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MATHEMATICAL TRIPOS Part II
Monday, 1 June, 2009 9:00 am to 12:00 pm
PAPER 1
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,.. .,J 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
1G Number Theory State the Chinese Remainder Theorem. Determine all integers x satisfying the congruences x ≡ 2 mod 3, x ≡ 2 mod 5, x ≡ 6 mod 7.
2F Topics in Analysis (i) Let n > 1 and let x 1 ,... , xn be distinct points in [− 1 , 1]. Show that there exist numbers A 1 ,... , An such that
∫ (^1)
− 1
P (x) dx =
∑^ n
j=
Aj P (xj ) (∗)
for every polynomial P of degree 6 n − 1.
(ii) Explain, without proof, how one can choose the points x 1 ,... , xn and the numbers A 1 ,... , An such that (∗) holds for all polynomials P of degree 6 2 n − 1.
3F Geometry of Group Actions Explain what is meant by stereographic projection from the 2-dimensional sphere to the complex plane.
Prove that u and v are the images under stereographic projection of antipodal points on the sphere if and only if u¯v = −1.
Part II, Paper 1
7E Dynamical Systems Let ˙x = f (x) be a two-dimensional dynamical system with a fixed point at x = 0. Define a Lyapunov function V (x) and explain what it means for x = 0 to be Lyapunov stable. Determine the values of β for which V = x^2 + βy^2 is a Lyapunov function in a sufficiently small neighbourhood of the origin for the system
x˙ = −x + 2y + 2xy − x^2 − 4 y^2 , y ˙ = −y + xy.
What can be deduced about the basin of attraction of the origin using V when β = 2?
8B Further Complex Methods Find all second order linear ordinary homogenous differential equations which have a regular singular point at z = 0, a regular singular point at z = ∞, and for which every other point in the complex z-plane is an analytic point.
[You may use without proof Liouville’s theorem.]
Part II, Paper 1
9E Classical Dynamics Lagrange’s equations for a system with generalized coordinates qi(t) are given by
d dt
∂L
∂ q˙i
∂L
∂qi
where L is the Lagrangian. The Hamiltonian is given by
H =
j
pj q˙j − L,
where the momentum conjugate to qj is
pj =
∂L
∂ q˙j
Derive Hamilton’s equations in the form
q˙i =
∂H
∂pi
, p˙i = −
∂H
∂qi
Explain what is meant by the statement that qk is an ignorable coordinate and give an associated constant of the motion in this case. The Hamiltonian for a particle of mass m moving on the surface of a sphere of radius a under a potential V (θ) is given by
H =
2 ma^2
p^2 θ +
p^2 φ sin^2 θ
where the generalized coordinates are the spherical polar angles (θ, φ). Write down two constants of the motion and show that it is possible for the particle to move with constant θ provided that pφ^2 =
ma^2 sin^3 θ cos θ
dV dθ
Part II, Paper 1 [TURN OVER
SECTION II
11F Geometry of Group Actions Define frieze group and crystallographic group and give three examples of each, identifying them as abstract groups as well as geometrically. Let G be a discrete group of isometries of the Euclidean plane which contains a trans- lation. Prove that G contains no element of order 5.
12H Coding and Cryptography (i) State and prove Gibbs’ inequality. (ii) A casino offers me the following game: I choose strictly positive numbers a 1 ,... , an with
∑n j=1 aj^ = 1.^ I give the casino my entire fortune^ f^ and roll an^ n-sided die. With probability pj the casino returns u− j 1 aj f for j = 1, 2 ,... , n. If I intend to play the game many times (staking my entire fortune each time) explain carefully why I should choose a 1 ,... , an to maximise
∑n j=1 pj^ log(u − 1 j aj^ ). [You should assume n > 2 and uj , pj > 0 for each j.] (iii) Determine the appropriate a 1 ,... , an. Let
∑n i=1 ui^ =^ U^. Show that, if^ U <^ 1, then, in the long run with high probability, my fortune increases. Show that, if U > 1, the casino can choose u 1 ,... , un in such a way that, in the long run with high probability, my fortune decreases. Is it true that, if U > 1, any choice of u 1 ,... , un will ensure that, in the long run with high probability, my fortune decreases? Why?
