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9 Dec 2005: A matrix A is said to be partition regular (PR) over a subset S of the positive integers if whenever S is finitely coloured, there exists a vector x, with all ... Those matrices which are partition regular over the positive integers were completely

25 Nov 2005: Leta K. Show that Xp a is irreducible over K if and only if it is irreducible over K′. ... Suppose further that L = K(x, y), where y is algebraic over K.

21 May 2005: 2. Let A and B be n n matrices over a field F. ... 3. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices.

10 Oct 2005: In whichof these cases is U a vector space over R?(a) x1 > 0.(b) either x1 = 0 or x2 = 0.(c) x1 x2 = 0.(d) x1 x2 = 1.(e) x1 ... 8. For what values of a and b does the system of simultaneous linear equations.

21 May 2005: 1. Which of the following symmetric matrices are congruent to the identity matrix (a) over Q, (b) over Rand (c) over C? (. ... Try to get away with the minimum calculation.]. 2. Find the rank and signature of the following quadratic forms over R.

21 May 2005: Prove that a finite-dimensional division algebra over R isisomorphic to one of R, C or H. ... 1.12 Question. Find examples of an irreducible representations of a group over the field of real numbers whose algebraof endomorphisms is (a) R (b) C.

21 May 2005: first part. 1. The square matrices A and B over the field F are congruent if B = P tAP for some invertible matrixP over F. ... Which, if any, over Q? [Try to get away with the minimum calculation.]. (.

21 May 2005: Show that:. (a) X is path connected. (b) X Y is homotopy equivalent to Y. ... 6. Show that the following are equivalent for a path connected space X:(i) X is simply connected.(ii) Every continuous map f: S1 X can be extended over B2.(iii)

18 May 2005: More-over, each point of the action spectrum arises as a singularity of Υ for an appropriatechoice of ϕ. ... PATERNAIN. Integrating Pestov’s identity over SM against the Liouville measure dµ, we get.

21 May 2005: Find the dual basis for R3. 4. Let V be a 4-dimensional vector space over R, and let {ξ1,ξ2,ξ3,ξ4} be the basis of V dual to the ... Verify that (DS) = SD and (SD) = DS. 16. For A an n m and B an m n matrix over the field F, let τA(B) denote