Volume 32 - Issue 2 - December 2007

The challenge for the piano maker

R. S. Anderssen and W. D. Stuart


For bowed and wind instruments, responsibility for maintaining sustained harmonicity rests with the player. Guaranteeing the corresponding sound quality for percussive instruments, such as the piano, rests with the instrument maker (see Dain (2002, pp. 16, 18)). Traditionally, piano strings are horizontally zig-zag clamped to the bridge. In the Australian manufactured Stuart & Sons pianos, vertical zig-zag clamping gives a unique and enhanced harmonicity (see Dain (2002, pp. 19, 22)). To explain this, it is necessary to turn to the nonlinear string equation which, by allowing for minute changes in the lengths of the strings as they vibrate, couples their natural tranverse motion with the longitudinal vibrations within and along the strings (Bank and Sujbert (2005)). The structure of this nonlinear equation identifies why vertically zig-zag clamped piano strings emit a sound different from that of horizontally clamped strings. As a corollary, this explains Weinreich's seminal observations about the elliptically polarized motions of piano strings (Weinreich (1977), (1979)). These findings highlight the need to turn to nonlinear models when subtle matters arise, as, for example, in a comparison of different modifications to the same class of instruments such as violins, horns, or pianos.

What can be said about the remainder in a divergent series?

R. Willink


The most familiar results about truncation error in power-series expansions apply only to convergent series. In this paper we place a bound on the truncation error in some divergent Taylor series and give a corresponding bound on the expected value of a differentiable function of an unbounded random variable. We also discuss what can be said about the remainder in a truncated asymptotic expansion.

Lost in space

W. L. Champion, T. M. Mills and Simon J. Smith


In 1921, Georg Pólya proved the remarkable result that, in one or two dimensions, the probability that a random walk will return to its starting point is 1; however, in three or more dimensions, there is a positive probability that a random walk will not return to its starting point. In this paper we present a self-contained, elementary exposition of Pólya's theorem using only undergraduate-level mathematics.

Computing the order of strength in bridge team tournaments where each team does not compete against every other team

G. Dirksen and M. Stieglitz


In bridge team tournaments there often is not enough time for each team to compete against every other team. Nevertheless, in order to obtain a final ranking order we modify a procedure proposed by Zermelo (1929) for chess tournaments. This algorithm can easily be implemented on any computer. Previous tests have proved that the differences of the so-called ‘international match points’ occurring during rank computation are normally distributed with constant variance, whereas their expectations depend on the corresponding pair of teams competing against each other. We propose an algorithm to estimate this expectation from the known differences of the International Match Points, including those for teams that have not met during the tournament.

On feedback quality control with EWMA

K. Triantafyllopoulos


This paper discusses some aspects of exponentially weighted moving average (EWMA) schemes in feedback adjustment for process improvement. A sensitivity analysis shows that the choice of the parameters should be carried out very carefully. An example indicates that different measures of dispersion yield different optimal parameter values. In particular, it is suggested that a robust statistic based on the median is more appropriate than the usual mean square error (MSE). We also consider an autoregressive time series model for modelling the deviations from target and this model is compared with the usual EWMA model. The problem of applying feedback adjustment to a controllable process is discussed, and it is shown via a simulation that the correlation structure of the data of interest plays a critical role in whether the MSE after adjustment is lower than the MSE before adjustment.

On the optimal control of a multidimensional simple epidemic process

E. G. Kyriakidis and A. Pavitsos


We consider a deterministic simple epidemic process in which the susceptibles are exposed to n + 1 diseases. It is assumed that one disease causes serious symptoms while the others are relatively harmless. Policies for isolating infectives with the serious disease are considered and the optimal policy that minimises the future cost for every initial state is found. For the corresponding stochastic model, the optimal policy is found by implementing a suitable dynamic programming algorithm, and is compared numerically with the optimal policy for the deterministic model.

On the spread of carrier-borne epidemics under the influence of carrier immigration

Jennifer Sun and Randall J. Swift


In this article, a model for the stochastic carrier-borne epidemic with carrier immigration is presented. It extends the work of Weiss (1965) for the carrier-borne epidemic model. A simplified model is presented in detail along with an outline for the model in its full generalization. Several illustrative numerical examples are given.

A simple optimisation method for clinical trials

W. F. Scott


A simple method of minimising the effects of prognostic factors in clinical trials is put forward. The method makes use of multiple regression procedures and may easily be put into practice using, say, SPSS®. We give a practical example involving lung cancer patients.

Transient solution for a queue with state- and time-dependent rates

P. R. Parthasarathy and R. Sudhesh


The transient system size probabilities are obtained in closed form for a queue with state- and time-dependent arrival and service rates by using generating functions. An exact expression for the expected system size is derived and numerical illustrations are presented.