Our unitary Hessenberg matrix A is now factored as a descending sequence of core transformations (n = 8)

We can use a turnover to pass a core transformation through a descending sequence as follows:

Exactly two of the core transformations from the descending sequence participate in the turnover.

1], and only B1, the bulge, disturbs the descending sequence of core transformations.

2, 1 in a

descending sequence, and then inserting n - 1 and n somewhere along the sequence, with n - 1 on the left.

And last week, when my eleven-year-old student was playing intermediate-level repertoire, I said, "With which motive in the next phrase does the

descending sequence begin that causes this phrase to decrescendo?

Describing the novel's basic structure, consisting of 'West' (nine chapters in ascending numeric sequent), 'Bridge', and 'East' (nine chapters in

descending sequence, which is usually interpreted as a pyramid), Fiddian proposes a more complex arrangement of wheels, corresponding to pre-Columbian cosmogony, with the bridge serving as a fulcrum between cyclical and linear times.

Moreover, we can construct a special pattern (called an X-pattern), such that we start on top of the matrix with a descending sequence of rank expanding Givens transformations, and on the bottom with an upgoing rank decreasing sequence Givens transformations.

This leads in a simple manner to the QR-decomposition of the matrix Q in which we first perform an upgoing sequence of Givens transformations (removing the low rank part), followed by a descending sequence of Givens transformations (expanding the part of rank zero).

v1(1);u1]; Perform the descending sequence of Givens transformations for i=1:n1-1 M=[R1(i,i:n);u1(i+1)*v1(i),R1(i+1,i+1:n)]; [c,s,M] = Givensexp(M,v1(i:i+1)); b1(i:i+1)=[c,s;-conj(s),c]*b1(i:i+1); % Update the representation of the matrices R and u u1(i:i+1)=M(1:2,1).

Based on these operations we can interchange the order of the upgoing and descending sequences of Givens transformations.

In this kind of vocalization the object should be to produce on one vowel the sudden contractions and expansions of the vocal chink from note to note in ascending and

descending sequences with increasing or decreasing power, sometimes in strict time, sometimes ad libitum, but always firmly and clearly.