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19 Jan 2021: a) Show that there is a surjection π1Sg G, for some g.(b) Show that G is a subgroup of Mod(Sh), for some h. ... a) Prove that TαTβ(α) = β. (b) Prove the braid relation: TαTβTα = TβTαTβ.

6 Feb 2021: Prove that f is continuous on (a,b). Must it be continuous at a andb too? ... 13. Let f : R R be a function which has the intermediate value property: Iff(a) < c < f(b), then f(x) = c for some x between a and b.

4 Feb 2021: BBB = {x : (B B)(x B)}. (a) Let B P() and C P(). ... B C) (BB. B) (CC. C). (b) This part seems to have caused problems for some.

15 Oct 2021: V 0[|E|2 |B|2. ] V µTµ0[F ]. 1. 4V 0. [|E|2 |B|2. ... T(X,Y ) = TσµνXµY νeσ. for some Crfunctions Tσµν := eσ [T(eµ,eν)].

8 Nov 2021: Show that H is normal in G if and only if H is a unionof some conjugacy classes of G. ... 1. |G|gG|Fix(g)|. Deduce that if G acts transitively and |X| > 1, then there is some g G with no xed point.How many distinct ways are there to

16 Mar 2021: F i //. Ep //. p. B. {b0} // B B. ... for some monic polynomial p(x) H(B; Z)[x]. Homotopy Theory Example sheet 3.

15 Jan 2021: some a,b 0. Hence prove that the class group of K is cyclic and find itsorder.

15 Oct 2021: 0. for some ai. 2 R. Show that there is a unique f : R! ... Show that:f(x) = g(x) p. k1(x). for some polynomial pk1 of order k 1.

26 Apr 2021: I hope you were not expecting to do it by induction on a or b!),However, one needs to take some thought. ... But of course GCD(a b, b) = GCD(a, b) so we get.

10 Nov 2021: 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),

15 Oct 2021: for some b > 0, and invoke. the contraction mapping principle]. ...  CbT(1 T) ||w v||X. for some constant C depending on b but not T.

8 Jan 2021: After allT might prove φ(a) for every a in Γ and yet prove φ(b) for some constant bnot in Γ. ... We need some definitions. DEFINITION 4. A theory T is Γn-complete iff whenever φ(x1,.

15 Oct 2021: i) If x S, then there exists some B β with x B.ii) If B1, B2 β, then for every x B1 B2 there exists B β with:. ... x B B B1 B2. b) Conversely, suppose that one is given a set S and a collection β of subsets of Ssatisfying i), ii) above.

6 Feb 2021: Prove that f is continuous on (a,b). Must it be continuous at a andb too? ... 13. Let f : R R be a function which has the intermediate value property: Iff(a) < c < f(b), then f(x) = c for some x between a and b.

20 Jan 2021: 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,.

15 Oct 2021: Show that ifeb|k|. f̂(k) 2 L2(R) for some b > 0 then f is real analytic at each point in R. ... B to mean that there exists a universal constant C such that A  CB.

15 Oct 2021: 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.

15 Oct 2021: for some a < b. ... Lu = u in I, u(a) = u(b) = 0, (?). for some 2 R. Show that the Wronskian:. W(x) = p(x)u0(x)ũ(x) u(x)ũ0(x).

7 Oct 2021: ii) M R n N x S |fn(x)| 6 M. (b) In this case assume that S is a metric space. ... Prove that there is some subinterval [a,b] of [0, 1] with a < b on which f isbounded.

7 Oct 2021: ii) M R n N x S |fn(x)| 6 M. (b) In this case assume that S is a metric space. ... Prove that there is some subinterval [a,b] of [0, 1] with a < b on which f isbounded.

11 Feb 2021: Motivation. Let F : (Schemes)opp (Sets) be a functor represented by some(noetherian) scheme M. ... A ĈO, and representable if F (A) = Hom(R,A) for some R ĈO.

15 Oct 2021: b. a) Show that the sequences (f(xj. )). 1j=0. and (f(yj. )). ... sup. x2(a,b). f. 0i. (x).  M. for some M independent of i.

30 Oct 2021: Example. Let a < b and R > 0 be real numbers, let z = (z0,z1,. ... Assume that for some K > 0 we have. |ψ(t,x) ψ(t,y)|6 K‖xy‖ for all t [a,b] and all x,y BR(z).

15 Jan 2021: some a,b 0. Hence prove that the class group of K is cyclic and find itsorder.

19 Jan 2021: a) Show that there is a surjection π1Sg G, for some g.(b) Show that G is a subgroup of Mod(Sh), for some h. ... a) Prove that TαTβ(α) = β. (b) Prove the braid relation: TαTβTα = TβTαTβ.

7 Oct 2021: algebra. 1.2. 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 ... 1.9. Let (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0.

19 Jan 2021: for every x X there exists some B B such that x B; and. ... for all B1,B2 B and for all x B1 B2 there exists some B3 B suchthat x B3 B1 B2.

