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30 Aug 2019: H(,B) are natural in B B. Part III, Paper 22. 3. ... B(KB′,B)B(ρ. B′,B). B(B′,B). is a bijection. (iii) ρB : B KB is an isomorphism in B.

30 Aug 2019: a) Use suffix notation to prove that. a (b c) = c (a b). ... r (a b b c c a) = a (b c),.

30 Aug 2019: Suppose that if a A and b B then (o(a), o(b)) = 1. ... Show that thereexists a generalised character ϑ of G with ϑ(a) = 1 if a A and ϑ(b) = 0 if b B.

30 Aug 2019: 2. (i) State and prove the Erdős–Ko–Rado Theorem. (ii) Two set systems, A and B, are called cross-intersecting if A B 6= whenever A Aand B B. ... f(r, s) elements such thatA B X 6= for all A A and B B.

30 Aug 2019: u B),. B = 0. The flow is such that all physical quantities are independent of t and z, when referred toCartesian coordinates (x,y,z). ... 1µ0ρ. B B ν2v,. (. t 2Ax. y. )B 2ABx ey v B = B v η2B,.

19 Jun 2019: µ(B) =. B. dp dθ. for B (0,) [0, 2π). A random countable set Φ R2 is defined to consist of allintersections of pairs of lines in Π.

30 Aug 2019: ρ. t+ u ρ = ρ u ,. p. t+ u p = γp u ,. ρ. (ut. u u). = ρΦ p 1µ0. ( B) B ,. Bt. = (u B) ,. B = 0 ,. 2Φ = 4πGρ. ... Σ =. ρ dz. (b) Assuming that the perturbations are also isothermal, derive the linearized equationsgoverning displacements of

30 Aug 2019: u. t+ uu. )= ρΦ p 1. µ0(B) B ,. B. ... t= E ,. B = µ0J , B = 0. under a Galilean transformation to a frame of reference moving with uniform relativevelocity v.

30 Aug 2019: You may assume that div (a b) = b. curl a a. ... a (a. )b a div b b div a.]. Paper 64.

30 Aug 2019: The region y < 0 is insulating. A line current J cos ωt flows in a wire parallel to thez-axis and located at x = 0, y = b (b > 0). ... b) Suppose that the frequency ω is large, so that (η/ω)1/2 b.