Search

28 May 2007: T (x) = B(x) v for all x R2. When T is an isometry with det B = 1, show that T is either a reflection or a glide reflection.2. ... Define the Schottky group corresponding to K pairs of discs all of which are disjoint.

16 Nov 2007: 6. Show that the set of all finite subsets of N is countable. ... 0 for somea, b, c Q, not all zero.

21 Nov 2007: 3. Show that the function ψ(A,B) = tr(ABT ) is a symmetric positive-definite bilinear form on the spaceMatn(R) of all n n real matrices.Show that the map ... sn forms a basis for Pn.(iii) For all 1 k n, sk spans the orthogonal complement of Pk1 in Pk.(iv

3 Jun 2007: One dimensional projective complex space P(C2) is the set of all one-dimensional subspaces of C2. ... Find all the meromorphic 1-forms (differentials) on C. 3. 3.3 Möbius Transformations.

5 Feb 2007: 8) An integral domain R is called a euclidian domain if there is a map N : R{0} Z>0 such thatfor all a,b R, b 6= 0, there are ... Show that 3, 7, 125. and 1 25 are all irreducible elements in OK.

25 Oct 2007: . 4. Let λ F. Evaluate the determinant of the n n matrix A with each diagonal entry equal to λ and all. ... other entries 1. [Note that the sum of all columns of A has all entries equal.].

11 Oct 2007: Which of the following sets of functions form a vector subspace of RR?(a) The set C of continuous functions.(b) The set {f C : |f (t)| 1 for all t ... i Ui. Given an example where Ui Uj = {0} for all i 6= j, yet U1.

3 Jun 2007: log |zn| is still convergent when we sum over all of the zeros. ... 0}. Also, since all the partial products of the Blaschkeproduct have modulus 1 on D, we have ||g|| 6 ||f||.

3 Jun 2007: a) F is bounded : there exists c with ||f|| < c for all f F. ... f (z) f (zo)| < ε for all f F and all z U.

3 Jun 2007: There is an open neighbourhoodU of ζ with c1 < b(z) < c2 for all z U. ... The difference u g iscontinuous and subharmonic on all of R, including zo.