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17 Jun 2019: You may assume that (ab) = a.b b.a (b.)a (a.)b. ]. Show that, in a two-dimensional flow, vortex lines keep their strength and movewith the fluid. ... another vortex ofcirculation Γ at the image point, distance b from the wall on the other side.

17 Jun 2019: φ = 1 on r = a for all θ,. φ = 2 on r = b for all θ. ... a) Prove that each pn, n > 1, has n distinct zeros in the interval (a,b).

17 Jun 2019: Show that if a = a′, b = b′ and c = c′, thenthere is a hyperbolic isometry taking ABC to A′B′C′. ... Is there always such an isometry ifinstead the triangles have one angle the same and a = a′, b = b′?

17 Jun 2019: i) (A B) (B A) is finite and. (ii) there is a finite set F A B such that f(x) = g(x) for all x A B F. ... n. n. k=1. ak = a. (b) Suppose ak > 0 for every k.

17 Jun 2019: Must there exist a A and b B such that d(a, b) 6 d(x, y)for all x A and y B? ... B = b[r (kẑ lŷ)ẑ r px̂(ŷ r) nẑ(x̂ r)],. where b 6= 0, r = (x, y, z) and x̂, ŷ, ẑ are unit vectors in the x,

17 Jun 2019: You may assume. 2 14π|x|. = δ3(x) ,. and for fixed vectors a, b. ... Let b < min(a,r/C, 1/K). Show that there isa unique C1 function x : [b,b] [x0 r,x0 r] such that.

30 Aug 2019: that if A,B D,A B then we have B A D. ... 1. set is. In 1-D it is defined byµ((a,b]) = ba,.

17 Jun 2019: gi > (Pg)i. jI. pijgj > 0 i,(a). gi = 1 i B(b). ... c) gi > (P̃g)i i,. where. p̃ij =. {. pij (i / B),δij (i B).

19 Jun 2019: b) there exists a positive random variable ν, P (ν > 0) = 1, with E ‖νA1‖ < suchthat A0 = E (νA1). ... 2θ, or otherwise, derive an. expression for E(eθTa,b. )for each b, < b <. Hence, or otherwise, show that, for t > 0,. P (Ta,b 6 t) = e2abΦ(.

17 Jun 2019: Show that if B is R-closed then c 6 b B implies c B.(b) Suppose that R satisfies the following condition. ... B C] = {a A |b 6 a (b B b C)}.