Digital Signatures using Edwards Curves

A digital signature is a mathematical means of verifying that an e-document is authentic, that it has come from the claimed sender and that it has not been tampered with or corrupted during transit. Digital signatures are a standard component of cryptographic systems. They use asymetric cryptography that is based on key pairs, consisting of … Continue reading Digital Signatures using Edwards Curves →

Complexity: are easily-checked problems also easily solved?

From the name of the Persian polymath Al Khwarizmi, who flourished in the early ninth century, comes the term algorithm. An algorithm is a set of simple steps that lead to the solution of a problem. An everyday example is a baking recipe, with instructions on what to do with ingredients (input) to produce a … Continue reading Complexity: are easily-checked problems also easily solved? →

Will mathematicians be replaced by computers?

There are ongoing rapid advances in the power and versatility of AI or artificial intelligence. Computers are now producing results in several fields that are far beyond human capability. The trend is unstoppable, and is having profound effects in many areas of our lives. Will mathematicians be replaced by computers? [TM195 or search for “thatsmaths” at irishtimes.com]. … Continue reading Will mathematicians be replaced by computers? →

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 →

Algorithms: Recipes for Success

The impact of computing on society is ever-increasing. Web-based commerce continues to grow and artificial intelligence now pervades our lives. To make wise choices, we need to understand how computers operate and how we can deploy them most constructively. Listen to any computer scientist and soon you will hear the word “algorithm” [TM168 or search for … Continue reading Algorithms: Recipes for Success →

A Toy Example of RSA Encryption

The RSA system has been presented many times, following the excellent expository article of Martin Gardner in the August 1977 issue of Scientific American. There is no need for yet another explanation of the system; the essentials are contained in the Wikipedia article RSA (cryptosystem), and in many other articles. The purpose of this note … Continue reading A Toy Example of RSA Encryption →

Can Mathematics Keep Us Secure?

The National Security Agency is the largest employer of mathematicians in America. Mathematics is a core discipline at NSA and mathematicians work on signals intelligence and information security (US citizenship is a requirement for employment). Why is NSA so interested in mathematics? [See TM096, or search for “thatsmaths” at irishtimes.com]. Many actions are easy to … Continue reading Can Mathematics Keep Us Secure? →

Computers Speaking in Irish

Most of us use computer terminals, tablets and smart phones, absorbing information quickly and easily. How do the many thousands of Irish people who are blind or visually impaired manage to interact with computers? For them, entering data by keyboard or voice is easy, but special software is needed to convert the text on screen … Continue reading Computers Speaking in Irish →

Big Data: the Information Explosion

The world is awash with data. Large data sets have been available for many decades but in recent years their volumes have grown explosively. With mobile devices and internet connections data capture is simple and with powerful computers the analysis of “big data” is feasible [see TM092, or search for “thatsmaths” at irishtimes.com]. But there are … Continue reading Big Data: the Information Explosion →

Richardson’s Fantastic Forecast Factory

Modern weather forecasts are made by calculating solutions of the mathematical equations that express the fundamental physical principles governing the atmosphere [TM083, or search for “thatsmaths” at irishtimes.com] The solutions are generated by complex simulation models with millions of lines of code, implemented on powerful computer equipment. The meteorologist uses the computer predictions to produce … Continue reading Richardson’s Fantastic Forecast Factory →

It’s a Small – Networked – World

Networks are everywhere in the modern world. They may be physical constructs, like the transport system or power grid, or more abstract entities like family trees or the World Wide Web. A network is a collection of nodes linked together, like cities connected by roads or people genetically related to each other. Such a system … Continue reading It’s a Small – Networked – World →

New Tricks: No Clicks

The quality of music recordings on compact discs or CDs is excellent. In the age of vinyl records, irritating clicks resulting from surface scratches were almost impossible to avoid. Modern recording media are largely free from this shortcoming. But this is curious: there are many reasons why CD music can be contaminated: dirt on the … Continue reading New Tricks: No Clicks →

