Think of a Number: What are the Odds that it is Even?

Pick a positive integer at random. What is the chance of it being 100? What or the odds that it is even? What is the likelihood that it is prime? Since the set $latex {\mathbb{N}}&fg=000000$ of natural numbers is infinite, there are difficulties in assigning probabilities to subsets of $latex {\mathbb{N}}&fg=000000$. We require the probability … Continue reading Think of a Number: What are the Odds that it is Even? →

Resolution of Paradox: a Gateway to Mathematical Progress

A paradox is a statement that appears to contradict itself, or that is counter-intuitive. The analysis of paradoxes has led to profound developments in mathematics and logic. One of the richest sources of paradox is the concept of infinity. Hermann Weyl, one of the most brilliant mathematicians of the twentieth century, defined mathematics as “the … Continue reading Resolution of Paradox: a Gateway to Mathematical Progress →

Berry’s Paradox and Gödel’s Incompleteness Theorem

  A young librarian at the Bodleian Library in Oxford devised an intriguing paradox. He defined a number by means of a statement of the form THE SMALLEST NATURAL NUMBER THAT CANNOT BE DEFINED IN FEWER THAN TWENTY WORDS. This appears to indicate a specific positive integer, which we denote $latex {\mathcal{B}}&fg=000000$. But there is … Continue reading Berry’s Paradox and Gödel’s Incompleteness Theorem →

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? →

Cornelius Lanczos – Inspired by Hamilton’s Quaternions

In May 1954, Cornelius Lanczos took up a position as senior professor in the School of Theoretical Physics at the Dublin Institute for Advanced Studies (DIAS). The institute had been established in 1940 by Eamon de Valera, with a School of Theoretical Physics and a School of Celtic Studies, reflecting de Valera's keen interest in … Continue reading Cornelius Lanczos – Inspired by Hamilton’s Quaternions →

The Geography of Europe is Mapped in our Genes

It may seem too much to expect that a person's geographic origin can be determined from a DNA sample. But, thanks to a mathematical technique called principal component analysis, this can be done with remarkable accuracy. It works by reducing multi-dimensional data sets to just a few variables  [TM189; or search for “thatsmaths” at irishtimes.com ]. … Continue reading The Geography of Europe is Mapped in our Genes →

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 →

Pooling Expertise to Tackle Covid-19

Our lives have been severely restricted in recent months. We are assured that the constraints have been imposed following “the best scientific advice”, but what is the nature of this advice? Among the most important scientific tools used for guidance on the Covid-19 outbreak are mathematical models  [TM188; or search for “thatsmaths” at irishtimes.com ]. A … Continue reading Pooling Expertise to Tackle Covid-19 →

Changing the way that we look at the world

Albrecht Dürer was born in Nuremberg in 1471, third of a family of eighteen children. Were he still living, he would be celebrating his 549th birthday today. Dürer's artistic genius was clear from an early age, as evidenced by a self-portrait he painted when just thirteen [TM187; or search for “thatsmaths” at irishtimes.com ]. In 1494, … Continue reading Changing the way that we look at the world →

A New Perspective on Perspective

The development of perspective in the early Italian Renaissance opened the doors of perception just a little wider. Perspective techniques enabled artists to create strikingly realistic images. Among the most notable were Piero della Francesca and Leon Battista Alberti, who invented the method of perspective drawing. For centuries, artists have painted scenes on a sheet … Continue reading A New Perspective on Perspective →

John Horton Conway: a Charismatic Genius

John Horton Conway was a charismatic character, something of a performer, always entertaining his fellow-mathematicians with clever magic tricks, memory feats and brilliant mathematics. A Liverpudlian, interested from early childhood in mathematics, he studied at Gonville & Caius College in Cambridge, earning a BA in 1959. He obtained his PhD five years later, after which … Continue reading John Horton Conway: a Charismatic Genius →

