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22 Nov 2007: Let b : [0, T ] Rd Rd be continuous and suppose that, for someK < , for all 0 6 t 6 T ,. |b(t, x) b(t, y)| 6 K|x y|, x, ... ẋt = b(t, xt), 0 6 t 6 T. Proof. Note that, by continuity, C = supt6T |b(t, x0)| <. Set xt(0) = x0 for all t > 0and define

18 Jan 2007: k l. h. c d. Problem 7: What is the least length of wire needed to connect all the cities? ... Problem 7: What is the least length of wire needed to connect all the cities?

22 Jun 2007: In compensation,. it is known that θ(λc) = 0 in all dimensions, [9]. ... above, uniformly in L, whenever θ is sufficiently small, and perhaps for all θ < θcwhere θc = 2 is the critical point.

21 Sep 2007: the operational characteristics of all the links in the light of the routes used by these and all othertravellers;. • ... All these purposes require interpretation of the outputs of modelling in the light of sound under-standing of the scope and

22 Nov 2007: We say that b is fully controllable in time t if, for all x0, x Rd,there exists a control (us)06s6t such that xt = x. ... Then vT eAsB = 0 for all s and sovT G(t)v = 0 for all t > 0.

18 Jan 2007: f(n) < g(n) for all n:f(n) = (g(n))means 9 > 0 s.t. ... such that(a)j(Xn1) (Xn)j B almost surely,and(b)E[(Xn1) (Xn) j Xn = x] B for all x.Then for all 0 > liminfn!1 "(Xn) n (1

22 Nov 2007: We use the matrix norm |A| =sup{|Ax| : |x| = 1}, for which |Ax| 6 |A||x| for all x Rd, |A| = |AT | and |AB| 6 |A||B|.Then A is a ... Since f is fully controllable or stabilizable, V (x) < for all x Rd.

22 Nov 2007: 40. where the infimum is taken over all continuous feasible controls starting from (s, x), andV (s, x) = if there are no such controls. ... We seek tomaximize. T. 0. eαs. usds. subject to ẋt = βxt ut and xt > 0 for all 0 6 t 6 T.

22 Nov 2007: Note that, in the time-unconstrained case, if b, c and C are time-independent, then µt = 0for all t. ... for all a A, with equality when a = u(t, x), for all (t, x) S̃.

22 Nov 2007: m, andb : S A Rd. We assume that these maps are all continuously differentiable on Rd. ... Provided that σ1,. , σm and b are all differentiable, withbounded derivative, the equation has a unique maximal local solution, just as in the deterministic case.