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5 Oct 2015: A. A’. (a) (b). Figure 1: Examples of (a) a contractible loop, (b) a non-contractible loop. ... A. A’. BB’. A. A’. B. B’(a) (b) (c). Figure 2: A contractible loop, as shown by the continuous deformations of (a) to (c).

8 Nov 2015: Let E B and F B be closedsubspaces of a Banach space B. ... Pk(T z)Pk))2nPmH. (6.2). 13. and the fact that for self-adjoint A,B B(H) we have dH(σ(A),σ(B)) ‖AB‖ it suffices to show

28 Sep 2015: lation point] (b) Invariant under f [Obvious from definition] (c) Closed [think about not being. ... a) ẋ = x2, ẏ = y [Saddle-node], (b)ẋ = x3, ẏ = y [Nonlinear Saddle].

5 May 2015: Θ′(h) = dexp1Θ(h)A(h) =. m=0. Bmm! admΘ(h)A(h), (2.4). where Bm are Bernoulli numbers and the adjoint map is recursively defined by ad0A(B) =. B, ... adk1A (B) = [A, adkA(B)]. Magnus (1954) resorted to solving the dexpinv equation.

8 Nov 2015: Let E B and F B be closedsubspaces of a Banach space B. ... Pk(T z)Pk))2nPmH. (6.2). 13. and the fact that for self-adjoint A,B B(H) we have dH(σ(A),σ(B)) ‖AB‖ it suffices to show