Classroom note: Sawtooth functions

Using MAPLE enables students to consider many examples which would be very tedious to work out by hand. This applies to graph plotting as well as to algebraic manipulation. The challenge is to use these observations to develop the students’ understanding of mathematical concepts. In this note an interesting relationship arising from inverse trigonometric functions is analysed. To understand what is going on students have to develop an understanding of how to deal with inverses where a function is not 1–1, by restricting the domain. The piece of work developed here also provides some interesting exercises in proof by induction.     

Classroom note: On the telescoping sum

The telescoping sum constitutes a powerful technique for summing series. In this note, this technique is illustrated by a series of problems starting off with some simple ones in arithmetic, then some in trigonometry, famous families of numbers, Apéry-like formulas, and finally ending with a class of problems that are solved by computer.     

Classroom note: Polynomial solutions of the Laguerre equation and other differential equations near a singular point

It can be shown that if a differential equation is analytic near a point, then it is always possible to select a forcing term along with initial conditions that will ensure the solution to the new non-homogeneous equation is a polynomial that is the finite, truncated portion of the (infinite) series solution of the original equation. It turns out that this result can be extended to expansions about a singular point. The conditions under which such a polynomial truncation can be accomplished about a singular point are presented in the Appendix. A brief algorithm is described that enables one to choose the appropriate forcing term and initial conditions. Following this, an example involving Laguerre's equation is presented.     

Classroom note: A model for considering multicollinearity

In this work a simple and convenient model is proposed for studying features of the multicollinearity effect in regression analysis. Using some reasonable approximations a multivariate regression is reduced to a kind of bivariate regression by each predictor. This approach yields some criteria for identifying the cases of evident multicollinearity, and leads to a better understanding of properties of multiple regression.     

Classroom note: Building simple hidden Markov models

Hidden Markov models (HMMs) are widely used in bioinformatics, speech recognition and many other areas. This note presents HMMs via the framework of classical Markov chain models. A simple example is given to illustrate the model. An estimation method for the transition probabilities of the hidden states is also discussed.     

Classroom note: On postulates of a group

A semigroup G is a group if it has a left identity and every element has a left inverse. The purpose of this note is to weaken this condition further in two different ways.     

Classroom note: The integrity of certain series

In this note, using the theory of Pell equation, the authors discuss the integrity of certain series involving generalized Fibonacci and Lucas numbers.     

Classroom note: The cardinality of even perfect numbers

We provide an existential proof of the fact that there are infinitely many even perfect numbers. For this we shall use some basic concepts of arithmetic and other known results (without proof) of classical Number Theory.     

Constructing new differentially 4-uniform permutations from known ones

In this paper we study a new construction of differentially 4-uniform permutations from known ones and the inverse function. We focus on constructing methods of [20]. We split a finite field into its subfield and remainder, and, we choose known differentially 4-uniform permutations over the subfield and the inverse function over the entire field. As a result, we obtain two families of differentially 4-uniform permutations.     

Classroom note: Inattentive drivers: making the solution method the model

A simple car following model based on the solution of coupled ordinary differential equations is considered. The model is solved using Euler's method and this method of solution is itself interpreted as a mathematical model for car following. Examples of possible classroom use are given.     

Classroom note: The number c in Cauchy's average value theorem

The location of the number c arising from Cauchy's Average Value Theorem is described when the size of the interval is small.     

A note on new exact solutions for the Kawahara equation using Exp-function method

Exact solutions of the Kawahara equation by Assas [L.M.B. Assas, New Exact solutions for the Kawahara equation using Exp-function method, J. Comput. Appl. Math. 233 (2009) 97-102] are analyzed. It is shown that all solutions do not satisfy the Kawahara equation and consequently all nontrivial solutions by Assas are wrong.     

Classroom note: A Generalization of Leibniz rule for higher derivatives

This note provides a simple method to extend the usual Leibniz rule for higher derivatives of the product of two functions to several functions, which is within the reach of freshman calculus students.     

Classroom note: Evaluating a class of series using Taylor's theorem

A class of infinite series is evaluated with the aid of Taylor's theorem and a comparison is made with other methods.     

Classroom note: A simple proof of an interesting Fibonacci generalization

It is reasonably well known that the ratios of consecutive terms of a Fibonacci series converge to the golden ratio. This note presents a simple, complete proof of an interesting generalization of this result to a whole family of ‘precious metal ratios’.     

Classroom note: Estimation of the cross-product of two mean vectors

The problem of estimating the cross-product of two mean vectors in three-dimensional Euclidian space is considered. Two ‘natural’ estimators are developed, both of which turn out to be biased. A third, unbiased estimator, resulting from a jackknife procedure, is also investigated. It is shown that, under normality, the latter is best among all the unbiased estimators of this quantity.     

Classroom note: Compact,convex, and symmetric sets are discs

Define the centre of a parallelogram to be the intersection of its diagonals. It was shown in an earlier paper that the intersection of arbitrarily many parallelograms with the same centre is the unit disc about that centre in a metric defined using ideas from Linear Algebra. In this note, it is shown that this characterizes compact, convex sets, which are symmetric about a point.     

Classroom note: Waiting for a sequence pattern in independent multinomial observations

The expected value (mean) is one of the most important measures of central tendency used both in statistics and in the mathematical theory of probability. Using the method of geometric transforms (probability generating functions), the expected number of observations is obtained until the sequence pattern 123…N123…N…123…N, which is an outcome in a sequence of independent multinomial observations. The results and the flow graph analysis obtained agree with and provide the generalization of similar problems considered earlier in this journal.