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MATHEMATICAL TRIPOS Part II Alternative A
Wednesday 4 June 2003 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.
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 A, B, C,... , J according to the letter affixed to each question. (For example, 16E, 19E should be in one bundle and 4F, 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’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) What is meant by a Poisson process of rate λ? Show that if (Xt)t> 0 and (Yt)t> 0 are independent Poisson processes of rates λ and μ respectively, then (Xt + Yt)t> 0 is also a Poisson process, and determine its rate.
(ii) A Poisson process of rate λ is observed by someone who believes that the first holding time is longer than all subsequent holding times. How long on average will it take before the observer is proved wrong?
2D Principles of Dynamics
(i) The trajectory x(t) of a non-relativistic particle of mass m and charge q moving in an electromagnetic field obeys the Lorentz equation
mx¨ = q(E +
x˙ c
∧ B).
Show that this equation follows from the Lagrangian
L =
m x˙^2 − q
φ −
x˙ · A c
where φ(x, t) is the electromagnetic scalar potential and A(x, t) the vector potential, so that
E = −
c
A˙ − ∇φ and B = ∇ ∧ A.
(ii) Let E = 0. Consider a particle moving in a constant magnetic field which points in the z direction. Show that the particle moves in a helix about an axis pointing in the z direction. Evaluate the radius of the helix.
Paper 2
5A Electromagnetism
(i) A plane electromagnetic wave has electric and magnetic fields
E = E 0 ei(k·r−ωt)^ , B = B 0 ei(k·r−ωt)^ (∗)
for constant vectors E 0 , B 0 , constant positive angular frequency ω and constant wave- vector k. Write down the vacuum Maxwell equations and show that they imply
k · E 0 = 0 , k · B 0 = 0 , ωB 0 = k × E 0.
Show also that |k| = ω/c, where c is the speed of light.
(ii) State the boundary conditions on E and B at the surface S of a perfect conductor. Let σ be the surface charge density and s the surface current density on S. How are σ and s related to E and B?
A plane electromagnetic wave is incident from the half-space x < 0 upon the surface x = 0 of a perfectly conducting medium in x > 0. Given that the electric and magnetic fields of the incident wave take the form (∗) with
k = k(cos θ, sin θ, 0) (0 < θ < π/2)
and E 0 = λ (− sin θ, cos θ, 0) ,
find B 0.
Reflection of the incident wave at x = 0 produces a reflected wave with electric field E′ 0 ei(k
′·r−ωt)
with k′^ = k(− cos θ, sin θ, 0).
By considering the boundary conditions at x = 0 on the total electric field, show that
E′ 0 = −λ (sin θ, cos θ, 0).
Show further that the electric charge density on the surface x = 0 takes the form
σ = σ 0 eik(y^ sin^ θ−ct)
for a constant σ 0 that you should determine. Find the magnetic field of the reflected wave and hence the surface current density s on the surface x = 0.
Paper 2
6D Dynamics of Differential Equations
(i) What is a Liapunov function?
Consider the second order ODE
x˙ = y , y˙ = −y − sin^3 x.
By finding a suitable Liapunov function of the form V (x, y) = f (x) + g(y), where f and g are to be determined, show that the origin is asymptotically stable. Using your form of V , find the greatest value of y 0 such that a trajectory through (0, y 0 ) is guaranteed to tend to the origin as t → ∞.
[Any theorems you use need not be proved but should be clearly stated.]
(ii) Explain the use of the stroboscopic method for investigating the dynamics of equations of the form ¨x + x = f (x, x, t˙ ), when || 1. In particular, for x = R cos(t + θ), x˙ = −R sin(t + θ) derive the equations, correct to order ,
R˙ = −〈f sin(t + θ)〉 , R θ˙ = −〈f cos(t + θ)〉, (∗)
where the brackets denote an average over the period of the unperturbed oscillator.
Find the form of the right hand sides of these equations explicitly when f = Γx^2 cos t − 3 qx, where Γ > 0, q 6 = 0. Show that apart from the origin there is another fixed point of (∗), and determine its stability. Sketch the trajectories in (R, θ) space in the case q > 0. What do you deduce about the dynamics of the full equation?
[You may assume that 〈cos^2 t〉 = 12 , 〈cos^4 t〉 = 38 , 〈cos^2 t sin^2 t〉 = 18 .]
