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# Volume 43 - Issue 2 - December 2018

**• Editorial: All good things
**

**• Connecting immunology and epidemiology**

Edme Soho and Stephen Wirkus

pp. 63–70

**Abstract**

Interaction of the immune system and the pathogens (population of bacteria, viruses, etc.) is considered as a dynamical system. Also, the spread of the disease in the population starts at the individual level. To understand the effect of the endemic environment on the variability of the immune system response, mostly the vulnerability, a model using an integro-differential is introduced from epidemiology to immunology in an effort to assess the impact of the pathogen's spread within the population on the immune system response of the individual. The model gives reasonable results and these suggest that the proportion of infected individuals in the population have an impact on the immune system behavior of an individual.

**• A multivariate log-normal moment closure technique for the**

**stochastic predator**

**–prey**

Tanawat Trakoolthai, Diana Curtis and Jennifer Switkes

pp. 71–81

**Abstract**

The deterministic Lotka--Volterra model is a simple predator–prey model that classically portrays the interaction between two species, leading to closed curves in the predator–prey phase plane. Using the probability generating function, we develop a corresponding stochastic version of this model, which has the form of a simple birth–death process. This stochastic model involves the expected values of the populations, which are governed by a system of differential equations almost identical in form to the deterministic system. However, we find that the stochastic model is no longer a closed system. To gain a more intuitive understanding of this model, we turn to a moment closure approximation technique, which captures the main features of the stochastic model. Assuming that the distribution of the two populations is approximately multivariate log-normal, we use a moment closure technique to obtain a closed system of differential equations for the expected values, multiplicative variances, and multiplicative covariance of the populations.

**•**

**The generating functions of Stirling numbers of the second kind**

**derived probabilistically**

George Kesidis, Takis Konstantopoulos and Michael A. Zazanis

pp. 82–87

**Abstract**

Stirling numbers of the second kind,

*S*(

*n, r*), denote the number of partitions of a finite set of size

*n*into

*r*disjoint nonempty subsets. The aim of this short article is to shed some light on the generating functions of these numbers by deriving them probabilistically. We do this by linking them to Markov chains related to the classical coupon collector problem; coupons are collected in discrete time (ordinary generating function) or in continuous time (exponential generating function). We also review the shortest possible combinatorial derivations of these generating functions.

**• Equally likely dice sums
** Alec Lewald and Amber Rosin

pp. 88–95

**Abstract**

For

*m n*-sided dice, the sums

*m*,

*m*+1,

*m*+2, ...,

*mn*are called the

*standard sums*, and there is an

*equal relabeling*if the dice can be labeled with positive integers in such a way that the standard sums are equally likely to occur. We will consider the case when

*n*=

*pq*for distinct primes

*p*<

*q*. The

*m*for which (

*n*=

*pq*)-sided dice have an equal relabeling have previously been characterized, and it has been shown that all such equal relabelings require stupid dice (dice with 1s on every face). We find the minimum and maximum number of stupid dice that are possible in an equal relabeling of

*pq*-sided dice and show that such a relabeling is possible for any number of stupid dice between the maximum and minimum.

**• Polynomial extensions of two Gibonacci delights and their graph-**

**theoretic confirmations**

Thomas Koshy

pp. 96–108

**Abstract**

We extend two charming Gibonacci identities to their corresponding polynomial versions, investigate their geometric and Jacobsthal implications, and confirm the two charming Gibonacci and Jacobsthal polynomial identities using graph-theoretic tools.

**• Optimal slack times for a bus route schedule with passenger**

**information**

**Jillian Cannons and Joshua Robles**

pp. 109–124

**Abstract**

Public transit timetabling aims to determine the set of departure and arrival times for all trips and all routes in the network. Extra time, known as slack time, is often added into a schedule to mitigate the effects of the random nature of bus travel times. In this paper we develop a stochastic linear program to minimize the expected total schedule deviation for a single bus line. We then incorporate two operational strategies, namely, the drivers' schedule recovery strategy and the holding control strategy, into the linear program. Finally, numerical comparisons of the optimal schedules produced for hypothetical urban and express bus routes are given.

**• ****Testing equality of players in a round-robin tournament****
** F. Thomas Bruss and Thomas S. Ferguson

pp. 125–136

**Abstract**

In a round-robin tournament with

*n*players, each player plays every other player once, resulting in \binom{

*n*}{2} games. Let

*X*

_{ij}denote the score by which player

*i*beats player

*j*, with

*X*

_{ji}=-

*X*

_{ij}for all

*i*\ne

*j*. If we take

*X*

_{ii }= 0 for all

*i*then

*S*

_{i}= \sum_{

*j*= 1}^

*n X*

_{ij}denotes the total score of player

*i*for

*i*= 1,2,...,

*n*. To test the hypothesis, H

_{0}, that the players are equally skillful, in the sense that the

*X*

_{ij}for

*i*<

*j*are independent and identically distributed with mean 0 and common variance, we suggest rejecting v if

*V*= \sum_{

_{n}*i*= 1}^

*n S*

_{i}

^{2}is too large. We show that an associated statistic,

*W*

_{n}, a generalization of the circular triads statistic of Kendall and Babington Smith (1940), is easier to work with and more stable. We establish the asymptotic normality of

*V*and

_{n}*W*

_{n}under general conditions. As an illustration, the results are applied to data from the Greek Soccer League 2016–2017.

**• ****Computer construction conjecture for symmetric Hadamard matrices****
** N. A. Balonin and Jennifer Seberry

pp. 137–143

**Abstract**

We consider and compare methods for computer construction of variously structured plug-in matrices (suitable matrices) used to construct skew-Hadamard matrices from the Goethals–Seidel array and symmetric Hadamard matrices from the Balonin–Seberry array. We call symmetric analogue matrices of suitable matrices for computer construction of skew-Hadamard matrices \textit{luchshie matrices} (luchshie is the Russian plural for `best'). We provide tables of known inequivalent luchshie matrices of order 4

*n*,

*n*\leq 53, and symmetric Hadamard matrices of order 4

*n*,

*n*< 400. We propose the conjecture that there exist luchshie (\pm 1) matrices of order odd

*t*for all

*t*. Hence, there exists a symmetric Hadamard matrix of order 4

*t*for every odd

*t*.

**• Index to Volume 43**

p. 144