## On Onwards And Downwards - student

When last they met, the Baron challenged Sir R----- to evade capture whilst moving rooks across and down a chessboard. Beginning with a single rook upon the first file and last rank, the Baron should have advanced it to the second file and thence downwards in rank in response to which Sir R----- should have progressed a rook from beneath the board by as many squares and if by doing so had taken the Baron's would have won the game. If not, Sir R----- could then have chosen either rook, barring one that sits upon the first rank, to move to the next file in the same manner with the Baron responding likewise. With the game continuing in this fashion and ending if either of them were to take a rook moved by the other or if every file had been played upon, the Baron should have had a coin from Sir R----- if he took a piece and Sir R----- one of the Baron's otherwise.

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## Archimedean Review - a.k.

In the last couple of posts we've been taking a look at Archimedean copulas which define the dependency between the elements of vector values of a multivariate random variable by applying a generator function φ to the values of the cumulative distribution functions, or CDFs, of their distributions when considered independently, known as their marginal distributions, and applying the inverse of the generator to the sum of the results to yield the value of the multivariate CDF.
We have seen that the densities of Archimedean copulas are rather trickier to calculate and that making random observations of them is trickier still. Last time we found an algorithm for the latter, albeit with an implementation that had troubling performance and numerical stability issues, and in this post we shall add an improved version to the ak library that addresses those issues.

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## Finally On Natural Analogarithms - student

Over the course of the year my fellow students and I have spent much of our spare time investigating the properties of the set of infinite dimensional vectors associated with the roots of rational numbers by way of the former's elements being the powers to which the latter's prime factors are raised, which we have dubbed -space.
We proceeded to define functions of such numbers by applying operations of linear algebra to their -space vectors; firstly with their magnitudes and secondly with their inner products. This time, I shall report upon our explorations of the last operation that we have taken into consideration; the products of matrices and vectors.

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## Archimedean View - a.k.

Last time we took a look at how we could define copulas to represent the dependency between random variables by summing the results of a generator function φ applied to the results of their cumulative distribution functions, or CDFs, and then applying the inverse of that function φ-1 to that sum.
These are known as Archimedean copulas and are valid whenever φ is strictly decreasing over the interval [0,1], equal to zero when its argument equals one and have nth derivatives that are non-negative over that interval when n is even and non-positive when it is odd, for n up to the number of random variables.
Whilst such copulas are relatively easy to implement we saw that their densities are a rather trickier job, in contrast to Gaussian copulas where the reverse is true. In this post we shall see how to draw random vectors from Archimedean copulas which is also much more difficult than doing so from Gaussian copulas.

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## Onwards And Downwards - baron m.

Greetings Sir R-----! Might I suggest that you take one of these spiced beef pies and a mug of mulled cider to stave off this winter chill? And perhaps a wager to fire the blood?

Good man! Good man!

I propose a game that ever puts me in mind of my ill-fated expedition to recover for the glory of the Empress of Russia the priceless Amulet of Yendor from the very depths of Hell.

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## Archimedean Skew - a.k.

About a year and a half ago we saw how we could use Gaussian copulas to define dependencies between the elements of a vector valued multivariate random variable whose elements, when considered in isolation, were governed by arbitrary cumulative distribution functions, known as marginals. Whilst Gaussian copulas are quite flexible, they can't represent every possible dependency between those elements and in this post we shall take a look at some others defined by the Archimedean family of copulas.

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## On The Rich Get Richer - student

The Baron's latest wager set Sir R----- the task of surpassing his score before he reached eight points as they each cast an eight sided die, each adding one point to their score should the roll of their die be less than or equal to it. The cost to play for Sir R------ was one coin and he should have had a prize of five coins had he succeeded.

A key observation when figuring the fairness of this wager is that if both Sir R----- and the Baron cast greater than their present score then the state of play remains unchanged. We may therefore ignore such outcomes, provided that we adjust the probabilities of those that we have not to reflect the fact that we have done so.

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## New Directions Of Interpolation - a.k.

We have spent a few months looking at how we might interpolate between sets of points (xi, yi), where the xi are known as nodes and the yi as values, to approximate values of y for values of x between the nodes, either by connecting them with straight lines or with cubic curves.
Last time, in preparation for interpolating between multidimensional vector nodes, we implemented the ak.grid type to store ticks on a set of axes and map their intersections to ak.vector objects to represent such nodes arranged at the corners of hyperdimensional rectangular cuboids.
With this in place we're ready to take a look at one of the simplest multidimensional interpolation schemes; multilinear interpolation.

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## Further Still On Natural Analogarithms - student

For several months now my fellow students and I have been exploring -space, being the set of infinite dimensional vectors whose elements are the powers of the prime factors of the roots of rational numbers, which we chanced upon whilst attempting to define a rational valued logarithmic function for such numbers.
We have seen how we might define functions of roots of rationals employing the magnitude of their associated -space vectors and that the iterative computation of such functions may yield cyclical sequences, although we conspicuously failed to figure a tidy mathematical rule governing their lengths.
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.

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## Cuboid Space Division - a.k.

Over the last few months we have been taking a look at algorithms for interpolating over a set of points (xi,yi) in order to approximate values of y between the nodes xi. We began with linear interpolation which connects the points with straight lines and is perhaps the simplest interpolation algorithm. Then we moved on to cubic spline interpolation which yields a smooth curve by specifying gradients at the nodes and fitting cubic polynomials between them that match both their values and their gradients. Next we saw how this could result in curves that change from increasing to decreasing, or vice versa, between the nodes and how we could fix this problem by adjusting those gradients.
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 y, like complex numbers or matrices, but not to similarly generalise cubic spline interpolation.

The obvious next question is whether or not we can also generalise the nodes to other arithmetic types; in particular to vectors so that we can interpolate between nodes in more than one dimension.

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### Gallimaufry

 AKCalc ECMA Endarkenment Turning Sixteen

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