# Volume 36 - Issue 1 - June 2011

A hexagonal tiles recursion motivated by a problem in biology
Christine M. O'Keefe, Sergiy Pereverzyev, Jr., Robert S. Anderssen
pp. 19

Abstract
In modelling the genetic signalling, communication, and switching (GSCS) which occurs between cells, the goal is the identification of possible mechanisms that can explain the known biology. An appropriate model system is the initiation and positioning of trichomes (hairs) on Arabidopsis leaves. A model is proposed that approximates the epidermal leaf cells (or groups of these cells) by hexagonal tiles and assumes that the signalling occurs only between certain adjacent tiles. For the signalling and communication between the tiles, a simple algebraic recursion, called the hexagonal tiles recursion, is used to model the accumulation of the relevant signal. The switching of the state of a tile, which represents an epidermal cell that becomes a trichome, is controlled by a threshold condition. The goal of this paper is the construction of an explicit solution for the hexagonal tiles recursion.

On the lower bounds of a symmetric inequality involving roots
Tuan Le
pp. 1018

Abstract
In their article in the journal Mathematical Reflections, Campos Daniel and Verdiyan Verdan (2009) proposed an interesting and difficult inequality, which has appeared on several occasions in the MathLinks forum (http://www.artofproblemsolving.com/Forum/index.php), without any references to its solution. In this paper, we present two original solutions to this inequality problem. We also consider various generalizations of this inequality from various perspectives. Finally, we apply the general results to solve some other difficult inequalities.

Random processes, social groups, and households. II
C. W. Lloyd-Smith, V. E. Jennings
pp. 1929

Abstract
This paper demonstrates how Poisson processes can be modified so that the observed distributions of household and family size in a human population are modelled more accurately than in recent work using a homogeneous Poisson model. Poisson-like models with size-dependent rates are found to arise as steady state distributions of queueing systems. These have state-dependent Poisson inputs, each of which is associated with a family or household. Numerical results are given to demonstrate the accuracy of the Poisson model. A fixed point property for the Poisson distribution is also proved.

Bernoulli numbers and probability distributions: some connections
D. S. Broca
pp. 3036

Abstract
Bernoulli numbers are a remarkable sequence of rational numbers which find application in seemingly disparate areas of mathematics. In this paper the definition and basic properties of Bernoulli numbers are reviewed. These are then applied to computing the mean, variance, and shape characteristics of some well-known probability distributions.

On Halley's correction to the Newton-Raphson method
Tanja Van Hecke
pp. 3742

Abstract
Probability density functions of statistical variables are fitted by choosing the best values for the unknown parameters based on sample values of the variable. The maximum likelihood approach often leads to a system of nonlinear equations to be solved for the parameters. We discuss the numerical methods of Newton-Raphson and Halley to obtain such parameter estimates. We present a two-dimensional implementation of Halley's method, and discuss the computation cost in one and more dimensions in the case of a defect detection function.

Testing if a distribution is a mixture of gammas
pp. 4346

Abstract
Suppose that we observe X = sXti with probability pi, 1 <= i <= c, where sumi=1c pi = 1 and Xt ~ Ft, a one-parameter family. Here, we provide the first parametric test of the hypothesis that Ft has some particular form; in particular that Ft is a gamma distribution.

Further aspects of gambling with the Kelly criterion
Ravi Phatarfod
pp. 4756

Abstract
We consider two aspects of gambling with the Kelly criterion. Firstly, we show that for a wide range of final results for a series of games, the Kelly bettor would be worse off compared to a flat bets bettor. Secondly, we consider, for the Kelly bettor, situations similar to the gambler's (or his opponent's) ruin in the classical gambler's ruin problem. Here the end points are a reduction of the gambler's capital to a fraction, or its growth to a certain multiple, of its original value.

An analysis of the disadvantage to players of multiple decks in the
game of twenty-one
Leslie M. Golden
pp. 5769

Abstract
To counter the advantage that players obtain by using card counting systems in the popular casino game of blackjack, casinos utilize multiple decks of cards instead of a single 52-card deck. Intuitively, players discover that their advantage decreases markedly. An analysis based on the central limit theorem and a Monte Carlo simulation quantifies the disadvantage both with the use of multiple decks and as the dealer progresses through the deck pack. Strategies to minimize the effect of the multiple decks are presented. A step-wise betting strategy based on these findings is found to increase the player's expected winnings by a factor of up to 2.6.

Obituary: David Blackwell
Randall Swift
pp. 7071

Letter to the Editor: On moments of the geometric distribution
Stuart Friedman, Kamta Rai
pp. 7274