7H Geometry of Surfaces (i) What are geodesic polar coordinates at a point P on a surface M with a Riemannian metric ds^2?
Assume that ds^2 = dr^2 + H(r, θ)^2 dθ^2 ,
for geodesic polar coordinates r, θ and some function H. What can you say about H and dH/dr at r = 0?
(ii) Given that the Gaussian curvature K may be computed by the formula K = −H−^1 ∂^2 H/∂r^2 , show that for small R the area of the geodesic disc of radius R centred at P is πR^2 − (π/12)KR^4 + a(R),
where a(R) is a function satisfying lim R→ 0
a(R)/R^4 = 0.
Paper 2 [TURN OVER
10I Algorithms and Networks
(i) Consider a network with node set N and set of directed arcs A equipped with functions d+^ : A → Z and d−^ : A → Z with d−^6 d+. Given u : N → R we define the differential ∆u : A → R by ∆u(j) = u(i′) − u(i) for j = (i, i′) ∈ A. We say that ∆u is a feasible differential if d−(j) 6 ∆u(j) 6 d+(j) for all j ∈ A.
Write down a necessary and sufficient condition on d+, d−^ for the existence of a feasible differential and prove its necessity.
Assuming Minty’s Lemma, describe an algorithm to construct a feasible differential and outline how this algorithm establishes the sufficiency of the condition you have given.
(ii) Let E ⊆ S × T , where S, T are finite sets. A matching in E is a subset M ⊆ E such that, for all s, s′^ ∈ S and t, t′^ ∈ T ,
(s, t), (s′, t) ∈ M implies s = s′ (s, t), (s, t′) ∈ M implies t = t′^.
A matching M is maximal if for any other matching M ′^ with M ⊆ M ′^ we must have M = M ′. Formulate the problem of finding a maximal matching in E in terms of an optimal distribution problem on a suitably defined network, and hence in terms of a standard linear optimization problem.
[You may assume that the optimal distribution subject to integer constraints is integer- valued.]
11I Principles of Statistics (i) Outline briefly the Bayesian approach to hypothesis testing based on Bayes factors.
(ii) Let Y 1 , Y 2 be independent random variables, both uniformly distributed on (θ − 12 , θ + 12 ). Find a minimal sufficient statistic for θ. Let Y(1) = min{Y 1 , Y 2 }, Y(2) = max{Y 1 , Y 2 }. Show that R = Y(2) − Y(1) is ancilliary and explain why the Conditionality Principle would lead to inference about θ being drawn from the conditional distribution of 12 {Y(1) + Y(2)} given R. Find the form of this conditional distribution.
Paper 2 [TURN OVER
12I Computational Statistics and Statistical Modelling
(i) Suppose Y 1 ,... , Yn are independent Poisson variables, and
E(Yi) = μi , log μi = α + βti , for i = 1,... , n ,
where α, β are two unknown parameters, and t 1 ,... , tn are given covariates, each of dimension 1. Find equations for ˆα, βˆ, the maximum likelihood estimators of α, β, and show how an estimate of var( βˆ) may be derived, quoting any standard theorems you may need.
(ii) By 31 December 2001, the number of new vCJD patients, classified by reported calendar year of onset, were 8 , 10 , 11 , 14 , 17 , 29 , 23
for the years 1994 ,... , 2000 respectively.
Discuss carefully the (slightly edited) R output for these data given below, quoting any standard theorems you may need.
year
year [1] 1994 1995 1996 1997 1998 1999 2000
tot [1] 8 10 11 14 17 29 23
first.glm _ glm(tot ~ year, family = poisson)
summary(first.glm) Call:
glm(formula = tot ~ year, family = poisson)
Coefficients Estimate Std. Error z value Pr(>|z|)
(Intercept) -407.81285 99.35366 -4.105 4.05e-
year 0.20556 0.04973 4.133 3.57e-
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 20.7753 on 6 degrees of freedom
Residual deviance: 2.7931 on 5 degrees of freedom
Number of Fisher Scoring iterations: 3
Paper 2
14C Quantum Physics
(i) A system of N distinguishable non-interacting particles has energy levels Ei with degeneracy gi, 1 6 i < ∞, for each particle. Show that in thermal equilibrium the number of particles Ni with energy Ei is given by
Ni = gie−β(Ei−μ),
where β and μ are parameters whose physical significance should be briefly explained.
