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MATHEMATICAL TRIPOS Part II Alternative B
Thursday 5 June 2003 1.30 to 4.
PAPER 3
Before you begin read these instructions carefully.
The number of marks for each question is the same. Additional credit will be given for a substantially complete answer.
Write legibly and on only one side of the paper.
Begin each answer on a separate sheet.
At the end of the examination:
Tie your answers in separate bundles, marked A, B, C,... , J according to the letter affixed to each question. (For example, 3A, 22A should be in one bundle and 1J, 14J in another bundle.)
Attach a completed cover sheet to each bundle.
Complete a master cover sheet listing all questions attempted.
It is essential that every cover sheet bear the candidate’s examination number and desk number.
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
1J Markov Chains
(i) Consider the continuous-time Markov chain (Xt)t> 0 with state-space { 1 , 2 , 3 , 4 } and Q-matrix
Q =
Set
Yt =
Xt if Xt ∈ { 1 , 2 , 3 } 2 if Xt = 4
and
Zt =
Xt if Xt ∈ { 1 , 2 , 3 } 1 if Xt = 4.
Determine which, if any, of the processes (Yt)t> 0 and (Zt)t> 0 are Markov chains.
(ii) Find an invariant distribution for the chain (Xt)t> 0 given in Part (i). Suppose X 0 = 1. Find, for all t > 0, the probability that Xt = 1.
2G Functional Analysis
(i) Let p be a point of the compact interval I = [a, b] ⊂ R and let δp : C(I) → R be defined by δp(f ) = f (p). Show that
δp : (C(I), || · ||∞) → R
is a continuous, linear map but that
δp : (C(I), || · || 1 ) → R
is not continuous.
(ii) Consider the space C(n)(I) of n-times continuously differentiable functions on the interval I. Write
||f ||( ∞n) =
∑^ n
k=
||f (k)||∞ and ||f ||( 1 n )=
∑^ n
r=
||f (k)|| 1
for f ∈ C(n)(I). Show that (C(n)(I), || · ||( ∞n) ) is a complete normed space. Is the space
(C(n)(I), || · ||( 1 n )) also complete?
Let f : I → I be an n-times continuously differentiable map and define
μf : C(n)(I) → C(n)(I) by g 7 → g ◦ f.
Show that μf is a continuous linear map when C(n)(I) is equipped with the norm || · ||( ∞n).
Paper 3
4D Dynamics of Differential Equations
(i) Define the Poincar´e index of a curve C for a vector field f (x), x ∈ R^2. Explain why the index is uniquely given by the sum of the indices for small curves around each fixed point within C. Write down the indices for a saddle point and for a focus (spiral) or node, and show that the index of a periodic solution of ˙x = f (x) has index unity.
A particular system has a periodic orbit containing five fixed points, and two further periodic orbits. Sketch the possible arrangements of these orbits, assuming there are no degeneracies.
(ii) A dynamical system in R^2 depending on a parameter μ has a homoclinic orbit when μ = μ 0. Explain how to determine the stability of this orbit, and sketch the different behaviours for μ < μ 0 and μ > μ 0 in the case that the orbit is stable.
Now consider the system
x˙ = y , y˙ = x − x^2 + y(α + βx)
where α, β are constants. Show that the origin is a saddle point, and that if there is an orbit homoclinic to the origin then α, β are related by
∮ y^2 (α + βx)dt = 0
where the integral is taken round the orbit. Evaluate this integral for small α, β by approximating y by its form when α = β = 0. Hence give conditions on (small) α, β that lead to a stable homoclinic orbit at the origin. [Note that ydt = dx.]
5F Representation Theory
If ρ 1 : G 1 → GL(V 1 ) and ρ 2 : G 2 → GL(V 2 ) are representations of the finite groups G 1 and G 2 respectively, define the tensor product ρ 1 ⊗ ρ 2 as a representation of the group G 1 × G 2 and show that its character is given by
χρ 1 ⊗ρ 2 (g 1 , g 2 ) = χρ 1 (g 1 )χρ 2 (g 2 ).
Prove that
(a) if ρ 1 and ρ 2 are irreducible, then ρ 1 ⊗ ρ 2 is an irreducible representation of G 1 × G 2 ;
(b) each irreducible representation of G 1 × G 2 is equivalent to a representation ρ 1 ⊗ ρ 2 where each ρi is irreducible (i = 1, 2).
