Posts Tagged 'Algorithms'



The Ups and Downs of Hailstone Numbers

Hailstones, in the process of formation, make repeated excursions up and down within a cumulonimbus cloud until finally they fall to the ground. We look at sequences of numbers that oscillate in a similarly erratic manner until they finally reach the value 1. They are called hailstone numbers.
Continue reading ‘The Ups and Downs of Hailstone Numbers’

CT Scans and the Radon Transform

Last December, Dublin’s Tallaght Hosptal acquired a new CT scanner, a Toshiba Aquilon Prime, the first of its type in the country. The state-of-the-art scanner is housed in a room with a ‘sky ceiling’ that allows patients to enjoy an attractive outdoor image during the scanning process.

This equipment, which cost €600,000 will undoubtedly result in timely treatment of patients and the saving of lives. The process of generating images from CT scans is described in the latest That’s Maths column (TM016) in the Irish Times.

Continue reading ‘CT Scans and the Radon Transform’

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’

Santa’s TSP Algorithm

This week’s That’s Maths column ( TM011 ) discusses the challenge faced by Santa Claus: he has about a billion homes to visit in one night, so he needs to be smart in picking his route. The challenge he faces is called the Travelling Salesman Problem, or TSP. Continue reading ‘Santa’s TSP Algorithm’

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. Continue reading ‘Carving up the Globe’

Packing & Stacking

In That’s Maths this week (TM004), we look at the problem of packing goods of fixed size and shape in the most efficient way. Packing problems, concerned with storing objects as densely as possible in a container, have a long history, and have broad applications in engineering and industry.

Johannes Kepler conjectured that the standard method used by grocers to pile oranges and gunners to stack cannon balls is the most efficient, but this conjecture was proved only recently by Thomas Hales. The mathematics involved in packing problems includes computational techniques, differential geometry and optimization algorithms.

The Foams and Complex Systems Group in Trinity College Dublin have recently discovered some new dense packings of spheres in cylindrical columns. An International Workshop on Packing Problems took place in TCD on 2-5 Sept. 2012. For more information, look here.

 

Google PageRank

This week’s That’s Maths article, at TM002, describes how Google’s PageRank software finds all those links when you enter a search word, by solving an enormous problem in linear algebra.

A comprehensive description of PageRank is given in the book Google’s PageRank and Beyond: The Science of Search Engine Rankings, by Amy N. Langville & Carl D. Meyer This book won an AAP Award in 2006  for Best Professional/Scholarly Book in Computer & Information Science.


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