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19 Jan 2022: 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

29 Nov 2022: 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.

22 Jan 2022: 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

26 Oct 2022: Show that if V (Ti) V (Tj) 6= for all i, j [k] thenV (T1) V (Tk) 6=. ... 12) Let G = (V, E) be a graph. Show that there is a partition V = AB so all the vertices in the graphsG[A] and G[B] are of even degree.

11 Oct 2022: The least such n iscalled the order of g.). (b) Show that there exists a positive integer n such that gn = e for all g G. ... θ(g) = e for all g D2n. Find all homomorphisms D2n Cnwhen n is even.

28 Oct 2022: 5. Show that T B(H) is normal iff ‖Tv‖ = ‖Tv‖ for all v H. ... 13. Let U B(H) unitary. Show that for all x H, the sequence n1n1.

4 Nov 2022: 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.

22 Jan 2022: Set. A = {ω : ω An for infinitely many n}, B = {ω : ω An for all sufficiently large n}. ... Calculate the probability thatm given people will all be on the committee (a) directly, (b) using the inclusion-exclusion formula.Deduce that (. nmr m. )=.

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

13 Jan 2022: Find thelikelihood ratio test of size α. (b) The scientist repeats the test in part (a) for each characteristic j = 1,. ... a random set C(Y ) such that Pβ,σ2 (b C(Y )) = 1 α for all β,σ2.

13 Oct 2022: 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;.

26 Apr 2022: a) Show that the unique minimizer is x = 0. (b) Find constants 0 < m < M and L > 0 such that m f ′′(x) M and |f ′′′(x)| L for all x. ... Which of the basic solutions are feasible?Are all the basic solutions non-degenerate?

19 Nov 2022: 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?

4 Oct 2022: Suppose that F(,σ2) isincreasing 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

27 Nov 2022: 4. Find all bases for which 39 is an Euler pseudoprime. ... More generally, showthat if p and 2p1 are both prime numbers, then N = p(2p1) is a pseudoprimefor precisely half of all bases.

10 Jan 2022: Use this to compute the rank of RM(d,r). Show that all but two codewords in RM(d, 1)have the same weight. ... 9 Prove the following results.(i) If K is a field containing F2, then (a b)2 = a2 b2 for all a,b K.(ii) If P F2[X] and K

15 Oct 2022: b) Let X Bin(n,θ), where θ Θ = [0, 1]. Find all prior distributions π on Θ for whichthe maximum likelihood estimator is a Bayes rule for estimating θ in ... a) Show that the James-Stein estimator θ̃(1) dominates all estimators θ̃(c),c 6= 1.(b)

26 Apr 2022: All the other edges of the network are of the form (ai, bj) for some i, j = 1,. , ... A is connected to all ai and B is connected to all ci with capacity 1edges.

3 Feb 2022: a) Find P(Ai Aj) for all 1 i 6= j n. ... 14. Can you construct non-negative integer valued random variables X and (Xn)n1 with E[X],E[Xn]all finite, and such that.

24 Oct 2022: 5. (a) Suppose that f Matn,n(F) is such that f(AB) = f(BA) for all A,B Matn,n(F) and f(I) = n.Show that f is the ... f(A) = trA for all A Matn,n(F).(b) Now let V be a non-zero finite dimensional real vector space.