1 Nov 2021: Show that M M is trivial.(b) Show that for all n we have TSn R = Rn1 over Sn. ... 4.† Let α be a nowhere-zero 1-form on a manifold X, and let β be a p-form.(a) Show that if β = α γ for some (p

26 Oct 2021: Some more questions. 13. (a) A topological space is said to be second countable if there is a countable family Bof open sets such that every open set is a union ... of some members of B.

15 Oct 2021: g2Zngv. Show that if 2 D(Rn) with supp K for some compact K Rn then. ... u[] =X. g2Agv[],. for some finite set A Zn which depends only on K.

14 Oct 2021: Show that ifeb|k|f̂(k) L2(R) for some b > 0 then f is real analytic at each point in R. ... u1. 1We write A B to mean that there exists a universal constant C such that A CB.

29 Oct 2021: finitely generated A-module, then Supp(B A M) =. (f)1(Supp(M)). [Note: You may find useful the identity on tensor products (M A B) B C = M A C ... a) Show that x is a zero divisor in A if and only if x p for some p D(A).(b) Let S be a multiplicatively

28 Oct 2021: Some more questions. 13. (a) A topological space is said to be second countable if there is a countable family Bof open sets such that every open set is a union ... of some members of B.

15 Oct 2021: Bj,γ,δ. zγxδ, j = 1,. ,n1,. andc(z,x′) =. γ,δ. cγ,δzγxδ,. where these power series converge if |z| |x′| < s for some small s > 0. ... cγ,δzγxδ,. 4. with these power series converging for |x| |x′| < s for some (possibly smaller) s.

8 Aug 2021: Replace‘Similarly, we can can take a = sin φ, b = cos φ for some real φ.’ by. ... Similarly, we can can take c = sin φ, d = cos φ for some real φ.’.

26 Feb 2021: 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.

8 Oct 2021: 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 I is the pp identitymatrix. ... a) Show that the James-Stein estimator θ̃(1) dominates all estimators θ̃(c),c 6= 1.(b) Let θ̂ be the

18 Mar 2021: 1 Unramified up to twist: τ = (ψ ψ, 0) for some characterψ : IK C. ... 4 Irreducible: τ = (r0, 0) where r0 is the restriction of some irreducibleWeil-Deligne representation.

8 Oct 2021: Suppose that for some θ0 in the interior of Θ and every ε > 0 small enough, wehave S(θ0ε) < 0 < S(θ0ε), and also 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

18 Jan 2021: Proof. Let f : J0(N) A be the quotient map. There exists a mapg : A J0(N) such that fg = [n] for some n > 0. ... and for a given f, it’s possible that some Vf,p are of the form (trivial) (cyclotomic) while others aren’t.

8 Oct 2021: The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... for some λ,µ,ν R (which should turn out in this example to be integers).Does there exist a linear transformation which reduces the pair of

3 Mar 2021: Prove that for anyn 2 there is some pair (A,B) and a constellation on (A,B) made of precisely n circles. ... 2). (c) Deduce that there is some δ > 0 such that, in any hyperbolic triangle, the union of the δ-neighbourhoods of twoof the sides completely

16 Mar 2021: Z. Rn. eix eiy2. ||2sd 6 Cn, |x y|2. b) Show that if s = n2 k for some k 2 Z>0, 0 < < 1, then. ... d) () A simple crystal consists of identical atoms placed at the lattice points of some.

15 Oct 2021: some function which maps an interval (a,b) with a < b to thereal line R. ... Thenf(x) = C for some C R. Proof. Pick c,d with a < c < d < b.

15 Oct 2021: supx(a,b). f ′i(x) Mfor some M independent of i. Show that F is uniformly equicontinuous. ... supx[a,b]. |fi(x)| K. for some K independent of i. Then F has a subsequence which converges uniformly.

15 Oct 2021: 1.1 Convolutions 3. ii) Since zj 0, there exists some ρ > 0 such that zj Bρ(0) for all j. ... 1.1 Convolutions 5. Lemma 4. Suppose f L1loc.(Rn) and g Ck0 (Rn) for some k 0.

15 Oct 2021: interval t (s,s), and that moreover the solution depends continuously on s.Later we’ll state more precisely some sufficient conditions on a,b,c such that we can ... and that for any y > 1supR|h′| a shock has formed for some x.

15 Oct 2021: With some reasonable furtherconditions (see the Appendix) we can in fact pick out τ uniquely. ... Lemma 1.12. Suppose f L1loc.(Rn) and g Ck0 (Rn) for some k 0.

15 Oct 2021: Supposethat the xi satisfy ||xi|| < r for all i and some r > 0. ... Similarly, (dj)j=0 is monotone decreasingand bounded below so dj d for some d.

15 Oct 2021: v) We need some topology on D′() such that the above operations are continuous. ... By Lemma 2.9 there exists some compactK and N N, C > 0 such that:.