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MATHEMATICAL TRIPOS Part II
Thursday, 3 June, 2010 1:30 pm to 4:30 pm
PAPER 3
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 (i) Let M and N be positive integers, such that N is not a perfect square. If M <
N , show that every solution of the equation
x 2 − N y 2 = M
in positive integers x, y comes from some convergent of the continued fraction of
N.
(ii) Find a solution in positive integers x, y of
x 2 − 29 y 2 = 5.
2F Topics in Analysis Let A = {z ∈ C : 1/ 2 6 |z| 6 2 } and suppose that f is complex analytic on an open subset containing A.
(i) Give an example, with justification, to show that there need not exist a sequence of complex polynomials converging to f uniformly on A.
(ii) Let R ⊂ C be the positive real axis and B = A \ R. Prove that there exists a sequence of complex polynomials p 1 , p 2 , p 3 ,... such that pj → f uniformly on each compact subset of B.
(iii) Let p 1 , p 2 , p 3 ,... be the sequence of polynomials in (ii). If this sequence converges uniformly on A, show that
C f^ (z)^ dz^ = 0 , where^ C^ =^ {z^ ∈^ C^ :^ |z|^ = 1}^.
3F Geometry of Group Actions Let U be a “triangular” region in the unit disc D bounded by three hyperbolic geodesics γ 1 , γ 2 , γ 3 that do not meet in D nor on its boundary. Let Jk be inversion in γk and set A = J 2 ◦ J 1 ; B = J 3 ◦ J 2.
Let G be the group generated by the M¨obius transformations A and B. Describe briefly a fundamental set for the group G acting on D.
Prove that G is a free group on the two generators A and B. Describe the quotient surface D/G.
Part II, Paper 3
7D Dynamical Systems Let I = [ 0, 1). The sawtooth (Bernoulli shift) map F : I → I is defined by
F (x) = 2 x [ mod 1 ].
Describe the effect of F using binary notation. Show that F is continuous on I except at x = 12. Show also that F has N -periodic points for all N > 2. Are they stable?
Explain why F is chaotic, using Glendinning’s definition.
8E Further Complex Methods Let Γ(z) and ζ(z) denote the gamma and the zeta functions respectively, namely
Γ(z) =
0
x z−^1 e−x^ dx , Re z > 0 ,
ζ(z) =
∑^ ∞
m=
mz^
, Re z > 1.
By employing a series expansion of (1 − e−x)−^2 , prove the following identity ∫ (^) ∞
0
xz (ex^ − 1)^2
dx = Γ(z + 1)
[
ζ(z) − ζ(z + 1)
]
, Re z > 1.
Part II, Paper 3
9D Classical Dynamics Euler’s equations for the angular velocity ω = (ω 1 , ω 2 , ω 3 ) of a rigid body, viewed in the body frame, are I 1 dω 1 dt = (I 2 − I 3 ) ω 2 ω 3
and cyclic permutations, where the principal moments of inertia are assumed to obey I 1 < I 2 < I 3.
Write down two quadratic first integrals of the motion.
There is a family of solutions ω(t), unique up to time–translations t → (t − t 0 ), which obey the boundary conditions ω → (0, Ω, 0) as t → −∞ and ω → (0, −Ω, 0) as t → ∞ , for a given positive constant Ω. Show that, for such a solution, one has
L^2 = 2EI 2 ,
where L is the angular momentum and E is the kinetic energy.
By eliminating ω 1 and ω 3 in favour of ω 2 , or otherwise, show that, in this case, the second Euler equation reduces to ds dτ
= 1 − s^2 ,
where s = ω 2 /Ω and τ = Ωt
[
(I 1 − I 2 )(I 2 − I 3 )/I 1 I 3
] 1 / 2
. Find the general solution s(τ ).
[You are not expected to calculate ω 1 (t) or ω 3 (t).]
10D Cosmology Consider a homogenous and isotropic universe with mass density ρ(t), pressure P (t) and scale factor a(t). As the universe expands its energy changes according to the relation dE = −P dV. Use this to derive the fluid equation
ρ˙ = − 3 a˙ a
ρ +
P
c 2
Use conservation of energy applied to a test particle at the boundary of a spherical fluid element to derive the Friedmann equation
( a˙ a
8 π 3 Gρ − k a^2 c^2 ,
where k is a constant. State any assumption you have made. Briefly state the significance of k.
