Let C be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon ∆. It is classical that the geometric genus of C equals the number of lattice points in the interior of ∆. In this paper we give similar combinatorial interpretations for the gonality, the Clifford index and the Clifford dimension, by removing a technical assumption from a recent...

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. moreover, w...

A distribution whose normalization constant is an A-hypergeometric polynomial is called an A-hypergeometric distribution. Such a distribution is in turn a generalization of the generalized hypergeometric distribution on the contingency tables with fixed marginal sums. In this paper, we will see that an A-hypergeometric distribution with a homogeneous matrix of two rows, especially, that associa...

We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the “elastic” operator. In the homogeneous case, we investigate the phase spaces in which the initial value problem gives rise to a semigroup, and the further regularity of solutions. In the nonhomogeneous case, we study how the regularity of solutions depends on th...

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 ...