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26 Nov 2008: Provethat A contains an infinite set B such that no point of B is a convex combination of otherpoints of B. ... 6. Show that every graph G has a partition of its vertex-set as X Y such that the numberof edges from X to Y is at least 12 e(G).

6 Mar 2008: vertices into sets B andR such that the only edges lie between vertices in different sets.(i) Show that if any finite subgraph of a graph is bipartite, then so is ... Suppose that any such formula is equivalent modulo T to aquantifier-free formula.

27 Feb 2008: extensions. We show that the Cartan map from K-theory to G-theory of such. ... Regarding the assumption on B, we ask the following. Question. Let H be a finite dimensional Hopf algebra such that the Cartan mapc : K0(H) G0(H) is injective and let

4 Feb 2008: Let P be the set of all p Q such that 2 < p2 < 3. ... c) If A and B are closed subsets of R such that 1 A, 1 B and [1, 1] A B, thenA B 6=.

12 Aug 2008: 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.

19 Mar 2008: 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

28 Feb 2008: In this and the following questions, youare not being asked to find such an isomorphism explicitly.)Decompose V 23 , Λ. ... b) Explicitly construct such a representation ρ : SU (2) SO(3) by showing that SU (2)acts on the vector space of matrices of the

10 Mar 2008: Show that for any b C, there exists a sequence of points zn D(a,R) with zn 6= a such thatzn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.

19 Mar 2008: 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

17 Oct 2008: 11 fn 6 0. Is it possible to find such a sequence with |fn(x)| 1 for all n and for all x? ... On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a, b] such that fn(xn) 6 f

19 Mar 2008: 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 not converge to f (x). ... derivative’ limh0+ ((f (h) f (0))/h),and similarly for f ′(1)), but such that f ′ is unbounded on [0, 1

10 Mar 2008: Prove that the set S of points x [0, 1] such that x Aj for infinitely many j is dense.Must S be open? ... b) If f : (0, 1) R is continuous and bounded, prove that there exists a sequence of polynomialsQn such that Qn are uniformly bounded and Qn f

19 Mar 2008: 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

28 Feb 2008: n}, let the non-negative numbers A(n)k and the“nodes” x(n)k [a, b] be given such that for each polynomial P , the error. ... such that for every f C([0, 1]), there exists a unique series.

14 Oct 2008: If so, howmany such ‘prime triplets’ are there?(b) Suppose that four out of five consecutive odd integers, greater than 10, are prime. ... But such a numberwould clearly be very interesting indeed. Therefore, all positive integers are interesting.

13 Nov 2008: a linear func-tional θ V such that θ(x) 6= θ(y).(b) Suppose that V is finite dimensional. ... If no such subspace exists,show that there exists a basis v, α(v),.

4 Feb 2008: 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 Aug 2008: anda distinguishedobject e ofL, such that thefollowing hold:. • (a o b) o c = a o (b o c) (associativity). • ... axbaandaxbb. • If m is such that vi a andm b thenm a x b.

28 Aug 2008: anda distinguishedobject e ofL, such that thefollowing hold:. • (a o b) o c = a o (b o c) (associativity). • ... axbaandaxbb. • If m is such that vi a andm b thenm a x b.

23 Feb 2008: 4) Let X be a space, x X, f : S1 X a map such that x is inthe image of f, and Y = X f B. ... 1) Showthat there is an element z C such that γ(z) and x′ in C′.

12 Feb 2008: Prove that for each x [1, 1], there existsζ (1, 1) such that. ... 14) Let be a bounded, open subset of Rn and g : (a, b) R be a function such that(i) g(, t) is bounded and continuous in for each t (a,

19 Aug 2008: Definition 4.2 For any pseudo-monad T on a 2-category C and for any bicat-egory B, define the bicategory Cocone((C, T ), B) such that an object consistsof. • ... ᾱ. D(HA, HB). H? D(HA, αB)- D(HA, HB). D(αA, KB)? such that for every pair of

13 Aug 2008: Furthermore. (u|A(B)|B = (u|B(C)|B. is a play in B. Conversely given a play r in A ( B anda play s in B ( C such that ... B. These morphisms are uniquelyrepresented by relations F : A - B such that.

17 Nov 2008: In this and the following questions, you arenot being asked to find such an isomorphism explicitly.). ... Here O(n) denotesthe subgroup of GLn(R) preserving the standard (positive definite) symmetric bilinear form.(b) Explicitly construct such a

16 Jul 2008: So there exists a basis v1,. , vd for L(D1) such that L(D1Pi) has basis v1,. , ... One such involution is given in the statement of thelemma. Remark 3.6.

21 Aug 2008: are those of B and to the treebelow" each such opening move, we graft onto it a copy ofthe forest of questions of A.This is a good place to consider ... We presuppose an innite collection of names whichare ranged over by symbols such as x;x0;xi;f, a;b;

21 Aug 2008: B. 85]) such that:. (i) the base category B has nite products,. ... x:t is not necessarily. dened in U. For example, in any c-pca U where there are elements a; b 2 U such that a b is.

