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23 Feb 2008: fundamental group of some space. (5) Let L be the “infinite ladder of circles” given by the subset ofR2 consisting of the union of the circle of radius 1/2 aroundeach ... in C which maps to 0 in C′. (1) Show that x maps to 0 in Dand hence that x is

27 Aug 2008: L3;! A B! A (!R)! AFig. 1. (Multiplicative) Intuitionistic Linear Logicpackets, using some predened set of inference rules. ... Some thought that this represented a mismatch betweenthe semantics and syntax of linear logic.

17 Oct 2008: Show that if fn f uniformly and (xm) isa sequence of points in [a, b] with xm x then fn(xn) f (x). ... Prove that there is some subinterval [a, b] of [0, 1] with a < b on which f is bounded.

4 Feb 2008: 6) Let f : B B be a continuous map from the open disc B = {(x, y) R2 : x2 y2 < 1} into it-self. ... 10) Suppose f : R R has the intermediate value property that if a, b R, f (a) < c < f (b),then f (y) = c for some y between a and b.

3 Nov 2008: 3. Let a,b,c be three positive integers. Are the following true or false? ... b) Now let F = Zp be the field of integers mod p, for some prime p, and suppose d dividesp 1.

21 Nov 2008: f(z) =a(z z0) bc(z z0) d. ,. for some a,b,c,d,z0 C. ... N. m,n=0. am,nf(z)mg(z)n 0, for some am,n C. Hence show that any two meromorphic functions f,g on a complextorus C/Λ are

17 Nov 2008: δ is abelian.(c) Assume that δ(g) = 1 for some g G. ... irreducible. 3 Given any prime p. Find an example of a representation of some finite group oversome field of characteristic p, which is not completely reducible.

14 Oct 2008: Hence x1 = x2 = = xk1, so theresult is true for n = k 1.(b) Clearly, some positive integers are interesting: for example, 6 is the smallest perfectnumber, 1729 is the smallest number ... Suppose some positive integers were not interesting: then, by the

28 Nov 2008: 1. The square matrices A and B over the field F are congruent if B = P T AP for some invertible matrixP over F. ... λn.(i) Show that if Sk = I for some k 1 then S2 = I.(ii) Find the eigenvalues of the endomorphism of Matn(R) given by X 7 SXT S.

27 Aug 2008: L3;! A B! A (!R)! AFig. 1. (Multiplicative) Intuitionistic Linear Logicpackets, using some predened set of inference rules. ... Some thought that this represented a mismatch betweenthe semantics and syntax of linear logic.