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6 Feb 2021: Which of(1)–(4) must be true? (1) If f is increasing then f ′(x) 0 for all x (a,b).(2) If f ′(x) 0 for all x (a,b) ... then f is increasing.(3) If f is strictly increasing then f ′(x) > 0 for all x (a,b).(4) If f ′(x) > 0 for all x (a,b) then f

9 Nov 2021: a) Compute dim SnV and dim ΛnV for all n. (b) Let g G and λ1,. , ... even. 7. Find all the characters of S5 obtained by inducing irreducible representations of S4.

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;.

6 Feb 2021: Which of(1)–(4) must be true? (1) If f is increasing then f ′(x) 0 for all x (a,b).(2) If f ′(x) 0 for all x (a,b) ... then f is increasing.(3) If f is strictly increasing then f ′(x) > 0 for all x (a,b).(4) If f ′(x) > 0 for all x (a,b) then f

15 Oct 2021: b. a. f(t)dt = 0, 0  f(x), for all x 2 [a, b]. ... R iscontinuous on [a, b], differentiable on (a, b) and satisfies F 0(x) = f(x) for all x 2 (a, b).

15 Oct 2021: Define τ by:. U τ for all x U, there exists B β such that x B and B U. ... xn xm B, for all n,m N.

20 Jan 2021: Set. A = {ω : ω An infinitely often}, B = {ω : ω An for all sufficiently large n}. ... P(Xn = 0) 12π. h. (b) Show further that, for all x R,.

15 Oct 2021: c) Show that if 0  f(x) for all x 2 [a, b] then:. ... 0 Z. b. a. f(x)dx. [Hint: Use part a)]. d) Show that if f(x)  g(x) for all x 2 [a, b] then:Z.

20 Jan 2021: a) Show that, for all p (0, ) and all x (0, ),. P(|X| x) E(|X|p)xp. (b) Show that, for all β 0,P(X x) E(eβX)eβx. ... 3. Let X be a Poisson random variable of parameter λ (0, ).(a) By optimizing the estimate of Question 2(b) over β, show that, for

15 Mar 2021: 1. |B|vB. d(v) > (α ε)|A|. Show there is B′ B with |B′| > ε|B| so that d(v) > α|A|, for all v B′. ... Is this bound sharp? (11) Prove that the matrix J (all of whose entries are 1) is a polynomial in the adjacency matrix of a.