Year 2006 - Volume 42
2020 - Present

Welcome to the first issue of Parabola Incorporating Function for 2006. The first two articles by Mike Hirschhorn are good illustrative examples of the sort of recreational games that you can play with mathematics.

Michael D. Hirshhorn
It is well-known that
loge(1+x) = x − (x2/2) + (x3/3) − + ⋯ for 0≤x≤1,
where the −+ ⋯ notation is used to indicate that successive terms are alternately subtracted and added.

Michael D. Hirshhorn
In this article I am going to assume you are a little bit familiar with the inverse tangent function, tan−1x=arctanx. There are several properties of this function that you need to know.

Michael A. B. Deakin
The story I want to tell in this issue concerns a development I first learned of a little over 30 years ago, but whose roots lie much deeper than that. It was in 1973 that I was first introduced to the mathematical development known as Catastrophe Theory.

Recently a colleague from the Optometry school came to me with a problem. He had designed a new shaped contact lens. Unlike other lens systems this was not designed as an optical lens, but rather as a shaping lens, designed to be put in overnight.

Q1201 Let x1 and x2 be the solutions of

Prove that x31 and x32 are the solutions of

Q1191 Let ABC be a triangle with sides a,b,c in the usual way and let r be its circumradius.
1. Show that 3r>(a+b+c)/2.
2. Let P be any point within ABC , and let r1 be the circumradius of ABP. Is it true that r< r?

As we go to press, media around the world have been reporting the latest round of awards of the coveted Fields Medal (popularly called the "Nobel Prize for Mathematics") which are awarded every four years.

Bill McKee
We sometimes see in newspapers or on television situations where a straight line is drawn so as to approximately fit some data points. This can always be done by eye, using human judgment, but the results would then tend to vary depending on the person drawing the line.

Michael A. B. Deakin
Two separate events happily combined to suggest the topic for this issue’s column. In the first place, I devoted my previous column to a somewhat controversial attempt to apply Mathematics to the "softer sciences" such as Biology and Linguistics.

M. P. Wand
Support vector machines emerged in the mid-1990s as a flexible and powerful means of classification. Classification is a very old problem in Statistics but, in our increasingly data-rich age, remains as important as ever.

Problem 1. An American football field is 100 yards long, and its width is half the average of its length and its diagonal. Find its area..

Prize Winners – Senior Division
First Prize

Graham Robert White James Ruse Agricultural High School

Q1211. Solve (2+√2)sin2x − (2+√2)cos2x + (2−√2)cos2x = (1+(1/√2)cos2x

Q1201. Let x1 and x2 be the solutions of x2−(a+d)x+ad−bc=0.
Prove that x31 and x32 are the solutions of x2−(a3+d3+3abc+3bcd)x+(ad−bc)3=0.

In this issue we are delighted to reproduce three award-winning articles on Cryptography by Professor Nuno Crato. These articles were awarded first prize in a special competition organized by the European Mathematical Society on Raising Public Awareness in Mathematics.

Nuno Crato
The safety of electronic commerce is secured by an innovative mathematical method: a public key that allows you to lock messages in a safe, but a safe that can only be open with another device - a secret key!

Nuno Crato
It sounds like science fiction, but it is a reality: the most bizarre properties of subatomic particles allow us to create unbreakable ciphers.

Nuno Crato
Internet communication is based on coding mechanisms that guarantee privacy in the exchange of messages. Mathematics makes this possible without having the people involved agree on a secret key.

Michael A. B. Deakin
Sometimes in these columns I will allow myself the liberty of straying outside the domain of History, properly so called, in order to look at somewhat wider issues.

James Franklin
The following ad appeared in The Australian on 25 March 2006.

Murray Batchelor and Bruce Henry
In the popular Hitch Hiker’s Guide to the Galaxy series Douglas Adams plays with the Ultimate Answer to the great mystery of Life, the Universe and Everything.

Q1221. Complete the mathematical equations below by inserting the least number of mathematical symbols from the table

Q1211. Solve
(2+√2)sin2x − (2+√2)cos2x + (2−√2)cos2x = (1+(√12))cos2x