Re: [math-fun] Greedily packing disks in a sector
James Propp wrote
(Is there software that makes problems like this easy to play with? If I had to write the code myself it'd take me days to do, and there's a good chance that my code would have mistakes that I'd miss.)
You might want to get in touch with any of the 'packomania' contributors. Their software (plural) might be overkill, as they're optimising rather than just greedily rule-following. David Cantrell's code is Mathematica-based (I was tempted to say "alas", but I hear some people like that platform). I'm surprised he's not on this list; he deserves to be, he's one of the few worthwhile contributors to sci.math nowadays. Phil -- () ASCII ribbon campaign () Hopeless ribbon campaign /\ against HTML mail /\ against gratuitous bloodshed [stolen with permission from Daniel B. Cristofani]
Anyone else see the article on the SVD in the current monthly? If so, I'm curious what you think of it. --Dan ((( The basic SVD theorem says that any KxL real matrix M can be expressed uniquely as M = P D Q where P (KxK) and Q (LxL) are orthogonal matrices, and D (KxL) is diagonal with D_11 >= D_22 >= . . . D_pp >= 0, where p = min{K,L}. OK, the "singular values" D_jj are unique. P and Q are unique only if the D_jj are all distinct. This has a geometric meaning: For any two subspaces V, W of some R^n, with dim(V) = K, dim(W) = L, then K and L have p = min{K,L} "principal angles" 0 <= theta_1 <= theta_2 <= . . . <= theta_p <= pi/2, with D_jj = cos(theta_j) for all j = 1,...,p, such that: (*) the theta_j's completely determine the relative positions of K and L in R^n. (Here theta_1 is the least angle between any v in V and w in W. Now take the orthogonal complements of v in V and of w in W: theta_2 is the least angle between vectors in these subspaces. Lather, rinse, repeat.) (*) means that for any other subspaces V', W' of R^n with dim(V') = dim(V), dim(W') = dim(W), and having the same principal angles, there is an isometry h: R^n -> R^n in O(n) with h(V) = V' and h(W) = W'. For n >= K+L, any sequence of principal angles 0 <= theta_1 <= theta_2 <= . . . <= theta_p <= pi/2 can arise. )))
I haven't seen the article ... was this post a summary of the contents? Whether or not, I don't understand the connection between V,W and P,Q here. The geometric interpretation as it stands is at best incomplete, at worst self-evidently fallacious: singular values of a general matrix M may exceed unity, yielding complex angles. WFL On 12/22/12, Dan Asimov <dasimov@earthlink.net> wrote:
Anyone else see the article on the SVD in the current monthly?
If so, I'm curious what you think of it.
--Dan
((( The basic SVD theorem says that any KxL real matrix M can be expressed uniquely as
M = P D Q
where P (KxK) and Q (LxL) are orthogonal matrices, and D (KxL) is diagonal with D_11 >= D_22 >= . . . D_pp >= 0, where p = min{K,L}.
OK, the "singular values" D_jj are unique. P and Q are unique only if the D_jj are all distinct.
This has a geometric meaning: For any two subspaces V, W of some R^n, with dim(V) = K, dim(W) = L, then K and L have p = min{K,L} "principal angles" 0 <= theta_1 <= theta_2 <= . . . <= theta_p <= pi/2, with D_jj = cos(theta_j) for all j = 1,...,p, such that:
(*) the theta_j's completely determine the relative positions of K and L in R^n.
(Here theta_1 is the least angle between any v in V and w in W. Now take the orthogonal complements of v in V and of w in W: theta_2 is the least angle between vectors in these subspaces. Lather, rinse, repeat.)
(*) means that for any other subspaces V', W' of R^n with dim(V') = dim(V),
dim(W') = dim(W), and having the same principal angles, there is an isometry h: R^n -> R^n in O(n) with h(V) = V' and h(W) = W'.
For n >= K+L, any sequence of principal angles 0 <= theta_1 <= theta_2 <= . . . <= theta_p <= pi/2 can arise. )))
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Dan Asimov -
Fred lunnon -
Phil Carmody