Continue reading ‘Topological Calculus: away with those nasty epsilons and deltas’

## Posts Tagged 'Analysis'

### Topological Calculus: away with those nasty epsilons and deltas

Published September 29, 2022 Occasional Leave a CommentTags: Analysis

### Dynamic Equations for Weather and Climate

Published September 8, 2022 Occasional Leave a CommentTags: Analysis, Geometry, Relativity

*“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 are convenient. Here is the longitude and the latitude. In Figure 1 we show the momentum equations as presented in the monograph of Lorenz (1967):

Continue reading ‘Dynamic Equations for Weather and Climate’

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.

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

Published August 11, 2022 Occasional ClosedTags: Analysis

It is simple to define a mapping from the unit interval into the unit square . Georg Cantor found a one-to-one map from ** onto** , showing that the one-dimensional interval and the two-dimensional square have the same cardinality. Cantor’s map was not continuous, but Giuseppe Peano found a continuous surjection from onto , that is, a

*curve that fills the entire unit square.*Shortly afterwards, David Hilbert found an even simpler space-filling curve, which we discussed in Part I of this post.

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”

Published August 4, 2022 Occasional ClosedTags: Analysis

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 located by giving its Cartesian coordinates , or its polar coordinates . In space, we may specify the location by giving three numbers .

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

### Mamikon’s Visual Calculus and Hamilton’s Hodograph

Published April 7, 2022 Occasional ClosedTags: Analysis, Mechanics

[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 the area of its tangent cluster’.* We shall illustrate how this theorem can help to solve a range of integration problems.

Continue reading ‘Mamikon’s Visual Calculus and Hamilton’s Hodograph’

### Infinitesimals: vanishingly small but not quite zero

Published March 31, 2022 Occasional ClosedTags: Analysis, History

A few weeks ago, I wrote about Hyperreals and Nonstandard Analysis , promising to revisit the topic. Here comes round two.

Continue reading ‘Infinitesimals: vanishingly small but not quite zero’

### The Chromatic Number of the Plane

Published March 24, 2022 Occasional ClosedTags: Analysis, Combinatorics, Topology

To introduce the problem in the title, we begin with a quotation from the Foreword, written by Branko Grünbaum, to the book by Alexander Soifer (2009): *The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators*:

*If each point of the plane is to be given a color, how many colors do we need if every two points at unit distance are to receive distinct colors?*

About 70 years ago it was shown that the least number of colours needed for such a colouring is one of 4, 5, 6 and 7. But which of these is the correct number? Despite efforts by many very clever people, some of whom had solved problems that appeared to be much harder, no advance has been made to narrow the gap

.

### Hyperreals and Nonstandard Analysis

Published March 10, 2022 Occasional ClosedTags: Analysis, Logic

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 the 1800s, Bolzano, Cauchy and Weierstrass developed the – definition of limits and continuity, which allowed derivatives and integrals to be defined without recourse to infinitesimal quantities. Continue reading ‘Hyperreals and Nonstandard Analysis’

### De Branges’s Proof of the Bieberbach Conjecture

Published December 30, 2021 Occasional ClosedTags: Analysis

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 and convince their colleagues by submitting their work to peer-reviewed journals. The work is then critically scrutinized and evaluated by mathematicians familiar with the relevant field, and is either accepted for publication, sent back for correction or revision or flatly rejected.

Continue reading ‘De Branges’s Proof of the Bieberbach Conjecture’

### The Square Root Spiral of Theodorus

Published October 14, 2021 Occasional ClosedTags: Analysis, Geometry

The square-root spiral is attributed to Theodorus, a tutor of Plato. It comprises a sequence of right-angled triangles, placed edge to edge, all having a common point and having hypotenuse lengths equal to the roots of the natural numbers.

The spiral is built from right-angled triangles. At the centre is an isosceles triangle of unit side and hypotenuse . Another triangle, with sides and and hypotenuse is stacked upon the first. This process continues, giving hypotenuse lengths for all .

### A Grand Unification of Mathematics

Published October 7, 2021 Irish Times ClosedTags: Algebra, Analysis, Group Theory, Number Theory

There are numerous branches of mathematics, from arithmetic, geometry and algebra at an elementary level to more advanced fields like number theory, topology and complex analysis. Each branch has its own distinct set of axioms, or fundamental assumptions, from which theorems are derived by logical processes. While each branch has its own flavour, character and methods, there are also strong overlaps and interdependencies. Several attempts have been made to construct a grand unified theory that embraces the entire field of maths [TM220 or search for “thatsmaths” at irishtimes.com].

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 ClosedTags: 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’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 ClosedTags: 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’

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 ClosedTags: 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’

### Will RH be Proved by a Physicist?

Published December 10, 2020 Occasional ClosedTags: 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!

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.