Part II, Paper 3 [TURN OVER
13A Mathematical Biology Consider an epidemic model in which S(x, t) is the local population density of susceptibles and I(x, t) is the density of infectives
∂S ∂t
= − rIS , ∂I ∂t
= D
∂ 2 I
∂x 2
where r, a, and D are positive. If S 0 is a characteristic population value, show that the rescalings I/S 0 → I, S/S 0 → S, (rS 0 /D)^1 /^2 x → x, rS 0 t → t reduce this system to
∂S ∂t
= − IS ,
∂I
∂t
∂ 2 I
∂x 2
where λ should be found.
Travelling wavefront solutions are of the form S(x, t) = S(z), I(x, t) = I(z), where z = x − ct and c is the wave speed, and we seek solutions with boundary conditions S(∞) = 1, S′(∞) = 0, I(∞) = I(−∞) = 0. Under the travelling-wave assumption reduce the rescaled PDEs to ODEs, and show by linearisation around the leading edge of the advancing front that the requirement that I be non-negative leads to the condition λ < 1 and hence the wave speed relation
c > 2(1 − λ)^1 /^2 , λ < 1.
Using the two ODEs you have obtained, show that the surviving susceptible population fraction σ = S(−∞) after the passage of the front satisfies
σ − λ ln σ = 1 ,
and sketch σ as a function of λ.
14D Dynamical Systems Describe informally the concepts of extended stable manifold theory. Illustrate your discussion by considering the 2-dimensional flow
x˙ = μx + xy − x^3 , y˙ = −y + y^2 − x^2 ,
where μ is a parameter with |μ| ≪ 1, in a neighbourhood of the origin. Determine the nature of the bifurcation.
Part II, Paper 3 [TURN OVER
15D Cosmology The number density for particles in thermal equilibrium, neglecting quantum effects, is
n = gs 4 π h^3
p^2 dp exp(−(E(p) − μ)/kT ) ,
where gs is the number of degrees of freedom for the particle with energy E(p) and μ is its chemical potential. Evaluate n for a non-relativistic particle.
Thermal equilibrium between two species of non-relativistic particles is maintained by the reaction a + α ↔ b + β ,
where α and β are massless particles. Evaluate the ratio of number densities na/nb given that their respective masses are ma and mb and chemical potentials are μa and μb.
Explain how a reaction like the one above is relevant to the determination of the neutron to proton ratio in the early universe. Why does this ratio not fall rapidly to zero as the universe cools?
Explain briefly the process of primordial nucleosynthesis by which neutrons are converted into stable helium nuclei. Letting
YHe = ρHe/ρ
be the fraction of the universe’s helium, compute YHe as a function of the ratio r = nn/np at the time of nucleosynthesis.
16G Logic and Set Theory Define the sets Vα , α ∈ ON. What is meant by the rank of a set?
Explain briefly why, for every α , there exists a set of rank α.
Let x be a transitive set of rank α. Show that x has an element of rank β for every β < α.
For which α does there exist a finite set of rank α? For which α does there exist a finite transitive set of rank α? Justify your answers.
[Standard properties of rank may be assumed.]
Part II, Paper 3
20H Algebraic Topology Suppose X is a finite simplicial complex and that H∗(X) is a free abelian group for each value of ∗. Using the Mayer-Vietoris sequence or otherwise, compute H∗(S^1 × X) in terms of H∗(X). Use your result to compute H∗(T n).
[Note that T n^ = S^1 ×... × S^1 , where there are n factors in the product.]
21H Linear Analysis State and prove the Stone-Weierstrass theorem for real-valued functions.
[You may use without proof the fact that the function s → |s| can be uniformly approximated by polynomials on [− 1 , 1].]
22G Riemann Surfaces Show that the analytic isomorphisms (i.e. conformal equivalences) of the Riemann sphere C∞ to itself are given by the non-constant M¨obius transformations.
State the Riemann–Hurwitz formula for a non-constant analytic map between compact Riemann surfaces, carefully explaining the terms which occur.
Suppose now that f : C∞ → C∞ is an analytic map of degree 2; show that there exist M¨obius transformations S and T such that
Sf T : C∞ → C∞
is the map given by z 7 → z^2.
