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3 Jul 2024: a and b, and writing for the group operation, we do not necessarily have ab=ba).

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.

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: a) Let G act on B B by g(B1,B2) = (gB1, gB2). ... B B | (B,F(B)) O(w)}. Show that X(w) can be identified with the variety L1(BwB)/B, for a rational Borelsubgroup B, and show that there exists a

22 Aug 2023: i) Order the vectors by magnitude. (ii) Use the scalar product to find the angles between the pairs of vectors (a) A and B, (b) B and C,leaving your answer ... iii) Find the lengths of the projections of the vectors (a) A onto B, (b) B onto A.

30 Aug 2019: µ0ρ(B b b B),. b. t= B u u B. Consider solutions of the form u(s, φ, z, t) = ũ(s, z)eσtimφ, etc., where m is aninteger. ... Then verify that. B b b B = Jẑ b 12imJb,.

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. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTSCover sheet ... δρ = ξ ρ ρ ξ ,δp = ξ p γp ξ ,. δB = ξ B B ξ B( ξ) ,. ρ2ξ. t2= δΠ g δρ

30 Aug 2019: Now suppose given a frame homomorphism h : B A. We say two nuclei j andk on A are B-equivalent if j(h(b)) = k(h(b)) for all b B. ... By considering the nucleus{o(h(b)) c(j(h(b))) | b B} (where j is an arbitrary nucleus), show that each B-.

30 Aug 2019: Bt u B = B u α B η2B,. where α is a constant and where the simple shear flow u = T 1(y, 0, 0). ... 2 A model for an “α2” dynamo, analogous to the Parker wave model, is given bythe equations (where A = A(x, t), B = B(x, t)):.

30 Aug 2019: ρ. t+ uρ = ρu ,. p. t+ up = γpu ,. ρ. (u. t+ uu. )= ρΦ p 1. µ0( B) B ,. B. t= (u B) ,. B = 0 ,2Φ = 4 π Gρ. ... iωu = p̃ 1µ0ρ. (imJb 2Jêz b) ,. iωb = imJu ,. where p̃ is a modified pressure and ρ is the fluid density.

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: 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: 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.

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.

17 Jun 2019: Hence show that the line integral P2P1. F dx. along any path in R3 between points P1 = (0,a, 0) and P2 = (b,b,b) vanishes for anyvalues of a and ... b. The two-dimensional vector field G is defined at all points in R2 except (0, 0) by.

17 Jun 2019: Find constantsα and β such that. xjTij = [( B) B]i. ... i) the union of all sets in B is X;. (ii) if x B1 B2 for two sets B1 and B2 in B, then there is a set B B withx