Mathematical Problems from a Paper in Statistics, Dynamical Systems, and Cosmology, Exams of Mathematics

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MATHEMATICAL TRIPOS Part II

Thursday, 4 June, 2009 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 For any integer x > 2, define θ(x) =

p 6 x log^ p, where the sum is taken over all primes p 6 x. Put θ(1) = 0. By studying the integer ( 2 n n

where n > 1 is an integer, prove that

θ(2n) − θ(n) < 2 n log 2.

Deduce that θ(x) < (4 log 2)x,

for all x > 1.

2F Topics in Analysis (a) If f : (0, 1) → R is continuous, prove that there exists a sequence of polynomials Pn such that Pn → f uniformly on compact subsets of (0, 1).

(b) If f : (0, 1) → R is continuous and bounded, prove that there exists a sequence of polynomials Qn such that Qn are uniformly bounded on (0, 1) and Qn → f uniformly on compact subsets of (0, 1).

3F Geometry of Group Actions Explain why there are discrete subgroups of the M¨obius group PSL 2 (C) which ab- stractly are free groups of rank 2.

4H Coding and Cryptography Define a binary code of length 15 with information rate 11/15 which will correct single errors. Show that it has the rate stated and give an explicit procedure for identifying the error. Show that the procedure works.

[Hint: You may wish to imitate the corresponding discussion for a code of length 7 .]

Part II, Paper 3

8B Further Complex Methods Suppose that the real function u(x, y) satisfies Laplace’s equation in the upper half complex z-plane, z = x + iy, x ∈ R, y > 0, where

u(x, y) → 0 as

x^2 + y^2 → ∞, u(x, 0) = g(x), x ∈ R.

The function u(x, y) can then be expressed in terms of the Poisson integral

u(x, y) =

π

−∞

yg(ξ) (x − ξ)^2 + y^2

dξ, x ∈ R, y > 0.

By employing the formula

f (z) = 2u

z + ¯a 2

z − ¯a 2 i

− f (a),

where a is a complex constant with Im a > 0, show that the analytic function whose real part is u(x, y) is given by

f (z) =

iπ

−∞

g(ξ) ξ − z

dξ + ic, Im z > 0 ,

where c is a real constant.

9E Classical Dynamics (a) Show that the principal moments of inertia of a uniform circular cylinder of radius a, length h and mass M about its centre of mass are I 1 = I 2 = M (a^2 /4 + h^2 /12) and I 3 = M a^2 /2, with the x 3 axis being directed along the length of the cylinder.

(b) Euler’s equations governing the angular velocity (ω 1 , ω 2 , ω 3 ) of an arbitrary rigid body as viewed in the body frame are

I 1

dω 1 dt

= (I 2 − I 3 )ω 2 ω 3 ,

I 2

dω 2 dt = (I 3 − I 1 )ω 3 ω 1

and

I 3

dω 3 dt = (I 1 − I 2 )ω 1 ω 2.

Show that, for the cylinder of part (a), ω 3 is constant. Show further that, when ω 3 6 = 0, the angular momentum vector precesses about the x 3 axis with angular velocity Ω given by

3 a^2 − h^2 3 a^2 + h^2

ω 3.

Part II, Paper 3

10D Cosmology (a) Write down an expression for the total gravitational potential energy Egrav of a spherically symmetric star of outer radius R in terms of its mass density ρ(r) and the total mass m(r) inside a radius r, satisfying the relation dm/dr = 4πr^2 ρ(r). An isotropic mass distribution obeys the pressure-support equation,

dP dr

Gmρ r^2

where P (r) is the pressure. Multiply this expression by 4πr^3 and integrate with respect to r to derive the virial theorem relating the kinetic and gravitational energy of the star

Ekin = − 12 Egrav ,

where you may assume for a non-relativistic ideal gas that Ekin = 32 〈P 〉V , with 〈P 〉 the average pressure.

