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13 Oct 2012: Let u =(. ac. ), v =. (bd. )for some integers a, b, c, d with ad bc = 1. ... Show that every vector. w Z Z can be written as mu nv for some integers m and n.

22 Jan 2015: Xn be independent Poisson random variables, with Xi having meaniθ, for some θ > 0. ... b) For some n > 2, let X1,. ,Xn be iid with Xi Exponential(θ).

9 Jun 2003: b. 3. for some pair of positive integers a,b}. (vi) X. ... n. , but for a general proof, we must specify some method that always works.

30 Jun 2024: JME Hyland. (1979). 753,. 442. SOME MISLEADING ARGUMENTS INVOLVING CONDITIONAL INDEPENDENCE. ... 1979). 41,. 1. (link to publication). Set colourings of graphs. B BOLLOBAS, A THOMASON. –

30 Jun 2024: 2006). 10,. 553. Some notes on approximations for the deficit at ruin in the compound Poisson model. ... doi: 10.1086/505480). Mutual information, synergy and some curious phenomena for simple channels.

19 Jan 2021: a) Show that there is a surjection π1Sg G, for some g.(b) Show that G is a subgroup of Mod(Sh), for some h. ... a) Prove that TαTβ(α) = β. (b) Prove the braid relation: TαTβTα = TβTαTβ.

26 Nov 2012: a) Suppose that D1f exists and is continuous in some open ball around (a, b), and that. ... b) Suppose instead that D1f exists and is bounded on some open ball around (a, b), and.

29 Nov 2013: 7. Let f: R2 R and (a,b) R2.(a) Suppose that D1f exists and is continuous in some open ball around (a,b), and that. ... b) Suppose instead that D1f exists and is bounded on some open ball around (a,b), and.

12 Nov 2019: 1. 5. Let G be any finite group. (a) Show that there is a surjection π1Sg G, for some g.(b) Show that G is a subgroup of Mod(Sh), for

23 Oct 2013: 4. Let (fn) be a sequence of real-valued continuous functions on a closed, bounded interval [a,b], and. ... f. Prove that there is some subinterval [a,b] of [0, 1] with a < b on which f is bounded.

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

1 Nov 2021: Show that M M is trivial.(b) Show that for all n we have TSn R = Rn1 over Sn. ... 4.† Let α be a nowhere-zero 1-form on a manifold X, and let β be a p-form.(a) Show that if β = α γ for some (p

29 Oct 2012: a sequence of points in [a, b] with xm x then fn(xn) f (x). ... f. Prove that there is some subinterval [a, b] of [0, 1] with a < b on which f is bounded.

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.

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

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.

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