**Abstract:** Continuity is defined relative to a topology. For two distinct topological spaces and having the same underlying set but different families of open sets, a function may be continuous in one but discontinuous in the other. Continue reading ‘The Signum Function may be Continuous’

## Archive for the 'Occasional' Category

### The Signum Function may be Continuous

Published September 9, 2021 Occasional Leave a CommentTags: Topology

### Real Derivatives from Imaginary Increments

Published August 26, 2021 Occasional Leave a CommentTags: Numerical Analysis

The solution of many problems requires us to compute derivatives. **Complex step differentiation** is a method of computing the first derivative of a real function, which circumvents the problem of roundoff error found with typical finite difference approximations.

Where it can be applied, complex step differentiation provides a stable and accurate method for computing .

Continue reading ‘Real Derivatives from Imaginary Increments’

**The Aperiodical** is described on its `About’ page as “*a meeting-place for people who already know they like maths and would like to know more*”. The Aperiodical coordinates the **Carnival of Mathematics (CoM)**, a monthly blogging roundup hosted on a different blog each month. Generally, the posts describe a collection of interesting recent items on mathematics from around the internet. This month, it is the turn of **thatsmaths.com** to host CoM.

Continue reading ‘Carnival of Mathematics’

### Simple Models of Atmospheric Vortices

Published July 29, 2021 Occasional Leave a CommentTags: Fluid Dynamics, Geophysics

Atmospheric circulation systems have a wide variety of structures and there is no single mechanistic model that describes all their characteristics. However, we can construct simple kinematic models that capture some primary aspects of the flow. For simplicity, we will concentrate on idealized extra-tropical depressions. We will not consider hurricanes and tropical storms in any detail, because the effects of moisture condensation and convection dominate their behaviour.

An isometry on a metric space is a one-to-one distance-preserving transformation on the space. The Euclidean group is the group of isometries of -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 , we have the group , comprising all combinations of translations, rotations and reflections of the plane.

The realization that the circumference of a circle is related in a simple way to the diameter came at an early stage in the development of mathematics. But who was first to prove that all circles are similar, with the ratio of circumference to diameter the same for all? Searching in Euclid’s *Elements*, you will not find a proof of this. It is no easy matter to define the length of a curve? It required the genius of Archimedes to prove that is constant, and he needed to introduce axioms beyond those of Euclid to achieve this; see earlier post here.

Continue reading ‘Approximating the Circumference of an Ellipse’

### Gauss Predicts the Orbit of Ceres

Published June 24, 2021 Occasional Leave a CommentTags: Astronomy, Gauss

### Al Biruni and the Size of the Earth

Published June 10, 2021 Occasional Leave a CommentTags: Geometry, History, Trigonometry

**Abu Rayhan al-Biruni (AD 973–1048)**

The 11th century Persian mathematician Abu Rayhan al-Biruni used simple trigonometric results to estimate the radius and circumference of the Earth. His estimate has been quoted as 6,340 km, which is within 1% of the mean radius of 6,371 km. While al-Biruni’s method was brilliant and, for its era, spectacular, the accuracy claimed must be regarded with suspicion.

Al-Biruni assumed that the Earth is a perfect sphere of (unknown) radius . He realised that because of the Earth’s curvature the horizon, as viewed from a mountain-top, would appear to be below the horizontal direction. This direction is easily obtained as being orthogonal to the vertical, which is indicated by a plumb line.

### Hanoi Graphs and Sierpinski’s Triangle

Published May 27, 2021 Occasional Leave a CommentTags: Algorithms, Fractals, Games

The **Tower of Hanoi** is a famous mathematical puzzle. A set of disks of different sizes are stacked like a cone on one of three rods, and the challenge is to move them onto another rod while respecting strict constraints:

- Only one disk can be moved at a time.
- No disk can be placed upon a smaller one.

### A Model for Elliptic Geometry

Published May 13, 2021 Occasional Leave a CommentTags: Geometry, Spherical Trigonometry

For many centuries, mathematicians struggled to derive Euclid’s fifth postulate as a theorem following from the other axioms. All attempts failed and, in the early nineteenth century, three mathematicians, working independently, found that consistent geometries could be constructed without the fifth postulate. Carl Friedrich Gauss (c. 1813) was first, but he published nothing on the topic. Nikolai Ivanovich Lobachevsky, around 1830, and János Bolyai, in 1832, published treatises on what is now called hyperbolic geometry.

