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19 Dec 2003: k}. Show that equality holds for some. fixed r if and only if B(p,r) is isometric to the ball of radius r in the constant curvature space formof curvature ... limr. V (p,r)wnrn. = 1. for some p M, must be isometric to the Euclidean space.

24 Nov 2003: Deducethat there exists a pair of irrational numbers a, b such that ab is rational. ... c there is some x with a < x < b and f(x) = c.

7 Oct 2003: 1 Here and in the following, when we require the existence of some structure in a. ... Pf =df ΣA Πf B. for some arrow f : B A of C.

9 Jun 2003: b. 3. for some pair of positive integers a,b}. (vi) X. ... n. , but for a general proof, we must specify some method that always works.

17 Jul 2003: Some examples of 5 and 7 descent for elliptic curves over Q 175. ... Some examples of 5 and 7 descent for elliptic curves over Q 177.

3 Sep 2003: What does that mean? Well, if f(x)=u and f(y)=v and z=axby for some a,b> 0 with ab=1 then f(z) ought to be close ... there must be some c between a and b such that f(c)=w.

11 Mar 2003: where B is a nn-symmetric matrix. Assume that for some t = t1 the two planesE0 and Γλ0,λ1,λ(t1) have a non trivial intersection. ... b) Write f = Π e. Suppose that e(z) E(f(z)) 6= {0} for some z Σ.

14 May 2003: Just so that we don't have to keep writing the phrase considered as a function on B', let us invent some notation. ... i. ,w. i. ), for some unknown bilinear function f, determined the value of f(v,w).