Part II, Paper 1 [TURN OVER
13I Statistical Modelling A three-year study was conducted on the survival status of patients suffering from cancer. The age of the patients at the start of the study was recorded, as well as whether or not the initial tumour was malignant. The data are tabulated in R as follows:
cancer age malignant survive die 1 <50 no 77 10 2 <50 yes 51 13 3 50-69 no 51 11 4 50-69 yes 38 20 5 70+ no 7 3 6 70+ yes 6 3
Describe the model that is being fitted by the following R commands:
total <- survive + die fit1 <- glm(survive/total ~ age + malignant, family = binomial,
Explain the (slightly abbreviated) output from the code below, describing how the hypothesis tests are performed and your conclusions based on their results.
summary(fit1) Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 2.0730 0.2812 7.372 1.68e-13 *** age50-69 -0.6318 0.3112 -2.030 0.0424 * age70+ -0.9282 0.5504 -1.686 0.. malignantyes -0.7328 0.2985 -2.455 0.0141 *
Null deviance: 12.65585 on 5 degrees of freedom Residual deviance: 0.49409 on 2 degrees of freedom AIC: 30.
Based on the summary above, motivate and describe the following alternative model:
age2 <- as.factor(c("<50", "<50", "50+", "50+", "50+", "50+")) fit2 <- glm(survive/total ~ age2 + malignant, family = binomial,
- weights = total) This question continues on the next page
Part II, Paper 1
15D Cosmology (i) In a homogeneous and isotropic universe, the scalefactor a(t) obeys the Fried- mann equation (^) ( a˙ a
kc^2 a^2
8 πG 3
ρ,
where ρ(t) is the matter density which, together with the pressure P (t), satisfies
ρ˙ = − 3 a˙ a
ρ + P/c^2
Use these two equations to derive the Raychaudhuri equation,
¨a a
4 πG 3
ρ + 3P/c^2
(ii) Conformal time τ is defined by taking dt/dτ = a, so that ˙a = a′/a ≡ H where primes denote derivatives with respect to τ. For matter obeying the equation of state P = wρc^2 , show that the Friedmann and energy conservation equations imply
H^2 + kc^2 =
8 πG 3 ρ 0 a−(1+3w),
where ρ 0 = ρ(t 0 ) and we take a(t 0 ) = 1 today. Use the Raychaudhuri equation to derive the expression H′^ + 12 (1 + 3w)[H^2 + kc^2 ] = 0.
For a kc^2 = 1 closed universe, by solving first for H (or otherwise), show that the scale factor satisfies a = α(sin βτ )^2 /(1+3w)
where α, β are constants. [Hint: You may assume that
dx/(1 + x^2 ) = − cot−^1 x + const.]
For a closed universe dominated by pressure-free matter (P = 0), find the complete parametric solution
a = 12 α(1 − cos 2βτ ), t =
α 4 β (2βτ − sin 2βτ ).
Part II, Paper 1
16G Logic and Set Theory Prove that if G : On × V → V is a definable function, then there is a definable function F : On → V satisfying
F (α) = G(α, {F (β) : β < α}).
Define the notion of an initial ordinal, and explain its significance for cardinal arithmetic. State Hartogs’ lemma. Using the recursion theorem define, giving justification, a function ω : On → On which enumerates the infinite initial ordinals. Is there an ordinal α with α = ωα? Justify your answer.
17F Graph Theory (i) State and prove Hall’s theorem concerning matchings in bipartite graphs. (ii) The matching number of a graph G is the maximum size of a family of independent edges (edges without shared vertices) in G. Deduce from Hall’s theorem that if G is a k-regular bipartite graph on n vertices (some k > 0) then G has matching number n/2. (iii) Now suppose that G is an arbitrary k-regular graph on n vertices (some k > 0). Show that G has a matching number at least (^4) kk− 2 n. [Hint: Let S be the set of vertices in a maximal set of independent edges. Consider the edges of G with exactly one endpoint in S.] For k = 2, show that there are infinitely many graphs G for which equality holds.
18H Galois Theory Define a K-isomorphism, ϕ : L → L′, where L, L′^ are fields containing a field K, and define AutK (L). Suppose α and β are algebraic over K. Show that K(α) and K(β) are K-isomorphic via an isomorphism mapping α to β if and only if α and β have the same minimal polynomial. Show that AutK K(α) is finite, and a subgroup of the symmetric group Sd, where d is the degree of α. Give an example of a field K of characteristic p > 0 and α and β of the same degree, such that K(α) is not isomorphic to K(β). Does such an example exist if K is finite? Justify your answer.
Part II, Paper 1 [TURN OVER
22H Linear Analysis (a) State and prove the Baire category theorem. (b) Let X be a normed space. Show that every proper linear subspace V ⊂ X has empty interior. (c) Let P be the vector space of all real polynomials in one variable. Using the Baire category theorem and the result from (b), prove that for any norm ‖ · ‖ on P, the normed space (P, ‖ · ‖) is not a Banach space.
