Category: Occasional

Low-pass Filtering and the Remarkable Integrals of Borwein and Borwein

In last week's post we looked at aspects of puzzles of the form ``What is the next number''. We are presented with a short list of numbers, for example $latex {1, 3, 5, 7, 9}&fg=000000$ and asked for the next number in the sequence. Arguments were given indicating why any number might be regarded as … Continue reading Low-pass Filtering and the Remarkable Integrals of Borwein and Borwein →

The Power of the 2-gon: Extrapolation to Evaluate Pi

  Richardson's extrapolation procedure yields a significant increase in the accuracy of numerical solutions of differential equations. We consider his elegant illustration of the technique, the evaluation of $latex {\pi}&fg=000000$, and show how the estimates improve dramatically with higher order extrapolation. [This post is a condensed version of a paper in Mathematics Today (Lynch, 2003).] … Continue reading The Power of the 2-gon: Extrapolation to Evaluate Pi →

From Sub-atomic to Cosmic Strings

The two great pillars of modern physics are quantum mechanics and general relativity. These theories describe small-scale and large-scale phenomena, respectively. While quantum mechanics predicts the shape of a hydrogen atom, general relativity explains the properties of the visible universe on the largest scales. A longstanding goal of physics is to construct a new theory … Continue reading From Sub-atomic to Cosmic Strings →

CND Functions: Curves that are Continuous but Nowhere Differentiable

A function $latex {f(x)}&fg=000000$ that is differentiable at a point $latex {x}&fg=000000$ is continuous there, and if differentiable on an interval $latex {[a, b]}&fg=000000$, is continuous on that interval. However, the converse is not necessarily true: the continuity of a function at a point or on an interval does not guarantee that it is differentiable … Continue reading CND Functions: Curves that are Continuous but Nowhere Differentiable →

Topological Calculus: away with those nasty epsilons and deltas

A new approach to calculus has recently been developed by Peter Olver of the University of Minnesota. He calls it ``Continuous Calculus'' but indicates that the name ``Topological Calculus'' is also appropriate. He has provided an extensive set of notes, which are available online (Olver, 2022a)]. Motivation Students embarking on a university programme in mathematics … Continue reading Topological Calculus: away with those nasty epsilons and deltas →

The 3-sphere: Extrinsic and Intrinsic Forms

The circle in two dimensions and the sphere in three are just two members of an infinite family of hyper-surfaces. By analogy with the circle $latex {\mathbb{S}^1}&fg=000000$ in the plane $latex {\mathbb{R}^2}&fg=000000$ and the sphere $latex {\mathbb{S}^2}&fg=000000$ in three-space $latex {\mathbb{R}^3}&fg=000000$, we can consider hyper-spheres in higher dimensional spaces. In particular, we will consider the … Continue reading The 3-sphere: Extrinsic and Intrinsic Forms →

Dynamic Equations for Weather and Climate

``I could have done it in a much more complicated way'', said the Red Queen, immensely proud. --- Lewis Carroll. Books on dynamic meteorology and oceanography usually have a full chapter devoted to the basic dynamical equations. Since the Earth's fluid envelop is approximately a thin spherical shell, spherical coordinates $latex {(\lambda,\varphi, r)}&fg=000000$ are convenient. … Continue reading Dynamic Equations for Weather and Climate →

Curl Curl Curl

Many of us have struggled with the vector differential operators, grad, div and curl. There are several ways to represent vectors and several expressions for these operators, not always easy to remember. We take another look at some of their properties here. We consider a vector $latex {\mathbf{V} = (u, v, w)^{\mathrm{T}}}&fg=000000$ which may be … Continue reading Curl Curl Curl →

Space-Filling Curves, Part II: Computing the Limit Function

The Approximating Functions It is simple to define a mapping from the unit interval $latex {I := [0,1]}&fg=000000$ into the unit square $latex {Q:=[0,1]\times[0,1]}&fg=000000$. Georg Cantor found a one-to-one map from $latex {I}&fg=000000$ onto $latex {Q}&fg=000000$, showing that the one-dimensional interval and the two-dimensional square have the same cardinality. Cantor's map was not continuous, but … Continue reading Space-Filling Curves, Part II: Computing the Limit Function →

