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MATHEMATICAL TRIPOS Part II Alternative A
Tuesday 4 June 2002 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. If you submit answers to Parts of more than six questions, your lowest scoring attempt(s) will be rejected.
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.
Begin each answer on a separate sheet.
Write legibly and on only one side of the paper.
At the end of the examination:
Tie your answers in separate bundles, marked C, D, E,... , M according to the letter affixed to each question. (For example, 3K, 7K should be in one bundle and 1M, 10M in another bundle.)
Attach a completed cover sheet to each bundle.
Complete a master cover sheet listing all Parts of all questions attempted.
It is essential that every cover sheet bear the candidate’s examination number and desk number.
1M Markov Chains
(i) In each of the following cases, the state-space I and non-zero transition rates qij (i 6 = j) of a continuous-time Markov chain are given. Determine in which cases the chain is explosive.
(a) I = { 1 , 2 , 3 ,.. .}, qi,i+1 = i^2 , i ∈ I, (b) I = Z, qi,i− 1 = qi,i+1 = 2i, i ∈ I. (ii) Children arrive at a see-saw according to a Poisson process of rate 1. Initially there are no children. The first child to arrive waits at the see-saw. When the second child arrives, they play on the see-saw. When the third child arrives, they all decide to go and play on the merry-go-round. The cycle then repeats. Show that the number of children at the see-saw evolves as a Markov Chain and determine its generator matrix. Find the probability that there are no children at the see-saw at time t.
Hence obtain the identity
∑^ ∞
n=
e−t^
t^3 n (3n)!
e−^
3 2 t^ cos
t.
2G Principles of Dynamics
(i) A number N of non-interacting particles move in one dimension in a potential V (x, t). Write down the Hamiltonian and Hamilton’s equations for one particle.
At time t, the number density of particles in phase space is f (x, p, t). Write down the time derivative of f along a particle’s trajectory. By equating the rate of change of the number of particles in a fixed domain V in phase space to the flux into V across its boundary, deduce that f is a constant along any particle’s trajectory.
(ii) Suppose that V (x) = 12 mω^2 x^2 , and particles are injected in such a manner that the phase space density is a constant f 1 at any point of phase space corresponding to a particle energy being smaller than E 1 and zero elsewhere. How many particles are present?
Suppose now that the potential is very slowly altered to the square well form
V (x) =
{ (^0) , −L < x < L ∞ elsewhere.
Show that the greatest particle energy is now
E 2 =
π^2 8
E 12
mL^2 ω^2
Paper 2
6F Dynamics of Differential Equations
(i) Define the terms stable manifold and unstable manifold of a hyperbolic fixed point x 0 of a dynamical system. State carefully the stable manifold theorem.
Give an approximation, correct to fourth order in |x|, for the stable and unstable manifolds of the origin for the system
( x˙ y ˙
x + x^2 − y^2 −y + x^2
(ii) State, without proof, the centre manifold theorem. Show that the fixed point at the origin of the system x˙ = y − x + ax^3 , y ˙ = rx − y − zy , z ˙ = −z + xy ,
where a is a constant, is non-hyperbolic at r = 1.
Using new coordinates v = x + y , w = x − y, find the centre manifold in the form
w = αv^3 +... , z = βv^2 + γv^4 +...
for constants α, β, γ to be determined. Hence find the evolution equation on the centre manifold in the form
v˙ =
(a − 1)v^3 +
(3a + 1)(a + 1) 128
(a − 1) 32
v^5 +....
Ignoring higher order terms, give conditions on a that guarantee that the origin is asymptotically stable.
7K Geometry of Surfaces
(i) Consider the surface
z =
(λx^2 + μy^2 ) + h(x, y),
where h(x, y) is a term of order at least 3 in x, y. Calculate the first fundamental form at x = y = 0.
