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21 Nov 2012: 12. Let S be a collection of subsets of N such that for every A, B S we have A B orB A. ... otherwords, for every a < b and every c there is an x with a < x < b such that f (x) = c.

3 Sep 2012: X n, r, R) such that, for every s [r, R], every B 1, U > 0, and everyn N,. ... If one imposes anupper bound on U then adaptation even holds for every B 1.

26 Oct 2012: 2. Every Faculty Board shall include members in each of the following classes (a), (b), and (c):. ... 3. For every Faculty Board the number of members in each category of class (a) and the number of members in classes (b), (c), (d), (e), and (f) shall be

3 Aug 2012: Since this is true for every n, it follows that. Λa = {dimMGA(B f) a}T =n. ... Pk(f)}. For every i {1,. ,Pk(f)} define. Ni = #{j 6= i : |(B f)(ti) (B f)(tj)| < 21k} and G =Pk(f)i=1.

6 Feb 2012: Find {a : f is continuous at a}.2. Write down the definition of “f (x) as x ”. Prove that f (x) as x if, andonly if, f (xn) for every sequence ... Suppose that f : [0, 1] R is continuous, that f (0) = f (1) = 0, and that for every x (0, 1)there

9 Jan 2012: b) Let d < D. For every p (0, 1), there exists M 1 such that for Pp -a.e.ω D , it is the case that for every η d , there ... b) For each j {0, 1,. , d}, every j -cluster with respect to G has volume atmost Rd.

22 Nov 2012: Show that there are Hausdorff subsetsof R with every dimension between 0 and 1. ... Deduce that the trace of every element in the group generated by A and B is determined by the 3numbers tr(A), tr(B) and tr(AB).

7 Oct 2012: a) X and Y are independent,. (b) the conditional distribution of X given Y = y is independent of y. ... b) Which states are recurrent and which are transient? 9. Show that every transition matrix on a finite state-space has at least one closed

10 Jan 2012: 22j r. for every m and for j j0, j0 = j0(s, r, B, ε). ... In particular if Cn does not depend on B, then Ef |Cn| cannot be of order rn(r)uniformly over (r, B, [0, 1]) for every B > 0, unless B can

23 Oct 2012: x = 1. Deduce, or prove directly, that every element of Zp has a cube root. ... If a, b and c are distinct positive integers, with say a < b < c, must every block of cconsecutive positive integers contains three distinct numbers whose product is a