23G Algebraic Geometry (i) Let X be a curve, and p ∈ X be a smooth point on X. Define what a local parameter at p is.
Now let f : X 99K Y be a rational map to a quasi-projective variety Y. Show that if Y is projective, f extends to a morphism defined at p.
Give an example where this fails if Y is not projective, and an example of a morphism f : C 2 \ { 0 } → P 1 which does not extend to 0.
(ii) Let V = Z(X 08 + X 18 + X 28 ) and W = Z(X 04 + X 14 + X 24 ) be curves in P^2 over a field of characteristic not equal to 2. Let φ : V → W be the map [X 0 : X 1 : X 2 ] 7 → [X 02 : X 12 : X 22 ]. Determine the degree of φ, and the ramification ep for all p ∈ V.
Part II, Paper 3
24H Differential Geometry (i) State and prove the Theorema Egregium.
(ii) Define the notions principal curvatures, principal directions and umbilical point.
(iii) Let S ⊂ R^3 be a connected compact regular surface (without boundary), and let D ⊂ S be a dense subset of S with the following property. For all p ∈ D, there exists an open neighbourhood Up of p in S such that for all θ ∈ [ 0, 2 π), ψp,θ(Up) = Up , where ψp,θ : R^3 → R^3 denotes rotation by θ around the line through p perpendicular to Tp S. Show that S is in fact a sphere.
25I Probability and Measure Let (Xn : n ∈ N) be a sequence of independent random variables with common density function f (x) = 1 π(1 + x^2 )
Fix α ∈ [0, 1] and set
Yn = sgn(Xn)|Xn|α, Sn = Y 1 +... + Yn.
Show that for all α ∈ [0, 1] the sequence of random variables Sn/n converges in distribution and determine the limit. [Hint: In the case α = 1 it may be useful to prove that E(eiuX^1 ) = e−|u|, for all u ∈ R.] Show further that for all α ∈ [0, 1 /2) the sequence of random variables Sn/
n converges in distribution and determine the limit. [You should state clearly any result about random variables from the course to which you appeal. You are not expected to evaluate explicitly the integral
m(α) =
0
xα π(1 + x^2 )
dx. ]
Part II, Paper 3 [TURN OVER
27J Principles of Statistics Define the normal and extensive form solutions of a Bayesian statistical decision problem involving parameter Θ, random variable X, and loss function L(θ, a). How are they related? Let R 0 = R 0 (Π) be the Bayes loss of the optimal act when Θ ∼ Π and no data can be observed. Express the Bayes risk R 1 of the optimal statistical decision rule in terms of R 0 and the joint distribution of (Θ, X).
The real parameter Θ has distribution Π, having probability density function π(·). Consider the problem of specifying a set S ⊆ R such that the loss when Θ = θ is L(θ, S) = c |S| − (^1) S (θ), where (^1) S is the indicator function of S, where c > 0, and where |S| =
S dx. Show that the “highest density” region^ S
∗ (^) := {θ : π(θ) > c} supplies a Bayes act for this decision problem, and explain why R 0 (Π) 6 0.
For the case Θ ∼ N (μ, σ^2 ), find an expression for R 0 in terms of the standard normal distribution function Φ.
Suppose now that c = 0.5 , that Θ ∼ N (0, 1) and that X|Θ ∼ N (Θ, 1 /9). Show that R 1 < R 0.
Part II, Paper 3 [TURN OVER
28J Optimization and Control Consider an infinite-horizon controlled Markov process having per-period costs c(x, u) > 0, where x ∈ X is the state of the system, and u ∈ U is the control. Costs are discounted at rate β ∈ (0, 1], so that the objective to be minimized is
E
[ ∑
t> 0
βtc(Xt, ut)
X 0 = x
]
What is meant by a policy π for this problem?
Let L denote the dynamic programming operator
Lf (x) ≡ inf u∈U
c(x, u) + βE
[
f (X 1 )
X 0 = x, u 0 = u
] }
Further, let F denote the value of the optimal control problem:
F (x) = inf π Eπ
[ ∑
t> 0
βtc(Xt, ut)
∣ (^) X 0 = x
]
where the infimum is taken over all policies π, and Eπ^ denotes expectation under policy π. Show that the functions Ft defined by
Ft+1 = LFt (t > 0), F 0 ≡ 0
increase to a limit F∞ ∈ [0, ∞]. Prove that F∞ 6 F. Prove that F = LF.
