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21 May 2005: . 1 0 3 01 3 1 20 0 1 01 2 1 1. ... 1. 11. Consider the matrix A =. 1 0 20 1 10 1 0.

21 May 2005: . . 1 1 00 3 20 1 0. .  ,. . . 1 1 10 3 20 1 0. .  ,. . . 1 1 11 3 11 1 1. ... 10. Consider the matrix A =. . . 1 0 20 1 10 1 0.

21 May 2005: Z[ 3d] OK. 1. 3Z[ 3d]. 20. (Tripos 01) Let K = Q(α) be a number field, where α OK.

21 May 2005: 20. Prove Hadamard’s Inequality: if A is a real n n matrix with |aij| k, then.

8 Nov 2005: 2. Between 10 and 20 there are 4 primes; does it ever happen again that there are 4 primesbetween two consecutive multiples of 10 (apart from between 0 and 10)?

21 May 2005: 20. Prove Hadamard’s Inequality: if A is a real n n matrix with |aij| k, then.

21 May 2005: 20) You are given m apparently identical coins, one of which may be a forgery.

21 May 2005: Show that E is finite dimensional. 20. Let E be a normed space.

4 Nov 2005: Show that E is finite dimensional. 20. Let E be a normed space.

21 May 2005: 16. 17. 18. 19. 20. 212223. 24. 25. D. C. D.

22 Nov 2005: 20. Suppose that (X,d) is a complete metric space. Let U X be a proper open subset withcomplement A = X U.

21 May 2005: 4 ANALYSIS II EXAMPLES 4. 20. [Tripos IB 93301(b)] Let (X,d) be a metric space without isolated points (i.e.

12 Jul 2005: where X has b3 = 0 and Y is aconnected sum of 20 copies of S3 S3. ... Global Anal. Geom. 26 (2004) 209–229.[19] C.Z. Tan, Ph.D. thesis, University of Cambridge, in preparation.[20] C.T.C.

30 Jun 2005: Th. and Dynam. Syst. 20 (2000), 1619–1626. [4] M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. ... linearity 15 (2002) 281–314. 20 G. P. PATERNAIN. [6] G. Contreras, Action potential and weak KAM solutions, Calc.

26 Sep 2005: In the case of Riemannian manifolds, the above operators were denoted in [20, 22]. ... iju:ji = I II III. 20 N.S. DAIRBEKOV AND G.P. PATERNAIN.

21 May 2005: . . . 20. Let P2 = P2(x,y) be the space of polynomials in x,y of degree 2 in each variable.

8 Dec 2005: Corollary 20. Whenever N is finitely coloured, some colour class containssolutions to all PR equations. ... So Corollary 20 says that if N = D1 D2 Dk thensome Di is PR.

21 May 2005: 20. Show that the dual of the space P of real polynomials is isomorphic to the space RN of all sequences ofreal numbers, via the mapping which sends a linear form

10 Nov 2005: 20) Show that |pG(1)| = (1)|G|pG(1) is the number of acyclic orientations of G.(An orientation of G is an assignment of a direction to each edge.).

21 May 2005: 20. A space-filling curve (Exercise 14, Chapter 7 of Rudin’s book).

21 May 2005: 20) Let G have n r 1 vertices and tr1(n) 1 edges.(i) Prove that for every p in the range r p n, G has a subgraph of order

18 Oct 2005: 4 ANALYSIS II EXAMPLES 1. 20. A space-filling curve (Exercise 14, Chapter 7 of Rudin’s book).

27 Oct 2005: 20) (i) Construct a [7, 8, 4]-code from Hamming’s code.(ii) Prove that A(n, d) 2A(n 1, d).(iii) Prove that if d is even then

21 May 2005: Show that satisfies the triangle inequality. Is it a metric? 40) Rephrase your solution to Question 20 (on coin weighing) in terms of conditionalentropy.

4 Apr 2005: for any M DG and any V M(K). 20 KONSTANTIN ARDAKOV AND SIMON WADSLEY.

14 Mar 2005: 20. graphs G(X,Y ), G(Y,Z) and G(Z,X) with densities p, q and r and let G be the tripartite.

21 May 2005: 20. An n n magic square is a square matrix whose rows, columns and two diagonals all sum to the samequantity.

30 Jan 2005: 0, 3, 1, 4, 7, 10, 13, 16, 19, 57, 60, 20, 17, 14, 11, 8, 5, 2, 6, 9, 12, 15, 18, 21,24, 27, 30, 33, 36, 39, 42,

12 Apr 2005: ThenBB f. B(ω) is. 20. non-zero only if HC(ω) = 1 for every C K1.