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27 Oct 2014: 0 3 1. . 3. Let A and B be nn matrices over a field F. ... Show that A can bediagonalised. 11. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices.

22 Sep 2014: It is characterised up to homotopyequivalence over BO(2n) by these properties. ... Secondly, let hAut(θ) denote the grouplike topological monoid of self-homotopy. equivalences of B over BO(2n), i.e.

27 Oct 2014: Concretely we have. Objects Over b B: x E(b′) with u : b′ b.Maps From y with v : c′ c to x with u : b′ b :. c′v. - c. y ... have. Objects Over b B: x E(b′) with u : b′ b.Maps From y with v : c′ c to x with u : b′ b :. f b′F. - c′v. -

4 Dec 2014: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

27 Oct 2014: 0 3 1. . 3. Let A and B be nn matrices over a field F. ... Show that A can bediagonalised. 11. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices.

15 Jan 2014: PART II REPRESENTATION THEORYSHEET 4. Unless otherwise stated, all vector spaces are finite-dimensional over C. ...  : a,b,x Fp. of matrices over the finite field Fp (p prime).

26 Nov 2014: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

15 Jan 2014: b) Find an example of a representation of some finite group over some field of charac-teristic p, which is not completely reducible. ... Determine all finite groups which have a faithful 2-dimensional representation over R.

18 Sep 2014: The amalgamated free product of A and B. over α and β is the direct limit of this diagram. ... It is also the HNN-extension of Z over the triv-ial group:.

18 Sep 2014: Then the amalgamatedfree product of G and H over φ is defined to be. ... of G over φ is defined to be. G?φ =〈. A,{t}.

10 Oct 2014: An orthogonal space of dimension N over k is a pair (V,〈, 〉), where:. ... The dimension of V is equal to the cardinality of B, so it suffices to show that B spans V over k.However, this is immediate from the relations (4.1).

18 Sep 2014: If b is of weight 12, use lemma 3.3 to subdivide B over the end-point of cnearest the midpoint of b. ... 1. Slide over B = b I all positive-weight bases c whose midpoint is movedaway from x as a result, apart from b and b.

19 Dec 2014: A filter F over κ is normal if for every {Xα F : α < κ}, the diagonal. ... ii) Prove Ulam’s theorem: there is no κ+ -saturated κ+ -complete ideal over κ+.

16 Jul 2014: Mark Gross Mirror symmetry and the SYZ conjecture. In this case, the SYZ fibration over B has a singular fibre over thesingular point of B, a pinched torus, and the total ... space of thefibration over B has two holomorphic disks, one as depicted:.

18 Jul 2014: by (det A)2. So it suffices to consider g = [I5,B] for B running over a set of. ... solubility over Knr is equivalent to solubility over K. 2. Remark 3.3.

27 May 2014: In particular C is a smooth curve of genus one defined over K. ... to reduce to the case where F1 and F2 are linearly independent over k.

19 Dec 2014: ZFC. (a) G is generic in P over M ;(b) GD 6= for every dense open set D P in M ;(c) GC 6= for every dense subset C P in ... The intended interpretationof the second-order variables ranges over arbitrary subsets of the domain.

15 Apr 2014: infinitely many 6-tuples of directly 7-congruent non-isogenouselliptic curves over Q. ... Since c6 is not identically zeroit follows that x, e, f are linearly independent over K(a,b,c), and likewise for.

18 Sep 2014: FillM (x) = supl(γ)x. AreaM (γ). where γ ranges over all suitable null homotopic loops. ... l(γ) = supa=t0<t1<.<tk=b. ki=1. d(γ(ti1),γ(ti)). where the supremum ranges over all finite partitions t0 < t1 <. < tk of [a,b].The path γ

13 Oct 2014: is a third (independent) small point with x-coordinate (w2 2)/(2w2 1).(7)For Family I, instead of factoring m2 mn n2, one could (say) loop over ... uses arithmetic over Z[1] to determine a matrix similar to Monsky’s.

16 Apr 2014: q B), A/p is geometricallyintegral over k (resp. B/q is geometrically integral over k). ... q B/(λ)), A/p isgeometrically integral over k (resp. B/q is geometrically integral over k).