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MATHEMATICAL TRIPOS Part II
Friday, 10 June, 2011 9:00 am to 12:00 pm
PAPER 4
Before you begin read these instructions carefully.
The examination paper is divided into two sections. Each question in Section II carries twice the number of marks of each question in Section I. Candidates may attempt at most six questions from Section I and any number of questions from Section II.
Complete answers are preferred to fragments.
Write on one side of the paper only and begin each answer on a separate sheet.
Write legibly; otherwise you place yourself at a grave disadvantage.
At the end of the examination:
Tie up your answers in bundles, marked A, B, C,.. ., K according to the code letter affixed to each question. Include in the same bundle all questions from Sections I and II with the same code letter.
Attach a completed gold cover sheet to each bundle.
You must also complete a green master cover sheet listing all the questions you have attempted.
Every cover sheet must bear your examination number and desk number.
STATIONERY REQUIREMENTS
Gold cover sheet Green master cover sheet
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
SECTION I
1I Number Theory
(i) Prove that there are infinitely many primes.
(ii) Prove that arbitrarily large gaps can occur between consecutive primes.
2F Topics in Analysis
(a) Let γ : [0, 1] → C \ { 0 } be a continuous map such that γ(0) = γ(1). Define the winding number w(γ; 0) of γ about the origin. State precisely a theorem about homotopy invariance of the winding number.
(b) Let B = {z ∈ C : |z| 6 1 } and let f : B → C be a continuous map satisfying
|f (z) − z| 6 1
for each z ∈ ∂B.
(i) For 0 6 t 6 1, let γ(t) = f (e^2 πit). If γ(t) 6 = 0 for each t ∈ [0, 1], prove that w(γ; 0) = 1. [Hint: Consider a suitable homotopy between γ and the map γ 1 (t) = e^2 πit, 0 6 t 6 1 .] (ii) Deduce that f (z) = 0 for some z ∈ B.
3G Geometry and Groups Define inversion in a circle Γ on the Riemann sphere. You should show from your definition that inversion in Γ exists and is unique.
Prove that the composition of an even number of inversions is a M¨obius transfor- mation of the Riemann sphere and that every M¨obius transformation is the composition of an even number of inversions.
4G Coding and Cryptography Describe a scheme for sending messages based on quantum theory which is not vulnerable to eavesdropping. You may ignore engineering problems.
Part II, Paper 4
Why are expressions freqF, locB, age15-19, and sexF not listed? Suppose that we plan to observe a group of 20 female, non-frequent, beach swimmers, aged 20-24. Give an expression (using the coefficient estimates from the model fitted above) for the expected number of ear infections in this group.
Now, suppose that we allow for interaction between variables age and sex. Give the R command for fitting this model. We test for the effect of this interaction by producing the following (abbreviated) ANOVA table:
Resid. Df Resid. Dev Df Deviance P(>|Chi|) 1 18 51. 2 16 44.319 2 7.3948 0.02479 *
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05. 0.1 1
Briefly explain what test is performed, and what you would conclude from it. Does either of these models fit the data well?
6B Mathematical Biology A neglected flower garden contains Mn marigolds in the summer of year n. On average each marigold produces γ seeds through the summer. Seeds may germinate after one or two winters. After three winters or more they will not germinate. Each winter a fraction 1 − α of all seeds in the garden are eaten by birds (with no preference to the age of the seed). In spring a fraction μ of seeds that have survived one winter and a fraction ν of seeds that have survived two winters germinate. Finite resources of water mean that the number of marigolds growing to maturity from S germinating seeds is N (S), where N (S) is an increasing function such that N (0) = 0, N ′(0) = 1, N ′(S) is a decreasing function of S and N (S) → Nmax as S → ∞.
Show that Mn satisfies the equation
Mn+1 = N (αμγMn + νγα^2 (1 − μ)Mn− 1 ).
Write down an equation for the number M∗ of marigolds in a steady state. Show graphically that there are two solutions, one with M∗ = 0 and the other with M∗ > 0 if
αμγ + νγα^2 (1 − μ) > 1.
