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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

13 Jan 2015: F = Fe Fm = q(E v B), (13). where E = E(r) and B = B(r). ... 1.4, arguing there that B = B(s)eφ by ‘symmetry consid-erations’. b) Long solenoid.

14 Oct 2015: A b B. |ψyb |φy. “. A|ψy. b. B|φy. and extend by linearity. ... The general multiparticle statesbelong to. V1 b V2 b. b VN , (5.3.1)with basis states.

3 Jun 2015: colo. nies. q. (b)(a) (c). (d ). Figure 1. Volvox geometry and flows. ... p6. p3. p2. p6. p3. p2. p6. p3. p2. b b b.

21 Jan 2015: Using (2.14a,b) to evaluate P1 and P2 in (2.11), we obtain. ... its asymptotic value isdependent on all the parameters M, B, D and XF.

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: With the result (1.75) for B we have also. Bt = Cγ5t = γ5C = B , B† = C†γ5 = B1 B = B1 , (1.76). ... 1.85). = ηT. p. [. a(pP)eip.x b(pP). †eip.x]. ,. (1.86).

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].

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