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12 Nov 2005: Let a be an integer with 1 a < p, and let σ Aut(L) be theunique automorphism such that σ(X) = aX. ... Use (ii) to show that there exists b K such that det(σiσj g1(b)) 6= 0.Deduce that if y = g1(b) then {σ(y) | σ G} is a

23 Nov 2005: 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.

21 May 2005: Tripos IB 96401(b), modified] (i) Let (X,d) be a nonempty complete metric space, and let f :X X be a continuous map such that, for any two points ... xn} is a finite set of points inX such that the balls B(xi,) cover X, consider the 2n 1 points of HX

21 May 2005: Why would such a thesis be unlikely to be consideredworth a Ph.D? ... xn} is a finite set of points inX such that the balls B(xi, ) cover X, consider the 2n 1 points of HX which are the nonemptysubsets of X0.].

21 May 2005: for some x,y Z with (x,y) = 1, then there exist a,b Z such that p = a2 b2. ... αn be algebraicintegers in K such that the discriminant D(α1,. ,αn) is a square-free non-zerointeger.

21 May 2005: 1. 2 ANALYSIS II EXAMPLES 1. (ii) If fn does not converge uniformly, show that we can find a convergent sequence xn x in[a,b] such that fn(xn) does ... derivative’ limh0+ ((f(h) f(0))/h),and similarly for f′(1)), but such that f′ is unbounded on [0,

21 May 2005: 1. 2 ANALYSIS II EXAMPLES 1. (ii) If fn does not converge uniformly, show that we can find a convergent sequence xn x in[a,b] such that fn(xn) does ... derivative’ limh0+ ((f(h) f(0))/h),and similarly for f′(1)), but such that f′ is unbounded on [0,

21 May 2005: 5. Show that if x 6= y are vectors in the finite dimensional vector space V , then there is a linear functionalθ V such that θ(x) 6= θ(y). ... fn for V such that when x =. m1xiei and y =. n1yjfj we. have ψ(x, y) =r. 1xkyk, where r is the rank of ψ.

21 May 2005: Show that there is a bijection f̄ : X/R Imf such that f factorizes as thecomposite. ... Identify a geometrical feature such that therotations preserving it form a subgroup of order 8.(ii) The group of symmetries of the tetrahedron is isomorphic to S4.

26 Apr 2005: theorem of algebra, prove that there is a complex number w such that. ... 6. Construct a sequence of polynomials Pn such that, for every positive real number.

14 Oct 2005: degree. sum is at least n, until no such pairs remain. Show that C(G) is well-defined(that is, the order in which pairs are joined is immaterial). ... Show that. G has a 1-factor and deduce that one can find a permutation matrix P and a real λ,0 < λ 1,

22 Nov 2005: iii) Give an example of disjoint closed sets A and B in R2 with points an A and bn B such. ... 9. [Tripos IB 96401(b), modified] (i) Suppose that (X,d) is a nonempty complete metric space, andf : X X a continuous map such that, for any x,y X, the

21 May 2005: a,b] such that fn(xn) does not converge to f(x). [Hint: Bolzano–Weierstrass.]. ... every point of [0,1] (here we interpret f ′(0) as the ‘one-sided derivative’ limh0+((f(h) f(0))/h),and similarly for f′(1)), but such that f ′ is

21 May 2005: there are permutation matricesP1,P2,. ,Pm such that A =. m. 1 λiPi andm. ... Show that. G has a 1-factor and deduce that one can find a permutation matrix P and a real λ,0 < λ 1, such that A λP = B = (bij)n1

18 Oct 2005: ii) Give an example of a sequence of uniformly continuous functions fn on R, such that fn con-. ... a,b] such that fn(xn) does not converge to f(x). 12. (i) Suppose f is defined and differentiable on a (bounded or unbounded) interval E R, andthat its

21 May 2005: iii) Give an example of disjoint closed sets A and B in R2 with points an A and bn B such. ... 10. [Tripos IB 96401(b), modified] (i) Suppose that (X,d) is a nonempty complete metric space, andf : X X a continuous map such that, for any x,y X, the

21 May 2005: Thenthere is a non-empty subinterval [a,b], with a < b, such that {fi} is uniformly bounded. ... λn, such that. T(v) =. i. λie(vi,v)vi,. where {λi} = σ(T).