(b) Consider a white dwarf supported by electron Fermi degeneracy pressure P ≈ h^2 n^5 /^3 /me, where me is the electron mass and n is the number density. Assume a uniform density ρ(r) = mpn(r) ≈ mp〈n〉, so the total mass of the star is given by M = (4π/3)〈n〉mp R^3 where mp is the proton mass. Show that the total energy of the white dwarf can be written in the form

Etotal = Ekin + Egrav = α R^2

β R

where α, β are positive constants which you should specify. Deduce that the white dwarf has a stable radius RWD at which the energy is minimized, that is,

RWD ∼

h^2 M −^1 /^3 Gmem^5 p/^3

Part II, Paper 3 [TURN OVER

14E Dynamical Systems Consider the dynamical system

x˙ = −ax − 2 xy, y ˙ = x^2 + y^2 − b,

where a > 0 and b > 0. (i) Find and classify the fixed points. Show that a bifurcation occurs when 4 b = a^2 > 0. (ii) After shifting coordinates to move the relevant fixed point to the origin, and setting a = 2

b − μ, carry out an extended centre manifold calculation to reduce the two-dimensional system to one of the canonical forms, and hence determine the type of bifurcation that occurs when 4b = a^2 > 0. Sketch phase portraits in the cases 0 < a^2 < 4 b and 0 < 4 b < a^2. (iii) Sketch the phase portrait in the case a = 0. Prove that periodic orbits exist if and only if a = 0.

Part II, Paper 3 [TURN OVER

15D Cosmology In the Zel’dovich approximation, particle trajectories in a flat expanding universe are described by r(q, t) = a(t)[q+Ψ(q, t)] , where a(t) is the scale factor of the universe, q is the unperturbed comoving trajectory and Ψ is the comoving displacement. The particle equation of motion is

¨r = −∇Φ −

ρ

∇P ,

where ρ is the mass density, P is the pressure (P ≪ ρc^2 ) and Φ is the Newtonian potential which satisfies the Poisson equation ∇^2 Φ = 4π Gρ.

(i) Show that the fractional density perturbation and the pressure gradient are given by

δ ≡ ρ − ρ¯ ¯ρ

≈ −∇q · Ψ , ∇P ≈ −¯ρ c^2 s a

∇^2 q Ψ ,

where ∇q has components ∂/∂qi, ¯ρ = ¯ρ(t) is the homogeneous background density and c^2 s ≡ ∂P/∂ρ is the sound speed. [You may assume that the Jacobian |∂ri/∂qj |−^1 = |a δij + a ∂ψi/∂qj |−^1 ≈ a−^3 (1 − ∇q · Ψ) for |Ψ| ≪ |q| .]

Use this result to integrate the Poisson equation once and obtain then the evolution equation for the comoving displacement:

Ψ¨ + 2 a˙ a

Ψ˙ − 4 π Gρ¯ Ψ − c

2 s a^2

∇^2 q Ψ = 0 ,

[You may assume that the integral of ∇^2 Φ = 4πG ¯ρ is ∇Φ = 4π G¯ρr/3 , that Ψ is irrotational and that the Raychaudhuri equation is ¨a/a ≈ − 4 π G¯ρ/3 for P ≪ ρc^2 .]

Consider the Fourier expansion δ(x, t) =

k δk^ exp(i^ k^ ·^ x) of the density perturba- tion using the comoving wavenumber k (k = |k|) and obtain the evolution equation for the mode δk: ¨δk + 2 a˙ a δ˙k − (4π G¯ρ − c^2 s k^2 /a^2 ) δk = 0. (∗)

(ii) Consider a flat matter-dominated universe with a(t) = (t/t 0 )^2 /^3 (background density ¯ρ = 1/(6π Gt^2 )) and with an equation of state P = βρ^4 /^3 to show that (∗) becomes

¨δk +^4 3 t

δ˙k − 1 t^2

( 23 − ¯v^2 s k^2 ) δk = 0 ,

where the constant ¯v^2 s ≡ (4β/3)(6π G)−^1 /^3 t^40 / 3. Seek power law solutions of the form δk ∝ tα^ to find the growing and decaying modes

δk = Ak tn+^ + Bk tn−^ where n± = − 16 ±

[

( 56 )^2 − ¯v^2 s k^2

] 1 / 2

Part II, Paper 3

18H Galois Theory Let K = Fp(x), the function field in one variable, and let G = Fp. The group G acts as automorphisms of K by σa(x) = x + a. Show that KG^ = Fp(y), where y = xp^ − x.