3 Mar 2008: eachx Sn has a neighborhood U such that U g(U ) = for any1 6= g G. ... σ = σ0, σ1,. , σr = τ such that each σi and σi1 intersectalong an (n 1)-simplex.

19 Aug 2008: C(B, AX ) = C(B, A)X. V -natural in B. For instance, taking V to be P oset, this allows us not only toconsider objects such as AA in a ... structures on categories, such as finiteproduct or finite coproduct, typically with many others too.

21 Aug 2008: B. 85]) such that:. (i) the base category B has nite products,. ... x:t is not necessarily. dened in U. For example, in any c-pca U where there are elements a; b 2 U such that a b is.

1 Feb 2008: Let γ : [a,b] C be a (continuous) curve. Suppose we have a = a0 < a1 < <an1 < an = b such that for each 0 i < m the restriction γi of γ to ... U is starlike if there exists a0 U such that for every b U the line segementfrom a0 to b lies in U.

27 Aug 2008: 6. Secondary reduction rules. 5 The Categorical ModelMuch work has been done on providing such (categorical) models of IntuitionisticLinear Logic. ... coalgebracarries naturally the structure of a commutative comonoid in such a way that coal-gebra maps

27 Aug 2008: 6. Secondary reduction rules. 5 The Categorical ModelMuch work has been done on providing such (categorical) models of IntuitionisticLinear Logic. ... coalgebracarries naturally the structure of a commutative comonoid in such a way that coal-gebra maps

19 Aug 2008: The first idea is to generalisefrom the category B to a category BA, parameterised by a small categoryA, such that B 1 = B. ... Xn] :(. i[n] Ai) // Set be the unique presheaf such that for all i [n],.

20 Feb 2008: A such that M and M′′ belong to B, then M′ also belongs to B[17, 12.4.2]. ... module and as such is killed by some power of the central element z.

8 Feb 2008: Show that such an enemy can find our common key if K = 3. ... n) is the output from a LFSR of length d. (c) The period of (xn)n>0 is the least integer r > 1 such that xnr = xn for all sufficiently largen.

22 Jan 2008: If further such an action is a partial function then the automatonis deterministic. ... See [7]and [8] for ramifications of the theory of such special traced monoidal categories.) WritePar for the traced monoidal category just described.

13 Aug 2008: logical operations.Such subcollections arise naturally when the PCA (A, ) has some additionalstructure respected by application. ... An extensional assembly is an object (X, | = |) of the topos S[PER] such that.

29 Sep 2008: 2-natural in C such that the diagram. [C,D]kB- [[B,C], [B,D]]. kA- [[A, [B,C]], [A, [B,D]]]. ... Note that what wecall pseudo-maps [3] call morphisms.) We write f = (f, f̄) : A B for such apseudo-map, the 2-cell usually being understood.

20 Jan 2008: b) Let N Zn be a submodule of rank n. Show that there is a matrix A Mn(Z) such thatA(Zn) = N and the index [Zn : N] of N in ... 8) An integral domain R is called a euclidian domain if there is a map N : R{0} Z>0 such thatfor all a,b R, b 6= 0, there are

2 Dec 2008: Does there exist such a bijection f which isorder-preserving (that is, satisfies f(x) < f(y) whenever x < y)? ... Let S be a collection of subsets of N such that for every A, B S we have eitherA B or B A.

27 Aug 2008: Wecan dene a procedure called normalisation which can systematically eliminate such de-tours from a deduction. ... Much work has been done on providing such (categorical) models ofIntuitionistic Linear Logic.

27 Aug 2008: Wecan dene a procedure called normalisation which can systematically eliminate such de-tours from a deduction. ... Much work has been done on providing such (categorical) models ofIntuitionistic Linear Logic.

13 Aug 2008: Thus concretely maps A B of Dial(I) consist of maps f : I U V andF : I U Y X in T such that α(i,u,F(i,u,y)) ... or equivalently. (f,F) : (U X•) V U• Y •. such that an entailment which I write. (

20 Feb 2008: Assume there is θ OK such that 1, θ,. , θn1 is anintegral basis of OK. ... dK| (. nn. n! )2 ( π4. )n. Deduce that |dK| > 1 for every number field K 6= Q.(b) Show that there are constants A > 1, c > 1, such that for every

28 Nov 2008: Then there exists a basis B for V such that. [ψ]B =. . 0 11 0 0. 0 11 0 0. 0 0 1. ... For each α End(V ) there exists a unique α End(V ) such that.

16 Oct 2008: We say A, B Matm,n(F ) are equivalentif there exist invertible matrices P and Q such that B = Q1AP. ... s} such that, if we delete each yij from Y and replace it by xj , theresulting set is still linearly independent.