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5 Dec 2019: compact. 5. Are the invertible operators dense in B(l2)? 6. Construct a self-adjoint operator T on a (non-zero) Hilbert space H such that T hasno eigenvalues. ... 7. Construct a compact operator T on a (non-zero) Hilbert space H such that T has

5 Feb 2019: Show that for any b C, there exists a sequence of pointszn D(a,r), with zn 6= a, such that zn a and f(zn) b, as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.

23 Jan 2019: But for such a fibre bundle we have. B. f(ch4kd(ξH) σ4kd) = 0,. ... Suppose that the fibration is in addition equipped with a map τ : E BOand a section s : B E, such that τ s : B BO is the constant map to thebasepoint.

25 Jan 2019: Can such a map be a Möbiustransformation? 10. Calculateγ z sin z dz when γ is the straight line joining 0 to i. ... functions of which theyare derivatives) on the domains indicated:. (a)1. z, (0 < |z| < ); (b) z.

27 Nov 2019: 12. Let S be a collection of subsets of N such that for every A, B S we have A B orB A. ... otherwords, for every a < b and every c there is an x with a < x < b such that f(x) = c.

16 Oct 2019: Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d.

16 Oct 2019: 7. Let 1 < p, q, r < be such that 1/p 1/q 1/r = 1. ... Must Y Z be dense in X? 9. Give an example of a vector space V and two inequivalent norms ‖ ‖A and ‖ ‖B onV such that the normed spaces (V,‖ ‖A)

6 Feb 2019: Show that for any b C, there exists a sequence of pointszn D(a,r), with zn 6= a, such that zn a and f(zn) b, as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.

14 Oct 2019: 1. Let E be a Lebesgue measurable subset of the real line with positive Lebesgue measure m(E).Show that for every > 0 there exists an open interval (a,b) such ... A Dirac mass at x X is themeasure δx on B such that δx(A) = 1 or 0 according as x A or x /

27 Nov 2019: Let a, b K such that thepolynomials f = xm a, g = xm b are irreducible. ... Show that G is solvable if and only if there is an i such that.

28 Feb 2019: 2. Let X and Y be N(0, 1) random variables such that cov(X,Y ) = 0. ... a random set C(Y ) such that Pβ,σ2 (b C(Y )) = 1 α for allβ,σ2.

12 Nov 2019: Show that φis isotopic to the identity. 8. Let α,β be a pair of simple closed curves on a surface S, such thati(α,β) = 1. ... Let α,β be astandard pair of simple closed curves on S such that i(α,β) = 1, andlet γ be the boundary curve.

22 Oct 2019: to the zero function but with 10fn(x) dx 6 0. Is it possible to find such a sequence with. ... Onthe other hand, show that if (fn) does not converge uniformly to f, then there is a convergentsequence xn x in [a, b] such that fn(xn) 6 f(x).

5 Nov 2019: A = {x M : (xn) in A such that xn x} ,. Define the interior of A (in M) to be the set. ... Deduce that if A and B are disjoint closed subsets of M, then there exist disjoint opensubsets U and V of M such that A U and B V.

17 May 2019: Show. there exist disjoint open sets U1 and U2 such that Ci Ui.8. ... continuous path. f : [0, 1] Xconnecting a and b such that f is a homeomorphism onto its image.

6 Oct 2019: map(B,M) = {families of (.) over B}. Then. H(M) = {characteristic classes of such families}. ... This “resolution” point of view applies to many further examples,such as Confn(Rk) or BDi (W).

25 Nov 2019: everywhere.(b) If f is a measurable function on X such that f = f T almost everywhere, show that there is. ... Xn). Assume that F = σ(X1,X2,. ). Show that for every > 0, and every A F,there is n 1 and B Fn such that P(A4B) <. 3. 13. (Markov shift)

28 Jan 2019: Can such a map be a Möbiustransformation? 10. Calculateγ z sin z dz when γ is the straight line joining 0 to i. ... functions of which theyare derivatives) on the domains indicated:. (a)1. z, (0 < |z| < ); (b) z.

29 Nov 2019: V such that A Uand B V. ... I) Mn such that g(A)2 = A for all A Dr(I).

23 Apr 2019: Show that f is integrable. What is. 10f(x)dx? 4. Give an example of a continuous function f : [0,) [0,) such that0f(x)dx. ... Show. that for every > 0 and every closed interval I [a,b] of positive length, there isa closed interval J I such that J has

27 Nov 2019: S aSbb | T, T bTaa | S | ǫ. (3) Give a CFG for each of the following CFL’s, and then transform each such CFG intoCNF (giving a justification for the transformation). ... b) Show that the language L := {anbncn | n 1} is not a CFL.