[State clearly any theorems you use.]

Is K/KG^ a separable extension? Now let H =

d a 0 1

: a ∈ Fp, d ∈ F∗ p

and let H act on K by

d a 0 1

x = dx + a. (The group structure on H is given by matrix

multiplication.) Compute KH^. Describe your answer in the form Fp(z) for an explicit z ∈ K.

Is KG/KH^ a Galois extension? Find the minimum polynomial for y over the field KH^.

19F Representation Theory Let G = SU(2). Let Vn be the complex vector space of homogeneous polynomials of degree n in two variables z 1 , z 2. Define the usual left action of G on Vn and denote by ρn : G → GL(Vn) the representation induced by this action. Describe the character χn afforded by ρn. Quoting carefully any results you need, show that (i) The representation ρn has dimension n + 1 and is irreducible for n ∈ Z> 0 ; (ii) Every finite-dimensional continuous irreducible representation of G is one of the ρn; (iii) Vn is isomorphic to its dual V (^) n∗.

Part II, Paper 3

20G Algebraic Topology (i) Suppose that (C, d) and (C′, d′) are chain complexes, and f, g : C → C′^ are chain maps. Define what it means for f and g to be chain homotopic. Show that if f and g are chain homotopic, and f∗, g∗ : H∗(C) → H∗(C′) are the induced maps, then f∗ = g∗. (ii) Define the Euler characteristic of a finite chain complex. Given that one of the sequences below is exact and the others are not, which is the exact one?

0 → Z^11 → Z^24 → Z^20 → Z^13 → Z^20 → Z^25 → Z^11 → 0 ,

0 → Z^11 → Z^24 → Z^20 → Z^13 → Z^20 → Z^24 → Z^11 → 0 , 0 → Z^11 → Z^24 → Z^19 → Z^13 → Z^20 → Z^23 → Z^11 → 0.

Justify your choice.

21H Linear Analysis (a) State the Arzela–Ascoli theorem, explaining the meaning of all concepts involved. (b) Prove the Arzela–Ascoli theorem. (c) Let K be a compact topological space. Let (fn)n∈N be a sequence in the Banach space C(K) of real-valued continuous functions over K equipped with the supremum norm ‖ · ‖. Assume that for every x ∈ K, the sequence fn(x) is monotone increasing and that fn(x) → f (x) for some f ∈ C(K). Show that ‖fn − f ‖ → 0 as n → ∞.

22G Riemann Surfaces (i) Let f (z) =

n=1 z 2 n. Show that the unit circle is the natural boundary of the function element (D(0, 1), f ), where D(0, 1) = {z ∈ C : |z| < 1 }. (ii) Let X be a connected Riemann surface and (D, h) a function element on X into C. Define a germ of (D, h) at a point p ∈ D. Let G be the set of all the germs of function elements on X into C. Describe the topology and the complex structure on G, and show that G is a covering of X (in the sense of complex analysis). Show that there is a one- to-one correspondence between complete holomorphic functions on X into C and the con- nected components of G. [You are not required to prove that the topology on G is second- countable.]

Part II, Paper 3 [TURN OVER

26J Applied Probability (a) Define the Poisson process (Nt, t > 0) with rate λ > 0, in terms of its holding times. Show that for all times t > 0, Nt has a Poisson distribution, with a parameter which you should specify. (b) Let X be a random variable with probability density function

f (x) =

λ^3 x^2 e−λx (^1) {x> 0 }. (∗)

Prove that X is distributed as the sum Y 1 + Y 2 + Y 3 of three independent exponential random variables of rate λ. Calculate the expectation, variance and moment generating function of X. Consider a renewal process (Xt, t > 0) with holding times having density (∗). Prove that the renewal function m(t) = E(Xt) has the form

m(t) = λt 3

p 1 (t) −

p 2 (t),

where p 1 (t) = P

Nt = 1 mod 3

, p 2 (t) = P

Nt = 2 mod 3

and (Nt, t > 0) is the Poisson process of rate λ. (c) Consider the delayed renewal process

X tD , t > 0

with holding times S 1 D , S 2 , S 3 ,... where (Sn, n > 1), are the holding times of (Xt, t > 0) from (b). Specify the distribution of S 1 D for which the delayed process becomes the renewal process in equilibrium. [You may use theorems from the course provided that you state them clearly.]