Space-Filling Curves, Part I: “I see it, but I don’t believe it”

We are all familiar with the concept of dimension: a point is zero-dimensional, a line is one-dimensional, a plane is two-dimensional and the space around us is three-dimensional. A position on a line can be specified by a single number, such as the distance from a fixed origin. In the plane, a point can be … Continue reading Space-Filling Curves, Part I: “I see it, but I don’t believe it” →

Poincare’s Square and Unbounded Gomoku

Henri Poincar'e was masterful in presenting scientific concepts and ideas in an accessible way. To explain that the Universe might be bounded and yet infinite, he imagined that the temperature of space decreased from the centre to the periphery in such a way that everything contracted with the distance from the centre. As travellers moved … Continue reading Poincare’s Square and Unbounded Gomoku →

Goldbach’s Conjecture and Goldbach’s Variation

Goldbach's Conjecture is one of the great unresolved problems of number theory. It simply states that every even natural number greater than two is the sum of two prime numbers. It is easily confirmed for even numbers of small magnitude. The conjecture first appeared in a letter dated 1742 from German mathematician Christian Goldbach to … Continue reading Goldbach’s Conjecture and Goldbach’s Variation →

Fairy Lights on the Farey Tree

The rational numbers $latex {\mathbb{Q}}&fg=000000$ are dense in the real numbers $latex {\mathbb{R}}&fg=000000$. The cardinality of rational numbers in the interval $latex {(0,1)}&fg=000000$ is $latex {\boldsymbol{\aleph}_0}&fg=000000$. We cannot list them in ascending order, because there is no least rational number greater than $latex {0}&fg=000000$. However, there are several ways of enumerating the rational numbers. The … Continue reading Fairy Lights on the Farey Tree →

Parity and Partition of the Rational Numbers. Part II: Density of the Three Parity Classes

In last week's post, we defined an extension of parity from the integers to the rational numbers. Three parity classes were found --- even, odd and none. This week, we show that, with an appropriate ordering or enumeration of the rationals, the three classes are not only equinumerate (having the same cardinality) but of equal … Continue reading Parity and Partition of the Rational Numbers. Part II: Density of the Three Parity Classes →

Parity and Partition of the Rational Numbers. Part I: The Three Parity Classes

We define an extension of parity from the integers to the rational numbers. Three parity classes are found --- even, odd and none. Using the 2-adic valuation, we partition the rationals into subgroups with a rich algebraic structure. The natural numbers $latex {\mathbb{N}}&fg=000000$ split nicely into two subsets, the odd and even numbers $latex \displaystyle … Continue reading Parity and Partition of the Rational Numbers. Part I: The Three Parity Classes →

A Finite but Unbounded Universe

Henri Poincaré described a beautiful geometric model with some intriguing properties. He envisioned a circular disk in the Euclidean plane, where distances were distorted to give it geometric properties quite different from those of Euclid's Elements. He supposed that the temperature varied linearly from a fixed value at the centre of the disk to absolute … Continue reading A Finite but Unbounded Universe →

Mamikon’s Visual Calculus and Hamilton’s Hodograph

[This is a condensed version of an article [5] in Mathematics Today] A remarkable theorem, discovered in 1959 by Armenian astronomer Mamikon Mnatsakanian, allows problems in integral calculus to be solved by simple geometric reasoning, without calculus or trigonometry. Mamikon's Theorem states that `The area of a tangent sweep of a curve is equal to … Continue reading Mamikon’s Visual Calculus and Hamilton’s Hodograph →

Infinitesimals: vanishingly small but not quite zero

A few weeks ago, I wrote about  Hyperreals and Nonstandard Analysis , promising to revisit the topic. Here comes round two. We know that 2.999… is equal to three. But many people have a sneaking suspicion that there is “something” between the number with all those 9’s after the 2 and the number 3, that is not … Continue reading Infinitesimals: vanishingly small but not quite zero →