31 Oct 2019: 5. (a) Suppose that f Matn,n(F) is such that f(AB) = f(BA) for all A,B Matn,n(F) and f(I) = n.Show that f is the ... fn for V such that. ψ.  mi=1. xiei,. nj=1. yjfj.

1 Nov 2019: if any), and why? (b) By replacing A in the construction of B with a suitably chosen set, construct a setC N such that neither C nor N C are ... Construct two explicit codes m, m′ such that Tm is recursive, and Tm′ is not r.e.Hint: Use the results of

29 Nov 2019: Use ψ to build, foreach interval I = [a,b] and > 0 a C-function ψI, on R such that. ... 1[a,b] ψI, 1[a,b+]. Use this to construct for each compact set K Rd and each open set U K a C-function ψK,Uon Rd such that.

11 Oct 2019: b) The model is to be reparametrised in such a way that it can be specialised to (i) two parallellines for the two groups, (ii) two lines with the same intercept, ... Z21 Z2np,where the Zi are independent and Zi N(λi,σ2) for some λi such that.

4 Dec 2019: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 14. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

22 Feb 2019: 6. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elementsa and b in R is an element d in R such that d divides both ... ii) Show that may be identified with the set Fq {} in such a way that GL2(Fq)acts on as the group of

11 Oct 2019: b) The model is to be reparametrised in such a way that it can be specialised to (i) two parallellines for the two groups, (ii) two lines with the same intercept, ... Z21 Z2np,where the Zi are independent and Zi N(λi,σ2) for some λi such that.

11 Nov 2019: vector space isomorphic to V ;. (ii) any point p B has a neighbourhood U, such that there is a diffeomorphism. ... A local section of the principal bundle P is a smooth map s : U P defined on aneighbourhood U B and such that π s = idU.

2 Nov 2019: r(t) such that limt0 r(t)/tk = 0. differential geometry 7. Hence. ... Thefollowing are equivalent:. (a) there exists a nowhere vanishing smooth differential n-form on M;(b) there exists a family of charts in the differentiable structure on M such that

1 May 2019: By definition, there exists some Zariskiopen affine subscheme Ux = Spec B(x) X containing x such that x1,. ... skd.Fix k such that x Spec Afk. Then B(x)fk A 1(X) is freely generated bysk1,.

11 Mar 2019: Thenf : A B is a graded ring homomorphism if it is a ring homomorphism such thatf(Aλ) Bλ for all λ R>0. ... ie functions f : M N such that f(ab) = f(a)f(b) for alla,b M and f(eM ) = eN.

4 Apr 2019: βn be an integral basis forI. Let B Mn(Z) be the matrix such that βj =. nk=1 Bkjαk. We have seen that disc(I) =. disc(OL) det(B)2, so ... It therefore suffices to show that the map is surjective. Let N = [OL : Z[α]].By hypothesis, p does not divide N,

10 Jan 2019: Then there is a point y / E such. that B(y,r) E 6= whenever r > 0. ... ii) Since F is compact we can find an R such that B(0,R) F.

8 Nov 2019: iii) If A is a finite set of cardinality r (such as {a1,. ... Whenever G is a group and thereare homomorphisms f A G, g B G such that f i = g j, there existsa unique homomorphism φ Γ G such that f = φ

9 Aug 2019: m1R called the exte-. rior differential, defined in such a way that the following equality holds for all b,b1,. ... ai1. ). Let b R such that. aib (at1,a2,. ,ai1. ). We write aib = ξ1at1 ξ2a2 ξi1ai1. In particular we have aib (at11 ,a2,.

9 Aug 2019: Consider A R S B such that S C is included in theinterior of A in S. ... Indeed, the singular points of fu are the pointsz B such that f(z) = u, and they are degenerate if and only if z is a singular point of f(and

10 Jan 2019: Theorem 4.2. If f : [a,b] [a,b] is a continuous function, then we canfind a w [a,b] such that f(w) = w. ... Supposethat we can find a a0 < a1 < < an b such that, writing σ = ‖fP‖we have either.

27 Oct 2019: 4) (a b) (cd) = (a c) (bd),. and such that each has a two-sided unit (1 and 1 respectively). ... x0,x1) sα0 (x0)holds for all pairs (α0,α1) of such α.

2 Dec 2019: countable union C :“. Ť. n Bn of sets from B such that µ˚pCzEq ă ε.The existence part of Caratheodory’s theorem then follows from. ... At some point we considered bounded subsets of Rd: replacethis notion by the word “contained in Xn for some

25 Nov 2019: Choose then a and b such that. 1. a. def=. 1. ... 29. Proof of Lemma 3.1.1. x K , rx > 0 such that B(x,rx).