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15 Oct 2021: g2Zngv. Show that if 2 D(Rn) with supp K for some compact K Rn then. ... u[] =X. g2Agv[],. for some finite set A Zn which depends only on K.

29 Oct 2021: finitely generated A-module, then Supp(B A M) =. (f)1(Supp(M)). [Note: You may find useful the identity on tensor products (M A B) B C = M A C ... a) Show that x is a zero divisor in A if and only if x p for some p D(A).(b) Let S be a multiplicatively

28 Oct 2021: Some more questions. 13. (a) A topological space is said to be second countable if there is a countable family Bof open sets such that every open set is a union ... of some members of B.

15 Oct 2021: Bj,γ,δ. zγxδ, j = 1,. ,n1,. andc(z,x′) =. γ,δ. cγ,δzγxδ,. where these power series converge if |z| |x′| < s for some small s > 0. ... cγ,δzγxδ,. 4. with these power series converging for |x| |x′| < s for some (possibly smaller) s.

26 Feb 2021: Euclidean domain. Could there be some other Euclidean function φ making Z[3]. ... iii) Show that if R is a PID, the gcd of elements a and b exists and can be written asra sb for some r,s R.

8 Aug 2021: Replace‘Similarly, we can can take a = sin φ, b = cos φ for some real φ.’ by. ... Similarly, we can can take c = sin φ, d = cos φ for some real φ.’.

8 Oct 2021: What happensif Θ = [a,b] for some 0 < a < b <? 9. Let X be multivariate normal N(θ,I), where θ Θ = Rp,p 3, and I is the pp identitymatrix. ... a) Show that the James-Stein estimator θ̃(1) dominates all estimators θ̃(c),c 6= 1.(b) Let θ̂ be the

8 Oct 2021: Suppose that for some θ0 in the interior of Θ and every ε > 0 small enough, wehave S(θ0ε) < 0 < S(θ0ε), and also that Sn has exactly one zero θ̂n ... f(,φ) : φ = Φ(θ) for some θ Θ} equals Φ(θ̂MLE).(b) Now consider a mapping Φ that is

18 Mar 2021: 1 Unramified up to twist: τ = (ψ ψ, 0) for some characterψ : IK C. ... 4 Irreducible: τ = (r0, 0) where r0 is the restriction of some irreducibleWeil-Deligne representation.

8 Oct 2021: The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... for some λ,µ,ν R (which should turn out in this example to be integers).Does there exist a linear transformation which reduces the pair of