Show that the M∗ = 0 steady-state solution is unstable to small perturbations in this case.
Part II, Paper 4
7C Dynamical Systems
(i) Explain the use of the energy balance method for describing approximately the behaviour of nearly Hamiltonian systems.
(ii) Consider the nearly Hamiltonian dynamical system
x¨ + ǫ x˙(−1 + αx^2 − βx^4 ) + x = 0 , 0 < ǫ ≪ 1 ,
where α and β are positive constants. Show that, for sufficiently small ǫ, the system has periodic orbits if α^2 > 8 β, and no periodic orbits if α^2 < 8 β. Show that in the first case there are two periodic orbits, and determine their approximate size and their stability. What can you say about the existence of periodic orbits when α^2 = 8β? [You may assume that ∫ (^2) π
0
sin^2 t dt = π ,
∫ (^2) π
0
sin^2 t cos^2 t dt = π 4
∫ (^2) π
0
sin^2 t cos^4 t dt = π 8
]
8E Further Complex Methods Let F (z) be defined by
F (z) =
0
e−zt 1 + t^2 dt, | arg z| <
π 2
Let F˜ (z) be defined by
F˜ (z) = P
∫ (^) ∞e− iπ 2
0
e−zζ 1 + ζ^2
dζ , 0 < arg z < π ,
where P denotes principal value integral and the contour is the negative imaginary axis. By computing F (z) − F˜ (z), obtain a formula for the analytic continuation of F (z) for π 2 6 arg z < π.
Part II, Paper 4 [TURN OVER
SECTION II
11I Number Theory
(i) Prove the law of reciprocity for the Jacobi symbol. You may assume the law of reciprocity for the Legendre symbol.
(ii) Let n be an odd positive integer which is not a square. Prove that there exists an odd prime p with
n p
12G Geometry and Groups Define a lattice in R^2 and the rank of such a lattice. Let Λ be a rank 2 lattice in R^2. Choose a vector w 1 ∈ Λ \ { 0 } with ||w 1 || as small as possible. Then choose w 2 ∈ Λ \ Zw 1 with ||w 2 || as small as possible. Show that Λ = Zw 1 + Zw 2.
Suppose that w 1 is the unit vector
. Draw the region of possible values for w 2.
Suppose that Λ also equals Zv 1 + Zv 2. Prove that
v 1 = aw 1 + bw 2 and v 2 = cw 1 + dw 2 ,
for some integers a, b, c, d with ad − bc = ±1.
13J Statistical Modelling Consider the general linear model Y = Xβ + ǫ, where the n × p matrix X has full rank p 6 n, and where ǫ has a multivariate normal distribution with mean zero and covariance matrix σ^2 In. Write down the likelihood function for β, σ^2 and derive the maximum likelihood estimators β,ˆ ˆσ^2 of β, σ^2. Find the distribution of βˆ. Show further that βˆ and ˆσ^2 are independent.
Part II, Paper 4 [TURN OVER
14C Dynamical Systems
(i) State and prove Lyapunov’s First Theorem, and state (without proof) La Salle’s Invariance Principle. Show by example how the latter result can be used to prove asymptotic stability of a fixed point even when a strict Lyapunov function does not exist.
(ii) Consider the system
x˙ = −x + 2y + x^3 + 2x^2 y + 2xy^2 + 2y^3 ,
y ˙ = −y − x − 2 x^3 +
x^2 y − 3 xy^2 + y^3.
Show that the origin is asymptotically stable and that the basin of attraction of the origin includes the region x^2 + 2y^2 < 2 /3.
Part II, Paper 4
16H Logic and Set Theory Define the sets Vα for ordinals α. Show that each Vα is transitive. Show also that Vα ⊆ Vβ whenever α 6 β. Prove that every set x is a member of some Vα.
For which ordinals α does there exist a set x such that the power-set of x has rank α? [You may assume standard properties of rank.]
