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20 Nov 2015: 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.(.

11 Feb 2015: 7. For a Serre fibration p : E B with connected fibre F over b0, and simply connectedbase, consider the maps of fibrations. ... Fi //. Ep //. p. B. {b0} // B B. and show using the naturality of the Serre spectral sequence that the compositions.

12 Feb 2015: i) Show that if B is finite over A, and C is finite over B, then C is finite over A. ... ii) Show that if B is integral over A, and C is integral over B, then C is integralover A.

9 Nov 2015: 14. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices. ... Deduce that if two n n real matrices P and Q are similar whenregarded as matrices over C, then they are similar as matrices over R.

26 Oct 2015: 3. Let V be a 4-dimensional vector space over R, and let {ξ1,ξ2,ξ3,ξ4} be the basis of V dual to the basis{x1,x2,x3,x4} for ... 9. Let A and B be nn matrices over a field F.

23 Nov 2015: 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.(.

19 Jan 2015: 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).

10 Nov 2015: 14. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices. ... Deduce that if two n n real matrices P and Q are similar whenregarded as matrices over C, then they are similar as matrices over R.

26 Oct 2015: 3. Let V be a 4-dimensional vector space over R, and let {ξ1,ξ2,ξ3,ξ4} be the basis of V dual to the basis{x1,x2,x3,x4} for ... 9. Let A and B be nn matrices over a field F.

27 Feb 2015: Nj=1 ‖γ(tj) γ(tj1)‖ where the sup is taken over all. finite partitions 0 = t0 < t1 <. < tN = 1. (i) Give an example for which (γ) =. If γ is continuously ... b) = inf (γ), wherethe inf is taken over all continuous γ: [0, 1] E with γ(0) = a,

2 Jun 2015: Let Kn be the function field of X(n) over K. Let B K3 /(K3 )3 be thesubgroup generated by all rational functions on X(3) with zeros and poles only ... To find some interesting examples over Q(T), we seta = b = 27j/(4(j 1728)) to obtain a model for Z(9),

15 Sep 2015: The proportion of plane cubic curves over Q thateverywhere locally have a point. ... 12] T. van den Bogaart and B. Edixhoven, Algebraic stacks whose number of points over finite fieldsis a polynomial, Number Fields and Function Fields- Two Parallel Worlds

15 Sep 2015: b) ρDn () is equal to the probability that a random n-ary quadratic form over R, with respect to thedistribution D, is indefinite; and. ... 3.1 Preliminaries on n-ary quadratic forms over ZpFix a prime p.

18 Feb 2015: H = B is the multiplicative group of a quaternion division algebra B over F (sothat Y is compact). ... over R or C).• (Log-)syntomic cohomology Hisyn(,sQp(j)) (for varieties over a p-adic.

11 Sep 2015: 1. Introduction. Let E be an elliptic curve over a number field K. ... solving conics over K, and over quadratic extensions of K. 20 TOM FISHER.

7 Aug 2015: Thentheir product AB is another formal model in A, and AB is finitely generated as amodule over A and B. ... b) The associated graded ring gr U(Li)K with respect to the π-adic filtration isk[t] k gr U(L)K, which is flat over gr U(L)K.

20 May 2015: Then RX is a vector space (over R). (3) If [a,b] is a closed interval in R then C([a,b], R) := {f R[a,b] | f is continuous}is ... Let A be an nn matrix over F. The following are equivalent(i) there is a matrix B such that BA = In;.

22 Sep 2015: f(t) dt. (iii) If a c b, b. a. f(t) dt+. ... log(xa)b = b log xa = ba log x = ab log x = log xab,.

2 Dec 2015: linear. In thisway L(U,V ) is a vector space over F. Definition. ... Let A be an nn matrix over F. The following are equivalent(a) there is a matrix B such that BA = In;(b) there is a matrix C such that AC =

10 Dec 2015: There is, up to isomorphism, aunique outer form of GLn over F which is split by E. ... Let Sp2k denote the symplectic group over F in 2k = n1 variables over F ,defined by the form.

3 Apr 2015: group over OF ; its functor of points is GQ(R) = (OQ OF R). ... action become isomorphic over F̌, so the results of[BZ95] apply without modification.

10 Mar 2015: where the sum is over all sequences a1,a2,a3,. ,aR of length R. ... More-. over, p(a)l(a) <. p(a) (1 logD p(a)) = 1. p(a) logD p(a) = 1.