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12 Aug 2008: 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.

12 Aug 2008: 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.

28 Nov 2008: 1. The square matrices A and B over the field F are congruent if B = P T AP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

26 Feb 2008: a) When the culture has been going for just over n minutes, what is the probabilitythat no white cells have yet appeared? ... b) What is the probability that the entire culture eventually dies out?

27 Feb 2008: Moreover, the restriction of a finitely generated A-module is finitelygenerated over B. ... action explained in the proof of Proposition 2.3. The latter module is just a directsum of dim V copies of ResAB(M) and is hence finitely generated over B.

20 Jan 2008: b) Determine which of the following polynomials are irreducible over Q: X3X 1, X3X 2,X3 X 3. ... 2,. 7) over Q. You may assume that G = {1,α,β,αβ}where.

12 Aug 2008: 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

28 Feb 2008: BT08. Part II Representation Theory Sheet 4. Unless otherwise stated, all vector spaces are finite dimensional over C. ... Q.8 (a) Let G be a compact group. Show that there is a continuous group homomorphismρ : G O(n) if and only if G has an

17 Nov 2008: PART II REPRESENTATION THEORYSHEET 4. Unless otherwise stated, all vector spaces are finite-dimensional over C. ...  : a,b,x Fp. of 3 3 upper unitriangular matrices over the finite field Fp of p elements (p prime).Show that G has p

4 Feb 2008: over Q(ζ). Determine the possible Galois groups of L over Q(ζ). ... ii) Find the Galois group of f(x) = x4 4x 2 over Q and over Q(i).

31 Oct 2008: 5. Let A and B be n n matrices over a field F. ... 13. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices.

21 Aug 2008: Example 1.13 Given an extensive cartesian closed category D with small products and sums, anda proper factorization system (E, M) over D s.t. ... Let P Cat(B) be the full and internally complete category of -closed partial equivalencerelations over N. =

21 Aug 2008: 26]. We let ;. i. 4. range over -redexes. Let (x:A)B. ... P. m. (U) over a right-absorptive c-pca U. Let I be a set, for any A; B 2 Prf(U).

21 Aug 2008: 26]. We let ;. i. 4. range over -redexes. Let (x:A)B. ... P. m. (U) over a right-absorptive c-pca U. Let I be a set, for any A; B 2 Prf(U).

16 Jul 2008: 1. Introduction. We work over an algebraically closed field k of arbitrary characteristic. ... It sufficesto note that there are no 4 3 matrices B over k for which.

16 Oct 2008: 15. Let Matm,n(F ) be the space of m by n matrices over a field F. ... Let V be a vector space of dimensionn over Fp. How many vectors are there in V?

21 Aug 2008: pcf-types,ranged over by A;Ai;B, etc., are dened by the followinggrammar in bnf:A ::= j o j A ) A:Every pcf-type A has a unique representation of ... We presuppose an innite collection of names whichare ranged over by symbols such as x;x0;xi;f, a;b; u;v,

6 Mar 2008: Assume that B is integral over A; thatis, every element x of B is is a zero of a monic polynomial in A[X].(i) Suppose that A, B are domains ... Show that the set C ofelements x B that are integral over A is a subring of B.

22 Aug 2008: A on rete des ription is as follows: On obje ts, W(A) is the set of nite multisets over A. ... Take : A - B and : B - C in Gam and let t be a sequen eof moves over A, B, and C with tjA;B 2 , tjB;C 2 so that

22 Aug 2008: A on rete des ription is as follows: On obje ts, W(A) is the set of nite multisets over A. ... Take : A - B and : B - C in Gam and let t be a sequen eof moves over A, B, and C with tjA;B 2 , tjB;C 2 so that

20 Feb 2008: The Grothendieck group K0(B) of B is the abelian group with generators [M]where M runs over all the objects of B and relations [M] = [M′] [M′′] for anyshort exact ... n1N are finite dimensional over k for all n. Hence. the associated graded module.

7 Aug 2008: For any monad T , an algebraic operation of arity (I, O) over T is a. ... Our task is then to find anew model over the monad TE T = T (E ) to incorporate exceptions.

13 Aug 2008: To handle a basic form of quantification we need toindex it over some category to represent the types over which we quantify. ... Just as we did for Dial, we can consider Dill as a category fibred over T.Objects are still of the form U.

29 Sep 2008: mined by the precedence (i,j): that is, i takes precedence over j. ... we write [X,A] for the cotensor of a T -algebra A over thecategory X; and we shall write [A,B] for the pseudo-closed structure on T -Algwhich we introduce

13 Nov 2008: 13. Let V be a vector space of finite dimension over a field F. ... 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 trAB.Show that, for

22 Aug 2008: Dialogue games over omputational arenas.5. Inno ent strategies.6. Context lemma for CA. ... Formally we dene a relation between losed terms andvalues indu tively over the following rules.

2 Feb 2008: We factor A and B over K as. A = (x1 α1x2 β1x3 γ1x4)(x1 α3x2 β3x3 γ3x4)B = (x1 α2x2 β2x3 γ2x4)(x1 α4x2 β4x3 γ4x4). ... Let S = (λ1A µ1B,λ2A µ2B) with λi,µi L′. Then κ L, whereas if Aand B are not dened over L then Gal(L′/L)

22 Aug 2008: Dialogue games over omputational arenas.5. Inno ent strategies.6. Context lemma for CA. ... Formally we dene a relation between losed terms andvalues indu tively over the following rules.

21 Aug 2008: Example 1.13 Given an extensive cartesian closed category D with small products and sums, anda proper factorization system (E, M) over D s.t. ... Let P Cat(B) be the full and internally complete category of -closed partial equivalencerelations over N. =