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MATHEMATICAL TRIPOS Part II
Wednesday 7 June 2006 9 to 12
PAPER 2
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 gold cover sheet to each bundle; write the code letter in the box marked ‘EXAMINER LETTER’ on the cover sheet.
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 sheets 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
1H Number Theory
Prove that all binary quadratic forms of discriminant -7 are equivalent to x^2 + xy + 2y^2.
Determine which prime numbers p are represented by x^2 + xy + 2y^2.
2G Topics in Analysis
(a) State Chebyshev’s equal ripple criterion.
(b) Let f : [− 1 , 1] → R be defined by
f (x) = cos 4πx ,
and let g be a polynomial of degree 7. Prove that there exists an x ∈ [− 1 , 1] such that |f (x) − g(x)| > 1.
3F Geometry and Groups
Determine whether the following elements of PSL 2 (R) are elliptic, parabolic, or hyperbolic. Justify your answers.
( 5 8 − 2 − 3
In the case of the first of these transformations find the fixed points.
4G Coding and Cryptography Let Σ 1 and Σ 2 be alphabets of sizes m and a. What does it mean to say that an a-ary code f : Σ 1 → Σ∗ 2 is decipherable? Show that if f is decipherable then the word lengths s 1 ,... , sm satisfy ∑m
i=
a−si^6 .
Find a decipherable binary code consisting of codewords 011, 0111, 01111, 11111, and three further codewords of length 2. How do you know the example you have given is decipherable?
Paper 2
8E Further Complex Methods
The function F (t) is defined, for Re t > −1, by
F (t) =
0
ute−u 1 + u
du
and by analytic continuation elsewhere in the complex t-plane. By considering the integral of a suitable function round a Hankel contour, obtain the analytic continuation of F (t) and hence show that singularities of F (t) can occur only at z = − 1 , − 2 , − 3 ,....
9C Classical Dynamics
Two point masses, each of mass m, are constrained to lie on a straight line and are connected to each other by a spring of force constant k. The left-hand mass is also connected to a wall on the left by a spring of force constant j. The right-hand mass is similarly connected to a wall on the right, by a spring of force constant , so that the potential energy is V = 12 k(η 1 − η 2 )^2 + 12 jη^21 + 12η^22 ,
where ηi is the distance from equilibrium of the ith^ mass. Derive the equations of motion. Find the frequencies of the normal modes.
10D Cosmology
The total energy of a gas can be expressed in terms of a momentum integral
E =
0
E(p) ¯n(p) dp ,
where p is the particle momentum, E(p) = c
p^2 + m^2 c^2 is the particle energy and ¯n(p) dp is the average number of particles in the momentum range p → p + dp. Consider particles in a cubic box of side L with p ∝ L−^1. Explain why the momentum varies as
dp dV
p 3 V
Consider the overall change in energy dE due to the volume change dV. Given that the volume varies slowly, use the thermodynamic result dE = −P dV (at fixed particle number N and entropy S) to find the pressure
P =
3 V
0
p E′(p) ¯n(p) dp.
Use this expression to derive the equation of state for an ultrarelativistic gas.
During the radiation-dominated era, photons remain in equilibrium with energy density γ ∝ T 4 and number density nγ ∝ T 3. Briefly explain why the photon temperature falls inversely with the scale factor, T ∝ a−^1. Discuss the implications for photon number and entropy conservation.
Paper 2
SECTION II
11G Topics in Analysis
(a) Let K be a closed subset of the unit disc in C. Let f : C → C be a rational function with all its poles of modulus strictly greater than 1. Explain why f can be approximated uniformly on K by polynomials. [Standard results from complex analysis may be assumed.]
(b) With K as above, define Λ to be the set of all λ ∈ C \ K such that the function (z − λ)−^1 can be uniformly approximated on K by polynomials. If λ ∈ Λ, prove that there is some δ > 0 such that μ ∈ Λ whenever |λ − μ| < δ.
