The envelope of straight lines normal to a plane curve C is its evolute; the envelope of lines tangent to C is the original curve, together with the entire tangent line at each inflexion of C. We introduce some standard techniques of singularity theory and use them to explain how the first of these envelopes turns into the second, as the (constant) angle between the set of lines forming the env...

A cyclic urn is an urn model for balls of types 0, . . . ,m − 1 where in each draw the ball drawn, say of type j, is returned to the urn together with a new ball of type j + 1 mod m. The case m = 2 is the well-known Friedman urn. The composition vector, i.e., the vector of the numbers of balls of each type after n steps is, after normalization, known to be asymptotically normal for 2 ≤ m ≤ 6. F...

The dynamics of tumour evolution are not well understood. In this paper we provide a statistical framework for evaluating the molecular variation observed in different parts of a colorectal tumour. A multi-sample version of the Ewens Sampling Formula forms the basis for our modelling of the data, and we provide a simulation procedure for use in obtaining reference distributions for the statisti...

In this paper we propose adaptive finite element methods for computing the band structure of 2D periodic photonic crystals and of photonic crystal fibres, modelled as spectral problems for Maxwell’s equations under either TM or TE polarisation. With the application of the Floquet transform the problem of computing the spectrum can be reduced to the computation of the discrete spectra of each me...

In 2006, Suzuki, and Akbari & Alipour independently presented a necessary and sufficient condition for edge-colored graphs to have a heterochromatic spanning tree, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors. In this paper, we propose f -chromatic graphs as a generalization of heterochromatic graphs. An edge-colored graph is f -chromatic if each col...

We develop a new and general notion of parametric measure models and statistical models on an arbitrary sample space Ω which does not assume that all measures of the model have the same null sets. This is given by a differentiable map from the parameter manifold M into the set of finite measures or probability measures on Ω, respectively, which is differentiable when regarded as a map into the ...

The distribution of the “mixing time” or the “time to stationarity” in a discrete time irreducible Markov chain, starting in state i, can be defined as the number of trials to reach a state sampled from the stationary distribution of the Markov chain. Expressions for the probability generating function, and hence the probability distribution of the mixing time starting in state i are derived an...

Let M be an R-module and 0 6= f ∈ M∗ = Hom(M, R). The graph Γf (M) is a graph with vertices Z f (M) = {x ∈ M \ {0} | xf(y) = 0 or yf(x) = 0 for some non-zero y ∈ M}, in which non-zero elements x and y are adjacent provided that xf(y) = 0 or yf(x) = 0, which introduced and studied in [3]. In this paper we associate an undirected submodule based graph ΓfN (M) for each submodule N of M with vertic...

In this paper we introduce the notion of classical quasi-primary submodules that generalizes the concept of classical primary submodules. Then, we investigate decomposition and minimal decomposition into classical quasi-primary submodules. In particular, existence and uniqueness of classical quasi-primary decompositions in finitely generated modules over Noetherian rings are proved. More...

Given a linear equation L, a set A ⊆ [n] is L-free if A does not contain any ‘non-trivial’ solutions to L. We determine the precise size of the largest L-free subset of [n] for several general classes of linear equations L of the form px+ qy = rz for fixed p, q, r ∈ N where p ≥ q ≥ r. Further, for all such linear equations L, we give an upper bound on the number of maximal L-free subsets of [n]...