Hyperreals and Nonstandard Analysis

Following the invention of calculus, serious concerns persisted about the mathematical integrity of the method of infinitesimals. Leibniz made liberal use of infinitesimals, with great effect, but his reasoning was felt to lack rigour. The Irish bishop George Berkeley criticised the assumptions underlying calculus, and his objections were not properly addressed for several centuries. In … Continue reading Hyperreals and Nonstandard Analysis →

Letters to a German Princess: Euler’s Blockbuster Lives On

The great Swiss mathematician Leonhard Euler produced profound and abundant mathematical works. Publication of his Opera Omnia began in 1911 and, with close to 100 volumes in print, it is nearing completion. Although he published several successful mathematical textbooks, the book that attracted the widest readership was not a mathematical work, but a collection of … Continue reading Letters to a German Princess: Euler’s Blockbuster Lives On →

De Branges’s Proof of the Bieberbach Conjecture

It is a simple matter to post a paper on arXiv.org claiming to prove Goldbach's Conjecture, the Twin Primes Conjecture or any of a large number of other interesting hypotheses that are still open. However, unless the person posting the article is well known, it is likely to be completely ignored. Mathematicians establish their claims … Continue reading De Branges’s Proof of the Bieberbach Conjecture →

Number Partitions: Euler’s Astonishing Insight

In 1740, French mathematician Philippe Naudé wrote to Leonhard Euler asking in how many ways a positive integer can be written as a sum of distinct numbers. In his investigations of this, Euler established the theory of partitions, for which he used the term partitio numerorum. Many of Euler's results in number theory involved divergent … Continue reading Number Partitions: Euler’s Astonishing Insight →

Set Density: are even numbers more numerous than odd ones?

In pure set-theoretic terms, the set of even positive numbers is the same size, or cardinality, as the set of all natural numbers: both are infinite countable sets that can be put in one-to-one correspondence through the mapping $latex {n \rightarrow 2n}&fg=000000$. This was known to Galileo. However, with the usual ordering, $latex \displaystyle \mathbb{N} … Continue reading Set Density: are even numbers more numerous than odd ones? →

Cantor’s Theorem and the Unending Hierarchy of Infinities

In 1891, Georg Cantor published a seminal paper, U"ber eine elementare Frage der Mannigfaltigkeitslehren --- On an elementary question of the theory of manifolds --- in which his ``diagonal argument'' first appeared. He proved a general theorem which showed, in particular, that the set of real numbers is uncountable, that is, it has cardinality greater … Continue reading Cantor’s Theorem and the Unending Hierarchy of Infinities →

The Spine of Pascal’s Triangle

We are all familiar with Pascal's Triangle, also known as the Arithmetic Triangle (AT). Each entry in the AT is the sum of the two closest entries in the row above it. The $latex {k}&fg=000000$-th entry in row $latex {n}&fg=000000$ is the binomial coefficient $latex {\binom{n}{k}}&fg=000000$ (read $latex {n}&fg=000000$-choose-$latex {k}&fg=000000$), the number of ways of … Continue reading The Spine of Pascal’s Triangle →

Embedding: Reconstructing Solutions from a Delay Map

M In mechanical systems described by a set of differential equations, we normally specify a complete set of initial conditions to determine the motion. In many dynamical systems, some variables may easily be observed whilst others are hidden from view. For example, in astronomy, it is usual that angles between celestial bodies can be measured … Continue reading Embedding: Reconstructing Solutions from a Delay Map →

The Signum Function may be Continuous

Abstract: Continuity is defined relative to a topology. For two distinct topological spaces $latex {(X,\mathcal{O}_1)}&fg=000000$ and $latex {(X,\mathcal{O}_2)}&fg=000000$ having the same underlying set $latex {X}&fg=000000$ but different families of open sets, a function may be continuous in one but discontinuous in the other. The signum function is defined on the real line as follows: $latex … Continue reading The Signum Function may be Continuous →

Real Derivatives from Imaginary Increments

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. For finite difference approximations, the choice of step size $latex {h}&fg=000000$ is crucial: if $latex {h}&fg=000000$ is … Continue reading Real Derivatives from Imaginary Increments →

Simple Models of Atmospheric Vortices

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 … Continue reading Simple Models of Atmospheric Vortices →