23G Riemann Surfaces (a) Let X = C ∪ {∞} be the Riemann sphere. Define the notion of a rational function r and describe the function f : X → X determined by r. Assuming that f is holomorphic and non-constant, define the degree of r as a rational function and the degree of f as a holomorphic map, and prove that the two degrees coincide. [You are not required to prove that the degree of f is well-defined.] Let A = {a 1 , a 2 , a 3 } and B = {b 1 , b 2 , b 3 } be two subsets of X each containing three distinct elements. Prove that X \ A is biholomorphic to X \ B. (b) Let Z ⊂ C^2 be the algebraic curve defined by the vanishing of the polynomial p(z, w) = w^2 − z^3 + z^2 + z. Prove that Z is smooth at every point. State the implicit function theorem and define a complex structure on Z, so that the maps g, h : Z → C given by g(z, w) = w, h(z, w) = z are holomorphic. Define what is meant by a ramification point of a holomorphic map between Riemann surfaces. Give an example of a ramification point of g and calculate the branching order of g at that point.
24G Algebraic Geometry Define what is meant by a rational map from a projective variety V ⊂ Pn^ to Pm. What is a regular point of a rational map? Consider the rational map φ : P^2 − → P^2 given by
(X 0 : X 1 : X 2 ) 7 → (X 1 X 2 : X 0 X 2 : X 0 X 1 ).
Show that φ is not regular at the points (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1) and that it is regular elsewhere, and that it is a birational map from P^2 to itself. Let V ⊂ P^2 be the plane curve given by the vanishing of the polynomial X 02 X 13 + X^21 X 23 + X 22 X 03 over a field of characteristic zero. Show that V is irreducible, and that φ determines a birational equivalence between V and a nonsingular plane quartic.
Part II, Paper 1 [TURN OVER
25H Differential Geometry (i) Define manifold and manifold with boundary for subsets X ⊂ RN^. (ii) Let X and Y be manifolds and f : X → Y a smooth map. Define what it means for y ∈ Y to be a regular value of f.
(iii) Let n > 0 and let Sn^ denote the set {(x^1 ,... , xn+1) ∈ Rn+1^ :
∑n+ i=1 (x i) (^2) = 1}.
Let Bn+1^ denote the set {(x^1 ,... , xn+1) ∈ Rn+1^ :
∑n+ i=1 (x i) (^2 6 1) }. Show that Sn (^) is an
n-dimensional manifold and Bn+1^ is an (n + 1)-dimensional manifold with boundary, with ∂Bn+1^ = Sn.
[You may use standard theorems involving regular values of smooth functions provided that you state them clearly.]
(iv) For n > 0, consider the map h : Sn^ → Sn^ taking x to −x. Show that h is smooth. Now let f be a smooth map f : Sn^ → Sn^ such that f ◦ h = f. Show that f is not smoothly homotopic to the identity map.
26J Probability and Measure Let (E, E, μ) be a measure space. Explain what is meant by a simple function on (E, E, μ) and state the definition of the integral of a simple function with respect to μ.
Explain what is meant by an integrable function on (E, E, μ) and explain how the integral of such a function is defined.
State the monotone convergence theorem. Show that the following map is linear
f 7 → μ(f ) : L^1 (E, E, μ) → R,
where μ(f ) denotes the integral of f with respect to μ.
[You may assume without proof any fact concerning simple functions and their integrals. You are not expected to prove the monotone convergence theorem.]
Part II, Paper 1
29J Stochastic Financial Models An investor must decide how to invest his initial wealth w 0 in n assets for the coming year. At the end of the year, one unit of asset i will be worth Xi, i = 1,... , n, where X = (X 1 ,... , Xn)T^ has a multivariate normal distribution with mean μ and non-singular covariance matrix V. At the beginning of the year, one unit of asset i costs pi. In addition, he may invest in a riskless bank account; an initial investment of 1 in the bank account will have grown to 1 + r > 1 at the end of the year.
(a) The investor chooses to hold θi units of asset i, with the remaining ϕ = w 0 − θ · p in the bank account. His objective is to minimise the variance of his wealth w 1 = ϕ(1 + r) + θ · X at the end of the year, subject to a required mean value m for w 1. Derive the optimal portfolio θ∗, and the minimised variance.
(b) Describe the set A ⊆ R^2 of achievable pairs (E[w 1 ], var(w 1 )) of mean and variance of the terminal wealth. Explain what is meant by the mean-variance efficient frontier as you do so.
(c) Suppose that the investor requires expected mean wealth at time 1 to be m. He wishes to minimise the expected shortfall E[(w 1 − (1 + r)w 0 )−] subject to this requirement. Show that he will choose a portfolio corresponding to a point on the mean-variance efficient frontier.
