## Archive for March, 2018

### Fourier’s Wonderful Idea – I

Breaking Complex Objects into Simple Pieces

“In a memorable session of the French Academy on the
21st of December 1807, the mathematician and engineer
Joseph Fourier announced a thesis which inaugurated a
new chapter in the history of mathematics. The claim of
Fourier appeared to the older members of the Academy,
including the great analyst Lagrange, entirely incredible.”

Introduction

Joseph Fourier (1768-1830)

The above words open the Discourse on Fourier Series, written by Cornelius Lanczos. What greatly surprised and shocked Lagrange and the other academicians was the claim of Fourier that an arbitrary function, defined by an arbitrarily capricious graph, can always be resolved into a sum of pure sine and cosine functions. There was good reason to question Fourier’s theorem. Since sine functions are continuous and infinitely differentiable, it was assumed that any superposition of such functions would have the same properties. How could this assumption be reconciled with Fourier’s claim?

### Sophus Lie

It is difficult to imagine modern mathematics without the concept of a Lie group.” (Ioan James, 2002).

Sophus Lie (1842-1899)

Sophus Lie grew up in the town of Moss, south of Oslo. He was a powerful man, tall and strong with a booming voice and imposing presence. He was an accomplished sportsman, most notably in gymnastics. It was no hardship for Lie to walk the 60 km from Oslo to Moss at the weekend to visit his parents. At school, Lie was a good all-rounder, though his mathematics teacher, Ludvig Sylow, a pioneer of group theory, did not suspect his great potential or anticipate his remarkable achievements in that field.

### Cubic Skulduggery & Intrigue

Solution of a cubic equation, usually called Cardano’s formula.

Babylonian mathematicians knew how to solve simple polynomial equations, in which the unknown quantity that we like to call x enters in the form of powers, that is, x multiplied repeatedly by itself. When only x appears, we have a linear equation. If x-squared enters, we have a quadratic. The third power of x yields a cubic equation, the fourth power a quartic and so on [TM135 or search for “thatsmaths” at irishtimes.com].

### Subtract 0 and divide by 1

We all know that division by zero is a prohibited operation, and that ratios that reduce to “zero divided by zero” are indeterminate. We probably also recall proving in elementary calculus class that

$\displaystyle \lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$

This is an essential step in deriving an expression for the derivative of ${\sin x}$.

### Reducing R-naught to stem the spread of Epidemics

We are reminded each year to get vaccinated against the influenza virus. The severity of the annual outbreak is not known with certainty in advance, but a major pandemic is bound to occur sooner or later. Mathematical models play an indispensable role in understanding and managing infectious diseases. Models vary in sophistication from the simple SIR model with just three variables to highly complex simulation models with millions of variables [TM134 or search for “thatsmaths” at irishtimes.com]. Continue reading ‘Reducing R-naught to stem the spread of Epidemics’