Algebra | ThatsMaths

Posts Tagged 'Algebra'

Finding Fixed Points

Published July 22, 2021 Occasional Closed
Tags: Algebra

An isometry on a metric space is a one-to-one distance-preserving transformation on the space. The Euclidean group ${E(n)}$ is the group of isometries of ${n}$ -dimensional Euclidean space. These are all the transformations that preserve the distance between any two points. The group depends on the dimension of the space. For the Euclidean plane ${\mathbb{E}^2}$ , we have the group ${E(2)}$ , comprising all combinations of translations, rotations and reflections of the plane.

Continue reading ‘Finding Fixed Points’

Dimension Reduction by PCA

Published June 11, 2020 Occasional Closed
Tags: Algebra, Numerical Analysis

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 buried in mountains of data.

Correlated-Variables

Continue reading ‘Dimension Reduction by PCA’

Bang! Bang! Bang! Explosively Large Numbers

Published March 26, 2020 Occasional Closed
Tags: Algebra, Number Theory

Typical Comic-book `bang’ mark [Image from vectorstock ].

Enormous numbers pop up in both mathematics and physics. The order of the monster group, the largest of the 26 sporadic groups, is

$\displaystyle 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000$

which is approximately ${8\times 10^{53}}$ . The number of atoms in the universe is estimated to be about ${10^{80}}$ . When we consider permutations of large sets, even more breadth-taking numbers emerge.

Continue reading ‘Bang! Bang! Bang! Explosively Large Numbers’

The Rambling Roots of Wilkinson’s Polynomial

Published January 30, 2020 Occasional Closed
Tags: Algebra, Numerical Analysis

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.

A 10th-order polynomial (blue) and a slightly perturbed version, with the coefficient of ${x^9}$ changed by one part in a million.

Continue reading ‘The Rambling Roots of Wilkinson’s Polynomial’

Adjoints of Vector Operators

Published January 23, 2020 Occasional Closed
Tags: Algebra, Analysis

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:

$\displaystyle \begin{array}{rcl} \mathbf{curl}\ \mathbf{grad}\ \chi &\equiv& 0\,, \quad \mbox{for all scalar functions\ }\chi \\ \mathrm{div}\ \mathbf{curl}\ \boldsymbol{\psi} &\equiv& 0\,, \quad \mbox{for all vector functions\ }\boldsymbol{\psi} \end{array}$

Question: Is there a connection between these identities?

CurlGradAndDivCurl

Continue reading ‘Adjoints of Vector Operators’

Some Fundamental Theorems of Maths

Published October 24, 2019 Occasional Closed
Tags: Algebra, Analysis, Geometry

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 then used to deduce the consequences of those axioms with each major result designated as a theorem.

As each new theorem is proved, it provides a basis for the establishment of further results. The most important and fruitful theorem in each area of maths is often named as the fundamental theorem of that area. Thus, we have the fundamental theorems of arithmetic, algebra and so on. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus.

Continue reading ‘Some Fundamental Theorems of Maths’

The Many Modern Uses of Quaternions

Published October 4, 2018 Irish Times Closed
Tags: Algebra, Hamilton

Hamiltons-Bridge-Plaque The story of William Rowan Hamilton’s discovery of new four-dimensional numbers called quaternions is familiar. The solution of a problem that had bothered him for years occurred to him in a flash of insight as he walked along the Royal Canal in Dublin. But this Eureka moment did not arise spontaneously: it was the result of years of intense effort. The great French mathematician Henri Poincaré also described how sudden inspiration occurs unexpectedly, but always following a period of concentrated research [TM148, or search for “thatsmaths” at irishtimes.com].

Continue reading ‘The Many Modern Uses of Quaternions’

The Flight of the Bumble Bee

Published August 23, 2018 Occasional Closed
Tags: Algebra, Puzzles

Alice and Bob, initially a distance l apart, walk towards each other, each at a speed w. A bumble bee flies from the nose of one to the nose of the other and back again, repeating this zig-zag flight at speed f until Alice and Bob meet. How far does the bumble bee fly?

