A Measure Of Borel Weight – a.k.

a.k. from thus spake 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.

A Borel Universe – a.k.

a.k. from thus spake 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.

A Decent Borel Code – a.k.

a.k. from thus spake 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.

Finally On A Calculus Of Differences – student

student from thus spake a.k.

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.

Further Still On A Calculus Of Differences – student

student from thus spake a.k.

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.