12G Coding and Cryptography Define a cyclic code. Show that there is a bijection between the cyclic codes of length n, and the factors of Xn^ − 1 in F 2 [X].
If n is an odd integer then we can find a finite extension K of F 2 that contains a primitive nth root of unity α. Show that a cyclic code of length n with defining set {α, α^2 ,... , αδ−^1 } has minimum distance at least δ. Show that if n = 7 and δ = 3 then we obtain Hamming’s original code.
[You may quote a formula for the Vandermonde determinant without proof.]
Paper 2 [TURN OVER
14E Dynamical Systems
Let F : I → I be a continuous one-dimensional map of an interval I ⊂ R. Explain what is meant by saying (a) that F has a horseshoe, (b) that F is chaotic (Glendinning’s definition).
Consider the tent map defined on the interval [0, 1] by
F (x) =
μx 0 6 x < (^12) μ(1 − x) 12 6 x 6 1
with 1 < μ 6 2.
Find the non-zero fixed point x 0 and the points x− 1 < 12 < x− 2 that satisfy
F 2 (x− 2 ) = F (x− 1 ) = x 0.
Sketch a graph of F and F 2 showing the points corresponding to x− 2 , x− 1 and x 0. Hence show that F 2 has a horseshoe if μ > 21 /^2.
Explain briefly why F 4 has a horseshoe when 2^1 /^4 6 μ < 21 /^2 and why there are periodic points arbitrarily close to x 0 for μ > 21 /^2 , but no such points for 2^1 /^4 6 μ < 21 /^2.
Paper 2 [TURN OVER
15D Cosmology
(a) Consider a homogeneous and isotropic universe filled with relativistic matter of mass density ρ(t) and scale factor a(t). Consider the energy E(t) ≡ ρ(t)c^2 V (t) of a small fluid element in a comoving volume V 0 where V (t) = a^3 (t)V 0. Show that for slow (adiabatic) changes in volume, the density will satisfy the fluid conservation equation ρ˙ = − 3
a˙ a
ρ + P/c^2
where P is the pressure.
(b) Suppose that a flat (k = 0) universe is filled with two matter components: (i) radiation with an equation of state Pr = 13 ρrc^2.
(ii) a gas of cosmic strings with an equation of state Ps = − 13 ρsc^2.
Use the fluid conservation equation to show that the total relativistic mass density behaves as ρ =
ρr a^4
ρs a^2
where ρr0 and ρs0 are respectively the radiation and string densities today (that is, at t = t 0 when a(t 0 ) = 1). Assuming that both the Hubble parameter today H 0 and the ratio β ≡ ρr0/ρs0 are known, show that the Friedmann equation can be rewritten as (^) ( a˙ a
H^20
a^4
a^2 + β 1 + β
Solve this equation to find the following solution for the scale factor
a(t) =
(H 0 t)^1 /^2 (1 + β)^1 /^2
[
H 0 t + 2β^1 /^2 (1 + β)^1 /^2
] 1 / 2
Show that the scale factor has the expected asymptotic behaviour at early times t → 0.
Hence show that the age of this universe today is
t 0 = H 0 − 1 (1 + β)^1 /^2
[
(1 + β)^1 /^2 − β^1 /^2
]
and that the time teq of equal radiation and string densities (ρr = ρs) is
teq = H 0 −^1
β^1 /^2 (1 + β)^1 /^2.
Paper 2
20G Number Fields
Let K = Q(
- and let ε = 5 +
- By Dedekind’s theorem, or otherwise, show that the ideal equations
2 = [2, ε + 1]^2 , 5 = [5, ε + 1][5, ε − 1], [ε + 1] = [2, ε + 1][5, ε + 1]
hold in K. Deduce that K has class number 2.
Show that ε is the fundamental unit in K. Hence verify that all solutions in integers x, y of the equation x^2 − 26 y^2 = ±10 are given by
x +
26 y = ±εn(ε ± 1) (n = 0, ± 1 , ± 2 ,.. .).
[It may be assumed that the Minkowski constant for K is 12 .]
