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MATHEMATICAL TRIPOS Part II Alternative A
Wednesday 2 June 2004 9 to 12
PAPER 2
Before you begin read these instructions carefully.
Each question is divided into Part (i) and Part (ii), which may or may not be related. Candidates may attempt either or both Parts of any question, but must not attempt Parts from more than SIX questions.
The number of marks for each question is the same, with Part (ii) of each question carrying twice as many marks as Part (i).
Additional credit will be given for a substantially complete answer to either Part.
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 your answers in separate bundles, marked A, B, C,... , J according to the letter affixed to each question. (For example, 13E, 14E should be in one bundle and 3F, 8F in another bundle.)
Attach a completed cover sheet to each bundle listing the Parts of questions at- tempted.
Complete a master cover sheet listing separately all Parts of all questions attempted.
It is essential that every cover sheet bear the candidate 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.
1I Markov Chains
(i) Let J be a proper subset of the finite state space I of an irreducible Markov chain (Xn), whose transition matrix P is partitioned as
P =
J Jc J A B Jc^ C D
If only visits to states in J are recorded, we see a J-valued Markov chain ( X˜n); show that its transition matrix is
P˜ = A + B
n> 0
DnC = A + B(I − D)−^1 C.
(ii) Local MP Phil Anderer spends his time in London in the Commons (C), in his flat (F ), in the bar (B) or with his girlfriend (G). Each hour, he moves from one to another according to the transition matrix P , though his wife (who knows nothing of his girlfriend) believes that his movements are governed by transition matrix P W^ :
P =
C F B G
C 1 / 3 1 / 3 1 / 3 0
F 0 1 / 3 1 / 3 1 / 3
B 1 / 3 0 1 / 3 1 / 3
G 1 / 3 1 / 3 0 1 / 3
P^
W =
C F B
C 1 / 3 1 / 3 1 / 3
F 1 / 3 1 / 3 1 / 3
B 1 / 3 1 / 3 1 / 3
The public only sees Phil when he is in J = {C, F, B}; calculate the transition matrix P˜ which they believe controls his movements.
Each time the public Phil moves to a new location, he phones his wife; write down the transition matrix which governs the sequence of locations from which the public Phil phones, and calculate its invariant distribution.
Phil’s wife notes down the location of each of his calls, and is getting suspicious
- he is not at his flat often enough. Confronted, Phil swears his fidelity and resolves to dump his troublesome transition matrix, choosing instead
P ∗^ =
C F B G
C 1 / 4 1 / 4 1 / 2 0
F 1 / 2 1 / 4 1 / 4 0
B 0 3 / 8 1 / 8 1 / 2
G 2 / 10 1 / 10 1 / 10 6 / 10
Will this deal with his wife’s suspicions? Explain your answer.
Paper 2
3F Functional Analysis
(i) Prove Riesz’s Lemma, that if V is a normed space and A is a vector subspace of V such that for some 0 6 k < 1 we have d(x, A) 6 k for all x ∈ V with ||x|| = 1, then A is dense in V. [Here d(x, A) denotes the distance from x to A.]
Deduce that any normed space whose unit ball is compact is finite-dimensional. [You may assume that every finite-dimensional normed space is complete.]
Give an example of a sequence f 1 , f 2 ,... in an infinite-dimensional normed space such that ||fn|| 6 1 for all n, but f 1 , f 2 ,... has no convergent subsequence.
(ii) Let V be a vector space, and let ||.|| 1 and ||.|| 2 be two norms on V. What does it mean to say that ||.|| 1 and ||.|| 2 are equivalent?
Show that on a finite-dimensional vector space all norms are equivalent. Deduce that every finite-dimensional normed space is complete.
Exhibit two norms on the vector space l^1 that are not equivalent.
In addition, exhibit two norms on the vector space l∞^ that are not equivalent.
4G Groups, Rings and Fields (i) State Gauss’ Lemma on polynomial irreducibility. State and prove Eisenstein’s criterion.
(ii) Which of the following polynomials are irreducible over Q? Justify your answers.
(a) x^7 − 3 x^3 + 18x + 12 (b) x^4 − 4 x^3 + 11x^2 − 3 x − 5
(c) 1 + x + x^2 +... + xp−^1 with p prime
[Hint: consider substituting y = x − 1 .] (d) xn^ + px + p^2 with p prime.
[Hint: show any factor has degree at least two, and consider powers of p dividing coefficients.]
Paper 2
5C Electromagnetism
(i) Write down the general solution of Poisson’s equation. Derive from Maxwell’s equations the Biot-Savart law for the magnetic field of a steady localised current distribu- tion.
(ii) A plane rectangular loop with sides of length a and b lies in the plane z = 0 and is centred on the origin. Show that when r = |r| a, b, the vector potential A(r) is given approximately by
A(r) =
μ 0 4 π
m ∧ r r^3
where m = Iabzˆ is the magnetic moment of the loop.