(ii) Calculate the second fundamental form, at x = y = 0, of the surface given in Part (i). Calculate the Gaussian curvature. Explain why your answer is consistent with Gauss’ “Theorema Egregium”.
Paper 2
8H Graph Theory
(i) Define the chromatic polynomial p(G ; t) of the graph G, and establish the standard identity p(G ; t) = p(G − e ; t) − p(G/e ; t),
where e is an edge of G. Deduce that, if G has n vertices and m edges, then
p(G ; t) = antn^ − an− 1 tn−^1 + an− 2 tn−^2 +... + (−1)na 0 ,
where an = 1, an− 1 = m and aj ≥ 0 for 0 ≤ j ≤ n.
(ii) Let G and p(G ; t) be as in Part (i). Show that if G has k components G 1 ,... , Gk then p(G ; t) =
∏k i=1 p(Gi^ ;^ t). Deduce that^ ak^ >^ 0 and^ aj^ = 0 for 0^ ≤^ j < k. Show that if G is a tree then p(G ; t) = t(t − 1)n−^1. Must the converse hold? Justify your answer.
Show that if p(G ; t) = p(Tr (n) ; t), where Tr (n) is a Tur´an graph, then G = Tr (n).
9H Coding and Cryptography (i) Explain the idea of public key cryptography. Give an example of a public key system, explaining how it works.
(ii) What is a general feedback register of length d with initial fill (X 0 ,... , Xd− 1 )? What is the maximal period of such a register, and why? What does it mean for such a register to be linear?
Describe and justify the Berlekamp-Massey algorithm for breaking a cypher stream arising from a general linear feedback register of unknown length.
Use the Berlekamp-Massey algorithm to find a linear recurrence in F 2 with first eight terms 1, 1, 0, 0, 1, 0, 1, 1.
Paper 2 [TURN OVER
12L Computational Statistics and Statistical Modelling
(i) Suppose that the random variable Y has density function of the form
f (y|θ, φ) = exp
[
yθ − b(θ) φ
]
where φ > 0. Show that Y has expectation b′(θ) and variance φb′′(θ).
(ii) Suppose now that Y 1 ,... , Yn are independent negative exponential variables, with Yi having density function f (yi|μi) = (^) μ^1 i e−yi/μi^ for yi > 0. Suppose further that
g(μi) = βT^ xi for 1 6 i 6 n, where g(·) is a known ‘link’ function, and x 1 ,... , xn are given covariate vectors, each of dimension p. Discuss carefully the problem of finding βˆ, the maximum-likelihood estimator of β, firstly for the case g(μi) = 1/μi, and secondly for the case g(μ) = log μi; in both cases you should state the large-sample distribution of βˆ.
[Any standard theorems used need not be proved.]
13E Foundations of Quantum Mechanics
(i) A Hamiltonian H 0 has energy eigenvalues Er and corresponding non-degenerate eigenstates |r〉. Show that under a small change in the Hamiltonian δH,
δ|r〉 =
s 6 =r
〈s|δH|r〉 Er − Es
|s〉,
and derive the related formula for the change in the energy eigenvalue Er to first and second order in δH.
(ii) The Hamiltonian for a particle moving in one dimension is H = H 0 + λH′, where H 0 = p^2 / 2 m + V (x), H′^ = p/m and λ is small. Show that
i ℏ
[H 0 , x] = H′
and hence that
δEr = −λ^2
i ℏ
〈r|H′x|r〉 = λ^2
i ℏ
〈r|xH′|r〉
to second order in λ.
Deduce that δEr is independent of the particular state |r〉 and explain why this change in energy is exact to all orders in λ.
Paper 2 [TURN OVER
14E Quantum Physics
(i) A simple model of a one-dimensional crystal consists of an infinite array of sites equally spaced with separation a. An electron occupies the nth site with a probability amplitude cn. The time-dependent Schr¨odinger equation governing these amplitudes is
iℏ
dcn dt
= E 0 cn − A(cn− 1 + cn+1) ,
where E 0 is the energy of an electron at an isolated site and the amplitude for transition between neighbouring sites is A > 0. By examining a solution of the form
cn = eikan−iEt/ℏ^ ,
show that E, the energy of the electron in the crystal, lies in a band
E 0 − 2 A ≤ E ≤ E 0 + 2A.
