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20 Jan 2007: φ(s,v) = α(s) r(n(s) cos v b(s) sin v)where n = n(s) and b = b(s) are the normal and binormal vectors of α. ... 9. Consider a surface of revolution parametrized by φ : (0, 2π) (a,b) R3, whereφ(u,v) = (f(v) cos u,f(v) sin u,g(v)).

21 Nov 2007: Show that f = gR f gLand hence, or otherwise, show that f is differentiable at B. ... Show that f is differentiable at (a, b). (b) Suppose instead that D1f exists and is bounded on some open ball around (a, b), and that for fixed x.

3 Jun 2007: 2. Let T : z 7 (az b)/(cz d) be a Möbius transformation.24. ... s s2 1 and s. s2 1. where. s =|a|2 |b|2 |c|2 |d|2.

26 Nov 2007: b, c (b,. ). a&b, c (a&b, {b}) (a&b) d, c ((a&b) d, {a&b}). ... 0 A , A. A. 0 B , B. B ( A) ( B),.

18 Sep 2007: If x, t [a, b]. f (t) =n. j=0. f (j)(x). ... b. a. f (x) dx b. a. g(x) dx. Lemma 12.6.

8 Nov 2007: B+. B. I+. H. I. II. At this point, let us make the following definitions:. ... 23. The geometry of these regionsis completely different, however, in particular, the sets B = B are timelike,and this allows for the repeating structure to develop.

19 May 2007: numbers of thesepairs belonging to B be g(B) and b(B) respectively. ... Then there must exist for this r a set B such thatg(B) γ1b(B) > αrm2, which implies that |B| > αr/2m and b(B) 6 γ|B|2.

5 Oct 2007: b) Use (a) to show that every finitely presented group is the fundamental groupof some space. ... Suppose that the maps A A′, B B′, D D′, and E E′ are isomorphisms.Show that the middle map γ : C C′ must be an isomorphism, as follows.

3 Jun 2007: It corresponds to the Möbius transformation C C ; z 7 (az b)/(cz d). ... Exercises. Let T : z 7 (az b)/(cz d) be a Möbius transformation.7.

25 Oct 2007: b) Let R([0, 1]. )denote the vector space of all integrable functions on [0, 1]. ... An interval is a set taking one of the forms (, ), (, b), (, b], [a, ), (a, ), [a, b], [a, b), (a, b]or (a, b) for some a, b R with a b.