### Can You Believe Your Eyes?

Published April 29, 2021 Occasional Leave a CommentTags: Numerical Analysis

Remember the old cowboy movies? As the stage-coach comes to a halt, the wheels appear to spin backwards, then forwards, then backwards again, until the coach stops. How can this be explained?

In Euclidean geometry, all lengths, areas and volumes are relative. Once a unit of length is chosen, all other lengths are given in terms of this unit. Classical geometry could determine the lengths of straight lines, the areas of polygons and the volumes of simple solids. However, the lengths of curved lines, areas bounded by curves and volumes with curved surfaces were mostly beyond the scope of Euclid. Only a few volumes — for example, the sphere, cylinder and cone — could be measured using classical methods.

### Circles, polygons and the Kepler-Bouwkamp constant

Published April 8, 2021 Occasional Leave a CommentTags: Analysis

If circles are drawn in and around an equilateral triangle (a regular trigon), the ratio of the radii is . More generally, for an N-gon the ratio is easily shown to be . Johannes Kepler, in developing his amazing polyhedral model of the solar system, started by considering circular orbits separated by regular polygons (see earlier post on the *Mysterium Cosmographicum* here).

Kepler was unable to construct an accurate model using polygons, but he noted that, if successive polygons with an increasing number of sides were inscribed within circles, the ratio did not diminish indefinitely but appeared to tend towards some limiting value. Likewise, if the polygons are circumscribed, forming successively larger circles (see Figure below), the ratio tends towards the inverse of this limit. It is only relatively recently that the limit, now known as the Kepler-Bouwkamp constant, has been established.

Continue reading ‘Circles, polygons and the Kepler-Bouwkamp constant’### Was Space Weather the cause of the Titanic Disaster?

Published April 1, 2021 Occasional Leave a CommentTags: Geophysics

Space weather, first studied in the 1950’s, has grown in importance with recent technological advances. It concerns the influence on the Earth’s magnetic field and upper atmosphere of events on the Sun. Such disturbances can enhance the solar wind, which interacts with the magnetosphere, with grave consequences for navigation. Space weather affects the satellites of the Global Positioning System, causing serious navigation problems [TM208 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Was Space Weather the cause of the Titanic Disaster?’### The Dimension of a Point that isn’t there

Published March 25, 2021 Occasional Leave a CommentTags: Topology

A slice of Swiss cheese has one-dimensional holes;

a block of Swiss cheese has two-dimensional holes.

What is the dimension of a point? From classical geometry we have the definition “A point is that which has no parts” — *also sprach Euclid*. A point has dimension zero, a line has dimension one, a plane has dimension two, and so on.

Continue reading ‘The Dimension of a Point that isn’t there’

### Differential Forms and Stokes’ Theorem

Published March 11, 2021 Occasional Leave a CommentTags: Analysis

The theory of exterior calculus of differential forms was developed by the influential French mathematician Élie Cartan, who did fundamental work in the theory of differential geometry. Cartan is regarded as one of the great mathematicians of the twentieth century. The exterior calculus generalizes multivariate calculus, and allows us to integrate functions over differentiable manifolds in dimensions.

The fundamental theorem of calculus on manifolds is called Stokes’ Theorem. It is a generalization of the theorem in three dimensions. In essence, it says that the change on the boundary of a region of a manifold is the sum of the changes within the region. We will discuss the basis for the theorem and then the ideas of exterior calculus that allow it to be generalized. Finally, we will use exterior calculus to write Maxwell’s equations in a remarkably compact form.

### Mamikon’s Theorem and the area under a cycloid arch

Published February 25, 2021 Occasional Leave a CommentTags: Analysis, Geometry

**The Cycloid**

The cycloid is the locus of a point fixed to the rim of a circular disk that is rolling along a straight line (see figure). The parametric equations for the cycloid are

where is the angle through which the disk has rotated. The centre of the disk is at .

* * * * *

**That’s Maths II: A Ton of Wonders**

by Peter Lynch now available.

Full details and links to suppliers at

http://logicpress.ie/2020-3/

>> Review in *The Irish Times <<*

* * * * *

Continue reading ‘Mamikon’s Theorem and the area under a cycloid arch’

A ring torus (or, simply, torus) is a surface of revolution generated by rotating a circle about a coplanar axis that does not intersect it. We let be the radius of the circle and the distance from the axis to the centre of the circle, with .

