## Archive for the ‘Research’ Category

### Surface Areas and Volumes of Hyperspheres

Thursday, 3 October 2013

I noticed that the formulae for the sizes of the surface (or rather, boundary) and volume (interior) of circles, spheres and hyperspheres seem to have discontinuities (or ‘jumps’) in the index of $\pi$.

However, this is only a result of simplifying a uniform formula for particular cases.

### The Amazing ‘fusc’ Function

Tuesday, 16 April 2013

I recently encountered a collection of related articles regarding an amazing explicit enumeration of the non-negative rational numbers. Whilst experimenting with the mathematical constructs described, I made some discoveries of my own (which I doubt are new, but I have not found them described elsewhere).

Below, are the references and links to those articles. They’re all worth a read.

### Subversion Configuration Suite

Wednesday, 4 July 2012

I’ve now uploaded my Subversion Configuration Suite (svn-meta) repository to Google Code so that the files are publicly available for anyone to use.

### Some Simple New Elementary Functions

Friday, 22 January 2010

In this post I introduce some new functions:

• $\mathrm{exph}(x)$
• $\mathrm{lnh}(x)$
• $\mathrm{cish}(x)$

### Generalised Means

Tuesday, 2 June 2009

There are many notions of average in mathematics and statistics. Well-known are the mean, median and mode.

Also well-known, amongst the means, are the arithmetic mean (A), geometric mean (G), harmonic mean (H) and quadratic mean or  root mean squared (Q or RMS). Recently, I have become interested in the notion of a generalised mean (or generalized mean) over positive (non-negative and non-zero) real numbers $\in \mathbb{R}^+$.

### Falling and Rising Roots

Friday, 8 May 2009

In my post on hypertetrahedral polytopic roots, I gave a few formulae for functions such as the triangular root.

Here I give some very similar formulae for some falling and rising roots.

### Further Calculator Statistics

Tuesday, 28 April 2009

This is further to my post on calculator statistics.

If the calculator also stores the six values $\Sigma x^3$, $\Sigma x^4$ and $\Sigma x^2y$, then it can calculate the linear regression to a parabola.

### Catalan and Fibonacci Formulae

Sunday, 26 April 2009

Using the difference table method described in my previous post, I discovered the following formulae.

The first is not new. I do not know whether the last two are:

$\bigtriangleup_d(n) = \dbinom{n+d-1}{d} = \sum_{k=1}^{d} \dbinom{d-1}{k-1} \dbinom{n}{k}$;

$C_n = \sum_{i=0}^{n} \dbinom{n}{i} \bigtriangleup(F_{i-1})$;

$\sum_{i=0}^{n-1} C_i = \sum_{i=0}^{n} \dbinom{n}{i} \bigtriangleup(F_{i-2})$.

Here, $C_n = \dfrac{1}{n+1}\dbinom{2n}{n}$ is the nth Catalan number, $F_n$ is the nth Fibonacci number, $\bigtriangleup_d(n)$ is the nth d-dimensional tetrahedral number, and $\bigtriangleup(n) = \bigtriangleup_2(n)$ is the nth triangular number.

### Hypertetrahedral Polytopic Roots

Thursday, 23 April 2009

The $n^{\rm th}$ triangular number is given by

$t = \bigtriangleup_2(n) = \bigtriangleup(n) = \dfrac{n(n+1)}{2} = \dbinom{n+1}{2}$.

For example, $\bigtriangleup(4)=10$.

   •
• •
• • •
• • • •

It turns out the inverse formula is (inventing some notation):

$n = \sqrt[\bigtriangleup]{t} = \dfrac{\sqrt{8t+1}-1}{2}$

Thus, for example, $\sqrt[\bigtriangleup]{10} = \dfrac{\sqrt{80+1}-1}{2} = 4$.

Can this be generalised?

### Dimensional Analysis of Statistical Formulae

Monday, 20 April 2009

Dimensional analysis (also used in Physics) can be used to check statistical formulae.