http://maths.straylight.co.uk for when the margin is too small Tue, 08 Jul 2008 16:05:41 +0000 http://backend.userland.com/rss092 en Mathematical study is often thought of as 'purer' than scientific research- instead of labs full of chemicals, fruit flies or lasers, our work could in theory proceed with nothing more than a chalkboard; rather than believing theories through weight of evidence and an absence of counterexamples, we prove theorems as ... http://maths.straylight.co.uk/archives/119 As a companion to my previous post, here's the list of valid forms of a connected maximal cyclotomic graph with an entry from the ring of integers of Q(√ -7): Uncharged Lines: Uncharged Squares: Uncharged Hexagons: Uncharged Cubes: T_2k Variants (Infinite Family): A chain of the form for any integer k. Charged Triangles: Charged Squares: or C_2k Variants (Infinite Family): A chain ... http://maths.straylight.co.uk/archives/121 To recap: I've been trying to completely classify the possible matrices/graphs subject to a constraint on their eigenvalues we're describing as cyclotomicity. This is a problem that can be posed in the ring of integers of any imaginary quadratic extension field, but for all but finitely many of them reduces ... http://maths.straylight.co.uk/archives/120 On Friday I gave a geometry club seminar, speaking about some of the computational aspects of discrete-logarithm cryptography in general and as implemented for elliptic curves. My notes supplement rather than completely describe the talk, being heavier on the formalities and lighter on the narrative. The topics covered are: ... http://maths.straylight.co.uk/archives/118 This entry first appeared as a writeup for Everything2. Greedy pig is a simple maths game for groups that serves as an introduction to probability. I used it recently as a warm-up activity for a maths hour with local primary school children (around 11 years old), where it was well-received. For ... http://maths.straylight.co.uk/archives/117 This May, I'll be travelling all the way to Canada for ANTS-VIII, the Eighth Algorithmic Number Theory Symposium; I'm tacking a couple of days holiday on the front as well, so should be good! http://maths.straylight.co.uk/archives/115 So it's been over two months since a post; more attentive readers will have noticed that there was one, but now there isn't. I've moved away from thinking about cryptography to generalising some number/graph theoretic results of my supervisor, concerning matrices with constrained eigenvalues. However, this creates a problem: unless ... http://maths.straylight.co.uk/archives/114 I promised some of my tutorial students a demonstration of how the 'highschool' approach to Euclid's algorithm can be reversed to give rise to the extended Euclidean algorithm - as opposed to the version in their lecture notes, which finds both gcd(a,b) and x,y such that ax+by=gcd(a,b) in one ... http://maths.straylight.co.uk/archives/112 The latest version of my ellnet class is ellnet2d_lowmem.spyx. It combines all the tricks I know of: The use of precomputed inverses for all steps, and precomputed squares/products for each step, as described by Stange, computation with a local vector to avoid overhead from function calls to keep the dictionary up-to-date, mixed block ... http://maths.straylight.co.uk/archives/111 (updated 10/i/08) In an earlier post I described Stange's algorithm for efficiently finding terms in elliptic nets (with a view to pairing computation). I also made the observation that a shorter block structure could be used for doubling- but once employed, it was not possible to perform a double-and-add. This meant ... http://maths.straylight.co.uk/archives/110