Bang! Bang! Bang! Explosively Large Numbers

Enormous numbers pop up in both mathematics and physics. The order of the monster group, the largest of the 26 sporadic groups, is $latex \displaystyle 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 &fg=000000$ which is approximately $latex {8\times 10^{53}}&fg=000000$. The number of atoms in the universe is estimated to be about $latex {10^{80}}&fg=000000$. When we consider permutations of large sets, even … Continue reading Bang! Bang! Bang! Explosively Large Numbers →

Covid-19: Modelling the evolution of a viral outbreak

There is widespread anxiety about the threat of the Covid-19 virus. Mathematics now plays a vital role in combating the spread of epidemics, and will help us to bring this outbreak under control. For centuries, mathematics has been used to solve problems in astronomy, physics and engineering. But now biology and medicine have become topics … Continue reading Covid-19: Modelling the evolution of a viral outbreak →

Samuel Haughton and the Twelve Faithless Hangmaids

In his study of humane methods of hanging, Samuel Haughton (1866) considered the earliest recorded account of execution by hanging (see Haughton's Drop on this site). In the twenty-second book of the Odyssey, Homer described how the twelve faithless handmaids of Penelope ``lay by night enfolded in the arms of the suitors'' who were vying … Continue reading Samuel Haughton and the Twelve Faithless Hangmaids →

Zhukovsky’s Airfoil

A simple transformation with remarkable properties was used by Nikolai Zhukovsky around 1910 to study the flow around aircraft wings. It is defined by $latex \displaystyle \omega = \textstyle{\frac{1}{2}}\displaystyle{\left(z +\frac{1}{z}\right)} &fg=000000$ and is usually called the Joukowsky Map. We begin with a discussion of the theory of fluid flow in two dimensions. Readers familiar with … Continue reading Zhukovsky’s Airfoil →

How many numbers begin with a 1? More than 30%!

The irregular distribution of the first digits of numbers in data-bases provides a valuable tool for fraud detection. A remarkable rule that applies to many datasets was accidentally discovered by an American physicist, Frank Benford, who described his discovery in a 1938 paper, "The Law of Anomalous Numbers" [TM181 or search for “thatsmaths” at irishtimes.com]. … Continue reading How many numbers begin with a 1? More than 30%! →

A Ring of Water Shows the Earth’s Spin

Around 1913, while still an undergraduate, American physicist Arthur Compton described an experiment to demonstrate the rotation of the Earth using a simple laboratory apparatus. Compton (1892--1962) won the Nobel Prize in Physics in 1927 for his work on scattering of EM radiation. This phenomenon, now called the Compton effect, confirmed the particle nature of … Continue reading A Ring of Water Shows the Earth’s Spin →

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 →

Adjoints of Vector Operators

We take a fresh look at the vector differential operators grad, div and curl. There are many vector identities relating these. In particular, there are two combinations that always yield zero results: $latex \displaystyle \begin{array}{rcl} \mathbf{curl}\ \mathbf{grad}\ \chi &\equiv& 0\,, \quad \mbox{for all scalar functions\ }\chi \\ \mathrm{div}\ \mathbf{curl}\ \boldsymbol{\psi} &\equiv& 0\,, \quad \mbox{for all … Continue reading Adjoints of Vector Operators →

The “extraordinary talent and superior genius” of Sophie Germain

When a guitar string is plucked, we don't see waves travelling along the string. This is because the ends are fixed. Instead, we see a standing-wave pattern. Standing waves are also found on drum-heads and on the sound-boxes of violins. The shape of a violin strongly affects the quality and purity of the sound, as … Continue reading The “extraordinary talent and superior genius” of Sophie Germain →

Grad, Div and Curl on Weather Maps: a Gateway to Vector Analysis

Vector analysis can be daunting for students. The theory can appear abstract, and operators like Grad, Div and Curl seem to be introduced without any obvious motivation. Concrete examples can make things easier to understand. Weather maps, easily obtained on the web, provide real-life applications of vector operators. Weather charts provide great examples of scalar … Continue reading Grad, Div and Curl on Weather Maps: a Gateway to Vector Analysis →

