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7 Dec 2022: Use (ii) to show that there exists b K such that det(σiσjg1(b)) 6=0. ... Deduce that if y = g1(b) then {σ(y) | σ G} is a K-basis for L.Such a basis {σ(y)} is said to be a normal basis for

19 Nov 2022: 13. Let S be a collection of subsets of N such that for every A,B S we have A B or B A.Can S be uncountable? ... Is there an uncountable collection T of subsets of N such that A B isfinite for all distinct A,B T?

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. ... 10. Let H and G be groups and let X G such that 〈X〉 = G.

28 Oct 2022: 2. Given (fi)iI an arbitrary collection of continuous functions [0, 1] R such thatsupiI |fi(x)| < at each x [0, 1], show that there is an interval [a,b] [0, ... 1] witha < b such that supx[a,b] supiI |fi(x)| <. 3. Let X be a closed subspace of 1. Assume

11 Oct 2022: Find a, b, c Q such thaty = a bx cx2. ... 15. Let K and L be subfields of a field M such that M/K is finite.

13 Oct 2022: 1. Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d.

22 Nov 2022: Let a, b K such thatthe polynomials f = Xm a, g = Xm b are irreducible. ... Use a suitable Lagrange resolvent to finda Q(ζ3) such that K = Q(ζ3, 3.

28 Oct 2022: 1. Let T B(H) with |||T ||| < 1. Show that there is R B(H) such that R2 = 1 T. ... 11. Let T B(B) compact and λ 6 (σp(T) {0}). Show that T λ is bounded below.

24 Oct 2022: Ln := {w ; there are x, y W such that |y| = n 1 and w = x1y)},. ... Show that:. (a) There is a nondeterministic automaton N with n 1 states such that L(N) = Ln and(b) if D is a deterministic automaton with fewer than 2n states,

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 ... 0. Is detA > 0? 14. Let A,B in Mn(R) such that X KerA{0} with BX ImA.

1 Feb 2022: b) We say that a measurable f : Rn! C is in weak-Lp(Rn), written f 2 Lp,w(Rn) ifthere exists a constant C such that. ... Deduce. N[. i=1. Bi. 6 3n. kX. j=1. Bij. 3. b) () Suppose {Bj : j 2 J} is an arbitrary collection of balls in Rn such that each

3 Feb 2022: 1. Let A,B F be two events such that P(B) = 0 or 1. ... Ie, at least one day d such that at least k people were born on day d.).

6 Feb 2022: Regarding T as a map from C into C by identifying R2with C in the usual way, show that there exist unique complex numbers A, B such that for everyz C, ... that there need notexist holomorphic f : U C such that u = Re (f) on all of U.

1 Feb 2022: there exists 2 C such that:. u[] =. Z. R(x)dx, for all 2 D(R). () ... Rm(x, y)dy. b) Show that there exists a sequence of smooth functions fn 2 C1c (Rn) such that Tfn!

14 Nov 2022: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 14. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

24 Oct 2022: Suppose thatα: V W is a linear map such that α(Y ) Z. ... Let f be an endomorphism of E. Show theexistence and uniqueness of (F,G) vector subspaces of E such that:(a) F G = E;(b) F,G are stable by

13 Jan 2022: Prove directly from the definition thatAX has an m-variate normal distribution AX Nm(Aµ,AΣAT ). 2. Let X and Y be N(0, 1) random variables such that ... a random set C(Y ) such that Pβ,σ2 (b C(Y )) = 1 α for all β,σ2.

25 Feb 2022: Can you justify that f is unique? [Note: such a function f is called the (discrete) harmonic extension of (av). ]. ... Explain why E[|T |] = ,and find constants C,α such that P(|T | = m) Cmα.

20 Jan 2022: a) Construct a continuous surjection p : T2 T2 from the torus to itself such that for every x T2, p1(x)consists of exactly two points, and such that p is ... Such a map is called acovering map.]. (b) Show that T2 also admits a covering map p̂ : T2 K to

26 Apr 2022: Show that ϕ is convex. 5. Suppose f : X R is twice-differentiable and such that 2f(x) is positive definite for allx X. ... 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)|