21H Algebraic Topology
State the simplicial approximation theorem. Compute the number of 0-simplices (vertices) in the barycentric subdivision of an n-simplex and also compute the number of n-simplices. Finally, show that there are at most countably many homotopy classes of continuous maps from the 2-sphere to itself.
22G Linear Analysis
Let X be a metric space. Define what it means for a subset E ⊂ X to be of first or second category. State and prove a version of the Baire category theorem. For 1 6 p 6 ∞, show that the set p is of first category in the normed spacer when r > p and `r is given its standard norm. What about r = p?
23F Riemann Surfaces
Define the terms Riemann surface, holomorphic map between Riemann surfaces, and biholomorphic map.
(a) Prove that if two holomorphic maps f, g coincide on a non-empty open subset of a connected Riemann surface R then f = g everywhere on R. (b) Prove that if f : R → S is a non-constant holomorphic map between Riemann surfaces and p ∈ R then there is a choice of co-ordinate charts φ near p and ψ near f (p), such that (ψ ◦ f ◦ φ−^1 )(z) = zn, for some non-negative integer n. Deduce that a holomorphic bijective map between Riemann surfaces is biholomorphic.
[The inverse function theorem for holomorphic functions on open domains in C may be used without proof if accurately stated.]
Paper 2
24H Differential Geometry
Let S ⊂ R^3 be a surface.
(a) Define the exponential map expp at a point p ∈ S. Assuming that expp is smooth, show that expp is a diffeomorphism in a neighbourhood of the origin in TpS. (b) Given a parametrization around p ∈ S, define the Christoffel symbols and show that they only depend on the coefficients of the first fundamental form.
(c) Consider a system of normal co-ordinates centred at p, that is, Cartesian co- ordinates (x, y) in TpS and parametrization given by (x, y) 7 → expp(xe 1 + ye 2 ), where {e 1 , e 2 } is an orthonormal basis of TpS. Show that all of the Christoffel symbols are zero at p.
25J Probability and Measure
(a) What is meant by saying that (Ω, A, μ) is a measure space? Your answer should include clear definitions of any terms used.
(b) Consider the following sequence of Borel-measurable functions on the measure space (R, L, λ), with the Lebesgue σ-algebra L and Lebesgue measure λ:
fn(x) =
1 /n if 0 6 x 6 en; 0 otherwise
for n ∈ N.
For each p ∈ [1, ∞], decide whether the sequence (fn)n∈N converges in Lp^ as n → ∞. Does (fn)n∈N converge almost everywhere? Does (fn)n∈N converge in measure? Justify your answers.
For parts (c) and (d), let (fn)n∈N be a sequence of real-valued, Borel-measurable functions on a probability space (Ω, A, μ).
(c) Prove that {x ∈ Ω : fn(x) converges to a finite limit} ∈ A. (d) Show that fn → 0 almost surely if and only if sup m>n
|fm| → 0 in probability.
Paper 2 [TURN OVER
28I Stochastic Financial Models
(a) In the context of a single-period financial market with n traded assets and a single riskless asset earning interest at rate r, what is an arbitrage? What is an equivalent martingale measure? Explain marginal utility pricing, and how it leads to an equivalent martingale measure.
(b) Consider the following single-period market with two assets. The first is a riskless bond, worth 1 at time 0, and 1 at time 1. The second is a share, worth 1 at time 0 and worth S 1 at time 1, where S 1 is uniformly distributed on the interval [0, a], where a > 0. Under what condition on a is this model arbitrage free? When it is, characterise the set E of equivalent martingale measures.
An agent with C^2 utility U and with wealth w at time 0 aims to pick the number θ of shares to hold so as to maximise his expected utility of wealth at time 1. Show that he will choose θ to be positive if and only if a > 2.
An option pays (S 1 − 1)+^ at time 1. Assuming that a = 2, deduce that the agent’s price for this option will be 1/4, and show that the range of possible prices for this option as the pricing measure varies in E is the interval (0, 12 ).
