Many people, myself included, are not entirely happy with “computer proofs”. Such proofs use a computer program to check a large number of cases. Instead of an elegant argument, it is a brute force approach. Part of the problem is that there is no way of checking the result without a computer. Also people have [...]
One of the more interesting titbits about the current Greek crisis was the quote from someone at De La Rue (ticker LSE:DLAR) that it takes 3 months to print a new set of banknotes. So how far have they got? Printing it is not the whole story. The immediate reaction of every Greek to the [...]
You might think it was fairly straightforward to check out the existence of projective planes computationally. Unfortunately, the numbers are fairly daunting. The first three cases not settled by Bruck-Ryser are orders 6, 10 and 12. One obvious approach is to look for a complete set of orthogonal Latin squares. A quick check at the [...]
The problems were here. Problem 1 (1) Show that every basis in a finite-dimensional vector space has the same size; (2) Show that we can extend any basis for a subspace \(U\) of \(V\) to a basis for \(V\); (3) Show that every subspace has a complementary subspace. Is it unique? (1): Suppose the vectors [...]
In order to tackle the next topic in finite projective planes, we need a lightning introduction to finite-dimensional vector spaces. The concept is just a generalisation of the familiar two or three dimensional vectors of elementary maths. Formally, we have a set \(V\) of “vectors” and a field \(F\). The elements of \(F\) are referred [...]
