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2 Oct 2023: 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

14 Oct 2021: Show that ifeb|k|f̂(k) L2(R) for some b > 0 then f is real analytic at each point in R. ... u1. 1We write A B to mean that there exists a universal constant C such that A CB.

12 Oct 2007: Show that if fn f uniformly and (xm) isa sequence of points in [a, b] with xm x then fn(xn) f (x). ... Prove that there is some subinterval [a, b] of [0, 1] with a < b on which f is bounded.

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

17 Oct 2008: Show that if fn f uniformly and (xm) isa sequence of points in [a, b] with xm x then fn(xn) f (x). ... Prove that there is some subinterval [a, b] of [0, 1] with a < b on which f is bounded.

12 Oct 2018: 5. Let (fn) be a sequence of real-valued continuous functions on a closed, bounded interval [a,b], and suppose. ... a) If Dn is the set of discontinuities of fn and D is the set of discontinuities of f, show that D n=1j=nDj.(b) Suppose that for some

23 Oct 2017: Hint: Bolzano-Weierstrass.). 3. If A and B are subsets of Rn, let A B = {a b |a A,b B}. ... vn} C such that{B(v1),. ,B(vn)} cover C. (b) Show that there exists some > 0 such that for every v C, B(v) Uα for some α.Deduce that

25 Oct 2007: An interval is a set taking one of the forms (, ), (, b), (, b], [a, ), (a, ), [a, b], [a, b), (a, b]or (a, b) for some a, b R with a b. ... 15. Let V be a real vector space with a countably infinite basis; that is to say, there is some sequence e1, e2,e3

8 Dec 2009: 5. (i) Suppose that f : R2 R is such that D1f = f/x is continuous in some open ball around(a,b), and D2f = f/y exists at (a,b). ... 11. Show that there is a continuous square-root function on some neighbourhood of I in Mn;that is, show that there is an

4 Feb 2010: J : z 7 az bcz d. for some complex numbers a, b, c, d with ad bc = 1. ... For which choices of a, b, c, d is this map J aninvolution, that is J 2 = I?