Search

5 Oct 2007: b) Use (a) to show that every finitely presented group is the fundamental groupof some space. ... Freeness of a group action is as definedin lectures; some people call this property “free and properly discontinuous”.).

25 Oct 2007: 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. ... 15. Let V be a real vector space with a countably infinite basis; that is to say, there is some sequence e1, e2,e3

17 Oct 2007: 3. Let a,b,c be three positive integers. Are the following true or false? ... b) Now let F = Zp be the field of integers mod p, for some prime p, and suppose d dividesp 1.

21 Nov 2007: Questions 7–10 are more challenging than others and some parts certainly go beyond limitsof the examination. ... neous cubic polynomial in the generalized Weierstrass normal form XZ24Y 3AX2Y BX3,for some complex constants A, B, A3 27B2 6= 0.

17 Oct 2007: Hence x1 = x2 = = xk1, so theresult is true for n = k 1.(b) Clearly, some positive integers are interesting: for example, 6 is the smallest perfectnumber, 1729 is the smallest number ... Suppose some positive integers were not interesting: then, by the

9 Apr 2007: To this end, for F some class of B-measurable real-valued functions,define the mapping. ... holds for some real k > 1/2. Assume further that. cn := supf F.

3 Jun 2007: a) Show that Z(T) is a subgroup of Möb.(b) Find which groups (up to isomorphism)can arise as Z(T) for some Möbius transformation T. ... Show thatevery Möbius transformation is equal to exp A for some matrix A.

3 Jun 2007: a) {0}. (b) Zω1 = {nω1 : n Z} for some ω1 C {0}. ... Suppose that we did,then ω = xω1 yω2 for some x, y R.

18 Jan 2007: em infu2[a;b] I(u):This is typical. The general conclusions are:1. The frequency of occurence of rare events depends inan exponential manner on some parameters of theproblem. ... Any surplus bandwith that is availablebecause some queues are empty is

21 Nov 2007: 1. The square matrices A and B over the field F are congruent if B = P T AP for some invertible matrixP over F. ... 8. Let S be a real symmetric matrix with Sk = I for some k 1.