Is every representation of G 1 × G 2 the tensor product of a representation of G 1 and a representation of G 2?
Paper 3
6F Galois Theory
Let f be a separable polynomial of degree n > 1 over a field K. Explain what is meant by the Galois group Gal(f /K) of f over K. Explain how Gal(f /K) can be identified with a subgroup of the symmetric group Sn. Show that as a permutation group, Gal(f /K) is transitive if and only if f is irreducible over K.
Show that the Galois group of f (X) = X^5 + 20X^2 − 2 over Q is S 5 , stating clearly any general results you use.
Now let K/Q be a finite extension of prime degree p > 5. By considering the degrees of the splitting fields of f over K and Q, show that Gal(f /K) = S 5 also.
7G Algebraic Topology Define a covering map. Prove that any covering map induces an injective homo- morphisms of fundamental groups.
Show that there is a non-trivial covering map of the real projective plane. Explain how to use this to find the fundamental group of the real projective plane.
Paper 3 [TURN OVER
9G Riemann Surfaces
Let L be the lattice Zω 1 + Zω 2 for two non-zero complex numbers ω 1 , ω 2 whose ratio is not real. Recall that the Weierstrass function ℘ is given by the series
℘(u) =
u^2
ω∈L−{ 0 }
(u − ω)^2
ω^2
the function ζ is the (unique) odd anti-derivative of −℘; and σ is defined by the conditions
σ′(u) = ζ(u)σ(u) and σ′(0) = 1.
(a) By writing a differential equation for σ(−u), or otherwise, show that σ is an odd function.
(b) Show that σ(u + ωi) = −σ(u) exp(ai(u + bi)) for some constants ai, bi. Use (a) to express bi in terms of ωi. [Do not attempt to express ai in terms of ωi.] (c) Show that the function f (u) = σ(2u)/σ(u)^4 is periodic with respect to the lattice L and deduce that f (u) = −℘′(u).
10H Algebraic Curves (a) Let X ⊆ An^ be an affine algebraic variety. Define the tangent space TpX for p ∈ X. Show that the set {p ∈ X | dim TpX > d}
is closed, for every d > 0.
(b) Let C be an irreducible projective curve, p ∈ C, and f : C{p} → Pn^ a rational map. Show, carefully quoting any theorems that you use, that if C is smooth at p then f extends to a regular map at p.
11H Logic, Computation and Set Theory (i) What does it mean for a function from Nk^ to N to be recursive? Write down a function that is not recursive. You should include a proof that your example is not recursive.
(ii) What does it mean for a subset of Nk^ to be recursive, and what does it mean for it to be recursively enumerable? Give, with proof, an example of a set that is recursively enumerable but not recursive. Prove that a set is recursive if and only if both it and its complement are recursively enumerable. If a set is recursively enumerable, must its complement be recursively enumerable?
[You may assume the existence of any universal recursive functions or universal register machine programs that you wish.]
Paper 3 [TURN OVER
12G Probability and Measure
Explain what is meant by the characteristic function φ of a real-valued random variable and prove that |φ|^2 is also a characteristic function of some random variable.
Let us say that a characteristic function φ is infinitely divisible when, for each n > 1, we can write φ = (φn)n^ for some characteristic function φn. Prove that, in this case, the limit ψ(t) = (^) nlim→∞ |φ 2 n(t)|^2
exists for all real t and is continuous at t = 0.
Using L´evy’s continuity theorem for characteristic functions, which you should state carefully, deduce that ψ is a characteristic function. Hence show that, if φ is infinitely divisible, then φ(t) cannot vanish for any real t.
13I Applied Probability State the product theorem for Poisson random measures.
Consider a system of n queues, each with infinitely many servers, in which, for i = 1,... , n − 1, customers leaving the ith queue immediately arrive at the (i + 1)th queue. Arrivals to the first queue form a Poisson process of rate λ. Service times at the ith queue are all independent with distribution F , and independent of service times at other queues, for all i. Assume that initially the system is empty and write Vi(t) for the number of customers at queue i at time t > 0. Show that V 1 (t),... , Vn(t) are independent Poisson random variables.
In the case F (t) = 1 − e−μt^ show that
E(Vi(t)) =
λ μ
P(Nt > i), t > 0 , i = 1,... , n ,
where (Nt)t> 0 is a Poisson process of rate μ.
