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14 Oct 2016: 0a) Write down a definition of multiplication for natural numbers. Prove that multipli-cation is distributive over addition: that is, for all natural numbers a, b and c we havea(b ... Hint: first define a1 for all natural numbers a. Then, assuming that

20 Jul 2016: q}}Q. and the cohomology in all other degrees vanishes. The Schur–Weyl decomposition HqQ =. λ Sλ Sλ(HQ) in terms of Schur. functors Sλ() shows that for a partition λ ... Inparticular, as all prime numbers are greater than 1 we find that H(Aut(F); H

21 Nov 2016: 3. (i) Show that the function ψ(A,B) = tr(ABT ) is a symmetric positive definite bilinear form on the spaceMatn(R) of all nn real matrices. ... sn forms a basis for Pn.(iii) For all 1 k n, sk spans the orthogonal complement of Pk1 in Pk.(iv) sk is an

26 Oct 2016: Then find all pairs of integers x and y with. 152x 90y = 2. ... 4. Find all solutions of the congruences:. (i) 7w 77 (40);(ii) 12x 30 (54);(iii) 3y 2 (17) and 4y 3 (19) (simultaneously);(iv) z 2 (3), z 3 (4),

23 Nov 2016: 9. Show that the collection of all finite subsets of N is countable. ... 11. Let S be a collection of subsets of N such that for all A, B S we have A B or B A.

14 Oct 2016: 0a) Write down a definition of multiplication for natural numbers. Prove that multipli-cation is distributive over addition: that is, for all natural numbers a, b and c we havea(b ... Hint: first define a1 for all natural numbers a. Then, assuming that

27 Oct 2016: 11. Show that A = {(xn) 1 | |xn| 1/n2 for all n} is a sequentially compact subset of(1,‖‖1), but that B = {(xn) 1 | |xn| 1/n for all n} ... Hint: show that Φ(I) contains all points of the form (a/2n,b/2n), wherea,b Z, 0 a,b 2n.).

28 Oct 2016: Then find all pairs of integers x and y with. 152x 90y = 2. ... 4. Find all solutions of the congruences:. (i) 7w 77 (40);(ii) 12x 30 (54);(iii) 3y 2 (17) and 4y 3 (19) (simultaneously);(iv) z 2 (3), z 3 (4),

7 Oct 2016: 3. For a, b, c, d G, use the associativity axiom to show that ((a b) c) d = a (b (c d)).You will find similarly that all possible ways to ... 6. (a) Show that the rotations of a regular triangle form a subgroup of all symmetries of thetriangle.

12 Dec 2016: Uv,Uw) = (v,w) for all v,w H. Prove the mean ergodic theorem of von Neumann: for every v H,. ... 8. For T B(H ) normal, i.e., TT = TT , show that ‖Tv‖ = ‖Tv‖ for all v H, and conclude that.

23 Nov 2016: 9. Show that the collection of all finite subsets of N is countable. ... 11. Let S be a collection of subsets of N such that for all A, B S we have A B or B A.

7 Nov 2016: b) All words w {a,b,c} consisting of some number of a’s (possibly none), followed bysome number of b’s (at least one), followed by some number of ... c) All words w {0, 1} which contain a 1 somewhere in the last 4 positions.

10 Nov 2016: 3. Show that the function f : Rn R given by f(v) = ‖v‖2 is differentiable at all nonzerov Rn. ... Show that f is continuousat (0, 0) and that it has directional derivatives in all directions there.

27 Oct 2016: 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) =

14 Oct 2016: b) Show that there exists a positive integer n such that an = e for all a G.(The least such positive n is the exponent of G.). ... a) (12)(1234)(12);(b) (123)(235)(345)(45). 13. What is the largest possible order of an element in S5?

10 Oct 2016: 0, 1] |. 10 f(x) dx = 0}? 5. Let 0 be the set of real sequences (xn) such that all but finitely many xn are 0. ... a) Show that |ϕ(s) ϕ(t)| |s t| for all s,t R.(b) Define f(x) =. n=0. (34. )nφ(4nx). Prove that f is well-defined and continuous.

11 Nov 2016: x1). Using similar calculations as forthe system of ODEs (Question 5) on all entries of bj, j = 0,. , ... g(z1,. ,zm,x1,. ,x1) =Cr. r (x1. x1) (z1 zm). is a majorant of all these entries.

8 Jan 2016: b) Find all 1-dimensional representations of G.(c) Let ψ : Fp C be a non-trivial 1-dimensional representation of the cyclic group. ... d) Prove that the collection of representations constructed in (b) and (c) gives a com-plete list of all irreducible

21 Nov 2016: 10. Let f : Rn Rn be a C1 map. Suppose that there is some constant µ < 1 such that‖Df|x I‖op µ for all x Rn. ... Showthat ‖x y‖ (1 µ)1‖f(x) f(y)‖ for all x,y Rn. Deduce that f is injective andthat f(Rn) is a closed subset of Rn.

18 Feb 2016: Part II Algebraic Geometry. Example Sheet III, 2016(For all questions, assume k is algebraically closed. ... 22 x. 33 k[x0,. ,x3]. Find the dimension of the tangent space at all the.