## Further On Natural Analogarithms - student

My fellow students and I have of late been thinking upon an equivalence between the roots of rational numbers and an infinite dimensional rational vector space, which we have named -space, that we discovered whilst defining analogues of logarithms that were expressed purely in terms of rationals.
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.

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## Chalk The Lines - a.k.

Given a set of points (xi,yi), a common problem in numerical analysis is trying to estimate values of y for values of x that aren't in the set. The simplest scheme is linear interpolation, which connects points with consecutive values of x with straight lines and then uses them to calculate values of y for values of x that lie between those of their endpoints.
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.

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Good heavens Sir R----- you look quite pallid! Come take a seat and let me fetch you a measure of rum to restore your humors.
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.

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## A Measure Of Borel Weight - a.k.

In the last few posts we have implemented a type to represent Borel sets of the real numbers, which are the subsets of them that can be created with countable unions of intervals with closed or open lower and upper bounds. Whilst I would argue that doing so was a worthwhile exercise in its own right, you may be forgiven for wondering what Borel sets are actually for and so in this post I shall try to justify the effort that we have spent on them.

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## On Quaker's Dozen - student

The Baron's latest wager set Sir R----- the task of rolling a higher score with two dice than the Baron should with one twelve sided die, giving him a prize of the difference between them should he have done so. Sir R-----'s first roll of the dice would cost him two coins and twelve cents and he could elect to roll them again as many times as he desired for a further cost of one coin and twelve cents each time, after which the Baron would roll his.
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.

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## A Borel Universe - a.k.

Last time we took a look at Borel sets of real numbers, which are subsets of the real numbers that can be represented as unions of countable sets of intervals Ii. We got as far as implementing the 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.

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## On Natural Analogarithms - student

Last year my fellow students and I spent a goodly portion of our free time considering the similarities of the relationships between sequences and series and those between derivatives and integrals. During the course of our investigations we deduced a sequence form of the exponential function ex, which stands alone in satisfying the equations

D f = f
f(0) = 1

where D is the differential operator, producing the derivative of the function to which it is applied.
This set us to wondering whether or not we might endeavour to find a discrete analogue of its inverse, the natural logarithm ln x, albeit in the sense of being expressed in terms of integers rather than being defined by equations involving sequences and series.

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## A Decent Borel Code - a.k.

A few posts ago we took a look at how we might implement various operations on sets represented as sorted arrays, such as the union, being the set of every element that is in either of two sets, and the intersection, being the set of every element that is in both of them, which we implemented with 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 Ii, known as Borel sets, and so it's worth adding a type to the ak library to represent them.

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## Quaker's Dozen - baron m.

Sir R-----, my fine friend! The coming of spring always puts one in excellent spirits, do you not find? Speaking of which, come join me in a glass of this particularly peaty whiskey with which we might toast her imminent arrival!

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!

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## The After Strife - a.k.

As well as required arithmetic operations, such as addition, subtraction, multiplication and division, the IEEE 754 floating point standard has a number of recommended functions. For example finite determines whether its argument is neither infinite nor NaN and isnan determines whether its argument is NaN; behaviours that shouldn't be particularly surprising since they're more or less equivalent to JavaScript's isFinite and isNaN functions respectively.
One recommended function that JavaScript does not provide, and which I should like to add to the ak library, is nextafter which returns the first representable floating point number after its first argument in the direction towards its second.

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### Gallimaufry

 AKCalc ECMA Endarkenment Turning Sixteen

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