Asymmetrical Primitive Centrifuge Configurations and Vanishing Sums of Roots of Unity

A centrifuge has some fixed number of positions into each of which a sample may be loaded. Equal-weight samples must be placed so that they’re perfectly balanced.

In the attached article, I present an asymmetric balanced irreducible centrifuge configuration, and relate configurations to sums of roots of unity.

PDF article:
centrifuge.pdf

On Badly Balanced Centrifuge Configurations

A high-speed centrifuge can be so sensitive to balance that you should weigh the samples to ensure that they are equal (within tolerance), and load them symmetrically, or in symmetrical combination.

In a certain manufacturer’s instructions for their 24-position centrifuge, configurations were suggested for 5 on 7 samples. These seemed dangerously out of balance to me.

I discuss this in the attached article.

PDF article:
centrifuge-imbalance.pdf

Frustum Formulae

In the attached article, we generalise the Heronian mean, and discover that this behaves as a logarithmic mean in the limit.

PDF article:
frustum-formulae.pdf

On the Visualisation of Inequalities amongst Homogeneous Means

In the attached article, I show how the relationships between binary means might be illustrated.

I describe ‘distillation’, which is composed of ‘reduction’ and ‘compression’.

I include plots for

• arithmetic mean
• geometric mean
• harmonic mean
• contraharmonic mean
• quadratic mean
• Heronian mean
• logarithmic mean
• first and second Seiffert means
• Neuman–Sándor mean
• power means
• Lehmer means
• Stolarsky means

and a few others.

PDF article:
mean-inequal-vis.pdf

On a Connection between the Birthday Puzzles

The first birthday puzzle asks: How many people need to be gathered together in order that there is an at-least evens chance of a birthday coincidence? The answer is just 23.

We might write this

The second birthday puzzle asks: How many other people need to be gathered together in order that there is an at-least evens chance of a coincidence with your birthday? The answer is 253.

Notice that

It turns out that this is not a coincidence.

The value of the first birthday puzzle is awkward to calculate. In the attached article, I show how the above binomial connection leads to a good closed-form approximation for the first birthday problem:

PDF article:
birthday-puzzle-connection.pdf

Indeterminate Forms and L’Hôpital’s Rule

Introduction

In an earlier article, I showed how to prove that , i.e. the zero power mean is the geometric mean, using L’Hôpital’s rule. It was also necessary to perform some transformations on a limit expression to allow L’Hôpital’s rule to be applied.

In this article, I show how this and various other common indeterminate forms may be transformed so that L’Hôpital’s rule may similarly be applied.

I also provide a table of the results of the derivations (or ‘differentiations’) for each case, as a useful reference.

P₀(x,y) = G(x,y)

Introduction

In my previous post, I showed why

that is, why

Here, I repeat the exercise for the simpler case where has just 2 members. I.e.

i.e.

Therefore this is just a special case of what was discussed in the previous post, but which may be somewhat easier to follow.

P₀ = G

Introduction

Recently, I wished to understand why the zero-power mean is the geometric mean

that is, why

and this sent me on much more of a mathematical adventure than I expected, involving learning about evaluating limits, L’Hôpital’s rule, and about indeterminate forms and their transformations.

On the Geometry of Hypertetrahedral Slices

The full article is attached as a PDF; please see the link at the end of this post.

Abstract

In this article, I investigate the (regular) hypertetrahedra and their measures, and orthogonal slices (or sections) through them.

I consider how to configure the vertices in order easily to calculate the measure.

Formulae are found for the distance between opposing facets of a hypertetrahedron, and I investigate the behaviour of these as the dimension increases.

I discuss the “shape” of slices through hypertetrahedra, showing these to be Cartesian products of hypertetrahedra of lower dimension. Then I briefly investigate slices through slices.

Formulae are found for the measure of a hypertetrahedron as an integral over (measures of) slices. From these, I find recurrence formulae, and from these, in turn, I find a closed form for the measure of a hypertetrahedron.

For example:
the measure (length) of the 1-dimensional hypertetrahedron of unit edge, which is just the unit line segment, is 1; for 2-dimensional, which is just the equilateral triangle, the measure (area) is ; for 3-dimensional, which is the regular tetrahedron, the measure (capacity or volume) is ; and so on.

I determine when (i.e. for which dimensions) the measure of a hypertetrahedron is rational, and show that when it is rational, it is in fact a unit fraction.

For example:
the measures of the 7- and 8-dimensional hypertetrahedra of unit edge are, respectively, and . The value is also rational in dimensions 17 and 24.

The sequence of dimensions for such rational measures begins

Finally, I briefly investigate the behaviour of the discrete analogue of hypertetrahedra, i.e. hypertetrahedral numbers.

The article includes some tables of calculated values.

A Survey of Methods for the Evaluation of Sums of Powers

The full article (over 100 pages) is attached as a PDF; please see the link at the end of this post.

Abstract

The simple series

where and are non-negative integers, is a sum of powers

We regard as being fixed and relatively small, and as being variable and potentially very large. That is, .

Sometimes, we also denote this series by or by .

Such series may be expanded in closed form as polynomials.

Given the expression

for some chosen , we wish to re-express it as a polynomial

Consider, for example,

When (i.e. one million), we have

This is fairly tedious to evaluate. But we can use any of various techniques to establish that

Now, rather than having terms to sum, we have just 4 terms. So

which can quickly be evaluated as

Our example’s coefficients are

As we consider different values for , the coefficients of these polynomials exhibit some remarkably and unexpectedly complicated behaviour. They seem rather erratic, following no obvious pattern. Likewise, the factors exhibit no obvious pattern, and indeed are not even always real.

Mathematicians have been studying these series for hundreds of years. There are many approaches; some old, and some very recent. The approaches are quite varied. In this article, I present the many approaches I have found in the literature and online.