Flight-of-BumbleBee-Music

Continue reading ‘The Flight of the Bumble Bee’

Marden’s Marvel

Published May 10, 2018 Occasional Closed
Tags: Algebra, Geometry

Although polynomial equations have been studied for centuries, even millennia, surprising new results continue to emerge. Marden’s Theorem, published in 1945, is one such — delightful — result.

Cubic with roots at x=1, x=2 and x=3.

Continue reading ‘Marden’s Marvel’

Cubic Skulduggery & Intrigue

Published March 15, 2018 Irish Times Closed
Tags: Algebra, History

Solution of a cubic equation, usually called Cardano’s formula.

Babylonian mathematicians knew how to solve simple polynomial equations, in which the unknown quantity that we like to call x enters in the form of powers, that is, x multiplied repeatedly by itself. When only x appears, we have a linear equation. If x-squared enters, we have a quadratic. The third power of x yields a cubic equation, the fourth power a quartic and so on [TM135 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Cubic Skulduggery & Intrigue’

Metallic Means

Published February 9, 2017 Occasional Closed
Tags: Algebra, Arithmetic, Recreational Maths

The golden mean occurs repeatedly in the pentagram [image Wikimedia Commons]

Everyone knows about the golden mean. It must be one of the most written-about numbers, certainly in recreational mathematics. It is usually denoted by ${\phi}$ and is the positive root of the quadratic equation

$\displaystyle x^2 - x - 1 = 0 \ \ \ \ \ (1)$

with the value

${\phi = (1+\sqrt{5})/2 \approx 1.618}$ .

There is no doubt that ${\phi}$ is significant in many biological contexts and has also been an inspiration for artists. Called the Divine Proportion, it was described in a book of that name by Luca Pacioli, a contemporary and friend of Leonardo da Vinci.

Continue reading ‘Metallic Means’

The Beginning of Modern Mathematics

Published January 26, 2017 Occasional Closed
Tags: Algebra, History

The late fifteenth century was an exciting time in Europe. Western civilization woke with a start after the slumbers of the medieval age. Johannes Gutenberg’s printing press arrived in 1450 and changed everything. Universities in Bologna, Oxford, Salamanca, Paris and elsewhere began to flourish. Leonardo da Vinci was in his prime and Christopher Columbus was discovering a new world.

Illustrations by Leonardo da Vinci in Pacioli’s De Divina Proportione.

Continue reading ‘The Beginning of Modern Mathematics’

Raphael Bombelli’s Psychedelic Leap

Published December 8, 2016 Occasional Closed
Tags: Algebra, History

The story of how Italian Renaissance mathematicians solved cubic equations has elements of skullduggery and intrigue. The method originally found by Scipione del Ferro and independently by Tartaglia, was published by Girolamo Cardano in 1545 in his book Ars Magna. The method, often called Cardano’s method, gives the solution of a depressed cubic equation t³ + p t + q = 0. The general cubic equation can be reduced to this form by a simple linear transformation of the dependent variable. The solution is given by

cardano-formula

Cardano assumed that the discriminant Δ = ( q / 2 )² + ( p / 3 )³, the quantity appearing under the square-root sign, was positive.

Raphael Bombelli made the psychedelic leap that Cardano could not make. He realised that Cardano’s formula would still give a solution when the discriminant was negative, provided that the square roots of negative quantities were manipulated in the correct manner. He was thus the first to properly handle complex numbers and apply them with effect.

Continue reading ‘Raphael Bombelli’s Psychedelic Leap’

Andrew Wiles wins 2016 Abel Prize

Published May 19, 2016 Irish Times Closed
Tags: Algebra, Number Theory

A recent post described the Abel Prize, effectively the Nobel Prize for Mathematics, and promised a further post when the 2016 winner was announced. This is the follow-up post [also at TM091, or search for “thatsmaths” at irishtimes.com].