### Euler’s Product: the Golden Key

Published January 28, 2021 Occasional Leave a CommentTags: Analysis, Euler

The Basel problem was solved by Leonhard Euler in 1734 [see previous post]. His line of reasoning was ingenious, with some daring leaps of logic. The Basel series is a particular case of the much more general **zeta function**, which is at the core of the Riemann hypothesis, the most important unsolved problem in mathematics.

Euler treated the Taylor series for as a polynomial of infinite degree. He showed that it could also be expressed as an infinite product, arriving at the result

This enabled him to deduce the remarkable result

which he described as an unexpected and elegant formula.

### The Basel Problem: Euler’s Bravura Performance

Published January 14, 2021 Occasional Leave a CommentTags: Analysis, Euler

The Basel problem was first posed by Pietro Mengoli, a mathematics professor at the University of Bologna, in 1650, the same year in which he showed that the alternating harmonic series sums to . The Basel problem asks for the sum of the reciprocals of the squares of the natural numbers,

It is not immediately clear that this series converges, but this can be proved without much difficulty, as was first shown by Jakob Bernoulli in 1689. The sum is approximately 1.645 which has no obvious interpretation.

* * * * *

**That’s Maths II: A Ton of Wonders**

by Peter Lynch has just appeared.

Full details and links to suppliers at

http://logicpress.ie/2020-3/

* * * * *

Continue reading ‘The Basel Problem: Euler’s Bravura Performance’

### Derangements and Continued Fractions for e

Published December 31, 2020 Occasional Leave a CommentTags: Combinatorics, Number Theory

We show in this post that an elegant continued fraction for can be found using derangement numbers. Recall from last week’s post that we call any permutation of the elements of a set an *arrangement*. A *derangement* is an arrangement for which every element is moved from its original position.

Continue reading ‘Derangements and Continued Fractions for e’

### Will RH be Proved by a Physicist?

Published December 10, 2020 Occasional Leave a CommentTags: Analysis, Number Theory

The Riemann Hypothesis (RH) states that all the non-trivial (non-real) zeros of the zeta function lie on a line, the *critical line*, . By a simple change of variable, we can have them lying on the real axis. But the eigenvalues of any hermitian matrix are real. This led to the *Hilbert-Polya Conjecture*:

The non-trivial zeros of are the

eigenvalues of a hermitian operator.

Is there a *Riemann operator*? What could this operater be? What dynamical system would it represent? Are prime numbers and quantum mechanics linked? Will RH be proved by a physicist?

This last question might make a purest-of-the-pure number theorist squirm. But it is salutary to recall that, of the nine papers that Riemann published during his lifetime, four were on physics!

### Laczkovich Squares the Circle

Published November 26, 2020 Occasional Leave a CommentTags: Analysis, Logic

The phrase `squaring the circle’ generally denotes an impossible task. The original problem was one of three unsolved challenges in Greek geometry, along with trisecting an angle and duplicating a cube. The problem was to construct a square with area equal to that of a given circle, using only straightedge and compass.

### Aleph, Beth, Continuum

Published November 12, 2020 Occasional Leave a CommentTags: Logic, Set Theory

Georg Cantor developed a remarkable theory of infinite sets. He was the first person to show that *not all infinite sets are created equal*. The number of elements in a set is indicated by its cardinality. Two sets with the same cardinal number are “the same size”. For two finite sets, if there is a one-to-one correspondence — or bijection — between them, they have the same number of elements. Cantor extended this equivalence to infinite sets.

### The p-Adic Numbers (Part 2)

Published October 29, 2020 Occasional Leave a CommentTags: Number Theory

Kurt Hensel, born in Königsberg, studied mathematics in Berlin and Bonn, under Kronecker and Weierstrass; Leopold Kronecker was his doctoral supervisor. In 1901, Hensel was appointed to a full professorship at the University of Marburg. He spent the rest of his career there, retiring in 1930.

Hensel is best known for his introduction of the *p*-adic numbers. They evoked little interest at first but later became increasingly important in number theory and other fields. Today, *p*-adics are considered by number theorists as being “just as good as the real numbers”. Hensel’s *p*-adics were first described in 1897, and much more completely in his books, *Theorie der algebraischen Zahlen*, published in 1908 and *Zahlentheorie* published in 1913.

### The p-Adic Numbers (Part I)

Published October 22, 2020 Occasional Leave a CommentTags: Number Theory

The motto of the Pythagoreans was “All is Number”. They saw numbers as the essence and foundation of the physical universe. For them, numbers meant the positive whole numbers, or natural numbers , and ratios of these, the positive rational numbers . It came as a great shock that the diagonal of a unit square could not be expressed as a rational number.

