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20 Jan 2016: m. (a, ,b ). z. M. (r, ,z). B B. (a) Show that, up to irrelevant constants and ignoring gravity, the Lagrangian forthe system is: [6]. ... 2. (b) Find the corresponding equations of motion for the particle and the bead.

10 May 2016: minimize1. 1 x1+. 1. 2 x2s.t. x1 x2 = b, x1, x2 0. ... A)λ (c), λ 0. So D is of the same form as P, but with c b, b c, and A A.

10 May 2016: minimize1. 1 x1+. 1. 2 x2s.t. x1 x2 = b, x1, x2 0. ... A)λ (c), λ 0. So D is of the same form as P, but with c b, b c, and A A.

8 Mar 2016: P(xt1 | Xt,Ut) = P(xt1 | xt,ut). (b) Separable (or decomposable) cost function. ... To see this, recall that in the D case,|c(x,u)| < B.

15 Mar 2016: x(b)) = b and f(x(b)) = φ(b) = inf{f(x) λ(b)T (h(x) b) : x Rn}. ... b). The second term is zero as well, because x(b) is feasible andthus (h(x(b)) b) = 0.

23 Mar 2016: n. 9. 0. A. A′. a. B. B′. b. 23. 23. ... P. (Xτ,Zτ {B,B′}, (Xt) F. )= P. (Xτ {A,A′},ζ = 0, (Xt) F. )

15 Nov 2016: Proposition 1.6. Let A and B be two finite subsets of Zd. ... 4. We intend to apply (1.6) with A = R1n and B = R2n.

10 Oct 2016: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... 1.8. Let B be a Borel subset of the interval [0, 1].

18 Mar 2016: respect to M. Example 3.9. Suppose that B is a standard Brownian motion. ... Then B̃ = B [B,M]is a P̃-Brownian motion. Proof. Since B̃ is a continuous P̃-local martingale and has [B̃]t = [B]t = t, it follows from the Lévycharacterization

14 Jan 2016: Consider the vector field b on H̄{0} defined by. b(z) =2. ... Fix y (1,) and consider a complex Brownian motion B starting from iy.