17F Graph Theory
(i) Given a positive integer k, show that there exists a positive integer n such that, whenever the edges of the complete graph Kn are coloured with k colours, there exists a monochromatic triangle. Denote the least such n by f (k). Show that f (k) 6 3 · k! for all k.
(ii) You may now assume that f (2) = 6 and f (3) = 17. Let H denote the graph of order 4 consisting of a triangle together with one extra edge. Given a positive integer k, let g(k) denote the least positive integer n such that, whenever the edges of the complete graph Kn are coloured with k colours, there exists a monochromatic copy of H. By considering the edges from one vertex of a monochromatic triangle in K 7 , or otherwise, show that g(2) 6 7. By exhibiting a blue-yellow colouring of the edges of K 6 with no monochromatic copy of H, show that in fact g(2) = 7. What is g(3)? Justify your answer.
Part II, Paper 4
18H Galois Theory Let K be a field of characteristic 0, and let P (X) = X^4 + bX^2 + cX + d be an irreducible quartic polynomial over K. Let α 1 , α 2 , α 3 , α 4 be its roots in an algebraic closure of K, and consider the Galois group Gal(P ) (the group Gal(F/K) for a splitting field F of P over K) as a subgroup of S 4 (the group of permutations of α 1 , α 2 , α 3 , α 4 ). Suppose that Gal(P ) contains V 4 = { 1 , (12)(34), (13)(24), (14)(23)}.
(i) List all possible Gal(P ) up to isomorphism. [Hint: there are 4 cases, with orders 4, 8, 12 and 24.]
(ii) Let Q(X) be the resolvent cubic of P , i.e. a cubic in K[X] whose roots are −(α 1 + α 2 )(α 3 + α 4 ), −(α 1 + α 3 )(α 2 + α 4 ) and −(α 1 + α 4 )(α 2 + α 3 ). Construct a natural surjection Gal(P ) → Gal(Q), and find Gal(Q) in each of the four cases found in (i).
(iii) Let ∆ ∈ K be the discriminant of Q. Give a criterion to determine Gal(P ) in terms of ∆ and the factorisation of Q in K[X].
(iv) Give a specific example of P where Gal(P ) is abelian.
Part II, Paper 4 [TURN OVER
21H Algebraic Topology State the Mayer–Vietoris theorem, and use it to calculate, for each integer q > 1, the homology group of the space Xq obtained from the unit disc B^2 ⊆ C by identifying pairs of points (z 1 , z 2 ) on its boundary whenever z 1 q = z 2 q. [You should construct an explicit triangulation of Xq .] Show also how the theorem may be used to calculate the homology groups of the suspension SK of a connected simplicial complex K in terms of the homology groups of K, and of the wedge union X ∨ Y of two connected polyhedra. Hence show that, for any finite sequence (G 1 , G 2 ,... , Gn) of finitely-generated abelian groups, there exists a polyhedron X such that H 0 (X) ∼= Z, Hi(X) ∼= Gi for 1 6 i 6 n and Hi(X) = 0 for i > n. [You may assume the structure theorem which asserts that any finitely-generated abelian group is isomorphic to a finite direct sum of (finite or infinite) cyclic groups.]
22G Linear Analysis State Urysohn’s Lemma. State and prove the Tietze Extension Theorem. Let X, Y be two topological spaces. We say that the extension property holds if, whenever S ⊆ X is a closed subset and f : S → Y is a continuous map, there is a continuous function f˜ : X → Y with f˜ |S = f. For each of the following three statements, say whether it is true or false. Briefly justify your answers.
- If X is a metric space and Y = [− 1 , 1] then the extension property holds.
- If X is a compact Hausdorff space and Y = R then the extension property holds.
- If X is an arbitrary topological space and Y = [− 1 , 1] then the extension property holds.
23H Algebraic Geometry Let X be a smooth projective curve over an algebraically closed field k. State the Riemann–Roch theorem, briefly defining all the terms that appear.
Now suppose X has genus 1, and let P∞ ∈ X.