If we arrange the rational numbers on a line, there are gaps everywhere. We can fill these gaps by introducing additional numbers, which are the limits of sequences of rational numbers. This process of *completion* gives us the real numbers , which include rationals, irrationals like and transcendental numbers like .

### From Impossible Shapes to the Nobel Prize

Published October 8, 2020 Occasional 1 CommentTags: Astronomy, Geometry

Roger Penrose, British mathematical physicist, mathematician and philosopher of science has just been named as one of the winners of the 2020 Nobel Prize in Physics. Penrose has made major contributions to general relativity and cosmology.

Penrose has also come up with some ingenious mathematical inventions. He discovered a way of defining a pseudo-inverse for matrices that are singular, he rediscovered an “impossible object”, now called the Penrose Triangle, and he discovered that the plane could be tiled in a non-periodic way using two simple polygonal shapes called kites and darts.Continue reading ‘From Impossible Shapes to the Nobel Prize’

### Doughnuts and Dumplings are Distinct: Homopoty-101

Published September 24, 2020 Occasional Leave a CommentTags: Topology

As everyone knows, a torus is different from a sphere. Topology is the study of properties that remain unchanged under continuous distortions. A square can be deformed into a circle or a sphere into an ellipsoid, whether flat like an orange or long like a lemon or banana.

Continue reading ‘Doughnuts and Dumplings are Distinct: Homopoty-101’

Mathematicians owe a great debt of gratitude to Donald Knuth. A renowned American computer scientist and mathematician, Knuth is an emeritus professor at Stanford University. He is author of many books, including the multi-volume work, *The Art of Computer Programming*.

Knuth is the author of the powerful and versatile mathematical typesetting system called **TeX**. The original version, designed and written by Knuth, was released in 1978.

TeX is a powerful system for typesetting mathematical formulae. It is ideal both for simple mathematical notes with few formulas and for more complex documents and books involving subtle and sophisticated mathematical typography. TeX is used by almost all research mathematicians. It is also popular in computer science, engineering, physics, statistics, and and many other sciences.

### Jung’s Theorem: Enclosing a Set of Points

Published August 27, 2020 Occasional Leave a CommentTags: Geometry

Let us imagine that we have a finite set of points in the plane (Fig. 1a). How large a circle is required to enclose them. More specifically, what is the minimum radius of such a bounding circle? The answer is given by Jung’s Theorem.

Continue reading ‘Jung’s Theorem: Enclosing a Set of Points’

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

Published August 13, 2020 Occasional Leave a CommentTags: Number Theory, Probability

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?

Continue reading ‘Think of a Number: What are the Odds that it is Even?’

### Resolution of Paradox: a Gateway to Mathematical Progress

Published August 6, 2020 Occasional Leave a CommentA 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 science of the infinite” [TM192 or search for “thatsmaths” at irishtimes.com].

Ever since there has been time to wonder, humankind has been intrigued by the enigma of infinity. The idea has perplexed philosophers and mystified mathematicians for millennia. The Greeks were contemplating infinity from the time of Pythagoras. Head-on confrontation of infinity led to contradictory conclusions; the paradoxes of Zeno are amongst the most noted of these.

Continue reading ‘Resolution of Paradox: a Gateway to Mathematical Progress’

### Berry’s Paradox and Gödel’s Incompleteness Theorem

Published July 30, 2020 Occasional Leave a CommentTags: Logic

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.

Continue reading ‘Berry’s Paradox and Gödel’s Incompleteness Theorem’

### Does Numerical Integration Reflect the Truth?

Published July 23, 2020 Occasional Leave a CommentTags: Numerical Analysis, Numerical Weather Prediction

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 big question is:

“*Does the numerical solution resemble the true solution of the equation?*”

The answer is: “*Not necessarily*”.

There are often specific criteria that must be satisfied to ensure that the answer `crunched out’ by the computer is a reasonable approximation to reality. Although the principles of numerical stability are quite general, they are best illustrated by simple examples. We will look at some of these below.

Continue reading ‘Does Numerical Integration Reflect the Truth?’

“Buridan’s Ass” is a paradox in philosophy, in which a hungry donkey, located at the mid-point between two bales of hay, is frozen in indecision about which way to go and faces starvation — he is unable to move one way or the other.

