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6 Feb 2012: Suppose that f ((x y)/2) (f (x) f (y))/2 for all x, y [a, b].Prove that f is continuous on (a, b). ... Which of (1)–(4) must betrue? (1) If f is increasing then f ′(x) 0 for all x (a, b).(2) If f ′(x) 0 for all x (a, b) then

21 Feb 2012: 1. Find all the characters of S5 obtained by inducing irreducible representations of S4. ... a) Compute dim SnV and dim ΛnV for all n. (b) Let g G and λ1,. ,

8 Nov 2012: 1. Find all the characters of S5 obtained by inducing irreducible representations of S4. ... a) Compute dim SnV and dim ΛnV for all n. (b) Let g G and λ1,. ,

7 Feb 2012: Suppose that f ((x y)/2) (f (x) f (y))/2 for all x, y [a, b].Prove that f is continuous on (a, b). ... Which of (1)–(4) must betrue? (1) If f is increasing then f ′(x) 0 for all x (a, b).(2) If f ′(x) 0 for all x (a, b) then

23 Oct 2012: 5. Find all of the Möbius transformations that commute with Mk for a fixed k. ... Are these all inversions? 9. How many square roots of a Möbius transformation are there?

30 Oct 2012: g(b c) = g({g(b), g(c)}). for all nonempty sets b, c a. ... 8. By a selection function for a set a, we mean a function s : Pa Pa such that s(b) bfor all b a, both inclusions being strict unless b is

23 Oct 2012: 4. Is it true that for all positive integers a, b, c, d we have (a, b)(c, d) = (ac, bd)? ... 6. The Fibonacci numbers F1, F2, F3,. are defined by: F1 = F2 = 1, and Fn = Fn1 Fn2 for all n > 2 (so eg.

29 Feb 2012: 8. By a selection function for a set a, we mean a function s : Pa Pa such that s(b) bfor all b a, both inclusions being strict unless b is ... Given a subset b Pα withmore than one element, consider the least β belonging to some but not all members of b

21 Feb 2012: 1. Find all the characters of S5 obtained by inducing irreducible representations of S4. ... a) Compute dim SnV and dim ΛnV for all n. (b) Let g G and λ1,. ,

8 Nov 2012: Show that all functions f from Rwith metric d′ to R with metric d are continuous. ... denote the vector space of all (Riemann) integrable functions on [0, 1].

23 Jan 2012: By considering a suitable subgroup of thegroup G of all maps from Zp to itself of the form x 7 ax b, where a, b Zp with a 6= 0,show ... Without knowledge of what this group is, whyis it obvious that the group of all symmetries of the dodecahedron cannot

10 Oct 2012: If b = 0, there is only one possible δ and this recovers our old definition. ... 2g3. (If one of the. outgoing boundary conditions on the pair of pants is contractible inM then the map βg ,b is also a monomorphism in all degrees.).

26 Nov 2012: continuous function g: Bε(I) Mn such that g(A)2 = A for all A Bε(I). ... b) Suppose instead we assume only that for all x, y X at least one of the three.

9 Nov 2012: 1. Find all the characters of S5 obtained by inducing irreducible representations of S4. ... a) Compute dim SnV and dim ΛnV for all n. (b) Let g G and λ1,. ,

31 Jan 2012: 1. Write down all possible Hasse diagrams for a poset with four elements. ... ordered by inclusion.(iv) The set of all partial orderings of A, ordered by inclusion.

30 Oct 2012: 1. Write down all possible Hasse diagrams for a poset with four elements. ... ordered by inclusion.(iv) The set of all partial orderings of A, ordered by inclusion.

12 Nov 2012: There are enough that you probably don’t want to list all the possibilities.]. ... Begin by observing that B preserves ker(A λI)a, for all λ and a. ].

3 Dec 2012: 3. (i) Show that the function ψ(A,B) = tr(ABT ) is a symmetric positive definite bilinear form on the spaceMatn(R) of all n n real matrices. ... Show that S2 = I. 8. An endomorphism α of a finite dimensional inner product space V is positive definite

26 Feb 2012: Show that the sum ofthe residues of g over all its singularities is zero. ... 8. Let p(z) = z5 z. Find all z such that |z| = 1 and p(z) = 0.

28 Nov 2012: matrices ρ(g) are all block upper triangular;(iii) for every basis v1,. ... Wi = {v V | ρ(g)v = θi(g)v for all g G}.