Suppose now that arrivals to the first queue stop at time T. Determine the mean number of customers at the ith queue at each time t > T.
Paper 3
15I Principles of Statistics
(i) Let X 1 ,... , Xn be independent, identically distributed random variables, with the exponential density f (x; θ) = θe−θx, x > 0.
Obtain the maximum likelihood estimator θˆ of θ. What is the asymptotic distribution of
n(θˆ − θ)?
What is the minimum variance unbiased estimator of θ? Justify your answer carefully.
(ii) Explain briefly what is meant by the profile log-likelihood for a scalar parameter of interest γ, in the presence of a nuisance parameter ξ. Describe how you would test a null hypothesis of the form H 0 : γ = γ 0 using the profile log-likelihood ratio statistic.
In a reliability study, lifetimes T 1 ,... , Tn are independent and exponentially distributed, with means of the form E(Ti) = exp(β + ξzi) where β, ξ are unknown and z 1 ,... , zn are known constants. Inference is required for the mean lifetime, exp(β + ξz 0 ), for covariate value z 0.
Find, as explicitly as possible, the profile log-likelihood for γ ≡ β + ξz 0 , with nuisance parameter ξ.
Show that, under H 0 : γ = γ 0 , the profile log-likelihood ratio statistic has a distribution which does not depend on the value of ξ. How might the parametric bootstrap be used to obtain a test of H 0 of exact size α?
[Hint: if Y is exponentially distributed with mean 1, then μY is exponentially distributed with mean μ.]
16J Stochastic Financial Models (i) What does it mean to say that the process (Wt)t> 0 is a Brownian motion? What does it mean to say that the process (Mt)t> 0 is a martingale?
Suppose that (Wt)t> 0 is a Brownian motion and the process (Xt)t> 0 is given in terms of W as Xt = x 0 + σWt + μt
for constants σ, μ. For what values of θ is the process
Mt = exp(θXt − λt)
a martingale? (Here, λ is a positive constant.)
(ii) In a standard Black–Scholes model, the price at time t of a share is represented as St = exp(Xt). You hold a perpetual American put option on this share, with strike K; you may exercise at any stopping time τ , and upon exercise you receive max{ 0 , K − Sτ }. Let 0 < a < log K. Suppose you plan to use the exercise policy: ‘Exercise as soon as the price falls to ea^ or lower.’ Calculate what the option would be worth if you were to follow this policy. (Assume that the riskless rate of interest is constant and equal to r > 0.) For what choice of a is this value maximised?
Paper 3
17B Dynamical Systems
Let f : I → I be a continuous one-dimensional map of the interval I ⊂ R. Explain what is meant by saying (a) that the map f is topologically transitive, and (b) that the map f has a horseshoe.
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
for 1 < μ 6 2. Show that if μ >
2 then this map is topologically transitive, and also that f 2 has a horseshoe.
Paper 3 [TURN OVER
19D Methods of Mathematical Physics
Let f (λ) =
γ
eλ(t−t
(^3) /3) dt, λ real and positive ,
where γ is a path beginning at ∞e−^2 iπ/^3 and ending at +∞ (on the real axis). Identify the saddle points and sketch the paths of constant phase through these points.
Hence show that f (λ) ∼ e^2 λ/^3
π/λ as λ → ∞.
20E Numerical Analysis
(i) The diffusion equation
∂u ∂t
∂x
a(x)
∂u ∂x
, 0 6 x 6 1 , t > 0 ,
with the initial condition u(x, 0) = φ(x), 0 6 x 6 1 and zero boundary conditions at x = 0 and x = 1, is solved by the finite-difference method
un m+1 = unm + μ[am− 12 unm− 1 − (am− 12 + am+ 12 )unm + am+ 12 unm+1], m = 1, 2 ,... , N,
where μ = ∆t/(∆x)^2 , ∆x = (^) N^1 +1 and unm ≈ u(m∆x, n∆t), aα = a(α∆x).
Assuming sufficient smoothness of the function a, and that μ remains constant as ∆x > 0 and ∆t > 0 become small, prove that the exact solution satisfies the numerical scheme with error O((∆x)^3 ).