Compute L(nP∞) for n 6 6. Show that φ 3 P∞ defines an isomorphism of X with a smooth plane curve in P^2 which is defined by a polynomial of degree 3.
Part II, Paper 4 [TURN OVER
24I Differential Geometry Define what is meant by a geodesic. Let S ⊂ R^3 be an oriented surface. Define the geodesic curvature kg of a smooth curve γ : I → S parametrized by arc-length. Explain without detailed proofs what are the exponential map expp and the geodesic polar coordinates (r, θ) at p ∈ S. Determine the derivative d(expp) 0. Prove that the coefficients of the first fundamental form of S in the geodesic polar coordinates satisfy
E = 1 , F = 0 , G(0, θ) = 0 , (
G)r (0, θ) = 1.
State the global Gauss–Bonnet formula for compact surfaces with boundary. [You should identify all terms in the formula.] Suppose that S is homeomorphic to a cylinder S^1 × R and has negative Gaussian curvature at each point. Prove that S has at most one simple (i.e. without self- intersections) closed geodesic. [Basic properties of geodesics may be assumed, if accurately stated.]
25K Probability and Measure
(i) State and prove Fatou’s lemma. State and prove Lebesgue’s dominated convergence theorem. [You may assume the monotone convergence theorem.] In the rest of the question, let fn be a sequence of integrable functions on some measure space (E, E, μ), and assume that fn → f almost everywhere, where f is a given integrable function. We also assume that
|fn|dμ →
|f |dμ as n → ∞.
(ii) Show that
f (^) n+ dμ →
f +dμ and that
f (^) n− dμ →
f −dμ, where φ+^ = max(φ, 0) and φ−^ = max(−φ, 0) denote the positive and negative parts of a function φ.
(iii) Here we assume also that fn > 0. Deduce that
|f − fn|dμ → 0.
Part II, Paper 4
27K Principles of Statistics What does it mean to say that a (1 × p) random vector ξ has a multivariate normal distribution?
Suppose ξ = (X, Y ) has the bivariate normal distribution with mean vector μ = (μX , μY ), and dispersion matrix
σXX σXY σXY σY Y
Show that, with β := σXY /σXX , Y − βX is independent of X, and thus that the conditional distribution of Y given X is normal with mean μY + β(X − μX ) and variance σY Y ·X := σY Y − σ^2 XY /σXX.
For i = 1,... , n, ξi = (Xi, Yi) are independent and identically distributed with the above distribution, where all elements of μ and Σ are unknown. Let
S =
SXX SXY
SXY SY Y
∑^ n
i=
(ξi − ξ)T(ξi − ξ) ,
where ξ := n−^1
∑n i=1 ξi. The sample correlation coefficient is r := SXY /
SXX SY Y. Show that the distribu- tion of r depends only on the population correlation coefficient ρ := σXY /
σXX σY Y. Student’s t-statistic (on n − 2 degrees of freedom) for testing the null hypothesis H 0 : β = 0 is
t :=
β √ SY Y ·X /(n − 2)SXX
where β̂ := SXY /SXX and SY Y ·X := SY Y − S XY^2 /SXX. Its density when H 0 is true is
p(t) = C
t^2 n − 2
)− 12 (n−1) ,
where C is a constant that need not be specified.
Express t in terms of r, and hence derive the density of r when ρ = 0. How could you use the sample correlation r to test the hypothesis ρ = 0?
