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11 Feb 2014: It follows that for such q, gq λI is positive definite. ... The set of all such t is closed andcontained in [a,b], so it has a minimum value t0.

4 Dec 2014: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 14. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

6 Mar 2014: Show that for any Hecke character χ, thereexist continuous χv : Kv C (v ΣK) such that. ... πv7. )= 1. Show that there is a unique Hecke character χ of K such that. •

22 Sep 2014: homotopy equivalences f : B B such that. θ f = θ. ... Homology stability with respect to higher handles. A further immediateconsequence of Theorem B is an analogous homological stability theorem for gluingon to W a θ-cobordism K : W P such that

3 Nov 2014: b) Let C1c ([0, 1]) be the set of functions f C1([0, 1]) such that f(x) = 0 for x in some neighborhood ofthe end points 0 and 1. ... If Sis a subset of C([a,b]) such that every sequence in S has a subsequence converging uniformly to a functionin S (i.e.

10 Nov 2014: linearfunctional θ V such that θ(x) 6= θ(y).(b) Suppose that V is finite dimensional. ... If no such subspaceexists, show that there exists a basis 〈v,α(v),.

7 Feb 2014: That is, for everya < b and every t there should exist x (a,b) such that f(x) = t. ... 14. Let f : R R be a function that has the intermediate value property: that is, iff(a) < c < f(b) then there exists x (a,b) such that f(x) =

20 Oct 2014: On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a,b] such that fn(xn) 6 ... a) Must there exist a sequence (fn) of continuous functions on [0, 1] such that fn f uniformly on [0, 1]?(b)?

25 Nov 2014: n=1x3n? 14. Let S PN be such that if A, B S then A B or B A. ... Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A, B T?

10 Oct 2014: a) Let a G. Show that there is a positive integer n such that an = e, the identity element.(The least such positive n is the order of a.). ... b) Show that there exists a positive integer n such that an = e for all a G.(The least such positive n is the