Suppose that Φ = LΦ > 0. Prove that Φ > F.
[You may assume that there is a function u∗ : X → U such that
LΦ(x) = c(x, u∗(x)) + βE
[
Φ(X 1 )
∣ (^) X 0 = x, u 0 = u∗(x)
]
though the result remains true without this simplifying assumption.]
Part II, Paper 3
31C Asymptotic Methods Consider the ordinary differential equation
y′′^ = (|x| − E) y ,
subject to the boundary conditions y(±∞) = 0. Write down the general form of the Liouville-Green solutions for this problem for E > 0 and show that asymptotically the eigenvalues En, n ∈ N and En < En+1, behave as En = O(n^2 /^3 ) for large n.
32E Integrable Systems Consider a vector field
V = α x
∂x
∂t
∂v
on R^3 , where α, β and γ are constants. Find the one-parameter group of transformations generated by this vector field.
Find the values of the constants (α, β, γ) such that V generates a Lie point symmetry of the modified KdV equation (mKdV)
vt − 6 v^2 vx + vxxx = 0 , where v = v(x, t).
Show that the function u = u(x, t) given by u = v^2 + vx satisfies the KdV equation and find a Lie point symmetry of KdV corresponding to the Lie point symmetry of mKdV which you have determined from V.
Part II, Paper 3
33C Principles of Quantum Mechanics What are the commutation relations between the position operator ˆx and momen- tum operator ˆp? Show that this is consistent with ˆx, pˆ being hermitian. The annihilation operator for a harmonic oscillator is
a =
(ˆx + ipˆ)
in units where the mass and frequency of the oscillator are 1. Derive the relation [a, a†] = 1. Write down an expression for the Hamiltonian
H = 12 pˆ^2 + 12 xˆ^2
in terms of the operator N = a†a.
Assume there exists a unique ground state | 0 〉 of H such that a| 0 〉 = 0. Explain how the space of eigenstates |n〉, is formed, and deduce the energy eigenvalues for these states. Show that a|n〉 = A|n− 1 〉 , a†|n〉 = B|n+1〉 , finding A and B in terms of n.
Calculate the energy eigenvalues of the Hamiltonian for two harmonic oscillators
H = H 1 + H 2 , Hi = 12 pˆi^2 + 12 ˆxi^2 , i = 1, 2.
What is the degeneracy of the nth^ energy level? Suppose that the two oscillators are then coupled by adding the extra term
∆H = λˆx 1 ˆx 2
to H, where λ ≪ 1. Calculate the energies for the states of the unperturbed system with the three lowest energy eigenvalues to first order in λ using perturbation theory.
[You may assume standard perturbation theory results.]
Part II, Paper 3 [TURN OVER
35C Statistical Physics
(i) Given the following density of states for a particle in 3 dimensions
g(ε) = KV ε 1 /^2
write down the partition function for a gas of N such non-interacting particles, assuming they can be treated classically. From this expression, calculate the energy E of the system and the heat capacities CV and CP. You may take it as given that P V = 23 E. [Hint: The formula
0 dy y
(^2) e−y^2 = √π/ 4 may be useful.]
(ii) Using thermodynamic relations obtain the relation between heat capacities and compressibilities CP CV
κT κS where the isothermal and adiabatic compressibilities are given by
κ = −
V
∂V
∂P
derivatives taken at constant temperature and entropy, respectively.
(iii) Find κT and κS for the ideal gas considered above.
Part II, Paper 3 [TURN OVER
36B Electrodynamics A particle of rest-mass m, electric charge q, is moving relativistically along the path xμ(s) where s parametrises the path.
Write down an action for which the extremum determines the particle’s equation of motion in an electromagnetic field given by the potential Aμ(x).
Use your action to derive the particle’s equation of motion in a form where s is the proper time.
Suppose that the electric and magnetic fields are given by
E = (0, 0 , E) , B = (0, B, 0).
where E and B are constants and B > E > 0.
Find xμ(s) given that the particle starts at rest at the origin when s = 0.
Describe qualitatively the motion of the particle.
Part II, Paper 3