Abel-Prize-Medal Next Tuesday, HRH Crown Prince Haakon will present the Abel Medal to Sir Andrew Wiles at a ceremony in Oslo. The Abel Prize, comparable to a Nobel Prize, is awarded for outstanding work in mathematics. Wiles has won the award for his “stunning proof of Fermat’s Last Theorem” with his research “opening a new era in number theory”. Wiles’ proof made international headlines in 1994 when he cracked one of the most famous and long-standing unsolved problems in mathematics.

Pierre de Fermat, a French lawyer and amateur mathematician, stated the theorem in 1637, writing in the margin of a maths book that he had “a truly marvellous proof”. But for more than 350 years no proof was found despite the efforts of many of the most brilliant mathematicians.

Continue reading ‘Andrew Wiles wins 2016 Abel Prize’

James Joseph Sylvester

Published September 3, 2015 Irish Times Closed
Tags: Algebra, History

J. J. Sylvester as a graduate of Trinity College Dublin.

James Joseph Sylvester (1814-1897) as a graduate of Trinity College Dublin.

James Joseph Sylvester was born in London to Jewish parents in 1814, just 201 years ago today. The family name was Joseph but, for reasons unclear, Sylvester – the name of an anti-Semitic Pope from the Roman period – was adopted later. [TM075; or search for “thatsmaths” at irishtimes.com ]

Sylvester’s mathematical talents became evident at an early age. He entered Cambridge in 1831, aged just seventeen and came second in the notorious examinations known as the Mathematical Tripos; the man who beat him achieved nothing further in mathematics!

Continue reading ‘James Joseph Sylvester’

Increasingly Abstract Algebra

Published June 25, 2015 Occasional Closed
Tags: Algebra

In the seventeenth century, the algebraic approach to geometry proved to be enormously fruitful. When René Descartes (1596-1650) developed coordinate geometry, the study of equations (algebra) and shapes (geometry) became inextricably interlinked. The move towards greater abstraction can make mathematics appear more abstruse and impenetrable, but it brings greater clarity and power, and can lead to surprising unifications.

Evariste Galois, Sophus Lie and Emmy Noether.

Continue reading ‘Increasingly Abstract Algebra’

Emmy Noether’s beautiful theorem

Published June 18, 2015 Irish Times Closed
Tags: Algebra, Mechanics

The number of women who have excelled in mathematics is lamentably small. Many reasons may be given, foremost being that the rules of society well into the twentieth century debarred women from any leading role in mathematics and indeed in science. But a handful of women broke through the gender barrier and made major contributions. [TM070: search for “thatsmaths” at irishtimes.com ]

Death of the philosopher Hypatia, by Louis Figuier
[Wikimedia Commons].

Continue reading ‘Emmy Noether’s beautiful theorem’

Mode-S: Aircraft Data improves Weather Forecasts

Published April 2, 2015 Irish Times Closed
Tags: Algebra, Geometry, Geophysics

A simple application of vectors yields valuable new wind observations for weather forecasting [see this week’s That’s Maths column (TM065) or search for “thatsmaths” at irishtimes.com].

The air speed is A (blue), the wind speed is W (black) and the ground speed is G (red). Since the ground speed is the resultant (vector sum) of air speed and wind speed, a simple vector subtraction gives the wind speed: W= G – A.

Continue reading ‘Mode-S: Aircraft Data improves Weather Forecasts’

The Steiner Minimal Tree

Published January 29, 2015 Occasional Closed
Tags: Algebra, Algorithms, Gauss, Maps, Topology

Steiner’s minimal tree problem is this: Find the shortest possible network interconnecting a set of points in the Euclidean plane. If the points are linked directly to each other by straight line segments, we obtain the minimal spanning tree. But Steiner’s problem allows for additional points – now called Steiner points – to be added to the network, yielding Steiner’s minimal tree. This generally results in a reduction of the overall length of the network.

A solution of Steiner 5-point problem with soap film [from Courant and Robbins].

