Maths Bite

MATHS BITE: Apéry’s Constant

Apéry’s constant is defined as the number

{\displaystyle {\begin{aligned}\zeta (3)&=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}\\&=\lim _{n\to \infty }\left({\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+\cdots +{\frac {1}{n^{3}}}\right)\end{aligned}}}

where ζ is the Riemann Zeta Function.

This constant is named after the French mathematician Roger Apéry who proved that it was irrational in 1978. However it is still unknown whether or not it is transcendental.

History

The Basel Problem asked about the convergence of the following sum:
Screen Shot 2017-06-10 at 2.16.05 PM.png

In the 18th century, Leonhard Euler proved that in fact it did – to π^2/6. However, the limit of the following sum remained unknown:Screen Shot 2017-06-10 at 2.19.28 PM.png

Although mathematicians made some progress, including Euler who calculated the first 16 decimal digits of the sum, it was not known whether the number was rational or irrational, until Apéry.

Furthermore, it is currently not known specifically whether any other particular ζ(n), for n odd, is irrational. “The best we’ve got is from Wadim Zudilin, in 2001, who showed that at least one of ζ(5), ζ(7), ζ(9), ζ(11) must be irrational, and Tanguy Rivoal, in 2000, who showed that infinitely many of the ζ(2k+1) must be irrational.”

M x

MATHS BITE: Dini’s Surface

Dini’s surface, named after Ulisse Dini, is a surface with constant negative curvature that can be created by twisting a pseudosphere (see picture below).

Pseudosphere

Dini’s surface is given by the following parametric equations:

{\displaystyle {\begin{aligned}x&=a\cos u\sin v\\y&=a\sin u\sin v\\z&=a\left(\cos v+\ln \tan {\frac {v}{2}}\right)+bu\end{aligned}}}

 

 

Dini’s Surface

Dini’s surface is pictured in the upper right-hand corner of a book by Alfred Gray (1997), as well as on the cover of volume 2, number 3 of La Gaceta de la Real Sociedad Matemática Española (1999).

M x

MATHS BITE: The Beginning of Chaos

In 1887, King Oscar II of Norway and Sweden offered a prize for the solution of the following maths problem:

Say you have a number of celestial bodies with known mass and you know the speed and direction they are moving in at some given point in time. Use Newton’s laws of motion and the universal law of gravitation to calculate the trajectories of the bodies.

After three years, the French mathematician Henri Poincaré, who restricted himself to the case where there are just three bodies, won the prize.

However, after winning the prize, Poincaré noticed a flaw in his solution putting him in an embarrassing position, as his manuscript was to be published for the King’s birthday within a few weeks’ time!

In his attempt to correct his work, Poincaré discovered that even this simple problem suffered from the Butterfly Effect: “sensitive dependence on initial conditions“. This means that the smallest variation in the initial values can build up over time to create massive discrepancies in the trajectories. Hence, you cannot reliably predict the motion of planets over time, as you can’t know the initial values with an infinite degree of accuracy.

With this discovery, Poincaré laid the foundations for Chaos theory.

M x

MATHS BITE: The Cantor Set

The Cantor Set is constructed in the following way:

Start with the interval [0,1]. Next, remove the open middle third interval, which gives you two line segments [0,1/3] and [2/3,1]. Again, remove the middle third for each remaining interval, which leaves you now with 4 intervals. Repeat this final step ad infinitum.

Cantor_set_binary_tree.svg.png

The points in [0,1] that do not eventually get removed in the procedure form the Cantor set.

How many points are there in the Cantor Set?

Consider the diagram below:

Screen Shot 2017-02-21 at 8.05.41 PM.png

An interval from each step has been coloured in red, and each red interval (apart from the top one) lies underneath another red interval. This nested sequence shrinks down to a point, which is contained in every one of the red intervals, and hence is a member of the Cantor set. In fact, each point in the Cantor set corresponds to a unique infinite sequence of nested intervals.

To label a point in the Cantor set according to the path of red intervals that is taken to reach it, label each point by an infinite sequence consisting of 0s and 1s.

A 0 in the nth position symbolises that the point lies in the left hand interval after the nth stage in the Cantor process.

A in the nth position symbolises that the point lies in the right hand interval after the nth stage in the Cantor process.

For example, the point 0 in [0,1] is represented by the sequence 0000…., the point 1 is represented by the sequence 1111…. and the point 1/3 is represented by the sequence 01111….

So, as there are infinite sequences consisting of 0s and 1s, there are an infinite number of elements in the Cantor set. If we place a point before any one of these infinite sequences, for example 0100010… becomes .0100010…, then we convert an infinite sequence of 0s and 1s to the binary expansion of a real number between 0 and 1. This means that the number of points in the Cantor set is the same as the number of points in the interval [0,1]. We conclude that the infinite process of removing middle thirds from the interval [0,1] has no effect on the number of points in [0,1]!

M x

MATHS BITE: Leyland Numbers

A Leyland number is an integer of the form x^y + y^x, where x and y are integers greater than 1. This condition is very important as, without it, every positive integer would be a Leyland number of the form x1 + 1x.

They are named after Paul Leyland, a British number theorist who studied the factorisation of integers and primality testing.

Leyland numbers are of interest as some of them are very large primes.

Leyland Primes

A Leyland prime is a Leyland number that is also prime. The first of such primes are:

17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193, …

which correspond to:

32+23, 92+29, 152+215, 212+221, 332+233, 245+524, 563+356, 3215+1532, …

The largest known Leyland prime is Screen Shot 2016-12-26 at 10.59.12 AM.png.

M x

MATHS BITE: Catenary

A Catenary is the idealised shape created when you suspend a chain by its ends and let it hang naturally under its own weight. Any hanging chain will naturally find its equilibrium as the forces of tension from the hooks holding the chain up and the force of gravity pulling it downwards balance out.

Robert Hooke was the first to study the catenary mathematically; in 1675 he announced that he had solved the problem of the optimal shape of an arch by publishing the solution as an encrypted anagram: “As hangs the flexible line, so but inverted will stand the rigid arch.”

Its equation was obtained in 1691 by Leibniz, Huygens and Johann Bernoulli in response to a challenge put out by Jacob Bernoulli to find the equation of the ‘chain-curve’. The curve has a U-like shape and although it is similar in appearance to a parabola, it is not quite a parabola.

Source: IntMath

The equation of the catenary is
{\displaystyle y=a\cosh \left({\frac {x}{a}}\right)={\frac {a\left(e^{\frac {x}{a}}+e^{-{\frac {x}{a}}}\right)}{2}}}

File:Catenary-pm.svg

Catenaries for different values of a | Source: Wikipedia

In 1744, Euler proved that the catenary is the curve which gives the surface of minimum surface area for the given bounding circles when rotated about the x-axis.

M x