Hence show that the magnetic field B(r) at a great distance from an arbitrary small plane loop of area A, situated in the xy-plane near the origin and carrying a current I, is given by
B(r) =
μ 0 IA 4 πr^5
3 xz, 3 yz, 2 r^2 − 3 x^2 − 3 y^2
6B Nonlinear Dynamical Systems
(i) A linear system in R^2 takes the form ˙x = Ax. Explain (without detailed calculation but by giving examples) how to classify the dynamics of the system in terms of the determinant and the trace of A. Show your classification graphically, and describe the dynamics that occurs on the boundaries of the different regions on your diagram.
(ii) A nonlinear system in R^2 has the form x˙ = f (x), f (0) = 0. The Jacobian (linearization) A of f at the origin is non-hyperbolic, with one eigenvalue of A in the left-hand half-plane. Define the centre manifold for this system, and explain (stating carefully any results you use) how the dynamics near the origin may be reduced to a one-dimensional system on the centre manifold.
A dynamical system of this type has the form
x˙ = ax^3 + bxy + cx^5 + dx^3 y + exy^2 + f x^7 + gx^5 y y ˙ = −y + x^2 − x^4
Find the coefficients for the expansion of the centre manifold correct up to and including terms of order x^6 , and write down in terms of these coefficients the equation for the dynamics on the centre manifold up to order x^7. Using this reduced equation, give a complete set of conditions on the coefficients a, b, c,... that guarantee that the origin is stable.
Paper 2 [TURN OVER
10J Algorithms and Networks
(i) Define the minimum path and the maximum tension problems for a network with span intervals specified for each arc. State without proof the connection between the two problems, and describe the Max Tension Min Path algorithm of solving them.
(ii) Find the minimum path between nodes S and S′^ in the network below. The span intervals are displayed alongside the arcs.
[0, ∞) [−∞, 2]
[− 1 , 11]
[− 5 , 5] [− 1 , 2]
[0, 2]
[− 6 , 2] [− 3 , 3]
[− 2 , 3] [0, 4]
[− 6 , 0]
[− 1 , 6]
[− 1 , 3]
[− 1 , 9]
[− 2 , 0]
S
S′
Paper 2 [TURN OVER
11J Principles of Statistics
(i) In the context of a decision-theoretic approach to statistics, what is a loss function? a decision rule? the risk function of a decision rule? the Bayes risk of a decision rule? the Bayes rule with respect to a given prior distribution?
Show how the Bayes rule with respect to a given prior distribution is computed. (ii) A sample of n people is to be tested for the presence of a certain condition. A single real-valued observation is made on each one; this observation comes from density f 0 if the condition is absent, and from density f 1 if the condition is present. Suppose θi = 0 if the ith person does not have the condition, θi = 1 otherwise, and suppose that the prior distribution for the θi is that they are independent with common distribution
P (θi = 1) = p ∈ (0, 1), where p is known. If Xi denotes the observation made on the ith person, what is the posterior distribution of the θi?
Now suppose that the loss function is defined by
L 0 (θ, a) ≡
∑^ n
j=
(αaj (1 − θj ) + β(1 − aj )θj )
for action a ∈ [0, 1]n, where α, β are positive constants. If πj denotes the posterior probability that θj = 1 given the data, prove that the Bayes rule for this prior and this loss function is to take aj = 1 if πj exceeds the threshold value α/(α + β), and otherwise to take aj = 0.
In an attempt to control the proportion of false positives, it is proposed to use a different loss function, namely,
L 1 (θ, a) ≡ L 0 (θ, a) + γI{∑^ aj > 0 }
θj aj ∑ aj
where γ > 0. Prove that the Bayes rule is once again a threshold rule, that is, we take action aj = 1 if and only if πj > λ, and determine λ as fully as you can.
12I Computational Statistics and Statistical Modelling
(i) Suppose we have independent observations Y 1 ,... , Yn, and we assume that for i = 1,... , n, Yi is Poisson with mean μi, and log(μi) = βT^ xi, where x 1 ,... , xn are given covariate vectors each of dimension p, where β is an unknown vector of dimension p, and p < n. Assuming that {x 1 ,... , xn} span Rp, find the equation for βˆ, the maximum
likelihood estimator of β, and write down the large-sample distribution of βˆ.
(ii) A long-term agricultural experiment had 90 grassland plots, each 25m × 25m, differing in biomass, soil pH, and species richness (the count of species in the whole plot). While it was well-known that species richness declines with increasing biomass, it was not known how this relationship depends on soil pH, which for the given study has possible values “low”, “medium” or “high”, each taken 30 times. Explain the commands input, and interpret the resulting output in the (slightly edited) R output below, in which “species” represents the species count.
(The first and last 2 lines of the data are reproduced here as an aid. You may assume that the factor pH has been correctly set up.)
Paper 2
pHmid -0.33146 0.09217 -3.596 0.
Biomass -0.10713 0.01249 -8.577 < 2e-
pHlow:Biomass -0.15503 0.04003 -3.873 0.
pHmid:Biomass -0.03189 0.02308 -1.382 0.