### Cornelius Lanczos – Inspired by Hamilton’s Quaternions

Published July 16, 2020 Irish Times ClosedTags: Analysis, Applied Maths, Mechanics, Relativity

In May 1954, Cornelius Lanczos took up a position as senior professor in the School of Theoretical Physics at the Dublin Institute for Advanced Studies (DIAS). The institute had been established in 1940 by Eamon de Valera, with a School of Theoretical Physics and a School of Celtic Studies, reflecting de Valera’s keen interest in mathematics and in the Irish language. Later, a School of Cosmic Physics was added. DIAS remains a significant international centre of research today [TM191 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Cornelius Lanczos – Inspired by Hamilton’s Quaternions’

### The Online Encyclopedia of Integer Sequences

Published June 25, 2020 Occasional ClosedTags: 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’

### Exponential Growth must come to an End

Published April 16, 2020 Irish Times ClosedTags: Analysis, Applied Maths

In its initial stages, the Covid-19 pandemic grew at an exponential rate. What does this mean? The number of infected people in a country is growing exponentially if it increases by a fixed multiple R each day: if N people are infected today, then R times N are infected tomorrow. The size of the growth-rate R determines how rapidly the virus is spreading. An example should make this clear [TM185 or search for “thatsmaths” at irishtimes.com].

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 ClosedTags: 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 ClosedTags: 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.

### Elliptic Trigonometry: Fun with “sun”, “cun” and “dun”

Published November 14, 2019 Occasional ClosedTags: Analysis, Trigonometry

** Introduction **

The circular functions arise from ratios of lengths in a circle. In a similar manner, the elliptic functions can be defined by means of ratios of lengths in an ellipse. Many of the key properties of the elliptic functions follow from simple geometric properties of the ellipse.

Originally, Carl Gustav Jacobi defined the elliptic functions , , using the integral

He called the *amplitude* and wrote . It can be difficult to understand what motivated his definitions. We will define the elliptic functions , , in a more intuitive way, as simple ratios associated with an ellipse.

Continue reading ‘Elliptic Trigonometry: Fun with “sun”, “cun” and “dun”’

### Some Fundamental Theorems of Maths

Published October 24, 2019 Occasional ClosedTags: 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.

If you love mathematics and have never studied complex function theory, then you are missing something wonderful. It is one of the most beautiful branches of maths, with many amazing results. Don’t be put off by the name: complex does not mean complicated. With elementary calculus and a basic knowledge of imaginary numbers, a whole world of wonder is within your grasp.

In the early nineteenth century, Augustin-Louis Cauchy (1789–1857) constructed the foundations of what became a major new branch of mathematics, the theory of functions of a complex variable.

Given a function of a real variable, we often have to find the values of for which the function is zero. A simple iterative method was devised by Isaac Newton and refined by Joseph Raphson. It is known either as Newton’s method or as the Newton-Raphson method. It usually produces highly accurate approximations to the roots of the equation .

### Bernard Bolzano, a Voice Crying in the Wilderness

Published June 13, 2019 Occasional ClosedTags: Analysis, History

Bernard Bolzano, born in Prague in 1781, was a Bohemian mathematician with Italian origins. Bolzano made several profound advances in mathematics that were not well publicized. As a result, his mathematical work was overlooked, often for many decades after his death. For example, his construction of a function that is continuous on an interval but nowhere differentiable, did not become known. Thus, the credit still goes to Karl Weierstrass, who found such a function about 30 years later. Boyer and Merzbach described Bolzano as “a voice crying in the wilderness,” since so many of his results had to be rediscovered by other workers.

Continue reading ‘Bernard Bolzano, a Voice Crying in the Wilderness’

### Really, 0.999999… is equal to 1. Surreally, this is not so!

Published January 10, 2019 Occasional ClosedTags: Analysis, Number Theory

The value of the recurring decimal 0.999999 … is a popular topic of conversation amongst amateur mathematicians of various levels of knowledge and expertise. Some of the discussions on the web are of little value or interest, but the topic touches on several subtle and deep aspects of number theory.

Continue reading ‘Really, 0.999999… is equal to 1. Surreally, this is not so!’

In an earlier post, we discussed Grandi’s series, originally studied by the Italian monk Dom Guido Grandi around 1703. It is the series

This is a divergent series: the sequence of partial sums is , which obviously does not converge, but alternates between and .

### Grandi’s Series: Divergent but Summable

Published July 12, 2018 Occasional ClosedTags: Analysis, History, Puzzles

** Is the Light On or Off? **

Suppose a light is switched on for a half-minute, off for a quarter minute, on for one eighth of a minute and so on until precisely one minute has elapsed. Is the light on or off at the end of this (infinite) process? Representing the two states “on” and “off” by and , the sequence of states over the first minute is . But how do we ascertain the final state from this sequence? This question is sometimes known as Thomson’s Lamp Puzzle.

**Solving PDEs by a Roundabout Route**

Joseph Fourier, born just 250 years ago, introduced a wonderful idea that revolutionized science and mathematics: any function or signal can be broken down into simple periodic sine-waves. Radio waves, micro-waves, infra-red radiation, visible light, ultraviolet light, X-rays and gamma rays are all forms of electromagnetic radiation, differing only in frequency [TM136 or search for “thatsmaths” at irishtimes.com].

**Breaking Complex Objects into Simple Pieces**

“In a memorable session of the French Academy on the

21st of December 1807, the mathematician and engineer

Joseph Fourier announced a thesis which inaugurated a

new chapter in the history of mathematics. The claim of

Fourier appeared to the older members of the Academy,

including the great analyst Lagrange, entirely incredible.”

