This paper investigates a general identity which expresses an apparently complicated and intriguing sum of fractions as an elegant and straightforward sum of simple terms.

A method for the numerical evaluation of the error term in Gaussian quadrature rules is derived by means of Chebyshev polynomials of the first kind.

This article considers skills and abilities of young children with number in the areas of cardinality and ordinality. Responses to number tasks by a sample of 3- and 4-year-old children are described to outline their differing approaches to determining the numerosity of a set, and to establish their application of ordinality language to describe…

An alternative formulation of the multigroup common factor model with minimal uniqueness constraints is considered. This alternative formulation is based on a simple identification constraint that is related to the standard maximum likelihood constraint used in single-group common factor analysis. It is argued that the alternative formulation…

Eggerth and Andrew (2006 [this issue]) provide a valuable contribution to researchers and practitioners using the C index to measure congruence when person or environment codes contain fewer than three letters. However, the computational procedures outlined by Eggerth and Andrew may be more complex than they need be. A simpler alternative to…

The matrix diagonalization method is used to solve a limit problem.

Formulas presented for the calculation of [Summation of n over j=1] j[superscript k] (n, k [is a member of] N) do not have a closed form; they are in the form of recursive or complex formulas. Here an attempt is made to present a simple formula in which it is only necessary to compute the numerical coefficients in a recursive form, and the…

This article considers optimal sample allocation between the treatment and control condition in multilevel designs when the costs per sampling unit vary due to treatment assignment. Optimal unequal allocation may reduce the cost from that of a balanced design without sacrificing any power. The optimum sample allocation ratio depends only on the…

Kappa (Cohen, 1960) is a popular agreement statistic used to estimate the accuracy of observers. The response of kappa to differing baserates was examined and methods for estimating the accuracy of observers presented. Results suggest that setting a single value of kappa as "minimally acceptable" (i.e., a criterion value) is not useful in ensuring…

The concept of risk has entered into physics courses in various guises. It is treated explicitly in the "Advancing Physics" [1] course and implicitly at GCSE through "Ideas and Evidence" discussions. This could easily lead to such ideas as the balance between treatment and risk in radiotherapy and the likelihood of an asteroid strike. In this…

The absorbance decay of hexacyanoferrate(III) as a function of time shows a progressive deviation from zero to first order. This variation follows an experimental rate law that has been analyzed. The change in reaction order is due to a change in the relative rate of substrate oxidation with respect to that of catalyst regeneration. (Contains 2…

This article describes the combinatorial thinking involved in computing one's score in the game of cribbage. It deals only with counting the hand; it does not describe the game in its entirety.

This article employs the Archimedean method of estimating the value of pi within an inscribed pentagon. We show how to write these approximations in terms of the golden ration.

Methods for planning sample size (SS) for the standardized mean difference so that a narrow confidence interval (CI) can be obtained via the accuracy in parameter estimation (AIPE) approach are developed. One method plans SS so that the expected width of the CI is sufficiently narrow. A modification adjusts the SS so that the obtained CI is no…

Accurate and accessible expressions for the probability of having a matching birthday are obtained by using Stirling's formula and Taylor series. For interest, the results are applied to the planets of our solar system.