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# Volume 32 - Issue 1 - June 2007

### Statistical analysis of golden-ratio forms in piano sonatas by Mozart and Haydn

**Jesper Rydén**

Abstract

The golden ratio is occasionally referred to when describing issues of form in various arts. Among musicians, Mozart (1756–1791) is often considered a master of form. Introducing a regression model, we carry out a statistical analysis of possible golden-ratio forms in the musical works of Mozart. The model allows for statistical hypothesis testing and comparison with similar compositions by another master composer, Haydn (1732–1809), of the same epoch.

### A chain of curlicues

**Nelson M. Blachman**

Abstract

The upward-sweeping FM signal represented by exp(it1.4) is sampled at nonnegative integer values of the time t starting with 0. An interesting chain of increasingly more Cornu-like spirals in the complex plane is traced out when the sums of growing numbers of these samples are connected by unit-length line segments. These spirals are shown to result from the aliasing effect of the sampling and the low-pass effect of connecting the dots.

### On the gaps between consecutive primes

**Howard Skogman**

Abstract

In this article we prove some consequences of Dirichlet's theorem on primes in arithmetic progressions. In particular, we consider the gap between consecutive prime numbers. We prove that for a fixed positive integer n, the analytic density of primes that are n less than the next consecutive prime is zero. As a corollary, we have that the set of primes p with the next consecutive prime less than or equal to p + n also has analytic density zero. The paper concludes with some numerical investigations designed to inspire curiosity and conjectures.

### A Parrondo paradox in reliability theory

**Antonio Di Crescenzo**

Abstract

Parrondo's paradox arises in sequences of games in which a winning expectation may be obtained by playing the games in a random order, even though each game in the sequence may be lost when played individually. We present a suitable version of Parrondo's paradox in reliability theory involving two systems in series, the units of the first system being less reliable than those of the second. If the first system is modified so that the distributions of its new units are mixtures of the previous distributions with equal probabilities, then under suitable conditions the new system is shown to be more reliable than the second in the `usual stochastic order' sense.

### Some aspects of gambling with the Kelly criterion

**Ravi Phatarfod**

Abstract

In this paper we consider the problem of gambling with the Kelly criterion, i.e. gambling so as to maximize the expected exponential rate of capital growth. We consider gambling on games of chance such as horse races, as well as gambling involved in the buying and selling of shares on the stock market. For both these situations we obtain results which in some way are surprising and run counter to intuition.

### The effect of alcohol on neuron firing

**Jeannine T. Abiva, Erika T. Camacho, Edna S. Joseph, Arpy K. Mikaelian, Charles R. Rogers, John Shelton and Stephen A. Wirkus**

Abstract

Neurons are responsible for transmitting messages throughout the body via long-distance electrical signals known as action potentials. These depend on the active transport of sodium and potassium ions across the neuron cell membrane. The effect of various drugs on the process of neuron firing is a current research interest. The Hodgkin–Huxley equations, a system of four nonlinear ordinary differential equations, mathematically model the influx and efflux of these ions across the cell membrane. In the presence of alcohol, the release of potassium ions is accelerated. We propose a modified version of these equations, which incorporates the effect of alcohol, and examine its implications through mathematical analysis in dynamical systems. We investigate the qualitative behavior and interpret the results of the steady-state solutions in the fast and fast–slow phase planes.

### More general, further unified, yet more accessible sharing problem results

**Aaron Childs**

Abstract

In this paper we present results of the sharing problem when the population is finite and sampling is without replacement. Our results generalize and further unify the results for sampling with replacement given by Sobel and Frankowski (1994). By using the Dirichlet probability generating function introduced by Sobel and Childs (2002), we unify the probability results with the expected waiting time results. Furthermore, computations for this more general situation using the Dirichlet probability generating function have the advantage that they do not require any special functions to compute, and their construction is quite intuitive.

### A compound gamma distribution with application to drought data

**Saralees Nadarajah**

Abstract

Gamma distributions are some of the most popular models for hydrological processes. If X and Y are gamma distributed independent random variables with the same scale parameter, then it is well known that their sum, X + Y, is also gamma distributed. In this paper, the distribution of S = X + Y is considered, where X and Y are independent and have the compound gamma distribution. The distribution of S is referred to as a generalized compound gamma distribution. Various properties of the distribution are derived, including its moments. A detailed application to drought data from the state of Nebraska is illustrated.

### Identification of ARMA models with GARCH errors

**M. Ghahramani and A. Thavaneswaran**

Abstract

Financial returns are often modelled as autoregressive time series with innovations having conditional heteroscedastic variances, especially with GARCH processes. In this paper, we have extended the results of Thavaneswaran et al. (2005) on the kurtosis and the correlation for a class of stationary ARMA(p, q) models with GARCH errors. The kurtosis can be used to make a suitable choice of the marginal distribution of the innovations. The autocorrelation of the squared process will be useful in identifying the order of the GARCH processes.