Part II, Paper 1
30B Partial Differential Equations Consider the initial value problem for the so-called Liouville equation
ft + v · ∇xf − ∇V (x) · ∇vf = 0, (x, v) ∈ R^2 d, t ∈ R,
f (x, v, t = 0) = fI (x, v), for the function f = f (x, v, t) on R^2 d^ × R. Assume that V = V (x) is a given function with V , ∇xV Lipschitz continuous on Rd.
(i) Let fI (x, v) = δ(x − x 0 , v − v 0 ), for x 0 , v 0 ∈ Rd^ given. Show that a solution f is given by f (x, v, t) = δ(x − xˆ(t, x 0 , v 0 ), v − vˆ(t, x 0 , v 0 )), where (ˆx, ˆv) solve the Newtonian system
xˆ˙ = ˆv, xˆ(t = 0) = x 0 , vˆ^ ˙ = −∇V (ˆx), ˆv(t = 0) = v 0.
(ii) Let fI ∈ L^1 loc(R^2 d), fI > 0. Prove (by using characteristics) that f remains non- negative (as long as it exists).
(iii) Let fI ∈ Lp(R^2 d), fI > 0 on R^2 d. Show (by a formal argument) that
‖f (·, ·, t)‖Lp (^) (R 2 d) = ‖fI ‖Lp(R 2 d)
for all t ∈ R, 1 6 p < ∞.
(iv) Let V (x) = 12 |x|^2. Use the method of characteristics to solve the initial value problem for general initial data.
Part II, Paper 1 [TURN OVER
32B Integrable Systems Let H be a smooth function on a 2n–dimensional phase space with local coordinates (pj , qj ). Write down the Hamilton equations with the Hamiltonian given by H and state the Arnold–Liouville theorem. By establishing the existence of sufficiently many first integrals demonstrate that the system of n coupled harmonic oscillators with the Hamiltonian
H =
∑^ n
k=
(p^2 k + ω k^2 q^2 k),
where ω 1 ,... , ωn are constants, is completely integrable. Find the action variables for this system.
33C Principles of Quantum Mechanics The position and momentum for a harmonic oscillator can be written
ˆx =
2 mω
( a + a†^ ), ˆp =
( (^) ℏmω 2
i( a†^ − a ),
where m is the mass, ω is the frequency, and the Hamiltonian is
H =
2 m
ˆp^2 +
mω^2 ˆx^2 = ℏω
a†a +
Starting from the commutation relations for a and a†, determine the energy levels of the oscillator. Assuming a unique ground state, explain how all other energy eigenstates can be constructed from it. Consider a modified Hamiltonian
H′^ = H + λℏω ( a^2 + a†^2 ),
where λ is a dimensionless parameter. Calculate the modified energy levels to second order in λ, quoting any standard formulas which you require. Show that the modified Hamiltonian can be written as
H′^ =
2 m
αˆp^2 +
mω^2 βˆx^2 ,
where α and β depend on λ. Hence find the modified energies exactly, assuming |λ| < 12 , and show that the results are compatible with those obtained from perturbation theory.
Part II, Paper 1 [TURN OVER
34D Applications of Quantum Mechanics Consider the scaled one-dimensional Schr¨odinger equation with a potential V (x) such that there is a complete set of real, normalized bound states ψn(x), n = 0, 1 , 2 ,.. ., with discrete energies E 0 < E 1 < E 2 <.. ., satisfying
d^2 ψn dx^2
Show that the quantity
〈E〉 =
−∞
dψ dx
dx,
where ψ(x) is a real, normalized trial function depending on one or more parameters α, can be used to estimate E 0 , and show that 〈E〉 > E 0.
Let the potential be V (x) = |x|. Using a suitable one-parameter family of either Gaussian or piecewise polynomial trial functions, find a good estimate for E 0 in this case.
How could you obtain a good estimate for E 1? [ You should suggest suitable trial functions, but DO NOT carry out any further integration.]
35C Electrodynamics The action for a modified version of electrodynamics is given by
I =
d^4 x
FabF ab^ −
m^2 AaAa^ + μ 0 JaAa
where m is an arbitrary constant, Fab = ∂aAb − ∂bAa and Ja^ is a conserved current.
(i) By varying Aa, derive the field equations analogous to Maxwell’s equations by demanding that δI = 0 for an arbitrary variation δAa.
(ii) Show that ∂aAa^ = 0. (iii) Suppose that the current Ja(x) is a function of position only. Show that
Aa(x) = μ 0 4 π
d^3 x′^ Ja(x′) |x − x′|
e−m|x−x ′| .
Part II, Paper 1
