Nonlinear Equations - UFRJ

[SEC. 8.6: INEQUALITIES ABOUT THE CONDITION NUMBER 119
For the particular case of homogeneous systems, we consider f i ◦
U(t) ∗ (·) ∈ H di in function of t. We will compute its derivative at t =
0. We write down f i (x) as a tensor, using the notation of Exercise 5.3:
f i (x) =
∑
T j1···j di
x j1 x j2 · · · x jdi
so
0≤j k ≤n
We can pick coordinates so that
[ ]
cos t − sin t
U(t) =
⊕ I
sin t cos t n−k
Its derivative at t = 0 is
˙U =
[ ]
0 −1
⊕ 0
1 0 n−k .
So the derivative of f i at zero is
⎧
f˙
i (x) =
∑
x
D∑ ⎨ −T 1 j1···j di x 0
x j1 x j2 · · · x jdi if j k = 0
x
T 0 j1···j
⎩
di x 1
x j1 x j2 · · · x jdi if j k = 1
0≤j k ≤n k=1 0 otherwise.
Rearranging terms and writing J = [j 0 , . . . , j di ],
⎧
f˙
i (x) =
∑
∑d i ⎨ −T J+ek if j k = 0
x j1 x j2 · · · x jdi T i−ek if j k = 1
⎩
0≤j k ≤n
k=1 0 otherwise.
Comparing the two sides,
‖ ˙ f i ‖ ≤ d i ‖f i ‖.
‖ḟ‖ ≤ D‖f‖.
Theorem 8.23. Under the assumptions of Proposition 8.22, Let G
be a compact, connected symmetry group acting smoothly and transitively
on M/H, such that the induced action into the F i is by isometries.
Let D be the number of 8.22. Let f, g ∈ F, ‖f‖ = ‖g‖ = 1 and
x, y ∈ M/H. Then,
1
1 + u + v µ(f, x) ≤ µ(g, y) ≤ 1
1 − u − v
µ(f, x)