(ii) For the problem defined in Part (i), assume that there exist 0 < a− < a+ < ∞ such that a− 6 a(x) 6 a+, 0 6 x 6 1. Prove that the method is stable for 0 < μ 6 1 /(2a+).
[Hint: You may use without proof the Gerschgorin theorem: All the eigenvalues of the matrix A = (ak,l)k,l=1,...,M are contained in
⋃m k=1 Sk, where
Sk =
z ∈ C : |z − ak,k| 6
∑^ m
l^ l=1 6 =k
|ak,l|
, k = 1, 2 ,... , m. ]
Paper 3 [TURN OVER
21C Foundations of Quantum Mechanics
(i) What are the commutation relations satisfied by the components of an angular momentum vector J? State the possible eigenvalues of the component J 3 when J^2 has eigenvalue j(j + 1)ℏ^2.
Describe how the Pauli matrices
σ 1 =
, σ 2 =
0 −i i 0
, σ 3 =
are used to construct the components of the angular momentum vector S for a spin (^12) system. Show that they obey the required commutation relations.
Show that S 1 , S 2 and S 3 each have eigenvalues ± 12 ℏ. Verify that S^2 has eigenvalue 3 4 ℏ
(ii) Let J and |jm〉 denote the standard operators and state vectors of angular momentum theory. Assume units where ℏ = 1. Consider the operator
U (θ) = e−iθJ^2.
Show that U (θ)J 1 U (θ)−^1 = cos θJ 1 − sin θJ 3 U (θ)J 3 U (θ)−^1 = sin θJ 1 + cos θJ 3.
Show that the state vectors U
( (^) π 2
|jm〉 are eigenvectors of J 1. Suppose that J 1 is measured for a system in the state |jm〉; show that the probability that the result is m′ equals |〈jm′|ei^
π 2 J 2 |jm〉|^2.
Consider the case j = m = 12. Evaluate the probability that the measurement of J 1 will result in m′^ = − 12.
Paper 3
23C Applications of Quantum Mechanics
Consider the two Hamiltonians
H 1 =
p^2 2 m
H 2 =
p^2 2 m
ni∈Z
V (|r − n 1 a 1 − n 2 a 2 − n 3 a 3 |) ,
where ai are three linearly independent vectors. For each of the Hamiltonians H = H 1 and H = H 2 , what are the symmetries of H and what unitary operators U are there such that U HU −^1 = H?
For H 2 derive Bloch’s theorem. Suppose that H 1 has energy eigenfunction ψ 0 (r) with energy E 0 where ψ 0 (r) ∼ N e−Kr^ for large r = |r|. Assume that K|ai| 1 for each i. In a suitable approximation derive the energy eigenvalues for H 2 when E ≈ E 0. Verify that the energy eigenfunctions and energy eigenvalues satisfy Bloch’s theorem. What happens if K|ai| → ∞?
24B Fluid Dynamics II A steady two-dimensional jet is generated in an infinite, incompressible fluid of density ρ and kinematic viscosity ν by a point source of momentum with momentum flux in the x direction F per unit length located at the origin.
Using boundary layer theory, analyse the motion in the jet and show that the x-component of the velocity is given by
u = U (x)f ′(η),
where η = y/δ(x), δ(x) = (ρν^2 x^2 /F )^1 /^3 and U (x) = (F 2 /ρ^2 νx)^1 /^3.
Show that f satisfies the differential equation
f ′′′^ +
(f f ′′^ + f ′
2 ) = 0.
Write down the appropriate boundary conditions for this equation. [You need not solve the equation.]
Paper 3
25E Waves in Fluid and Solid Media
Derive the wave equation governing the velocity potential for linearised sound in a perfect gas. How is the pressure disturbance related to the velocity potential? Write down the spherically symmetric solution to the wave equation with time dependence eiωt, which is regular at the origin.
A high pressure gas is contained, at density ρ 0 , within a thin metal spherical shell which makes small amplitude spherically symmetric vibrations. Ignore the low pressure gas outside. Let the metal shell have radius a, mass m per unit surface area, and elastic stiffness which tries to restore the radius to its equilibrium value a 0 with a force −κ(a−a 0 ) per unit surface area. Show that the frequency of these vibrations is given by
ω^2
m +
ρ 0 a 0 θ cot θ − 1
= κ where θ = ωa 0 /c 0.
Paper 3
