1. Introduction By " discreteness " of a spatial model we generally understand that in any bounded neighbourhood, or (bounded) region, there are only finitely many elements of the carrier of the model. Thus the model should be (at least) locally finite. Another way of putting it is that the bounded regions should not be infinitely subdivisible. As is well known, Aristotle argued (Physics: III,6...

Allowing for variations in behaviour across respondents is one of the most fundamental principles in discrete choice modelling, given that the assumption of a purely homogeneous population cannot in general be seen to be valid. Two approaches have classically been used to address this problem; the use of deterministic segmentations of the population, and the use of a random continuous represent...

Discrete choice models are employed (as the name suggests) to explain a discrete choice that an individual makes. For example, such a choice might be whether or not to join the labor force, whether or not to buy a car, whether or not to continue on to graduate school, whether or not to accept a job with a consulting …rm, or whether or not to sign up with an internet provider. Later in the cours...

We investigate a probabilistic cellular automaton model which has been introduced recently. This model describes single-lane traac ow on a ring and generalizes the asymmetric exclusion process models. We study the equilibrium properties and calculate the so-called fundamental diagrams ((ow vs. density) for parallel dynamics. This is done numerically by computer simulations of the model and by m...

The model introduced in this article is designed to provide a consistent representation for both the real-world and pricing measures for the credit process. We find that good agreement with historical and market data can be achieved across all credit ratings simultaneously. The model is characterized by an underlying stochastic process that takes on values on a discrete lattice and represents c...

This paper introduces a class of Schur-constant survival models, of dimension n, for arithmetic non-negative random variables. Such a model is defined through a univariate survival function that is shown to be n-monotone. Two general representations are obtained, by conditioning on the sum of the n variables or through a doubly mixed multinomial distribution. Several other properties including ...

The mathematical theory of single outbreak epidemic models really began with the work of Kermack and Mackendrick about decades ago. This gave a simple answer to the long-standing question of why epidemics woould appear suddenly and then disappear just as suddenly without having infected an entire population. Therefore it seemed natural to expect that theoreticians would immediately proceed to e...

Physical modelling is the process of finding a mathematical description that is consistent with observations of the physical world. Many models, from the heat equation to the Schrödinger equation, are formulated in the continuous language of differential equations. Space and time are thought of as a continuum, hence the models potentially specify detail down to arbitrarily fine scales. Discrete...

Discrete choice models have played an important role in transportation modeling for the last 25 years. They are namely used to provide a detailed representation of the complex aspects of transportation demand, based on strong theoretical justiications. Moreover, several packages and tools are available to help practionners using these models for real applications, making discrete choice models ...