** Introduction **

The above words open the *Discourse on Fourier Series*, written by Cornelius Lanczos. What greatly surprised and shocked Lagrange and the other academicians was the claim of Fourier that an arbitrary function, defined by an arbitrarily capricious graph, can always be resolved into a sum of pure sine and cosine functions. There was good reason to question Fourier’s theorem. Since sine functions are continuous and infinitely differentiable, it was assumed that any superposition of such functions would have the same properties. How could this assumption be reconciled with Fourier’s claim?

We all know that division by zero is a prohibited operation, and that ratios that reduce to “zero divided by zero” are indeterminate. We probably also recall proving in elementary calculus class that

This is an essential step in deriving an expression for the derivative of .

### The Evolute: Envelope of Normals

Published February 22, 2018 Occasional ClosedTags: Analysis, Geometry

Every curve in the plane has several other curves associated with it. One of the most interesting and important of these is the evolute.

### Torricelli’s Trumpet & the Painter’s Paradox

Published April 13, 2017 Occasional ClosedTags: Analysis, Geometry, Recreational Maths

Evangelista Torricelli, a student of Galileo, is remembered as the inventor of the barometer. He was also a talented mathematician and he discovered the remarkable properties of a simple geometric surface, now often called *Torricelli’s Trumpet*. It is the surface generated when the curve for is rotated in 3-space about the x-axis.

Continue reading ‘Torricelli’s Trumpet & the Painter’s Paradox’

Children sometimes amuse themselves searching for the biggest number. After trying millions, billions and trillions, they realize that there is no end to the game: however big a number may be, we can always add 1 to produce a bigger number: the set of counting numbers is infinite. The concept of infinity has intrigued philosophers since antiquity, and it leads to many surprises and paradoxical results [TM110 or search for “thatsmaths” at irishtimes.com].

The English mathematician Brook Taylor (1685-1731) introduced the calculus of finite differences in his *Methodus Incrementorum Directa et Inversa*, published in 1715. This work contained the famous formula known today as Taylor’s formula. In 1772, Lagrange described it as “the main foundation of differential calculus” (Wikipedia: Brook Taylor). Taylor also wrote a treatise on linear perspective (see Fig. 1).

It is noteworthy that the series for , and were known to mathematicians in India about 400 years before Taylor’s time.

Continue reading ‘Taylor Expansions from India’

Which is greater, or ? Of course, it depends on the values of x and y. We might consider a particular case: Is or ?

It is well-known that an ellipse is the locus of all points such that the sum of their distances from two fixed points, the foci, is constant. Thus, a gardener may map out an elliptical flower-bed by driving two stakes into the ground, looping a rope around them and pulling it taut with a pointed stick, tracing out a curve on the ground.

### Random Harmonic Series

Published July 28, 2016 Occasional ClosedTags: Analysis, Number Theory, Probability

We consider the convergence of the random harmonic series

where is chosen randomly with probability of being either plus one or minus one. It follows from the Kolmogorov three-series theorem that the series is “almost surely” convergent.

### Sigmoid Functions: Gudermannian and Gompertz Curves

Published April 28, 2016 Occasional ClosedTags: Analysis, modelling

The Gudermannian is named after Christoph Gudermann (1798–1852). The Gompertz function is named after Benjamin Gompertz (1779–1865). These are two amongst several *sigmoid functions*. Sigmoid functions find applications in many areas, including population dynamics, artificial neural networks, cartography, control systems and probability theory. We will look at several examples in this class of functions.

Continue reading ‘Sigmoid Functions: Gudermannian and Gompertz Curves’

We can construct a beautiful fractal set by defining an operation of iterating exponentials and applying it to the numbers in the complex plane. The operation is tetration and the fractal is called the power tower fractal or sometimes the tetration fractal. A detail of the set is shown in the figure here.

### The Imaginary Power Tower: Part II

Published March 31, 2016 Occasional ClosedTags: Analysis, Recreational Maths

This is a continuation of last week’s post: LINK

The complex power tower is defined by an `infinite tower’ of exponents:

The sequence of successive approximations to this function is

If the sequence converges it is easy to solve numerically for a given .

In Part I we described an attempt to fit a logarithmic spiral to the sequence . While the points of the sequence were close to such a curve they did not lie exactly upon it. Therefore, we now examine the asymptotic behaviour of the sequence for large .

### The Imaginary Power Tower: Part I

Published March 24, 2016 Occasional ClosedTags: Analysis, Recreational Maths

The function defined by an `infinite tower’ of exponents,

is called the *Power Tower function*. We consider the sequence of successive approximations to this function:

As , the sequence converges for . This result was first proved by Euler. For an earlier post on the power tower, click here.

We will construct a sequence of functions on the unit interval such that it converges uniformly to zero while the arc-lengths diverge to infinity.

Next week there will be a post on tuning pianos using a method based on entropy. In preparation for that, we consider here how the entropy of a probability distribution function with twin peaks changes with the separation between the peaks.