## Posts Tagged 'Analysis'

### 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’