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...

We show that the no betting characterisation of the existence of common priors over finite type spaces extends only partially to improper priors in the countably infinite state space context: the existence of a common prior implies the absence of a bounded agreeable bet, and the absence of a common improper prior implies the existence of a bounded agreeable bet. However, a type space that lacks...

For a zero mean, proper, complex random vector x, the Hermitian covariance Exx is a complete second-order characterization. However, if the vector x is improper, it is correlated with its complex conjugate, meaning Exx 6= 0. This improper or complementary covariance must be accounted for in a complete second-order characterization. The improper covariance has been exploited for widely linear (W...

Improper choosability of planar graphs has been widely studied. In particular, Škrekovski investigated the smallest integer gk such that every planar graph of girth at least gk is k-improper 2-choosable. He proved [9] that 6 ≤ g1 ≤ 9; 5 ≤ g2 ≤ 7; 5 ≤ g3 ≤ 6 and ∀k ≥ 4, gk = 5. In this paper, we study the greatest real M(k, l) such that every graph of maximum average degree less than M(k, l) is ...