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19 Jan 2022: Prove that f is continuous on (a,b). Must it be continuous at a andb too? ... 13. Let f : R R be a function which has the intermediate value property: Iff(a) < c < f(b), then f(x) = c for some x between a and b.

22 Nov 2022: 11. Let K be a field containing a primitive mth root of unity for some m > 1. ... Show that f and g have the same splittingfield if and only if b = cmar for some c K and r N with gcd(r,m) = 1.

21 Oct 2022: Using this, we complete the proof. Preliminaries. We start by fixing some notation. ... Thus, any simplexappearing in Br(Fn) is contained in some Vi. Since B.

8 Nov 2022: Show that H is normal in G if and only if H is a unionof some conjugacy classes of G. ... 1. |G|gG|Fix(g)|. Deduce that if G acts transitively and |X| > 1, then there is some g G with no xed point.How many distinct ways are there to

24 Nov 2022: 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.

13 Jan 2022: b) For some n 2, let X1,. ,Xniid Exp(θ). Find a minimal sufficient statistic T , and compute. ... 4. For some n 2 let X1,. ,Xniid Unif(θ, 2θ) for some θ > 0.

22 Jan 2022: Prove that f is continuous on (a,b). Must it be continuous at a andb too? ... 13. Let f : R R be a function which has the intermediate value property: Iff(a) < c < f(b), then f(x) = c for some x between a and b.

28 Feb 2022: some a,b 0. Hence prove that the class group of K is cyclic and find itsorder.

5 Oct 2022: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... 1.9. Let (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0.

12 Oct 2022: We say X is bounded if there is some M R such that for all z X we have ‖z‖ 6 M. ... Show that there. is some subinterval [a,b] of [0, 1] with a < b on which (fn) is uniformly bounded.