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27 Nov 2019: Show that f and g have the same splitting field if andonly if b = cmar for some c K and r N with gcd(r,m) = 1. ... Deduce that every finite abelian group is the Galois group of some Galois extensionK/Q.

6 Oct 2019: Why cohomology? A moduli space M is anything which classifies some kind of families:. ... It always depends on the specifics of the groups Gn, though thereare some general principles.

15 Nov 2019: a) All words w {0, 1} consisting of either the string 01 repeated some number oftimes (possibly none), or the string 010 repeated some number of times (possiblynone). ... b) All words w {a, b, c} consisting of some number of a’s (possibly none),

16 Oct 2019: 6. Let (An : n N) be a sequence of events in some probability space (, F, P). ... P(Xn = 0) 12π. h. (b) Show further that, for all x R,.

28 Feb 2019: b) For some n > 2, let X1,. ,Xniid Exp(θ). Find a minimal sufficient statistic T, and. ... 4. For some n 2 let X1,. ,Xniid U[θ, 2θ], for some θ > 0.

12 Nov 2019: 1. 5. Let G be any finite group. (a) Show that there is a surjection π1Sg G, for some g.(b) Show that G is a subgroup of Mod(Sh), for

22 Oct 2019: 8. Let fn, n N, and f be real-valued continuous functions on the closed bounded interval[a, b]. ... Prove that there is some subinterval [a, b] of [0, 1] with a < b on which f isbounded.

5 Nov 2019: Assume that for some K > 0 we have ‖ϕ(t, x) ϕ(t, y)‖ 6 K‖xy‖ forall t [a, b] and all x, y BR(y0). ... Assume further that. sup{‖ϕ(t, x)‖ : t [a, b], x BR(y0).

10 Jan 2019: δ is abelian.(c) Assume that δ(g) = 1 for some g G. ... irreducible. 3 Find an example of a representation of some finite group over some field of charac-teristic p, which is not completely reducible.

11 Feb 2019: How many ringsare there of order 4? [You should find that all but one of these rings is isomorphic toF2[X]/(X2 aX b) for some a,b F2, where F2 = ... Prove that everyfinite Boolean ring is isomorphic to a power-set ring P(S) for some set S.

23 Jan 2019: As described above, by a theorem of Borel[Bor74] the maps (3.1) are isomorphisms, so ch4kd(ξH) is equal to some poly-nomial p(σ1,σ2,. , ... Hi(B; Q) 0.Now s(τc)|E = 0 so (τc)|E = u (E)x for some x H|c|d(B; Q).

14 Oct 2019: 2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b R}. ... 10. Let (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0.

13 Nov 2019: What happensif Θ = [a,b] for some 0 < a < b <? 9. Let X be multivariate normal N(θ,I) where θ Θ = Rp,p 3, and where I is the ppidentity ... a) Show that the James-Stein estimator θ̃(1) dominates all estimators θ̃(c),c 6= 1.b) Let θ̂ be the

5 Feb 2019: Prove that there exists some c [0, 1]with f(c) = c. ... Show that thereis some c (a,b) with f ′(c) = k.

11 Nov 2019: Some more terminology: B is called the base and E the total space of this vector bundle. ... The ψβα’s can be taken from some vector bundle E over B, then P will be‘constructed from E’.

31 Oct 2019: Suppose for some θ0 in the interior of Θ and every ε > 0 small enough we haveS(θ0 ε) < 0 < S(θ0 ε), and that Sn has exactly one zero θ̂n ... f(,φ) : φ = Φ(θ) for some θ Θ} equals Φ(θ̂MLE).b) Now consider a mapping Φ that is not

2 Nov 2019: Theorem 1.27. Every (smooth) vector field on S2m vanishes at some point. ... A contradiction. Some page references. to Warner, and Guillemin–Pollack, and Gallot–Hulin–Lafontaine.N.B.

