Low-pass Filtering and the Remarkable Integrals of Borwein and Borwein

In last week's post we looked at aspects of puzzles of the form ``What is the next number''. We are presented with a short list of numbers, for example $latex {1, 3, 5, 7, 9}&fg=000000$ and asked for the next number in the sequence. Arguments were given indicating why any number might be regarded as … Continue reading Low-pass Filtering and the Remarkable Integrals of Borwein and Borwein →

What’s the Next Number?

We are all familiar with simple mathematical puzzles that give a short sequence and ask ``What is the next number in the sequence''. Simple examples would be $latex \displaystyle \begin{array}{rcl} && 1, 3, 5, 7, 9, 11, \dots \\ && 1, 4, 9, 16, 25, \dots \\ && 1, 1, 2, 3, 5, 8, \dots … Continue reading What’s the Next Number? →

The Power of the 2-gon: Extrapolation to Evaluate Pi

Richardson's extrapolation procedure yields a significant increase in the accuracy of numerical solutions of differential equations. We consider his elegant illustration of the technique, the evaluation of $latex {\pi}&fg=000000$, and show how the estimates improve dramatically with higher order extrapolation. [This post is a condensed version of a paper in Mathematics Today (Lynch, 2003).] … Continue reading The Power of the 2-gon: Extrapolation to Evaluate Pi →

Following the Money around the Eurozone

Take a fistful of euro coins and examine the obverse sides; you may be surprised at the wide variety of designs. The eurozone is a monetary union of 19 member states of the European Union that have adopted the euro as their primary currency. In addition to these countries, Andorra, Monaco, San Marino and Vatican … Continue reading Following the Money around the Eurozone →

Real Derivatives from Imaginary Increments

The solution of many problems requires us to compute derivatives. Complex step differentiation is a method of computing the first derivative of a real function, which circumvents the problem of roundoff error found with typical finite difference approximations. For finite difference approximations, the choice of step size $latex {h}&fg=000000$ is crucial: if $latex {h}&fg=000000$ is … Continue reading Real Derivatives from Imaginary Increments →

Can You Believe Your Eyes?

Remember the old cowboy movies? As the stage-coach comes to a halt, the wheels appear to spin backwards, then forwards, then backwards again, until the coach stops. How can this be explained? The illusion is known as the stroboscopic effect. We don't see it in real life, where time flows smoothly and continuously. But, in … Continue reading Can You Believe Your Eyes? →

Does Numerical Integration Reflect the Truth?

Many problems in applied mathematics involve the solution of a differential equation. Simple differential equations can be solved analytically: we can find a formula expressing the solution for any value of the independent variable. But most equations are nonlinear and this approach does not work; we must solve the equation by approximate numerical means. The … Continue reading Does Numerical Integration Reflect the Truth? →

Dimension Reduction by PCA

We live in the age of ``big data''. Voluminous data collections are mined for information using mathematical techniques. Problems in high dimensions are hard to solve --- this is called ``the curse of dimensionality''. Dimension reduction is essential in big data science. Many sophisticated techniques have been developed to reduce dimensions and reveal the information … Continue reading Dimension Reduction by PCA →

The Monte-Carlo Method

Learning calculus at school, we soon find out that while differentiation is relatively easy, at least for simple functions, integration is hard. So hard indeed that, in many cases, it is impossible to find a nice function that is the integral (or anti-derivative) of a given one. Thus, given $latex {f(x)}&fg=000000$ we can usually find … Continue reading The Monte-Carlo Method →

The Rambling Roots of Wilkinson’s Polynomial

Finding the roots of polynomials has occupied mathematicians for many centuries. For equations up to fourth order, there are algebraic expressions for the roots. For higher order equations, many excellent numerical methods are available, but the results are not always reliable. James Wilkinson (1963) examined the behaviour of a high-order polynomial $latex \displaystyle p(x,\epsilon) = … Continue reading The Rambling Roots of Wilkinson’s Polynomial →

Bouncing Billiard Balls Produce Pi

There are many ways of evaluating $latex {\pi}&fg=000000$, the ratio of the circumference of a circle to its diameter. We review several historical methods and describe a recently-discovered and completely original and ingenious method. Historical Methods Archimedes used inscribed and circumscribed polygons to deduce that $latex \displaystyle \textstyle{3\frac{10}{71} < \pi < 3\frac{10}{70}} &fg=000000$ giving roughly … Continue reading Bouncing Billiard Balls Produce Pi →

Staying Put or Going with the Flow

The atmospheric temperature at a fixed spot may change in two ways. First, heat sources or sinks may increase or decrease the thermal energy; for example, sunshine may warm the air or radiation at night may cool it. Second, warmer or cooler air may be transported to the spot by the air flow in a … Continue reading Staying Put or Going with the Flow →

Marvellous Merchiston’s Logarithms

Log tables, invaluable in science, industry and commerce for 350 years, have been consigned to the scrap heap. But logarithms remain at the core of science, as a wide range of physical phenomena follow logarithmic laws [TM103 or search for “thatsmaths” at irishtimes.com]. The method of logarithms was first devised by John Napier, 8th Laird … Continue reading Marvellous Merchiston’s Logarithms →

Simulating the Future Climate

The Earth's climate is changing, and the consequences may be very grave. This week, That’s Maths in The Irish Times ( TM040 ) is about computer models for simulating and predicting the future climate. Liffey Bursts its Banks: St. Stephen's Green Flooded Again The above is an improbable but not entirely impossible future headline. Sea … Continue reading Simulating the Future Climate →

French Curves and Bézier Splines

A French curve is a template, normally plastic, used for manually drawing smooth curves. These simple drafting instruments provided innocent if puerile merriment to generations of engineering students, but they have now been rendered obsolete by computer aided design (CAD) packages, which enable us to construct complicated curves and surfaces using mathematical functions called Bézier … Continue reading French Curves and Bézier Splines →

Singularly Valuable SVD

In many fields of mathematics there is a result of central importance, called the "Fundamental Theorem" of that field. Thus, the fundamental theorem of arithmetic is the unique prime factorization theorem, stating that any integer greater than 1 is either prime itself or is the product of prime numbers, unique apart from their order. The … Continue reading Singularly Valuable SVD →

The Lambert W-Function

Follow on twitter: @thatsmaths In a recent post ( The Power Tower ) we described a function defined by iterated exponentiation: $latex \displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}} &fg=000000$ It would seem that when $latex {x>1}&fg=000000$ this must blow up. Surprisingly, it has finite values for a range of x>1. Below, we show that the power tower … Continue reading The Lambert W-Function →

Carving up the Globe

This week, That’s Maths (TM007) describes various ways of dividing up the sphere. This is an important problem in geometry, biology, chemistry, astronomy, meteorology and climate modelling. The problem of defining a uniform distribution of points on the sphere has challenged mathematicians for centuries. The vertices of the five Platonic solids achieve this but, in … Continue reading Carving up the Globe →