Part II, Paper 4
28K Optimization and Control Describe the type of optimal control problem that is amenable to analysis using Pontryagin’s Maximum Principle. A firm has the right to extract oil from a well over the interval [0, T ]. The oil can be sold at price £p per unit. To extract oil at rate u when the remaining quantity of oil in the well is x incurs cost at rate £u^2 /x. Thus the problem is one of maximizing ∫ (^) T
0
[
pu(t) −
u(t)^2 x(t)
]
dt ,
subject to dx(t)/dt = −u(t), u(t) > 0, x(t) > 0. Formulate the Hamiltonian for this problem. Explain why λ(t), the adjoint variable, has a boundary condition λ(T ) = 0. Use Pontryagin’s Maximum Principle to show that under optimal control
λ(t) = p −
1 /p + (T − t)/ 4
and
dx(t) dt
2 px(t) 4 + p(T − t)
Find the oil remaining in the well at time T , as a function of x(0), p, and T ,
Part II, Paper 4 [TURN OVER
31A Asymptotic Methods Determine the range of the integer n for which the equation
d^2 y dz^2
= zny
has an essential singularity at z = ∞. Use the Liouville–Green method to find the leading asymptotic approximation to two independent solutions of d^2 y dz^2 = z^3 y ,
for large |z|. Find the Stokes lines for these approximate solutions. For what range of arg z is the approximate solution which decays exponentially along the positive z-axis an asymptotic approximation to an exact solution with this exponential decay?
32D Principles of Quantum Mechanics The quantum-mechanical observable Q has just two orthonormal eigenstates | 1 〉 and | 2 〉 with eigenvalues −1 and 1, respectively. The operator Q′^ is defined by Q′^ = Q + ǫT , where T =
0 i −i 0
Defining orthonormal eigenstates of Q′^ to be | 1 ′〉 and | 2 ′〉 with eigenvalues q′ 1 , q 2 ′, respect- ively, consider a perturbation to first order in ǫ ∈ R for the states
| 1 ′〉 = a 1 | 1 〉 + a 2 ǫ| 2 〉 , | 2 ′〉 = b 1 | 2 〉 + b 2 ǫ| 1 〉 ,
where a 1 , a 2 , b 1 , b 2 are complex coefficients. The real eigenvalues are also expanded to first order in ǫ: q′ 1 = −1 + c 1 ǫ , q′ 2 = 1 + c 2 ǫ. From first principles, find a 1 , a 2 , b 1 , b 2 , c 1 , c 2. Working exactly to all orders, find the real eigenvalues q 1 ′, q′ 2 directly. Show that the exact eigenvectors of Q′^ may be taken to be of the form
Aj (ǫ)
−i(1 + Bq j′ )/ǫ
finding Aj (ǫ) and the real numerical coefficient B in the process. By expanding the exact expressions, again find a 1 , a 2 , b 1 , b 2 , c 1 , c 2 , verifying the perturbation theory results above.
Part II, Paper 4 [TURN OVER
33E Applications of Quantum Mechanics A particle of charge −e and mass m moves in a magnetic field B(x, t) and in an electric potential φ(x, t). The time-dependent Schr¨odinger equation for the particle’s wavefunction Ψ(x, t) is
iℏ
∂t
ie ℏ
φ
ℏ^2
2 m
ie ℏ
A
where A is the vector potential with B = ∇ ∧ A. Show that this equation is invariant under the gauge transformations
A(x, t) → A(x, t) + ∇f (x, t) , φ(x, t) → φ(x, t) − (^) ∂t∂ f (x, t) ,
where f is an arbitrary function, together with a suitable transformation for Ψ which should be stated.
Assume now that ∂Ψ/∂z = 0, so that the particle motion is only in the x and y directions. Let B be the constant field B = (0, 0 , B) and let φ = 0. In the gauge where A = (−By, 0 , 0) show that the stationary states are given by
Ψk(x, t) = ψk(x)e−iEt/ℏ^ ,
with ψk(x) = eikxχk(y). (∗)
Show that χk(y) is the wavefunction for a simple one-dimensional harmonic oscillator centred at position y 0 = ℏk/eB. Deduce that the stationary states lie in infinitely degenerate levels (Landau levels) labelled by the integer n > 0, with energy
En = (2n + 1) ℏeB 2 m
A uniform electric field E is applied in the y-direction so that φ = −Ey. Show that the stationary states are given by (∗), where χk(y) is a harmonic oscillator wavefunction centred now at
y 0 =
eB
ℏk − m
E
B
Show also that the eigen-energies are given by
En,k = (2n + 1) ℏeB 2 m
Why does this mean that the Landau energy levels are no longer degenerate in two dimensions?
Part II, Paper 4