A gas comprises a set of atoms with non-degenerate energy levels Ei, 1 6 i < ∞. Assume that the gas is dilute and the motion of the atoms can be neglected. For such a gas the atoms can be treated as distinguishable. Show that when the system is at temperature T , the number of atoms Ni at level i and the number Nj at level j satisfy
Ni Nj
= e−(Ei−Ej^ )/kT^ ,
where k is Boltzmann’s constant.
(ii) A system of bosons has a set of energy levels Wa with degeneracy fa, 1 6 a < ∞, for each particle. In thermal equilibrium at temperature T the number na of particles in level a is
na =
fa e(Wa−μ)/kT^ − 1
What is the value of μ when the particles are photons?
Given that the density of states ρ(ω) for photons of frequency ω in a cubical box of side L is
ρ(ω) = L^3
ω^2 π^2 c^3
where c is the speed of light, show that at temperature T the density of photons in the frequency range ω → ω + dω is n(ω)dω where
n(ω) =
ω^2 π^2 c^3
eℏω/kT^ − 1
Deduce the energy density, (ω), for photons of frequency ω.
The cubical box is occupied by the gas of atoms described in Part (i) in the presence of photons at temperature T. Consider the two atomic levels i and j where Ei > Ej and Ei − Ej = ℏω. The rate of spontaneous photon emission for the transition i → j is Aij. The rate of absorption is Bji (ω) and the rate of stimulated emission is Bij (ω). Show that the requirement that these processes maintain the atoms and photons in thermal equilibrium yields the relations Bij = Bji
and
Aij =
ℏω^3 π^2 c^3
Bij.
Paper 2
15A General Relativity
(i) What is a “stationary” metric? What distinguishes a stationary metric from a “static” metric?
A Killing vector field Ka^ of a metric gab satisfies
Ka;b + Kb;a = 0.
Show that this is equivalent to
gab,cKc^ + gacKc,b + gcbKc,a = 0.
Hence show that a constant vector field Ka^ with one non-zero component, K^4 say, is a Killing vector field if gab is independent of x^4.
(ii) Given that Ka^ is a Killing vector field, show that Kaua^ is constant along the geodesic worldline of a massive particle with 4-velocity ua. Hence find the energy ε of a particle of unit mass moving in a static spacetime with metric
ds^2 = hij dxidxj^ − e^2 U^ dt^2 ,
where hij and U are functions only of the space coordinates xi. By considering a particle with speed small compared with that of light, and given that U 1, show that hij = δij to lowest order in the Newtonian approximation, and that U is the Newtonian potential.
A metric admits an antisymmetric tensor Yab satisfying
Yab;c + Yac;b = 0.
Given a geodesic xa(λ), let sa = Yab x˙b. Show that sa is parallelly propagated along the geodesic, and that it is orthogonal to the tangent vector of the geodesic. Hence show that the scalar φ = sasa
is constant along the geodesic.
Paper 2 [TURN OVER
18B Nonlinear Waves and Integrable Systems
(i) Write down the shock condition associated with the equation
ρt + qx = 0,
where q = q(ρ). Discuss briefly two possible heuristic approaches to justifying this shock condition.
(ii) According to shallow water theory, waves on a uniformly sloping beach are described by the equations
∂h ∂t
∂x
(hu) = 0,
∂u ∂t
∂u ∂x
∂η ∂x
= 0, h = αx + η,
where α is the constant slope of the beach, g is the gravitational acceleration, u(x, t) is the fluid velocity, and η(x, t) is the elevation of the fluid surface above the undisturbed level.
Find the characteristic velocities and the characteristic form of the equations. What are the Riemann variables and how do they vary with t on the characteristics?
19E Numerical Analysis
(i) Explain briefly what is meant by the convergence of a numerical method for ordinary differential equations.
(ii) Suppose the sufficiently-smooth function f : R × Rd^ → Rd^ obeys the Lipschitz condition: there exists λ > 0 such that
||f (t, x) − f (t, y)|| 6 λ||x − y||, x, y ∈ Rd, t > 0.
Prove from first principles, without using the Dahlquist equivalence theorem, that the trapezoidal rule
yn+1 = yn +
h[f (tn, yn) + f (tn+1, yn+1)]
for the solution of the ordinary differential equation
y′^ = f (t, y), t > 0 , y(0) = y 0 ,
converges.
Paper 2
