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22 Dec 2023: B.A. in Mathematics, 2008- 2011. • First class all three years.

8 Nov 2023: 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.

3 Feb 2023: 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)

5 Mar 2023: 2 IB GEOMETRY EXAMPLES 3. (a) Show that Dσ|(u,0) is injective for all u, and hence Dσ|(u,v) is injective for all (u,v) with |v|sufficiently small. ... 8. Define an abstract Riemannian metric on the disc B(0, 1) R2 by du2dv21u2v2.

19 Oct 2023: Find all invariant distributions of the transition matrix in Exercise 9 of Example Sheet 1. ... i < b, where 0 < λi, µi < 1 for all i, and λi µi = 1 for 1 i < b.Show that this process is time-reversible in equilibrium.

3 Nov 2023: 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.

16 Nov 2023: 7. Find an injection from R2 to R. Is there an injection from the set of all real sequences to R? ... Is there an uncountable collection T of subsets of N such that A B isfinite for all distinct A,B T?

25 Nov 2023: 13. Show that T B(H) is positive if and only if 〈Tx,x〉> 0 for all x H. ... L(ab) = L(a)b, R(ab) = aR(b) and R(a)b = aL(b). Let M(A) be the set of all double centralizers for A.

12 Oct 2023: Write down its negation insymbolic form. (Here m, n, a, b should be understood as ranging over all natural numbers.). ... i) 1n = n 1 for all n;(ii) m1 = (m1)2 for all m > 1;.

11 Oct 2023: 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 ... Show that. (a) C is uncountable and has Lebesgue measure 0,(b) for all x [0, 1], the limit F(x) =

10 Oct 2023: Suppose thatF(,σ2) is increasing for all σ2 and F(m, ) is decreasing for all m. ... ψ(m,σ) = E[U(m σZ)].Show that(a) if U is increasing, then ψ(,σ) is increasing for all σ(b) if U is concave, then ψ concave, that

6 Mar 2023: 2 IB GEOMETRY EXAMPLES 3. (a) Show that Dσ|(u,0) is injective for all u, and hence Dσ|(u,v) is injective for all (u,v) with |v|sufficiently small. ... 8. Define an abstract Riemannian metric on the disc B(0, 1) R2 by du2dv21u2v2.

1 Mar 2023: Show that f is a polynomial, of degree k, if and only if there is aconstant M for which |f(z)| < M(1 |z|)k for all z. ... Show that if thereexists k such that |f(z)| |z|k for all z with |z| sufficiently large, then f is a rational function (i.e.

14 Nov 2023: Note that such a linear map is bounded.) Show that forT B(X,Y ) the following are all equivalent to T being weakly compact.(i) T(X) Y ;. (ii) T : Y ... x X : |x(xn)|6 1 for all n N} U. Deduce the following consequences.

8 Feb 2023: φ(x) =. [A(x) B(x)C(x) D(x). ]with A(y) = Ir, B(y) = C(y) = D(y) = 0. ... Hence show that A(x) is invertible for all x insome open neighbourhood U of y.

9 Oct 2023: 4. Find all bases for which 39 is an Euler pseudoprime. ... More generally, showthat if p and 2p 1 are both prime numbers, then N = p(2p 1) is a pseudoprimefor precisely half of all bases.

30 Oct 2023: . 3. Let λ F. Evaluate the determinant of the n n matrix A with each diagonal entry equal to λ and all. ... Now additionally assume ai,i 0, for all i; does this imply det(A) > 0?

19 Jan 2023: Operators, i.e., bounded linear maps, between Banach spaces; operatornorm; the space B(X,Y ) of all operators from X to Y. •

7 Feb 2023: 𝑓 𝑗 (𝑥)𝑑𝑥 𝐸. 𝑓 (𝑥)𝑑𝑥. for all bounded measurable sets 𝐸 R𝑛 of positive measure. ... i) Show that 𝑥𝑖 𝑥 if and only if (𝑦,𝑥𝑖) (𝑦,𝑥) for all 𝑦 𝐻.

6 Mar 2023: Show that the sum of the residuesof f at all its poles equals zero. ... 6. Let p(z) = z5 z. Find all z such that |z| = 1 and Im p(z) = 0.