26 Feb 2019: 71). Find an element of OK with norm 2a 3b 5 for some. ... a,b 0. Hence prove that the class group of K is cyclic and find its order.

4 Dec 2019: The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... amk50@cam.ac.uk - 1 - November 2019. for some λ,µ,ν R (which should turn out in this example to be integers).Does there exist a linear

1 May 2019: By replacing L by πmL for some positive integer m, we mayassume that Z is πnL-accessible for all n > 0 (see [4, Proposition 7.6]).Let B be an affine ... By definition, there exists some Zariskiopen affine subscheme Ux = Spec B(x) X containing x such

8 Nov 2019: Since every z S1 is in some such U, it follows that pis a covering map. ... We make some remarks on what Definition 3.1 means for exis-tence and uniqueness.

22 Feb 2019: Euclidean domain. Could there be some other Euclidean function φ making Z[3]. ... iii) Show that if R is a PID, the gcd of elements a and b exists and can be written asra sb for some r,s R.

4 Apr 2019: Thus we can write b = v/2 for some v Z and hence α = u. ... factor |det(B)| for some B GLn(Z); and this factor equals 1.Now let d Z be a negative square-free integer, and let L = Q(.

11 Mar 2019: Lecture 4. Aside on group commutators. We recall some general group theoretic facts. ... Since ω(G{1}) is a discrete subset of R (Lemma 3.13(b))there is some t < s maximal such that we can find x = gλ X with xGt =

10 Jan 2019: b) [Needs some topological knowledge.] First project S2 onto its equatorial plane by(x,y,z) 7 ζ = xiy. ... Show that χ(g) = 0 for some g G. [Hint: partition G into equivalence classes bycalling two elements of G equivalent if they generate the same

12 Apr 2019: x. In some sense, X(A) corresponds to the tangent bundle of X. ... 1 =ri=1. bki di,. for some d1,. ,dr D. Thus, b =ri=1 b.

9 Aug 2019: for some set Λ, where the fibres are considered with respect to f|B. ... for some set Λ. Proof. Apply Lemma 4.6 and Lemma 4.9.

9 Aug 2019: Hence,in some sense, algebraic de Rham cohomology goes back to the source of modern cohomologytheories. ... the case where R =k [x1,. ,xn] /I for some ideal I of k [x1,.

10 Jan 2019: Now P(z) = 0 if and only if z = ajfor some 1 j m. ... Thus there must exist an M suchthat EM has non-empty interior, so EM (a,b) for some non-empty interval(a,b).

17 Apr 2019: Factorisation of polynomials in Fp[T] is effective, so Proposition 2.4.7 gives analgorithm to compute pOK for some values of p. ... A binary quadratic form is a map q : Z2 Z of theform q(x,y) = ax2 bxy cy2 for some (a,b,c) Z3.

1 Jan 2019: We have Supp Ψn = a 1n Supp Ψ B for n larger than or equalto some n0 N. ... By Taylor’s formula, αϕ(x) =. Oa(|xa|N1α. )if |α| N. By Leibniz’s formula, we obtain |α (Ψnϕ) (x)| C′n|α|N1 for some.

12 Apr 2019: U opensatisfying some conditions. OX is called a sheaf. Example 2.2.6. ... In particular, if f and g coincide on some nonempty open subset of X, then f = g.

6 Jan 2019: if itis isomorphic to jJ A for some set J.Proposition 1.6.2. ... For every y Ni1, the coefficient of mi1 in y is some multiple b mi1 of ai1, soy bx Ni.

20 Nov 2019: Can you do it? 1.1. SOME PUZZLES TO GET YOU STARTED 11. ... down. Some heretical teachers try, but in vain. They attempt to bribe the.

2 Dec 2019: belongs to some open box B. But then BzC is open, hencea countable union of boxes Bn contained in B. ... At some point we considered bounded subsets of Rd: replacethis notion by the word “contained in Xn for some n”, where Xn is any family ofsets in