27I Principles of Statistics What is meant by an equaliser decision rule? What is meant by an extended Bayes rule? Show that a decision rule that is both an equaliser rule and extended Bayes is minimax. Let X 1 ,... , Xn be independent and identically distributed random variables with the normal distribution N (θ, h−^1 ), and let k > 0. It is desired to estimate θ with loss function L(θ, a) = 1 − exp{− 12 k(a − θ)^2 }. Suppose the prior distribution is θ ∼ N (m 0 , h− 0 1 ). Find the Bayes act and the Bayes loss posterior to observing X 1 = x 1 ,... , Xn = xn. What is the Bayes risk of the Bayes rule with respect to this prior distribution? Show that the rule that estimates θ by X^ =^ n−^1

∑n i=1 Xi^ is minimax.

Part II, Paper 3 [TURN OVER

28I Optimization and Control Two scalar systems have dynamics

xt+1 = xt + ut + ǫt, yt+1 = yt + wt + ηt,

where {ǫt} and {ηt} are independent sequences of independent and identically distributed random variables of mean 0 and variance 1. Let

F (x) = inf π E

[ ∞

t=

x^2 t + u^2 t

(2/3)t

x 0 = x

]

where π is a policy in which ut depends on only x 0 ,... , xt.

Show that G(x) = P x^2 +d is a solution to the optimality equation satisfied by F (x), for some P and d which you should find.

Find the optimal controls. State a theorem that justifies F (x) = G(x). For each of the two cases (a) λ = 0 and (b) λ = 1, find controls {ut, wt} which minimize

E

[ ∞

t=

x^2 t + 2λxtyt + y t^2 + u^2 t + w^2 t

(2/3 + λ/12)t

x 0 = x , y 0 = y

]

29J Stochastic Financial Models What is a Brownian motion? State the assumptions of the Black–Scholes model of an asset price, and derive the time-0 price of a European call option struck at K, and expiring at T.

Find the time-0 price of a European call option expiring at T , but struck at St, where t ∈ (0, T ), and St is the price of the underlying asset at time t.

Part II, Paper 3

31A Asymptotic Methods Consider the contour-integral representation

J 0 (x) = Re

iπ

C

eix^ cosh^ t^ dt

of the Bessel function J 0 for real x, where C is any contour from −∞ − iπ 2 to +∞ + iπ 2.

Writing t = u + iv, give in terms of the real quantities u, v the equation of the steepest-descent contour from −∞ − iπ 2 to +∞ + iπ 2 which passes through t = 0.

Deduce the leading term in the asymptotic expansion of J 0 (x), valid as x → ∞

J 0 (x) ∼

πx cos

x − π 4

32B Integrable Systems Consider the partial differential equation

∂u ∂t = un^

∂u ∂x

∂^2 k+1u ∂x^2 k+^

where u = u(x, t) and k, n are non-negative integers.

(i) Find a Lie point symmetry of (∗) of the form

(x, t, u) −→ (αx, βt, γu), (∗∗)

where (α, β, γ) are non-zero constants, and find a vector field generating this symmetry. Find two more vector fields generating Lie point symmetries of (∗) which are not of the form (∗∗) and verify that the three vector fields you have found form a Lie algebra.

(ii) Put (∗) in a Hamiltonian form.

Part II, Paper 3

33C Principles of Quantum Mechanics (i) Consider two quantum systems with angular momentum states | j m 〉 and | 1 q 〉. The eigenstates corresponding to their combined angular momentum can be written as

| J M 〉 =

q m

Cq mJ M | 1 q 〉| j m 〉 ,

where Cq mJ M are Clebsch–Gordan coefficients for addition of angular momenta one and j. What are the possible values of J and how must q, m and M be related for Cq mJ M 6 = 0? Construct all states | J M 〉 in terms of product states in the case j = 12.

