This is just an update on two topics looked over during the summer. In the case of the QR flow and Toda flow equivalence, I have had a hard time finding it due to some strange ways how Mathematica computes the QR composition. You can try yourself: the diagonal entries of the R matrix in the decomposition can have positive or negative signs. I had noticed there was a problem and eventually brushed over it because the theoretical proof was so clear. Last Friday, I decided to add the QR factorization routine from scratch, rather than hacking the built-in routine. Here it is
QR[A_]:=Module[{F,B,n,T},T=Transpose;F=T[A];n[x_]:=x/Sqrt[x.x];B={n[F[[1]]]};Do[v=F[[k]];
u=v-Sum[(v.B[[j]])*B[[j]],{j,k-1}];B=Append[B,n[u]],{k,2,Length[F]}];{T[B],B.A}];Lets look at some example. The assignment of signs to the diagonal entries is in the built in routine pretty random.
A = Table[RandomInteger[10], {4}, {4}];
{q, r} = QRDecomposition[1.0 A]; {Q, R} = QR[1.0 A];
Map[MatrixForm, Chop[{q, r}]]
Map[MatrixForm, Chop[{Q, R}]]An other update is to curvature, the topological index part of the story. The facet curvature of odd dimensional manifolds is not always constant zero. For the 3-dimensional projective plane = SO(3) for example it is is not. Update September 2: A PDF of the edge curvature paper.
