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# Volume 35 - Issue 2 - December 2010

**• Time series analysis to forecast temperature change****
** Tanja Van Hecke

pp. 63–69

**Abstract**

ARIMA models are often used to model the evolution in time of economic issues. We demonstrate that an ARIMA model is also valuable in the environmental field, where the evolution of climate change is causing many concerns. Can we confirm global warming by mathematical prediction theories?

**• A predator–prey model with spiteful hawk–dove predators****
** Noelle Sio, Jennifer Switkes

pp. 70–82

**Abstract**

In this paper, we focus on predator behavior in a predator–prey model. We assume that the predator–prey interactions take place on a slow time scale in comparison to predator behavioral dynamics, with the prey model equation taking a classic Lotka–Volterra form. Using replicator equations, we couple fast time scale classical hawk–dove predator dynamics with spiteful predation, that is, with predators willing to harm themselves in order to harm their counterparts even more. By assuming a mixture of spiteful and nonspiteful individuals, we observe the bifurcation effects of various proportions of spiteful individuals.

**• Random processes, social groups, and households****
** C. W. Lloyd-Smith, V. E. Jennings

pp. 83–95

**Abstract**

This paper uses random processes to develop mathematical models of size distributions for families and households. Some links are given to ideas in Cohen (1972) on social groups of primates, and with Goodman

*et al*. (1974), (1975) on family formation. Little's law from queueing theory yields relationships between relevant populations, birth rates, life expectancy, and other demographic variables. The superposition of point processes arising from these demographic variables generates Poisson inputs, each of which is associated with a family or household. Under steady state conditions, truncated Poisson distributions are found for household and family size. Numerical results are provided.

**• A generalization of the vernier algorithm****
** Rabe-R. von Randow

pp. 96–100

**Abstract**

After a short description of the vernier and brief biographies of the two scientists connected with it, we present an algorithm which generalizes the vernier algorithm and has a number of new and surprising properties not possessed by the vernier algorithm.

**• Collision of one-dimensional random walks****
** David K. Neal

pp. 101–110

**Abstract**

Let

*X*and

*Y*be simple random walks with initial heights

*j*

*and*

_{X}*j*

*, where*

_{Y}*j*<

_{X }*j*and

_{Y}*j*–

_{Y}*j*is even. The processes independently move upward or downward one unit at a time on each simultaneous step with the probability of

_{X}*X*moving upward being

*p*

*and the probability of*

_{X}*Y*moving upward being

*p*

*. We give the condition for which paths almost surely attain the same height, and for this condition we derive the average number of steps needed for paths to collide at the same height and the average height at the time of collision.*

_{Y}**• Gambler's ruin? Some aspects of coin tossing****
** Porter W. Johnson, David Atkinson

pp. 111–121

**Abstract**

What is the average number of coin tosses needed before a particular sequence of heads and tails first turns up? This problem is solved in our paper, starting with doubles; a tail, followed by a head, turns up on average after only four tosses, while six tosses are needed for two successive heads. The method is extended to encompass the triples head-tail-tail and head-head-tail, but head-tail-head and head-head-head are surprisingly more recalcitrant. However, the general case is finally solved by using a new algorithm, even for relatively long strings. It is shown that the average number of tosses is always an even integer.

**• The perimeter of an ellipse****
** Tirupathi R. Chandrupatla, Thomas J. Osler

pp. 122–131

**Abstract**

There is no simple way to calculate the perimeter of an ellipse. In this expository paper we review and compare four methods of evaluating this perimeter. These methods are known by the names Maclaurin, Gauss–Kummer, Cayley, and Euler. Maclaurin's method is derived in detail, while the other three are simply described. A short introduction to hypergeometric functions is included, since these functions are useful for finding the elliptic perimeter. We compare the usefulness of these four calculations, An appendix gives MICROSOFT EXCEL

^{®}programs for computing the perimeter.

**• Letter to the Editor: The inverse law of large numbers****
** Oscar Sheynin

pp. 132–133