(ii) A general stationary state for an electron in a hydrogen atom |n ℓ m〉 is specified by the principal quantum number n in addition to the labels ℓ and m corresponding to the total orbital angular momentum and its component in the 3-direction (electron spin is ignored). An oscillating electromagnetic field can induce a transition to a new state |n′^ ℓ′^ m′〉 and, in a suitable approximation, the amplitude for this to occur is proportional to 〈n′^ ℓ′^ m′| ˆxi |n ℓ m〉 , where ˆxi (i = 1, 2 , 3) are components of the electron’s position. Give clear but concise arguments based on angular momentum which lead to conditions on ℓ, m, ℓ′, m′^ and i for the amplitude to be non-zero. Explain briefly how parity can be used to obtain an additional selection rule.

[Standard angular momentum states | j m 〉 are joint eigenstates of J^2 and J 3 , obeying

J±| j m 〉 =

(j∓m)(j±m+1) | j m± 1 〉, J 3 | j m 〉 = m| j m 〉.

You may also assume that X± 1 = √^12 (∓ˆx 1 −^ ixˆ 2 ) and^ X 0 = ˆx 3 have commutation relations with orbital angular momentum L given by

[L 3 , Xq ] = qXq , [L±, Xq ] =

(1∓q)(2±q) Xq± 1.

Units in which ℏ = 1 are to be used throughout. ]

Part II, Paper 3 [TURN OVER

35D Statistical Physics Consider an ideal Bose gas in an external potential such that the resulting density of single particle states is given by

g(ε) = B ε^7 /^2 ,

where B is a positive constant. (i) Derive an expression for the critical temperature for Bose–Einstein condensation of a gas of N of these atoms. [Recall 1 Γ(n)

0

xn−^1 dx z−^1 ex^ − 1

∑^ ∞

ℓ=

zℓ ℓn^

]

(ii) What is the internal energy E of the gas in the condensed state as a function of N and T? (iii) Now consider the high temperature, classical limit instead. How does the internal energy E depend on N and T?

36C Electrodynamics A particle of charge of q moves along a trajectory ya(s) in spacetime where s is the proper time on the particle’s world-line. Explain briefly why, in the gauge ∂aAa^ = 0, the potential at the spacetime point x is given by

Aa(x) = μ 0 q 2 π

ds dya ds

θ

x^0 − y^0 (s)

δ

(xc^ − yc(s))(xd^ − yd(s))ηcd

Evaluate this integral for a point charge moving relativistically along the z-axis, x = y = 0, at constant velocity v so that z = vt. Check your result by starting from the potential of a point charge at rest

A = 0, φ = μ 0 q 4 π(x^2 + y^2 + z^2 )^1 /^2

and making an appropriate Lorentz transformation.

Part II, Paper 3 [TURN OVER

37E Fluid Dynamics II An axisymmetric incompressible Stokes flow has the Stokes stream function Ψ(R, θ) in spherical polar coordinates (R, θ, φ). Give expressions for the components uR and uθ of the flow field in terms of Ψ, and show that

∇ × u =

D^2 Ψ

R sin θ

where

D^2 Ψ =

∂^2 Ψ

∂R^2

sin θ R^2

∂θ

sin θ

∂θ

Write down the equation satisfied by Ψ.

Verify that the Stokes stream function

Ψ(R, θ) =

U sin^2 θ

R^2 −

aR +

a^3 R

represents the Stokes flow past a stationary sphere of radius a, when the fluid at large distance from the sphere moves at speed U along the axis of symmetry.

A sphere of radius a moves vertically upwards in the z direction at speed U through fluid of density ρ and dynamic viscosity μ, towards a free surface at z = 0. Its distance d from the surface is much greater than a. Assuming that the surface remains flat, show that the conditions of zero vertical velocity and zero tangential stress at z = 0 can be approximately met for large d/a by combining the Stokes flow for the sphere with that of an image sphere of the same radius located symmetrically above the free surface. Hence determine the leading-order behaviour of the horizontal flow on the free surface as a function of r, the horizontal distance from the sphere’s centre line.

What is the size of the next correction to your answer as a power of a/d? [Detailed calculation is not required.]

[Hint: For an axisymmetric vector field u,

∇ × u =

R sin θ

∂θ (uφ sin θ), −

R

∂R

(Ruφ),

R

∂R

(Ruθ) −

R

∂uR ∂θ

]

Part II, Paper 3