## Posts Tagged 'Analysis'

### Twin Peaks Entropy

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.

### Squaring the Circular Functions

The circular functions occur throughout mathematics. Fourier showed that, under very general assumptions, an arbitrary function can be decomposed into components each of which is a circular function. The functions get their name from their use in defining a circle in parametric form: if

$\displaystyle x = a\cos t \qquad\mbox{and}\qquad y = a\sin t$

then ${x^2 + y^2 = a^2}$, the usual equation for a circle in Cartesian coordinates. In the figure, we plot the familiar sinusoid, which has a period of ${2\pi}$.

### A Few Wild Functions

Sine Function: ${\mathbf{y=\sin x}}$

The function ${y=\sin x}$ is beautifully behaved, oscillating regularly along the entire real line ${\mathbb{R}}$ (it is also well-behaved for complex ${x}$ but we won’t consider that here).

The sine function, the essence of good behaviour.

### Which Way did the Bicycle Go?

“A bicycle, certainly, but not the bicycle,” said Holmes.

In Conan-Doyle’s short story The Adventure of the Priory School  Sherlock Holmes solved a mystery by deducing the direction of travel of a bicycle. His logic has been minutely examined in many studies, and it seems that in this case his reasoning fell below its normal level of brilliance.

As front wheel moves along the positive x-axis the back wheel, initially at (0,a), follows a tractrix curve (see below).

### Tap-tap-tap the Cosine Button

Tap any number into your calculator. Yes, any number at all, plus or minus, big or small. Now tap the cosine button. You will get a number in the range [ -1, +1 ]. Now tap “cos” again and again, and keep tapping it repeatedly (make sure that angles are set to radians and not degrees). The result is a sequence of numbers that converge towards the value 0.739085 … .

### Café Mathematics in Lvov

For 150 years the city of Lvov was part of the Austro-Hungarian Empire. After Polish independence following World War I, research blossomed and between 1920 and 1940 a sparkling constellation of mathematicians flourished in Lvov [see this week’s That’s Maths column in The Irish Times (TM063, or search for “thatsmaths” at irishtimes.com).

The Scottish Café, Lvov in earlier times (left), now Hotel Atlas in Lviv (right).

### The Birth of Functional Analysis

Stefan Banach (1892–1945) was amongst the most influential mathematicians of the twentieth century and the greatest that Poland has produced. Born in Krakow, he studied in Lvov, graduating in 1914 just before the outbreak of World War I. He returned to Krakow where, by chance, he met another mathematician, Hugo Steinhaus who was already well-known. Together they founded what would, in 1920, become the Polish Mathematical Society.

A coin and a postage stamp commemorating Stefan Banach.

### New Curves for Old: Inversion

Special Curves

A large number of curves, called special curves, have been studied by mathematicians. A curve is the path traced out by a point moving in space. To keep things simple, we assume that the point is confined to two-dimensional Euclidean space ${\mathbb{R}^2}$ so that it generates a plane curve as it moves. This, a curve results from a mapping ${\mathbf{\gamma} : [a,b]\longrightarrow \mathbb{R}^2}$. Continue reading ‘New Curves for Old: Inversion’

### Curves with Singularities

Many of the curves that we study are smooth, with a well-defined tangent at every point. Points where the derivative is defined — where there is a definite slope — are called regular points. However, many curves also have exceptional points, called singularities. If the derivative is not defined at a point, or if it does not have a unique value there, the point is singular.

Slinky traces a smooth helical curve in three dimensions.

Generally, if we zoom in close to a point on a curve, the curve looks increasingly like a straight line. However, at a singularity, it may look like two lines crossing or like two lines whose slopes converge as the resolution increases. Continue reading ‘Curves with Singularities’

### Invention or Discovery?

Is mathematics invented or discovered? As many great mathematicians have considered this question without fully resolving it, there is little likelihood that I can provide a complete answer here. But let me pose a possible answer in the form of a conjecture:

Conjecture: Definitions are invented. Theorems are discovered.

The goal is to prove this conjecture, or to refute it. Below, some arguments in support of the conjecture are presented. Continue reading ‘Invention or Discovery?’

