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Volume 42 - Issue 1 - April 2017

• Should we opt for the Black Friday discounted price or wait until the
    Boxing Day?
    Jiang Wu, Ričardas Zitikis
    pp. 1–12

Abstract
We derive an optimal strategy for minimizing the expected loss in a two-period economy when a pivotal decision needs to be made during the first time period and cannot be subsequently reversed. Our interest in the problem has been motivated by the classical shopper's dilemma during the Black Friday promotion period, and our solution crucially relies on the pioneering work of McDonnell and Abbott on the two-envelope paradox.

• An exotic binary code
   Rabe-R. von Randow
   pp. 13–19

Abstract
The Laetus barcode or pharmacode comprises a novel way of encoding binary numbers. After an expository introduction, this barcode is studied in detail and an alternative compact way of decoding and encoding pharmacode is given.

• A polynomial extension of Aurifeuille's identity
   Thomas Koshy
   pp. 20–24

Abstract
We establish a polynomial extension of Aurifeuille's identity and investigate its ramifications to the Pell and Jacobsthal families.

• Direct and indirect least-squares approximating polynomials for the
   first derivative function
   T. Van Hecke
   pp. 25–28

Abstract
Finite-difference methods are useful to give a discrete approximation of the derivative function f' based on a set of data points (x_i, f_i), i=0,1,2, . . . ,n. If a continuous function is required to represent the derivative function and only a scatter of data points is available, finite-difference formulae are insufficient. In this paper we describe two different approaches to derive an analytical description of the derivative function based on data points. Their performances are compared on several test functions where Monte Carlo simulations give statistics on the errors.

• A lecture on integration by parts
   John A. Rock
   pp. 29–37

Abstract
Integration by parts (IP) has developed a bad reputation. While it allows us to manage a wide variety of integrals when other methods fall short, its implementation can be thought of as plodding and confusing. However, readers familiar with the tabular method for IP know that it can significantly streamline computations and promote creativity. In this paper the flexibility of the tabular method is explored by approaching it from a different direction and examining a number of examples. This is done in order to showcase the notion that the tabular method can be applied whenever IP is used. The key idea is that each new row represents a new integral and, hence, tables are constructed one row at a time.

• A new look at Pólya's prime gap heuristics
   Anthony Breitzman, Sr.
   pp. 38–42

Abstract
In a very nice paper from 1959, Pólya explored heuristic methods related to prime gaps originally conjectured by Hardy and Littlewood in 1923. Pólya examined heuristics to justify the Hardy–Littlewood conjecture before testing their conjecture by computer program for gaps of size up to 70 and primes p up to 30 million. Following the result of Zhang (2014) we now know that there are an infinite number of primes for some gap with a maximum size of 246. The Hardy–Littlewood and Pólya results suggest a direct relationship between all even gaps, so that confirming the conjectured relationship would prove the twin prime conjecture as well as the broader Polignac conjecture. In this paper we extend Pólya's results to cover gaps up to 256 and extend the computer confirmation for primes up to one trillion. Along the way we revisit the related older results of Polignac, Hardy, and Littlewood, and the newer results of Zhang, Maynard, and others.

• Poisson windowing
   David L. Farnsworth
   pp. 43–50

Abstract
The statistical quality-control technique of windowing takes advantage of the additive property of the Poisson probability distribution and the geometry of the data. There are abundant applications of windowing to observations that naturally occur in arrays or that are produced in them, such as computer chips, which are manufactured many at a time on wafers. Standard statistical procedures are used.

• Stationary distributions of Markov chain models of the bonus-malus
   system
   Kenichi Hirose
   pp. 51–59

Abstract
For a certain type of Markov chain which models the bonus-malus system used in insurance, an algorithm has been developed to compute its stationary distribution. We show that, when penalty increments do not depend on the positive number of claims in the preceding period, finding the stationary distribution reduces to solving a difference equation. Examples are presented, and rates of convergence to stationarity are discussed.