Jean Buridan was a French philosopher who lived in the fourteenth century. He was not interested in donkeys, but in human morality. He wrote that if two courses of action are judged to be morally equal, we must suspend a decision until the right course of action becomes clear. The idea of the paradox can be found in the writings of the ancients, including Aristotle.

### The Online Encyclopedia of Integer Sequences

Published June 25, 2020 Occasional Leave a CommentTags: Analysis, Arithmetic, Number Theory

Suppose that, in the course of an investigation, you stumble upon a string of whole numbers. You are convinced that there must be a pattern, but you cannot find it. All you have to do is to type the string into a database called OEIS — or simply “Slone’s” — and, if the string is recognized, an entire infinite sequence is revealed. If the string belongs to several sequences, several choices are offered. OEIS is a great boon to both professional mathematicians and applied scientists in fields like physics, chemistry, astronomy and biology.

Continue reading ‘The Online Encyclopedia of Integer Sequences’

### Dimension Reduction by PCA

Published June 11, 2020 Occasional Leave a CommentTags: 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.

### The Monte-Carlo Method

Published May 28, 2020 Occasional Leave a CommentTags: Algorithms, Numerical Analysis

Learning calculus at school, we soon find out that while differentiation is relatively easy, at least for simple functions, integration is hard. So hard indeed that, in many cases, it is impossible to find a nice function that is the integral (or anti-derivative) of a given one. Thus, given we can usually find , whereas we may not be able to find .

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.

### Order in the midst of Chaos

Published April 30, 2020 Occasional 1 CommentTags: Combinatorics, Graph Theory

We open with a simple mathematical puzzle that is easily solved using only elementary reasoning. Imagine a party where some guests are friends while others are unacquainted. Then the following is always true:

*No matter how many guests there are at the party, there are
always two guests with the same number of friends present.*

If you wish, try proving this before reading on. The proof is outlined at the end of this post.

### John Horton Conway: a Charismatic Genius

Published April 23, 2020 Occasional Leave a CommentTags: Games, Group Theory, Number Theory, Recreational Maths, Topology

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 he was appointed Lecturer in Pure Mathematics.

In 1986, Conway moved to Princeton University, where he was Professor of Mathematics and John Von Neumann Professor in Applied and Computational Mathematics. He was awarded numerous honours during his career. Conway enjoyed emeritus status from 2013 until his death just two weeks ago on 11 April.

*A Mathematician’s Miscellany*. It was later analysed in detail by Sheldon Ross in his 1988 book

*A First Course in Probability*.

### Bang! Bang! Bang! Explosively Large Numbers

Published March 26, 2020 Occasional Leave a CommentTags: Algebra, Number Theory

which is approximately . The number of atoms in the universe is estimated to be about . When we consider permutations of large sets, even more breadth-taking numbers emerge.

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

### Samuel Haughton and the Twelve Faithless Hangmaids

Published March 12, 2020 Occasional Leave a CommentTags: Applied Maths

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 for Penelope’s hand in marriage. Her son Telemachus, with the help of his comrades, hanged all twelve handmaids on a single rope.

Continue reading ‘Samuel Haughton and the Twelve Faithless Hangmaids’

### Zhukovsky’s Airfoil

Published February 27, 2020 Occasional Leave a CommentTags: Applied Maths, Fluid Dynamics

A simple transformation with remarkable properties was used by Nikolai Zhukovsky around 1910 to study the flow around aircraft wings. It is defined by

and is usually called the *Joukowsky Map*. We begin with a discussion of the theory of fluid flow in two dimensions. Readers familiar with 2D potential flow may skip to the section *Joukowsky Airfoil*.

### A Ring of Water Shows the Earth’s Spin

Published February 13, 2020 Occasional Leave a CommentTags: Geophysics, Mechanics

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.

### The Rambling Roots of Wilkinson’s Polynomial

Published January 30, 2020 Occasional Leave a CommentTags: 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.

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

### Adjoints of Vector Operators

Published January 23, 2020 Occasional Leave a CommentTags: 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:

**Question: Is there a connection between these identities?**

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

Published January 9, 2020 Occasional Leave a CommentTags: Analysis, Numerical Weather Prediction

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.

Continue reading ‘Grad, Div and Curl on Weather Maps: a Gateway to Vector Analysis’

### Divergent Series Yield Valuable Results

Published December 26, 2019 Occasional Leave a CommentTags: Analysis

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.

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