Search

23 Mar 2007: Fix some subset B of [n]. How large a family A of subsets of [n] can we find such that the intersection of any two sets in A contains a cyclic ... all supersets of B.

18 Nov 2007: GROUPS EXAMPLES 4. G.P. Paternain Michaelmas 2007. The questions on this sheet are not all equally difficult and the harder ones are marked with ’s. ... 9. Let X be the set of all 2 2 real matrices with trace zero.

20 Oct 2007: Show, directly from the definition of holomorphic maps, that if f is holomorphic onR {p} then f is in fact holomorphic on all of R.(ii) Suppose that each of A = { ... algebraically related’: there is a polynomial Q in two variables, so that Q(f(z),g(z))

6 Oct 2007: Whatcan you say if n is even? Find all the homomorphisms from Cn to Cm. ... If we think of the planeas C you may assume that all isometries have the form z 7 az b or z 7 az̄ b, where a and b arecomplex numbers

3 Nov 2007: Let G be the set of all 3 3 matrices of the form 1 x y0 1 z. ... If we think of the plane as C you may assumethat all isometries have the form z 7 az b or z 7 az̄ b, where a and b are complex

19 Nov 2007: GROUPS EXAMPLES 4. G.P. Paternain Michaelmas 2007. The questions on this sheet are not all equally difficult and the harder ones are marked with ’s. ... 9. Let X be the set of all 2 2 real matrices with trace zero.

10 Oct 2007: Whatcan you say if n is even? Find all the homomorphisms from Cn to Cm. ... If we think of the plane. as C you may assume that all isometries have the form z 7 az b or z 7 az̄ b, where a and b arecomplex

6 Nov 2007: 7. Let G be the set of all 3 3 matrices of the form. ... If we think of the plane as C you may assumethat all isometries have the form z 7 az b or z 7 az̄ b, where a and b are complex

25 Oct 2007: b) Let R([0, 1]. )denote the vector space of all integrable functions on [0, 1]. ... 10. (a) Show that the space is complete. Show also that c0 = {x : xn 0}, the vector subspace of consisting of all sequences converging to 0, is complete.

17 Oct 2007: 6. Solve (i.e., find all solutions of) the following congruences:(a) 7x 77 (mod 40).(b) 12y 30 (mod 54).(c) 3z 2 (mod 17) and 4z 3 (mod 19). ... 13. Recall the Fibonacci numbers Fn from question 5.(a) Prove that (Fn,Fn1) = 1 for all n.(b) Let m 3.

21 Nov 2007: 5. Let Mn denote the space of nn real matrices with the operator norm ‖‖. Show that ‖AB‖ ‖A‖‖B‖for all A, B Mn. 6. Define f : Mn Mn by f ... Deduce the result that if f : U Rm isdifferentiable on U with Df|x = 0 for all x U then f is

5 Oct 2007: are isomorphisms for all i, then the maps HiK HiK′ are isomorphisms for all i. ... 9) (a) List all the closed surfaces, orientable or not, and compute their Euler char-acteristics.

20 Oct 2007: Show, directly from the definition of holomorphic maps, that if f is holomorphic onR {p} then f is in fact holomorphic on all of R.(ii) Suppose that each of A = { ... algebraically related’: there is a polynomial Q in two variables, so that Q(f(z),g(z))

5 Nov 2007: 5. Let Q(2) denote the set of all real numbers of the form a b. ... described is indeed an equivalence relation.(b) Define addition and multiplication of rational functions, and verify that they satisfythe usual laws of arithmetic, so that the set of all

5 Oct 2007: b) Let X2 be the 2-skeleton of X, the subcomplex containing all vertices, 1-simplices, and 2-simplices. ... Suppose that the maps A A′, B B′, D D′, and E E′ are isomorphisms.Show that the middle map γ : C C′ must be an isomorphism, as follows.

8 Nov 2007: Show that all functions f from Rwith metric d′ to R with metric d are continuous. ... Show that f has a fixed point. (b) Suppose instead we assume only that for all x, y X we have d(f (x), f (y)) kd(x, y), ord(f

2 Mar 2007: Prove that the Gaus-sian curvatures at the points (a,0,0),(0,b,0),(0,0,c) are all equal if and only if S is asphere. ... at the origin in C (possibly all of C), where h(r) > 0 for all r < δ.

17 Oct 2007: But such a numberwould clearly be very interesting indeed. Therefore, all positive integers are interesting. ... n 1}.]. 10. A binary operation , defined on the positive integers, satisfies the conditions(a) 1 n = n 1 for all n,(b) m 1 = (m 1) 2 for all

8 May 2007: 2. (a) Define a subset of the integers Z to be open either if it is empty or if for some k Zthe set S contains all integers k. ... b) For a function f : X Y , show f is continuous if and only if for all A X,f (cl(A)) cl(f (A)).

26 Oct 2007: Determine whether there are optimal binary codes with either (a) all but one codeword ofthe same length, or (b) each codeword a different length. ... Show that in the case where all of the odds are equal this maximum andthe entropy H(p1,.