Posts Tagged 'Computer Science'

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 cake (output). For a computer algorithm, the inputs are the known numerical quantities and the output is the required solution [TM204 or search for “thatsmaths” at].

Al Khwarizmi, Persian polymath (c. 780 – 850) [image, courtesy of Prof. Irfan Shahid].

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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].

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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 at dusk near Ballywilliam, Co Wexford. Photograph: Cyril Byrne.

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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 “thatsmaths” at].

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Face Recognition

As you pass through an airport, you are photographed several times by security systems. Face recognition systems can identify you by comparing your digital image to faces stored in a database. This form of identification is gaining popularity, allowing you to access online banking without a PIN or password.  [see TM146, or search for “thatsmaths” at].


Jimmy Wales, co-founder of Wikipedia, answering a question. Face detection indicated by squares.

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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.


L2R: Ron Rivest, Adi Shamir, Len Adleman (2003). Image from

The purpose of this note is to give an example of the method using numbers so small that the computations can easily be carried through by mental arithmetic or with a simple calculator.

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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].


Flag of the National Security Agency

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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 into a form for output to a loudspeaker or headphones, or to drive a refreshable Braille display [TM095, or search for “thatsmaths” at].


Braille display (

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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].


Google image search for “Big Data”

But there are challenges: many data sets are too large and too complex to be analysed or understood using traditional data processing methods. Our current armoury of analysis techniques is inadequate and new mathematical methods are needed.

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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]

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 localised forecasts and guidance for specialised applications.


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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 of nodes and links is what mathematicians call a graph [TM078; or search for “thatsmaths” at ].

Detail of a Twitter communications network. Image from:

Detail of a Twitter communications network.
Image from:

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New Tricks: No Clicks

The readable surface of a Compact Disc has a spiral track over 5 km in length.

The readable surface of a Compact Disc has a spiral track over 5 km in length.

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 disc surface, flaws in the plastic substrate, errors in burning on the recording, scratches and fingerprints, and so on [TM077; or search for “thatsmaths” at ]

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Hamming’s Smart Error-correcting Codes

Richard Hamming (1915 – 1998)

Richard Hamming (1915 – 1998)

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 way to detect errors is to send a message twice. If both versions agree, they are probably correct; if not, there is an error somewhere. But the discrepancy gives us no clue where the error lies. Sensing the message three times is better: if two versions agree, we assume they are correct and ignore the third version. But there is a serious overhead: the total data transmitted is three times the data volume; the information factor is 1/3.

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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, pricing, salaries, etc.

Theory of games and economic behavior. Centre: John von Neumann. Right: Oskar Morgenstern.

Theory of Games and Economic Behavior.
Centre: John von Neumann. Right: Oskar Morgenstern.

During the Cold War, Game Theory was the basis for many decisions concerning nuclear strategy that affected the well-being of the entire human race.

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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].

Dow-Jones Industrial Aversge for 6 May 2010. Graphic adapted from Sunday Times, 26 April, 2015.

Dow-Jones Industrial Average for the Flash-Crash on 6 May 2010.
Graphic adapted from Sunday Times, 26 April, 2015.

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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

EAN-13 barcode.

EAN-13 barcode.

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Information Theory

That’s Maths in The Irish Times this week (TM059, or Search for “thatsmaths” at is about data compression and its uses in modern technology.

Left: An equation form Shannon (1948), the paper that launched Information Theory.  Right: Claude Shannon (1916-2001) ©Alcatel-Lucent.

Left: An equation form Shannon (1948), the paper that launched Information Theory.
Right: Claude Shannon (1916-2001) ©Alcatel-Lucent.

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The Year of George Boole

This week’s That’s Maths column in The Irish Times (TM058, or search for “thatsmaths” at is about George Boole, the first Professor of Mathematics at Queen’s College Cork.

Boole-Year-UCC-Small 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.

Yogi Bear Curve. The Mathematica command to generate this is given below.

Yogi Bear Curve. The Mathematica command to generate this is given below.

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Digital Dentistry

That’s Maths in The Irish Times this week (TM049, or  Search for “thatsmaths” at is about applications of computer aided design and computer aided manufacture to making dental crowns.

High-precision digitally-driven mill carving a dental crown from a solid ceramic block [photo from].

High-precision digitally-driven mill carving a dental crown from a solid ceramic block
[photo from].

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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.

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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 splines.

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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 later. But the problems on the board were not homework assignments; they were two famous unsolved problems in statistics. The solutions earned Dantzig his Ph.D.
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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 need to compute earlier digits. This is known as the BBP formula.
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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 fundamental theorem of algebra states that every non-constant polynomial has at least one (complex) root. And the fundamental theorem of calculus shows that integration and differentiation are inverse operations, uniting differential and integral calculus.

The Fundamental Theorem of Linear Algebra
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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 That’s Maths column in this week’s Irish Times ( TM014 ) is about the use of computers for proving mathematical theorems, and also for simulating physical systems. Continue reading ‘Computer Maths’

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