21 May 2005: n=1 = (x1,x2,x3, ) such that for each n, |xn| 1/n. ... In such circumstances, T is a bounded linear operator. (What is the moral of this equivalence?)Let B(E,F) be the space of bounded linear operators equipped with the operator

4 Nov 2005: n=1 = (x1,x2,x3, ) such that for each n, |xn| 1/n. ... In such circumstances, T is a bounded linear operator. (What is the moral of this equivalence?)Let B(E,F) be the space of bounded linear operators equipped with the usual

21 May 2005: b) Show that the elements of S are in one-to-one correspondence with theelements of Σ (say A 7 σA) such that A B if and only if σA is ... Definition A possibly infinite simplicial complex consists of a set V (called thevertices) and a set S of finite

18 May 2005: Instead we will consider a second application ofTheorem B. Given any closed 2-form , fix a constant c 6= 0 such that the cohomology classof c is an integral class, ... a linear mapS(x,v) : TxM TxM such that E can be identified with the graph of S.

21 May 2005: Prove that there exists such a reparametrization Γ̃ = Γ ffor which ‖dΓ̃/dt‖ is constant, namely 1/l. ... 10) For a surface of revolution S, corresponding to a curve η : (a,b) R3 given byη(u) = (f(u),0,g(u)), where η is parametrized in such a

17 Jun 2005: this map.Then, there exists a positive constant C depending only on K,f,g such that for anyx K, T Ci and any y B(f(x),T/2) we ... M(B,n), so that M(fg) = M(f)M(g) and M(f) = f. In particular if f : A B is an injection and there exists g : B A such

21 May 2005: 2. Let R be an integral domain. The highest common factor of non-zero elements a and b in R is an elementd in R such that d divides both a ... be identified with the set Fq in such a way that GL2(Fq) acts on as the group ofMöbius transformations z 7 azb

8 Dec 2005: there exists a colour c1 such that c(F {a1}) = c1for all F B(r1)1. ... say with active coordinate set I, such. that for all a [m]n and all b, b′ L{L+}, we have c(a, b) = c(a, b′) = c′(a),say.

12 Jul 2005: ki Diff(Ũi), 1 i N ;. (4) a homomorphism Ψi : Γi Aut(T ki ) such that. ... On the other hand if a KOn then there exists a closed spin manifold X withan F-structure with one trivial Galois covering such that α[X] = a: if B is

21 May 2005: 2 ANALYSIS II EXAMPLES 3. 7. (i) Suppose that f : R2 R is such that D1f = f/x is continuous in a neighbourhood of(a,b), and D2f = f/y exists ... Show that f is differentiable at (a,b). (ii) Suppose that f : R2 R is such that D1f = f/x exists and is

12 Apr 2005: The pair is -regular if |d(A′,B′) d(A,B)| 6 for all subsets A′ Aand B′ B such that |A′|> |A| and |B′|> |B|. ... Proof. Define B [N]2 to be the set of all (x,y) such that x2y A.

4 Apr 2005: characteristic element of M to be any ξM K1((ΛG)T ) such that. ... Let A be a small abelian category. A full additivesubcategory B of A is admissible if whenever 0 M′ M M′′ 0 is a shortexact sequence in A such that M

14 Mar 2005: Proof. Define B [N]2 to be the set of all (x,y) such that xy A. ... It follows that there are functions a, b and c. taking values in [0, 1] such that.

21 May 2005: Complex analysis exercises 4. Sophia Demoulini Lent 05. 1. Let A C be an open set, F : A [0,1] C such that F is continuous (onA [0,1]) and z ... 8. Let D be a domain in C and γ : [0,1] D an arbitrary continuous path.(i) Show that there is an r > 0 such

22 Nov 2005: 2 ANALYSIS II EXAMPLES 3. 7. (i) Suppose that f : R2 R is such that D1f = f/x is continuous in a neighbourhood of(a,b), and D2f = f/y exists ... Show that f is differentiable at (a,b). (ii) Suppose that f : R2 R is such that D1f = f/x exists and is

21 May 2005: If A is the matrix of the first of these (say), find a non-singular matrix P such that P tAP is diagonalwith entries 1. ... 19. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a.