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23 Mar 2009: d : R{0} N. such that for all a R and b 6= 0, b R we can write. ... N by. N(a b. 2. )= a2 2b2. (=. a b22) (a, b Z).

5 Nov 2009: What does this have to dowith RPn? 4. Suppose Gn (n > 0) is a finitely generated abelian group, and that all but finitely manyGn are 0. ... Construct a finite cell complex X with Hn(X) ' Gn for all n > 0.

26 Oct 2009: Is it Lipschitz equivalent. to the uniform norm? (b) Let R[0, 1] denote the vector space of all integrable functions on [0, 1]. ... Usingthe fact that all norms on a finite-dimensional space are Lipschitz equivalent, deduce that α iscontinuous.

13 Oct 2009: with xn yn zn = wn. State explicitly what is meant by ‘sufficiently large’; that is, give some number N(depending on n) such that the result holds for all primes p ... Suppose that (i) |A| = 3 for each A A; and (ii) |A B| = 1for all distinct A, B A.

23 Feb 2009: 1. Which of the following propositions are tautologies, that is, are true for all valuations?(a) (A B) (B A). ... Show thatthe set of propositions that all believe is consistent and deductively closed.

13 Dec 2009: A map f : X Y is called cellular iff (X(n)) Y(n) for all n. ... d) Using the results of the previous part, show that 1 a = a for all a H(X).

15 Oct 2009: k=1 uk(x) to F (x) is uniform on [a, b]. Proof. ... It would suffice to assume e.g. that |fn(x)| 1for all n and x [a, b].

21 Jan 2009: Show thatf (x) = 0 for all x. 8. Let f : [a, b] R be bounded. ... iv) If f′(x) > 0 for all x (a, b) then f is strictly increasing.

23 Apr 2009: For R+, describe what happens as 0. 3. For the genus two Heegaard diagrams shown in Figure 1a and b, find all generatorsof ĈF. ... 11. Work out all the groups and maps in the exact triangle for n-surgery on the right-handedtrefoil.

5 Mar 2009: p. Show that the sumof residues of g at all its singularities is zero. ... 2π0. log |f (ρeiθ|dθ. Now suppose that w1,. , wn are all the zeros of f in the open disc B = {|z| < ρ}, repeatedaccording to multiplicities, but f never vanishes on