We have seen how we might define functions of roots of rationals employing the magnitude of their associated

The magnitude is not the only operation of linear algebra that we might bring to bear upon such roots, however, and we have lately busied ourselves investigating another. ]]>

I concluded by noting that, even with this improvement, the shape of a cubic spline interpolation is governed by choices that are not uniquely determined by the points themselves and that linear interpolation is consequently a more mathematically appropriate scheme, which is why I chose to generalise it to other arithmetic types for

The obvious next question is whether or not we can also generalise the

Come, let us drown our sorrows whilst we still have the means to do so and engage in a little sport to raise our spirits.

I have a fancy for a game that I used to play when I was the Russian ambassador to the Rose Tree Valley commune. Founded by the philosopher queen Zway Remington as a haven for downtrodden wealthy industrialists, it was the purest of pure meritocracies; no handouts to the idle labouring classes there! ]]>

We have also seen how extrapolating such polynomials beyond the first and last nodes can yield less than satisfactory results, which we fixed by specifying the first and last gradients and then adding new first and last nodes to ensure that the first and last polynomials would represent straight lines.

Now we shall see how cubic spline interpolation can break down rather more dramatically and how we might fix it. ]]>

In this post we shall see how we can define a smooth interpolation by connecting the points with curves rather than straight lines. ]]>

We were particularly intrigued by the possibility of defining functions of such numbers by applying linear algebra operations to their associated vectors, which we began with a brief consideration of that given by their magnitudes. We have subsequently spent some time further exploring its properties and it is upon our findings that I shall now report. ]]>

On the face of it implementing this would seem to be a pretty trivial business, but doing so both accurately and efficiently is a surprisingly tricky affair, as we shall see in this post. ]]>

To further improve your sanguinity might I suggest a small wager?

Splendid fellow!

I have in mind a game invented to commemorate my successfully quashing the Caribbean zombie uprising some few several years ago. Now, as I'm sure you well know, zombies have ever been a persistent, if sporadic, scourge of those islands. On that occasion, however, there arose a formidable leader from amongst their number; the zombie Lord J------ the Insensate. ]]>

The simplest way to reckon the fairness of this wager is to re-frame its terms; to wit, that Sir R----- should pay the Baron one coin to play and thereafter one coin and twelve cents for each roll of his dice, including the first. The consequence of this is that before each roll of the dice Sir R----- could have expected to receive the same bounty, provided that he wrote off any losses that he had made beforehand. ]]>

`ak.borelInterval`

type to represent an interval as a pair of `ak.borelBound`

objects holding its lower and upper bounds.With these in place we're ready to implement a type to represent Borel sets and we shall do exactly that in this post. ]]>

where

This set us to wondering whether or not we might endeavour to find a discrete analogue of its inverse, the natural logarithm

`ak.setUnion`

and `ak.setIntersection`

respectively.Such arrays are necessarily both finite and discrete and so cannot represent continuous subsets of the real numbers such as intervals, which contain every real number within a given range. Of particular interest are unions of countable sets of intervals

`ak`

library to represent them.
]]>
Might I tempt you with a little sport to quicken the blood still further?

It lifts my soul to hear it Sir!

I have in mind a game that I learned when in passage to the new world with a company of twelve Quakers. I was not especially relishing the prospect of yet another monotonous transatlantic crossing and so you can imagine my relief when I spied the boisterous party embarking, dressed in the finest silks and satins and singing a bawdy tavern ballad as they took turns at a bottle of what looked like a very fine brandy indeed! ]]>