Identify the Brillouin zone for this model and explain its significance.
(ii) In the above model the electron is now subject to an electric field E in the direction of increasing n. Given that the charge on the electron is −e write down the new form of the time-dependent Schr¨odinger equation for the probability amplitudes. Show that it has a solution of the form
cn = exp
i ℏ
∫ (^) t
0
(t′)dt′^ + i(k −
eEt ℏ
)na
where
(t) = E 0 − 2 A cos
(k −
eEt ℏ
)a
Explain briefly how to interpret this result and use it to show that the dynamical behaviour of an electron near the bottom of the energy band is the same as that for a free particle in the presence of an electric field with an effective mass m∗^ = ℏ^2 /(2Aa^2 ).
Paper 2
16G Theoretical Geophysics
(i) State the equations that relate strain to displacement and stress to strain in a linear, isotropic elastic solid.
In the absence of body forces, the Euler equation for infinitesimal deformations of a solid of density ρ is
ρ
∂^2 ui ∂t^2
∂σij ∂xj
Derive an equation for u(x, t) in a linear, isotropic, homogeneous elastic solid. Hence show that both the dilatation θ = ∇ · u and the rotation ω = ∇ ∧ u satisfy wave equations and find the corresponding wave speeds α and β.
(ii) The ray parameter p = r sin i/v is constant along seismic rays in a spherically symmetric Earth, where v(r) is the relevant wave speed (α or β) and i(r) is the angle between the ray and the local radial direction.
Express tan i and sec i in terms of p and the variable η(r) = r/v. Hence show that the angular distance and travel time between a surface source and receiver, both at radius R, are given by
∆(p) = 2
∫ R
rm
p r
dr (η^2 − p^2 )^1 /^2
, T (p) = 2
∫ R
rm
η^2 r
dr (η^2 − p^2 )^1 /^2
where rm is the minimum radius attained by the ray. What is η(rm)?
A simple Earth model has a solid mantle in R/ 2 < r < R and a liquid core in r < R/2. If α(r) = A/r in the mantle, where A is a constant, find ∆(p) and T (p) for P-arrivals (direct paths lying entirely in the mantle), and show that
T =
R^2 sin ∆ A
[You may assume that
du u
u − 1
= 2 cos−^1
u
.]
Sketch the T − ∆ curves for P and PcP arrivals on the same diagram and explain briefly why they terminate at ∆ = cos−1 1 4.
Paper 2
17C Mathematical Methods
(i) Show that the equation
x^4 − x^2 + 5x − 6 = 0 , || 1 ,
has roots in the neighbourhood of x = 2 and x = 3. Find the first two terms of an expansion in for each of these roots.
Find a suitable series expansion for the other two roots and calculate the first two terms in each case.
(ii) Describe, giving reasons for the steps taken, how the leading-order approximation for λ 1 to an integral of the form
I(λ) ≡
∫ B
A
f (t)eiλg(t)dt ,
where λ and g are real, may be found by the method of stationary phase. Consider the cases where (a) g′(t) has one simple zero at t = t 0 with A < t 0 < B; (b) g′(t) has more than one simple zero in A < t < B; and (c) g′(t) has only a simple zero at t = B. What is the order of magnitude of I(λ) if g′(t) is non-zero for A ≤ t ≤ B?
Use the method of stationary phase to find the leading-order approximation to
J(λ) ≡
0
sin[λ(2t^4 − t)] dt
for λ 1.
[You may use the fact that
−∞
eiu
2 du =
πeiπ/^4 .]
Paper 2 [TURN OVER