### Predator-Prey Models

Next week’s post will be about a model of the future of civilization! It is based on the classical predator-prey model, which is reviewed here.

Solution for X (blue) and Y (red) for 30 time units. X(0)=0.5, Y(0)=0.2 and k=0.5.

### The Unity of Mathematics

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.

### The Langlands Program

An ambitious programme to unify disparate areas of mathematics was set out some fifty years ago by Robert Langlands of the Institute for Advanced Study in Princeton. The “Langlands Program” (LP) is a set of deep conjectures that attempt to build bridges between certain algebraic and analytical objects.

Canadian mathematician Robert Langlands, who formulated a series of far-reaching conjectures [image from Wikimedia Commons].

### The Prime Number Theorem

God may not play dice with the Universe, but something strange is going on with the prime numbers  [Paul Erdös, paraphrasing Albert Einstein]

The prime numbers are the atoms of the natural number system. We recall that a prime number is a natural number greater than one that cannot be broken into smaller factors. Every natural number greater than one can be expressed in a unique way as a product of primes. Continue reading ‘The Prime Number Theorem’

### A Mathematical Dynasty

The idea that genius runs in families is supported by many examples in the arts and sciences. One striking case is the family of Johann Sebastian Bach, the most brilliant star in a constellation of talented musicians and composers.

In a similar vein, several generations of the Bernoulli family excelled in science and medicine. More that ten members of this Swiss family, over four generations, had distinguished careers in mathematics.

### Sonya Kovalevskaya

A brilliant Russian mathematician, Sonya Kovalevskaya, is the topic of the That’s Maths column this week (click Irish Times: TM029 and search for “thatsmaths”).

In the nineteenth century it was extremely difficult for a woman to achieve distinction in the academic sphere, and virtually impossible in the field of mathematics. But a few brilliant women managed to overcome all the obstacles and prejudice and reach the very top. The most notable of these was the remarkable Russian, Sonya Kovalevskya.

### Ternary Variations

Georg Cantor (1845-1918) was led, through his study of trigonometric series, to distinguish between denumerably infinite sets like the rationals and uncountable sets like the reals. He introduced a set that is an abstract form of what we now call Cantor’s Ternary Set. In fact, the ternary set had been studied some ten years earlier by the Dublin-born mathematician Henry Smith and, independently, by the Italian Vito Volterra. General sets of this form are now called Smith-Volterra-Cantor sets (SVC sets).

### The Lambert W-Function

In a recent post ( The Power Tower ) we described a function defined by iterated exponentiation:

$\displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}}$

It would seem that when ${x>1}$ this must blow up. Surprisingly, it has finite values for a range of x>1. Continue reading ‘The Lambert W-Function’

### The Power Tower

Look at the function defined by an `infinite tower’ of exponents:

$\displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}}$

It would seem that for x>1 this must blow up. But, amazingly, this is not so.

In fact, the function has finite values for positive x up to ${x=\exp(1/e)\approx 1.445}$. We call this function the power tower function. Continue reading ‘The Power Tower’

### Sharing a Pint

Four friends, exhausted after a long hike, stagger into a pub to slake their thirst. But, pooling their funds, they have enough money for only one pint.

Annie drinks first, until the surface of the beer is half way down the side (Fig. 1(A)). Then Barry drinks until the surface touches the bottom corner (Fig. 1(B)). Cathy then takes a sup, leaving the level as in Fig. 1(C), with the surface through the centre of the bottom. Finally, Danny empties the glass.

Question: Do all four friends drink the same amount? If not, who gets most and who gets least? Continue reading ‘Sharing a Pint’

### The Root of Infinity: It’s Surreal!

Can we make any sense of quantities like “the square root of infinity”? Using the framework of surreal numbers, we can.

• In Part 1, we develop the background for constructing the surreals.
• In Part 2, the surreals are assembled and their amazing properties described.

### The Popcorn Function

Continuity is not what it seems. In 1875, the German mathematician Carl Johannes Thomae defined a function P(x) with the following extraordinary property:

P(x) is  discontinuous if x is rational

P(x) is continuous if x is irrational.

A graph of this function on the interval (0,1) is shown below. Continue reading ‘The Popcorn Function’