On Lucky Sevens - student

The Baron's most recent game consisted of a race to complete a trick of four sevens, with the Baron dealing cards from a pristine deck, running from Ace to King once in each suit, and Sir R----- dealing from a well shuffled deck. As soon as either player held such a trick the game concluded and a prize was taken, eleven coins for the Baron if he should have four sevens and nine for Sir R----- otherwise.
The key to reckoning the equity of the wager is to note that it is unchanged should the Baron and Sir R----- take turns dealing out the rest of their cards one by one after the prize has been taken.

Full text...

submit to reddit  

Let's Talk About Sets - a.k.

In the last couple of posts we have seen various ways to partially or fully sort data and the kinds of queries that we can run against them once they have been. Such query operations make fully sorted arrays a convenient way to represent sets, or more accurately multisets which treat repeated elements as distinct from each other, and in this post we shall exploit this fact to implement some operations that we might wish to perform upon them.

Full text...

submit to reddit  

Finally On A Calculus Of Differences - student

My fellow students and I have spent much of our spare time this past year investigating the similarities between the calculus of functions and that of sequences, which we have defined for a sequence sn with the differential operator

  Δ sn = sn - sn-1

and the integral operator
  n
  Δ-1 sn = Σ si
  i = 1
where Σ is the summation sign, adopting the convention that terms with non-positive indices equate to zero.

We have thus far discovered how to differentiate and integrate monomial sequences, found product and quotient rules for differentiation, a rule of integration by parts and figured solutions to some familiar-looking differential equations, all of which bear a striking resemblance to their counterparts for functions. To conclude our investigation, we decided to try to find an analogue of Taylor's theorem for sequences.

Full text...

submit to reddit  

I Still Haven't Found What I'm Looking For - a.k.

Last time we took a look at a selection of sorting operations that we can use to sort arrays, or ranges of elements within them. After defining some useful comparison functions satisfying JavaScript's requirement of returning a negative number when the first argument compares smaller than the second, zero when they compare equal and a positive number otherwise, and a function to map negative integers to indices read from the end of arrays in the same way that Array.slice does, we first implemented ak.partition which divides elements into two ranges; those elements that satisfy some given condition followed by those elements that don't. We saw how this could be used to implement the quicksort algorithm but instead defined ak.sort to sort a range of elements using Array.sort, slicing them out beforehand and splicing them back in again afterwards if they didn't represent whole arrays. We did use it, however, to implement ak.nthElement which puts a the correctly sorted element in a given position position within a range, putting before it elements that are no greater and after it elements that are no smaller. Finally, we implemented ak.partialSort which puts every element in a range up to, but not including, a given position into its correctly sorted place with all of the elements from that position onwards comparing no less than the last correctly sorted element.
This time we shall take a look at some of the ways that we can query data after we have manipulated it with these functions.

Full text...

submit to reddit  

Lucky Sevens - baron m.

Greetings Sir R-----! This evening's chill wind might be forgiven some of its injurious assault upon me by delivering me some good company as I warm my bones. Come, shed your coat and join me in a glass of this rather delightful mulled cyder!

Might you be interested in a little sport whilst we recover?

Excellent!

This foul zephyr puts me in mind of the infantile conflict between King Oberon and Queen Titania that was in full force during my first visit to the faerie kingdom. I had arrived there quite by accident but fortunately my reputation was sufficient to earn me an invitation to dine at the King's table. That the fare was sumptuous beyond the dreams of mortal man goes without saying, but the conflict between the King and his consort cast something of a shadow upon the evening.

Full text...

submit to reddit  

We're All Sorted From A To Z - a.k.

Something that I miss when programming in JavaScript is the wide variety of array manipulation functions available in my primary language, C++. We have, in fact, already implemented one of them with ak.shuffle which randomly rearranges the elements of an array. We shall be needing another one of them in the not too distant future and so I have decided to take a short break from numerical computing to add those of them that I use the most frequently to the ak library, starting with a selection of sorting operations.

Full text...

submit to reddit  

On Share And Share Alike - student

When last they met, the Baron challenged Sir R----- to a wager in which, for a price of three coins and fifty cents, he would make a pile of two coins upon the table. Sir R----- was then to cast a four sided die and the Baron would add to that pile coins numbering that upon which it settled. The Baron would then make of it as many piles of equal numbers of no fewer than two coins as he could muster and take back all but one of them for his purse. After doing so some sixteen times, Sir R----- was to have as his prize the remaining pile of coins.

Full text...

submit to reddit  

Do The Evolution - a.k.

In the last few posts we have taken a look at genetic algorithms, which use simple models of biological evolution to search for global maxima of functions, being those points at which they return their greatest possible values.
These models typically represent the arguments of the function as genes within the binary chromosomes of individuals whose fitnesses are the values of the function for those arguments, exchange genetic information between them with a crossover operator, make small random changes to them with a mutation operator and, most importantly, favour the fitter individuals in the population for reproduction into the next generation with a selection operator.
We used a theoretical analysis of a simple genetic algorithm to suggest improved versions of the crossover operator, as well as proposing more robust schemes for selection and the genetic encoding of the parameters.
In this post we shall use some of them to implement a genetic algorithm for the ak library.

Full text...

submit to reddit  

Further Still On A Calculus Of Differences - student

For some time now my fellow students and I have been whiling away our spare time considering the similarities of the relationships between sequences and series and those between the derivatives and integrals of functions. Having defined differential and integral operators for a sequence sn with

  Δ sn = sn - sn-1

and
  n
  Δ-1 sn = Σ si
  i = 1
where Σ is the summation sign, we found analogues for the product rule, the quotient rule and the rule of integration by parts, as well as formulae for the derivatives and integrals of monomial sequences, being those whose terms are non-negative integer powers of their indices, and higher order, or repeated, derivatives and integrals in general.

We have since spent some time considering how we might solve equations relating sequences to their derivatives, known as differential equations when involving functions, and it is upon our findings that I shall now report.

Full text...

submit to reddit  

The Best Laid Schemata - a.k.

We have seen how we can exploit a simple model of biological evolution, known as a genetic algorithm, to search for global maxima of functions, being those points at which they return their greatest values.
This model treated the function being optimised as a non-negative measure of the fitness of individuals to survive and reproduce, replacing negative results with zero, and represented their chromosomes with arrays of bits which were mapped onto its arguments by treating subsets of them as integers that were linearly mapped to floating point numbers with given lower and upper bounds. It simulated sexual reproduction by splitting pairs of the chromosomes of randomly chosen individuals at a randomly chosen position and swapping their bits from it to their ends, and mutations by flipping randomly chosen bits from the chromosomes of randomly chosen individuals. Finally, and most crucially, it set the probability that an individual would be copied into the next generation to its fitness as a proportion of the total fitness of the population, ensuring that that total fitness would tend to increase from generation to generation.
I concluded by noting that, whilst the resulting algorithm was reasonably effective, it had some problems that a theoretical analysis would reveal and that is what we shall look into in this post.

Full text...

submit to reddit