TooMuchCoffeeMan's Blog

... are those torsion-free abelian groups such that any two elements are Z-linearly dependent, and apparently they are all realised as subgroups of Q. Clearly every subgroup of Q is of this form.

My interest in this condition comes from convolution algebras of the positive cones of such groups, e.g. Z_+ or Q_+. There, a certain calculation works whenever any two elements of the cone have a common non-zero divisor, and it's clear that if this property holds then the original group must have (at most) rank one. It turns out that the converse holds (use the Euclidean algorithm, together with the fact that the two elements have an HCF in Q, to show that said HCF lies in the original group).

See the Springer EOM entry (about half way down). There's also a Wikipedia entry (merely a stub at time of writing).

Interesting discussion going on at this post at the n-category cafe (hope the trackback works)

Collaborative work (as opposed to mere exchange of facts) in a blog format, which is nice to see.

Just a placeholder post: mustn't forget to re-read Tao's masterly exposition of amplification techniques for inequalities

I've often vaguely wondered if there isn't a homological perspective on the basic results of convex optimization: homology is about finding good generating sets for relations and relations between relations; and if we replace linear systems of equations with linear systems of inequalities this is precisely what convex optimization worries about.

Anyway: a link to Terence Tao's latest on how Farkas lemma gives an algorithmic/inductive proofs of the fin-dim separation theorem

Should, as ever, be worth reading.

[0711.3200] G. A. Elliott, "Towards a theory of classification"

Haven't read this properly but it has a general notion of classification functor. (We want to mod out by inner automorphisms, intuitively.)

Will be interesting to see if the n-category cafe pick up on this...

Have been meaning to follow the n-category café's series of posts on category-theoretic ways to look at probability theory.

Amusingly one reference in Corfield's original posts was a paper of Giry, which if I recall correctly used the language of (Kleisli?) monads to look at formal aspects of stochastic processes. (I had a photocopy of this paper years ago, when I was doing the CASM and was excited to find that two of my favourite courses might make contact...)

Too busy at the moment with other maths to look at this, but I should really track this.

So, another blog to play with.

Perhaps this should be for work and Since_it_is_not should be more metamathematical or explanatory?