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19 Jan 2021: Show that E has a subgroup isomorphic toE/M, and show that all such subgroups are conjugate. ... for all n. 3. 8. Let G be a group and let K be a subgroup of G.

1 Nov 2021: Show that M M is trivial.(b) Show that for all n we have TSn R = Rn1 over Sn. ... and Q7. compute HndR(RPn) for all n. 9. Let Σ be a compact 2-manifold-with-boundary, let F : Σ R2 be a smooth map, and let α bethe

19 Jan 2021: tβn. are joined by an edge if, after renumbering:. (a) αi is isotopic to βi for all i > 1;. ... b) if Sα2,.,αn is a one-holed torus then i(α1,β1) = 1;.

8 Oct 2021: b) Let X Bin(n,θ), where θ Θ = [0, 1]. Find all prior distributions π on Θ for whichthe maximum likelihood estimator is a Bayes rule for estimating θ in ... a) Show that the James-Stein estimator θ̃(1) dominates all estimators θ̃(c),c 6= 1.(b)

13 Oct 2021: foreach γ C there exist elements α(γ) A and β(γ) B, such that for all γ we have:. ... multiplication. An R-linear derivation OX,p R is an R-linearmap d : OX,p R such that d(fg) = d(f)g(p) f(p)d(g) for all f

10 Nov 2021: 5. (a) Suppose that f Matn,n(F) is such that f(AB) = f(BA) for all A,B Matn,n(F) and f(I) = n.Show that f is the ... f(A) = trA for all A Matn,n(F).(b) Now let V be a non-zero finite dimensional real vector space.

15 Oct 2021: uα =Dαu(x′,0). α!= 0, for all multi-indices α with αn = 0. ... We will proceed inductively tocompute all uα by making use of the equation.

10 Jun 2021: All the other edges of the network are of the form (ai, bj) for some i, j = 1,. , ... A is connected to all ai and B is connected to all ci with capacity 1 edges.The capacities of all other edges is infinite.

14 Oct 2021: Y := {v V : (v,u) = 0 for all u Y}.(a) Show that dim Y dim Y = dim V. ... y,λη) for all (x,ξ) M and λ [0,).

18 Jan 2021: Assume all normalized eigenforms f S2(N) have rational coefficients. Fix a prime p 6= N. ... Define the p-Eisenstein ideal as the ideal a of T generated by p andT ( 1) for all.