Continue reading ‘The Steiner Minimal Tree’

Old Octonions may rule the World

Published October 16, 2014 Irish Times Closed
Tags: Algebra, Hamilton, Number Theory

This week’s That’s Maths column in The Irish Times (TM055, or search for “thatsmaths” at irishtimes.com) is about octonions, new numbers discovered by John T Graves, a friend of William Rowan Hamilton.

Multiplication table for octonions, of the form z=a+bi+cj+dk+eE+fI+gJ+hK [Source: http://jmc2008.wurzel.org/index.php/Main/Logo]

Continue reading ‘Old Octonions may rule the World’

Algebra in the Golden Age

Published October 2, 2014 Irish Times Closed
Tags: Algebra, History

This week’s That’s Maths column in The Irish Times (TM054, or search for “thatsmaths” at irishtimes.com) is about the emergence of algebra in the Golden Age of Islam. The Chester Beatty Library in Dublin has several thousand Arabic manuscripts, many on mathematics and science.

Left: Societ stamp commemorating al-Khwārizmī's 1200th birthday. RIght: A page from al-Khwārizmī's Al-Jebr.

Left: Soviet Union postage stamp (1983) commemorating al-Khwārizmī’s 1200th birthday. RIght: A page from al-Khwārizmī’s Al-Jebr.

Continue reading ‘Algebra in the Golden Age’

Cartoon Curves

Published September 11, 2014 Occasional Closed
Tags: Algebra, Computer Science, Geometry, Recreational Maths

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.

Continue reading ‘Cartoon Curves’

The High-Power Hypar

Published May 29, 2014 Occasional Closed
Tags: Algebra, Gauss, Geometry, Recreational Maths

Maths frequently shows us surprising and illuminating connections between physical systems that are not obviously related: the analysis of one system often turns out to be ideally suited for describing another. To illustrate this, we will show how a surface in three dimensional space — the hyperbolic paraboloid, or hypar — pops up in unexpected ways and in many different contexts.

Warszawa Ochota railway station, a hypar structure
[Image Wikimedia Commons].

Continue reading ‘The High-Power Hypar’

The Unity of Mathematics

Published March 20, 2014 Irish Times Closed
Tags: Algebra, Analysis, Geometry, Number Theory

This week, That’s Maths in The Irish Times ( TM041 ) is about an ambitious program to unify mathematics.

Mathematics expands! Results once proven to be true remain forever true. They are not displaced by subsequent results, but absorbed in an ever-growing theoretical web. Thus, it is increasingly difficult for any individual mathematician to have a comprehensive understanding of even a single field of mathematics: the web of knowledge grows so fast that no-one can master it all.

Continue reading ‘The Unity of Mathematics’

Bézout’s Theorem

Published January 30, 2014 Occasional Closed
Tags: Algebra

Two lines in a plane intersect at one point, a line cuts a circle at two points, a cubic (an S-shaped curve) crosses the x-axis three times and two ellipses, one tall and one squat, intersect in four places.

Continue reading ‘Bézout’s Theorem’

The Watermelon Puzzle

Published November 14, 2013 Occasional Closed
Tags: Algebra, Puzzles, Recreational Maths

An amusing puzzle appears in a recent book by John A. Adam (2013). The answer is very surprising. The book argues in terms of simultaneous equations. A simpler argument, using the diagram below, should make all clear. Continue reading ‘The Watermelon Puzzle’

Monster Symmetry

Published May 2, 2013 Irish Times Closed
Tags: Algebra

The That’s Maths column in the Irish Times this week is about symmetry and group theory, and the possible link, through string theory, with the fundamental structure of the universe ( TM020 ).

In the arts, symmetry is intimately associated with aesthetic appeal. In science, it provides insight into the properties of physical systems.

Continue reading ‘Monster Symmetry’

Singularly Valuable SVD

Published February 14, 2013 Occasional Closed
Tags: Algebra, Algorithms, Computer Science, Numerical Analysis

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
Continue reading ‘Singularly Valuable SVD’

Google PageRank

Published August 2, 2012 Irish Times Closed
Tags: Algebra, Algorithms

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.

ThatsMaths