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10 May 2002: A formal definition could be something like this. A multivalued function from A to B is a subset H of AxB such that for each a in A there is at ... least one b in B such that (a,b) is in H.

7 Aug 2002: A formal statement of the result to be proved. Let X and Y be metric spaces and let f be a function from X to Y such that f. ... in U (for the reason I have just given) and such that [f(y) is in U] implies [d(f(x),f(y)) < a].

9 Sep 2002: In addition, if B. 21B23 = 0, then given a regular coadjoint orbit Oc1,c2 withc1 or c2 positive there exists 0 > 0 such that for all 0 < < 0 the topological entropyof ... Lemma 2.4 implies that B = 0. Thus:. Proposition 2.5. If there is a regular

21 Jun 2002: 3. We were inspired to find such an algorithm by the following observation onplane cubics. ... Thenthere is a rational covariant F : 2V W Y such that f = Fu.

6 Mar 2002: b in B (that is, some non-zero b (mod p)) such that the set of all bx. ... If, on the other hand, we cannot find such a b, then A=B and the theorem is proved.

13 Nov 2002: else prove that no such relation exists. 1. 11. Define a binary operation on Z2 by the formula (a,b) (c,d) = (ac,ad bc). ... a1,b1) (a2,b2). (ak,bk). 12. Let A1,A2,. be sets such that for every positive integer n we have A1.

9 Sep 2002: Let k be a positive number such that |K(P)| k for all 2-planes P. ... there exists a positive integer i0 such that forall i i0, πi(M) Q = 0.

29 Jan 2002: Once again let us consider a particular example, such as the equation x. ... What we then discovered was that if we take a new number b, given as a root of a polynomial with its coefficients in Q(a) (in such a way that

25 Apr 2002: 3. dy. 2. eyf in y and finally (the most important part) cleverly choose u and v in such a way that d. ... d. 2. equal to 3e, or it may be that, although such a choice exists, the dependence of u and v on a, b and c is so complicated that we

25 Nov 2002: We also know that CxB is a subset of AxB, and that the set-theoretic difference AxB-CxB consists of all ordered pairs (m,x) such that x is in B. ... me that if m> n then there is a unique positive integer t such that nt=m.

10 Sep 2002: Then either [a] is minimal or there is a non-empty subset B consisting of all [b] such that b is isomorphic to an initial segment I(t) of a. ... Pick [b] such that t is minimized (which can be done as a is a well-ordered set of reals).

12 Sep 2002: constant b < 4 such that R(k) bk for all sufficientlylarge k? ... The pair (A,B) is. said to be -uniform if there exists a real number α > 0 such that, whenever A′ A andB′ B are sets of cardinality at least |A|

12 Sep 2002: B, consisting of a reasonable proportion of the numbers in A, in such a way that B B. ... larger than A. Will we always be able to pick out a reasonably large set B in such a way.