Hamming’s Smart Error-correcting Codes

In the late 1940s, Richard Hamming, working at Bell Labs, was exasperated with the high level of errors occurring in the electro-mechanical computing equipment he was using. Punched card machines were constantly misreading, forcing him to restart his programs. He decided to do something about it. This was when error-correcting codes were invented. A simple … Continue reading Hamming’s Smart Error-correcting Codes →

Game Theory & Nash Equilibrium

Game theory deals with mathematical models of situations involving conflict, cooperation and competition. Such situations are central in the social and behavioural sciences. Game Theory is a framework for making rational decisions in many fields: economics, political science, psychology, computer science and biology. It is also used in industry, for decisions on manufacturing, distribution, consumption, … Continue reading Game Theory & Nash Equilibrium →

Modelling the Markets

Mathematics now plays a fundamental role in modelling market movements [see this week’s That’s Maths column (TM067) or search for “thatsmaths” at irishtimes.com]. The state of the stock market displayed on a trader's screen is history. Big changes can occur in the fraction of a second that it takes for information to reach the screen. … Continue reading Modelling the Markets →

Barcodes and QR Codes: Zebra stripes and Leopard spots

Barcodes and QR codes are described in this week’s That’s Maths column in The Irish Times (TM060, or search for “thatsmaths” at irishtimes.com). Virtually everything that you buy in your local supermarket has a curious little zebra-like pattern the size of a postage stamp printed on it. Barcodes, originally devised about forty years ago to … Continue reading Barcodes and QR Codes: Zebra stripes and Leopard spots →

Information Theory

That’s Maths in The Irish Times this week (TM059, or Search for “thatsmaths” at irishtimes.com) is about data compression and its uses in modern technology. The arrival of mobile phones was followed rapidly by "txtese", an abbreviation of language to enable messages to be written and transmitted rapidly using SMS (Short Message Service). The simplest … Continue reading Information Theory →

The Year of George Boole

This week’s That’s Maths column in The Irish Times (TM058, or search for “thatsmaths” at irishtimes.com) is about George Boole, the first Professor of Mathematics at Queen's College Cork. Mathematician and logician George Boole died just 150 years ago, on 8 December 1864, following a drenching as he was walking between his home and Queen's … Continue reading The Year of George Boole →

Cartoon Curves

The powerful and versatile computational software program called Mathematica is widely used in science, engineering and mathematics. There is a related system called Wolfram Alpha, a computational knowledge engine, that can do Mathematica calculations and that runs on an iPad. Mathematica can do numerical and symbolic calculations. Algebraic manipulations, differential equations and integrals are simple, … Continue reading Cartoon Curves →

Digital Dentistry

That’s Maths in The Irish Times this week (TM049, or Search for “thatsmaths” at irishtimes.com) is about applications of computer aided design and computer aided manufacture to making dental crowns. Next time you spot a kid immersed in a video game, pause before uttering a condemnatory remark. The spin-offs from computer gaming are benefiting us … Continue reading Digital Dentistry →

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 →

The Simpler the Better

This week’s That’s Maths in The Irish Times ( TM030 ) is about Linear Programming (LP) and about how it saves millions of Euros every day through optimising efficiency. A Berkeley graduate student, George Dantzig, was late for class. He scribbled down two problems written on the blackboard and handed in solutions a few days … Continue reading The Simpler the Better →

The remarkable BBP Formula

Information that is declared to be forever inaccessible is sometimes revealed within a short period. Until recently, it seemed impossible that we would ever know the value of the quintillionth decimal digit of pi. But a remarkable formula has been found that allows the computation of binary digits starting from an arbitrary position without the … Continue reading The remarkable BBP Formula →

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 →

Computer Maths

Will computers ever be able to do mathematical research? Automatic computers have amazing power to analyze huge data bases and carry out extensive searches far beyond human capabilities. They can assist mathematicians in checking cases and evaluating functions at lightning speed, and they have been essential in producing proofs that depend on exhaustive searches. The … Continue reading Computer Maths →