This article discusses prime numbers, defined as integers greater than 1 that are divisible only by only themselves and the number 1. A positive integer greater than 1 that is not a prime is called composite. The number 1 itself is considered neither prime nor composite. As the name suggests, prime numbers are one of the most basic but important…

The problem of rotating a matrix orthogonally to a best least squares fit with another matrix of the same order has a closed-form solution based on a singular value decomposition. The optimal rotation matrix is not necessarily rigid, but may also involve a reflection. In some applications, only rigid rotations are permitted. Gower (1976) has…

A simple, direct condition is formulated for determining the mode(s) of a probability mass function. This condition is then applied to the Poisson and hypergeometric mass functions.

Euler gave a simple method for showing that [zeta](2)=1/1[superscript 2] + 1/2[superscript 2] + 1/3[superscript 2] + ... = [pi][superscript 2]/6. He generalized his method so as to find [zeta](4), [zeta](6), [zeta](8),.... His computations became increasingly more complex as the arguments increased. In this note we show a different generalization…

This study provides an in-depth analysis of children's use of a taught mental calculation strategy. Three children (aged eight to nine years) who demonstrated contrasting spontaneous calculation approaches, were interviewed following the direct instruction of a calculation strategy. Their responses are explored in relation to constructivist and…

For the casino game Keno we determine optimal playing strategies. To decide such optimal strategies, both exact (hypergeometric) and approximate probability calculations are used. The approximate calculations are obtained via the Central Limit Theorem and simulation, and an important lesson about the application of the Central Limit Theorem is…

This note describes a procedure for estimating the range restriction component used in correcting correlations for unreliability and range restriction when an estimate of the reliability of a predictor is not readily available for the unrestricted sample. This procedure is illustrated with a few examples. (Contains 1 table.)

Some simple, nonoptimized coefficients (e.g., correlation weights, equal weights) were pitted against regression in extensive prediction competitions. After drawing calibration samples from large supersets of real and synthetic data, the researchers observed which set of sample-derived coefficients made the best predictions when applied back to…

This article presents two methods of estimating a study's replicability that researchers should consider reporting along with their statistical significant and effect size findings. One method of estimating the replicability of the findings deals with replication in the exact same system. The second method, which may contain subjective probability…

The emerging field of Evolutionary Computation (EC), inspired by neo-Darwinian principles (e.g. natural selection, mutation, etc.), offers developmental psychologists a wide array of mathematical tools for simulating ontogenetic processes. In this brief review, I begin by highlighting three of the approaches that EC researchers employ (Artificial…

We investigate the pendulum equation [theta] + [lambda][squared] sin [theta] = 0 and two approximations for it. On the one hand, we suggest that the third and fifth-order Taylor series approximations for sin [theta] do not yield very good differential equations to approximate the solution of the pendulum equation unless the initial conditions are…

In this paper we have converted the Laplace transform into an integral equation of the first kind of convolution type, which is an ill-posed problem, and used a statistical regularization method to solve it. The method is applied to three examples. It gives a good approximation to the true solution and compares well with the method given by…

Three level-implicit finite difference methods of order four are discussed for the numerical solution of the mildly quasi-linear second-order hyperbolic equation A(x, t, u)u[subscript xx] + 2B(x, t, u)u[subscript xt] + C(x, t, u)u[subscript tt] = f(x, t, u, u[subscript x], u[subscript t]), 0 less than x less than 1, t greater than 0 subject to…

An attempt is made to put the notion of sample quartiles on a mathematical footing in the light of ranks of observations, and equisegmentation property that the quartiles divide ordered sample observations into four segments leaving the same number of observations in each if all the observations are distinct. Sample quartiles provided by the…

It is shown that highly elliptical orbits, such as those of comets, can be explained well in terms of energy rather than forces. The principle of conservation of energy allows a comet's velocity to be calculated at aphelion and perihelion. An example asks students to calculate whether they can run fast enough to escape from a small asteroid.…