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 452.346 on 89 degrees of freedom
Residual deviance: 83.201 on 84 degrees of freedom
Number of Fisher Scoring iterations: 4
Paper 2
13E Foundations of Quantum Mechanics
(i) The creation and annihilation operators for a harmonic oscillator of angular frequency ω satisfy the commutation relation [a, a†] = 1. Write down an expression for the Hamiltonian H in terms of a and a†.
There exists a unique ground state | 0 〉 of H such that a| 0 〉 = 0. Explain how the space of eigenstates |n〉, n = 0, 1 , 2 ,... of H is formed, and deduce the eigenenergies for these states. Show that
a|n〉 =
n|n − 1 〉 , a†|n〉 =
n + 1|n + 1〉.
(ii) Write down the number operator N of the harmonic oscillator in terms of a and a†. Show that N |n〉 = n|n〉.
The operator Kr is defined to be
Kr =
a†r^ ar r!
, r = 0, 1 , 2 ,....
Show that Kr commutes with N. Show also that
Kr |n〉 =
{ (^) n! (n−r)! r! |n〉^ r^ ≤^ n^ ,
0 r > n.
By considering the action of Kr on the state |n〉 show that
∑^ ∞
r=
(−1)r^ Kr = | 0 〉〈 0 |.
Paper 2 [TURN OVER
15C General Relativity
(i) State and prove Birkhoff’s theorem.
(ii) Derive the Schwarzschild metric and discuss its relevance to the problem of gravitational collapse and the formation of black holes.
[Hint: You may assume that the metric takes the form
ds^2 = −eν(r,t)^ dt^2 + eλ(r,t)^ dr^2 + r^2 (dθ^2 + sin^2 θ dφ^2 ),
and that the non-vanishing components of the Einstein tensor are given by
Gtt =
e^2 ν+λ r^2
(−1 + eλ^ + rλ′), Grt = e(ν+λ)/^2
λ˙ r
, Grr =
eλ r^2
(1 − e−λ^ + rν′),
Gθθ = 14 r^2 e−λ
[
2 ν′′^ + (ν′)^2 +
r
(ν′^ − λ′) − ν′λ′
]
− 14 r^2 e−ν^
[
2 ¨λ + ( λ˙)^2 − λ˙ ν˙
]
Gtr = Grt and Gφφ = sin^2 θ Gθθ .]
16A Theoretical Geophysics
(i) Sketch the rays in a small region near the relevant boundary produced by reflection and refraction of a P -wave incident (a) from the mantle on the core-mantle boundary, (b) from the outer core on the inner-core boundary, and (c) from the mantle on the Earth’s surface. [In each case, the region should be sufficiently small that the boundary appears to be planar.]
Describe the ray paths denoted by SS, P cP , SKS and P KIKP.
Sketch the travel-time (T − ∆) curves for P and P cP paths from a surface source. (ii) From the surface of a flat Earth, an explosive source emits P -waves downwards into a stratified sequence of homogeneous horizontal elastic layers of thicknesses h 1 , h 2 , h 3 ,... and P -wave speeds α 1 < α 2 < α 3 <.. .. A line of seismometers on the surface records the travel times of the various arrivals as a function of the distance x from the source. Calculate the travel times, Td(x) and Tr (x), of the direct wave and the wave that reflects exactly once at the bottom of layer 1.
Show that the travel time for the head wave that refracts in layer n is given by
Tn =
x αn
n∑− 1
i=
2 hi αi
α^2 i α^2 n
Sketch the travel-time curves for Tr , Td and T 2 on a single diagram and show that T 2 is tangent to Tr.
Explain how the αi and hi can be constructed from the travel times of first arrivals provided that each head wave is the first arrival for some range of x.
Paper 2 [TURN OVER
17A Mathematical Methods
(i) Consider the integral equation
φ(x) = −λ
∫ (^) b
a
K(x, t)φ(t)dt + g(x), (†)
for φ in the interval a ≤ x ≤ b, where λ is a real parameter and g(x) is given. Describe the method of successive approximations for solving (†).
Suppose that |K(x, t)| ≤ M, ∀x, t ∈ [a, b].
By using the Cauchy-Schwarz inequality, or otherwise, show that the successive-approx- imation series for φ(x) converges absolutely provided
|λ| <
M (b − a)
(ii) The real function ψ(x) satisfies the differential equation
−ψ′′(x) + λψ(x) = h(x), 0 < x < 1 , (?)
where h(x) is a given smooth function on [0, 1], subject to the boundary conditions
ψ′(0) = ψ(0), ψ(1) = 0.
By integrating (?), or otherwise, show that ψ(x) obeys
ψ(0) =
0
(1 − t)h(t) dt −
λ
0
(1 − t)ψ(t) dt.
Hence, or otherwise, deduce that ψ(x) obeys an equation of the form (†), with
K(x, t) =
2 (1^ −^ x)(1 +^ t),^0 ≤^ t^ ≤^ x^ ≤^1 , 1 2 (1 +^ x)(1^ −^ t),^0 ≤^ x^ ≤^ t^ ≤^1 ,
and g(x) =
0
K(x, t)h(t) dt.
Deduce that the series solution for ψ(x) converges provided |λ| < 2.
Paper 2