Divergent Series Yield Valuable Results

Mathematicians have traditionally dealt with convergent series and shunned divergent ones. But, long ago, astronomers found that divergent expansions yield valuable results. If these so-called asymptotic expansions are truncated, the error is bounded by the first term omitted. Thus, by stopping just before the smallest term, excellent approximations may be obtained. Astronomical Series Many of … Continue reading Divergent Series Yield Valuable Results →

A New Mathematical Discovery from Neutrino Physics

Although abstract in character, mathematics has concrete origins: the greatest advances have been inspired by the natural world. Recently, a new result in linear algebra was discovered by three physicists trying to understand the behaviour of neutrinos [TM176 or search for “thatsmaths” at irishtimes.com]. Neutrinos are sub-atomic particles that interact only weakly with matter, so that they … Continue reading A New Mathematical Discovery from Neutrino Physics →

Archimedes and the Volume of a Sphere

One of the most remarkable and important mathematical results obtained by Archimedes was the determination of the volume of a sphere. Archimedes used a technique of sub-dividing the volume into slices of known cross-sectional area and adding up, or integrating, the volumes of the slices. This was essentially an application of a technique that was … Continue reading Archimedes and the Volume of a Sphere →

Airport Baggage Screening with X-Ray Tomography

When you check in your baggage for a flight, it must be screened before it is allowed on the plane. Baggage screening detects threats within luggage and personal belongings by x-ray analysis as they pass along a conveyor belt. Hold-baggage and passenger screening systems are capable of detecting contraband materials, narcotics, explosives and weapons [TM175 … Continue reading Airport Baggage Screening with X-Ray Tomography →

Elliptic Trigonometry: Fun with “sun”, “cun” and “dun”

Introduction The circular functions arise from ratios of lengths in a circle. In a similar manner, the elliptic functions can be defined by means of ratios of lengths in an ellipse. Many of the key properties of the elliptic functions follow from simple geometric properties of the ellipse. Originally, Carl Gustav Jacobi defined the elliptic … Continue reading Elliptic Trigonometry: Fun with “sun”, “cun” and “dun” →

The Vastness of Mathematics: No One Knows it All

No one person can have mastery of the entirety of mathematics. The subject has become so vast that the best that can be achieved is a general understanding and appreciation of the main branches together with expertise in one or two areas [TM174 or search for “thatsmaths” at irishtimes.com]. In the sciences, old theories fade away … Continue reading The Vastness of Mathematics: No One Knows it All →

Some Fundamental Theorems of Maths

Every branch of mathematics has key results that are so important that they are dubbed fundamental theorems. The customary view of mathematical research is that of establishing the truth of propositions or theorems by rigorous deduction from axioms and definitions. Mathematics is founded upon axioms, basic assumptions that are taken as true. Logical reasoning is … Continue reading Some Fundamental Theorems of Maths →

Maths and Poetry: Beauty is the Link

Mathematicians are not renowned for their ability to reach the deepest recesses of the human soul. This talent is usually associated with great artists and musicians, and a good poet can move us profoundly with a few well-chosen words [TM173 or search for “thatsmaths” at irishtimes.com]. William Rowan Hamilton, whose work we celebrate during Maths Week, was … Continue reading Maths and Poetry: Beauty is the Link →

Emergence of Complex Behaviour from Simple Roots

It is exhilarating to watch a large flock of birds swarming in ever-changing patterns. Swarming is an emergent behaviour, resulting from a set of simple rules followed by each individual animal, bird or fish, without any centralized control or leadership. A murmuration of starlings is a breathtaking sight, with thousands of birds moving in harmony, … Continue reading Emergence of Complex Behaviour from Simple Roots →

George Salmon, Mathematician & Theologian

As you pass through the main entrance of Trinity College, the iconic campanile stands before you, flanked, in pleasing symmetry, by two life-size statues. On the right, on a granite plinth is the historian and essayist William Lecky. On the left, George Salmon (1819–1904) sits on a limestone platform. Salmon was a distinguished mathematician and … Continue reading George Salmon, Mathematician & Theologian →