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20 Jul 2021: Let M be an A-module, and let MB denote the B-module withunderlying group M and multiplication bm = ϕ(b)m for b B, m M. ... Show that Supp s is a closed subset of X. (b) Let Z X be a closed subset of X.

20 Jul 2021: adj(AB) = adj(B) adj(A). (). (c) Let M be an arbitrary n n matrix. ... If such a B exists we write B =. A. Show that if A is.

20 Jul 2021: Write Pp for the correspondingprobability measure. (b) Define the percolation probability θ(p) and the critical probability pc. ... Show the following. (a) The process ξ is increasing in that ξAt ξBt if A B.(b) Moreover, ξ is additive in that ξABt =

20 Jul 2021: b) Find the solution of 2u = 16r2 on the two-dimensional domain D when. ... Deduce that O(2)= D(B), the generalised dihedral group defined in part (a).

20 Jul 2021: ρ. (u. t+ u u. )= ρΦ p 1. µ0(B) B ,. ρ. t+ u ρ = ρu ,. p. t+ u p = γpu ,. B. t= (uB) ,. 2Φ = 4πGρ. In cylindrical polar coordinates (r, ... You may assume the vector identity (ab) c = (a c)b (b c)a.](c) For an axisymmetric magnetostatic

20 Jul 2021: x 6 a, b 6 y 6 b, and c 6 z 6 c. ... n. 1. ) ck. for some integers c0,. ,ck. (b) State the binomial theorem.

16 Dec 2021: Show that there exist x and y such that. f(λa (1λ)b. )= ... x = a cos θ , y = b sin θ , 0 6 θ 6 2π.

20 Jul 2021: Express a, b, c and z′ in terms of n, m and z. ... b. (b) Fix b > 2. Let p be an odd prime not dividing b2 1, and let.