. The relevant bundle, SU(2)⊙Z2 → O(3), is a non trivial extension of the universal covering group SU(2) → SO(3). P̂ is the non relativistic limit of the corresponding Dirac matrix operator P = iγ0 and obeys P̂ 2 = −1. Then, from the direct product o...