29I Optimization and Control A policy π is to be chosen to maximize
F (π, x) = Eπ
t> 0
βtr(xt, ut)
∣∣ x 0 =^ x
where 0 < β 6 1. Assuming that r > 0, prove that π is optimal if F (π, x) satisfies the optimality equation.
An investor receives at time t an income of xt of which he spends ut, subject to 0 6 ut 6 xt. The reward is r(xt, ut) = ut, and his income evolves as
xt+1 = xt + (xt − ut)εt,
where (εt)t> 0 is a sequence of independent random variables with common mean θ > 0. If 0 < β 6 1 /(1 + θ), show that the optimal policy is to take ut = xt for all t.
What can you say about the problem if β > 1 /(1 + θ)?
Paper 2 [TURN OVER
30A Partial Differential Equations
Define a fundamental solution of a constant-coefficient linear partial differential operator, and prove that the distribution defined by the function N : R^3 → R
N (x) = (4π|x|)−^1
is a fundamental solution of the operator −∆ on R^3.
State and prove the mean value property for harmonic functions on R^3 and deduce that any two smooth solutions of
−∆u = f , f ∈ C∞(R^3 )
which satisfy the condition lim |x|→∞
u(x) = 0
are in fact equal.
31E Integrable Systems
Let φ(t) satisfy the singular integral equation
( t^4 + t^3 − t^2
) (^) φ(t) 2
(t^4 − t^3 − t^2 ) 2 πi
C
φ(τ ) τ − t
dτ = (A − 1)t^3 + t^2 ,
where C denotes the circle of radius 2 centred on the origin,
denotes the principal value integral and A is a constant. Derive the associated Riemann–Hilbert problem, and compute the canonical solution of the corresponding homogeneous problem.
Find the value of A such that φ(t) exists, and compute the unique solution φ(t) if A takes this value.
Paper 2
33D Applications of Quantum Mechanics
State and prove Bloch’s theorem for the electron wave functions for a periodic potential V (r) = V (r + l) where l =
i ni^ ai^ is a lattice vector. What is the reciprocal lattice? Explain why the Bloch wave-vector k is arbitrary up to k → k + g, where g is a reciprocal lattice vector.
Describe in outline why one can expect energy bands En(k) = En(k + g). Explain how k may be restricted to a Brillouin zone B and show that the number of states in volume d^3 k is 2 (2π)^3
d^3 k.
Assuming that the velocity of an electron in the energy band with Bloch wave-vector k is
v(k) =
∂k
En(k) ,
show that the contribution to the electric current from a full energy band is zero. Given that n(k) = 1 for each occupied energy level, show that the contribution to the current density is then
j = −e
(2π)^3
B
d^3 k n(k)v(k) ,
where −e is the electron charge.
34D Statistical Physics What is meant by the heat capacity CV of a thermodynamic system? By establishing a suitable Maxwell identity, show that
∂CV ∂V
T
= T
∂^2 P
∂T 2
V
In a certain model of N interacting particles in a volume V and at temperature T , the partition function is
Z =
N!
(V − aN )N^ (bT )^3 N/^2 ,
where a and b are constants. Find the equation of state and the entropy for this gas of particles. Find the energy and hence the heat capacity CV of the gas, and verify that the relation (∗) is satisfied.
Paper 2
35A General Relativity
The Schwarzschild metric is
ds^2 =
2 M
r
dr^2 + r^2
dθ^2 + sin^2 θ dφ^2
2 M
r
dt^2.
Writing u = 1/r, obtain the equation
d^2 u dφ^2
determining the spatial orbit of a null (massless) particle moving in the equatorial plane θ = π/2.
Verify that two solutions of (∗) are
(i) u =
3 M
, and
(ii) u =
3 M
M
cosh φ + 1
What is the significance of solution (i)? Sketch solution (ii) and describe its relation to solution (i).
Show that, near φ = cosh−^1 2, one may approximate the solution (ii) by
r sin(φ − cosh−^1 2) ≈
27 M ,
and hence obtain the impact parameter.
Paper 2 [TURN OVER
