Volume 39 - Issue 2 - December 2014

Cumulative incidence rates of cancer

C. T. Lenard, T. M. Mills and Ruth F. G. Williams

Abstract

There is widespread interest in the risk of being diagnosed with cancer. Internationally, especially in developed countries, governments collect and use data on the incidence of cancer for strategic planning to ensure that the nation has the resources that will be required to deal with the disease. Incidence data can also be used to assess the effectiveness of public health campaigns. However, there are several measures for quantifying the incidence of cancer. This paper examines one of them, namely the cumulative incidence rate. We present a review of the method for estimating the cumulative incidence rate of cancer in a population, and for comparing these rates in two populations. We explore the connection between the cumulative incidence rate and the cumulative risk of being diagnosed with cancer by a certain age, with details of the mathematical ideas that underpin these concepts. This expository paper has been written to be useful to researchers and policy makers who have an interest in the incidence of cancer, at a local or regional level, and who wish to understand the details associated with cumulative incidence rate and cumulative risk.

Models for homeless housing

Joe Gani, Natalie Gasca and Randall J. Swift

Abstract

This paper examines deterministic and stochastic population models for the homeless. United States Housing and Urban Development data are used to construct the transition rates in the models presented.

Can divergent series be of value?

Thomas J. Osler

Abstract

We introduce four methods of dealing with divergent series that have been shown to be of value in analysis. These are Cesaro summation, Abel summation, Borel summation, and asymptotic series. We mention how Euler used divergent series to discover the famous functional equation for the zeta function. This is an expository paper to gently introduce this subject to readers who are not familiar with divergent series.

Measuring the lack of monotonicity in functions

Danang Teguh Qoyyimi and Ricardas Zitikis

Abstract

Numerous problems in econometrics, insurance, reliability engineering, and statistics rely on the assumption that certain functions are non-decreasing. To satisfy this requirement, researchers frequently model the underlying phenomena using parametric and semi-parametric families of functions, thus effectively specifying the required shapes of the functions. To tackle these problems in a non-parametric way, in this paper we suggest indices for measuring the lack of monotonicity in functions. We investigate properties of the indices and offer a convenient computational technique for practical use.

Thresholds for seat apportionment methods

T. Van Hecke

Abstract

In this paper we investigate the different seat apportionment methods from a mathematical point of view. The number of votes counted when elections take place has to be transformed into a number of seats for each participating party. Different systems exist: the quota systems (namely the Hare quota, the Hagenbach-Bischoff quota, the Imperiali quota, or the Droop quota) and the highest average methods, but they give varying results. By modeling the thresholds for obtaining seats with a variety of methods used in contemporary elections, we show that it is possible to analyse the relation between the amplitude of the party and its possible benefits from the different methods.

'Codons' instead of Pythagorean triples: an extension of Pythagorean theory

Mels Sluyser

Abstract

An extension of Pythagorean triples is proposed, with a new generalized definition named codon, which satisfies not only right-angled triangles with integer sides but also those with noninteger sides. By introducing codons we show that right-angled triangles with sides an/2, bn/2, and cn/2 (where a, b, and c are integers) satisfy an + bn = cn with n = 1 or n = 2, but not with an integer n > 2.

On the labelling of polyhedral dice

Andrew D. Irving and Ebrahim L. Patel

Abstract

We extend previous work which modelled the rolling of a typical playing die using a Markov matrix. The conditional probability of a roll's result (i.e. the final uppermost face) varies according to a standard n-faced die's initial position. However, for some standard n-faced dice, we can label the faces in such a way that the conditional probability of a roll's outcome (i.e. the final score) does not vary according to the die's initial position. In such cases, we say that the die is fair under that labelling. Here, we derive general conditions that such labellings must satisfy. Using these conditions, we identify specific examples of fair polyhedral dice.

A higher look at vertical motion

Thomas J. Osler

Abstract

We extend the simple problem of vertical motion near the surface of a planet without air resistance by using a first-order approximation from the force of gravity. A simple result using no transcendental functions is obtained for this improved approximate solution. This nice approximate result is compared with the cumbersome exact solution.