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MATHEMATICAL TRIPOS Part II Alternative A
Friday 6 June 2003 9 to 12
PAPER 4
Before you begin read these instructions carefully.
Candidates must not attempt more than FOUR questions.
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, 2D, 6D should be in one bundle and 11I, 14I 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
Consider a pack of cards labelled 1,... , 52. We repeatedly take the top card and insert it uniformly at random in one of the 52 possible places, that is, either on the top or on the bottom or in one of the 50 places inside the pack. How long on average will it take for the bottom card to reach the top?
Let pn denote the probability that after n iterations the cards are found in increasing order. Show that, irrespective of the initial ordering, pn converges as n → ∞, and determine the limit p. You should give precise statements of any general results to which you appeal.
Show that, at least until the bottom card reaches the top, the ordering of cards inserted beneath it is uniformly random. Hence or otherwise show that, for all n,
|pn − p| 6 52(1 + log 52)/n.
Paper 4
3G Functional Analysis
(i) State the Monotone Convergence Theorem and explain briefly how to prove it.
(ii) For which real values of α is x−α^ log x ∈ L^1 ((1, ∞))? Let p > 0. Using the Monotone Convergence Theorem and the identity
xp(x − 1)
∑^ ∞
n=
xp+n+
prove carefully that ∫ (^) ∞
1
log x xp(x − 1)
dx =
∑^ ∞
n=
(n + p)^2
4F Groups, Rings and Fields
Write an essay on the theory of invariants. Your essay should discuss the theorem on the finite generation of the ring of invariants, the theorem on elementary symmetric functions, and some examples of calculation of rings of invariants.
5A Electromagnetism
Let E(r) be the electric field due to a continuous static charge distribution ρ(r) for which |E| → 0 as |r| → ∞. Starting from consideration of a finite system of point charges, deduce that the electrostatic energy of the charge distribution ρ is
W =
ε 0
|E|^2 dτ (∗)
where the volume integral is taken over all space.
A sheet of perfectly conducting material in the form of a surface S, with unit normal n, carries a surface charge density σ. Let E± = n · E± denote the normal components of the electric field E on either side of S. Show that
1 ε 0
σ = E+ − E−.
Three concentric spherical shells of perfectly conducting material have radii a, b, c with a < b < c. The innermost and outermost shells are held at zero electric potential. The other shell is held at potential V. Find the potentials φ 1 (r) in a < r < b and φ 2 (r) in b < r < c. Compute the surface charge density σ on the shell of radius b. Use the formula (∗) to compute the electrostatic energy of the system.
Paper 4
6D Dynamics of Differential Equations
Explain what is meant by a steady-state bifurcation of a fixed point x 0 (μ) of an ODE x˙ = f (x, μ), in Rn, where μ is a real parameter. Give examples for n = 1 of equations exhibiting saddle-node, transcritical and pitchfork bifurcations.
Consider the system in R^2 , with μ > 0,
x˙ = x(1 − y − 4 x^2 ) , y˙ = y(μ − y − x^2 ).
Show that the fixed point (0, μ) has a bifurcation when μ = 1, while the fixed points (± 12 , 0) have a bifurcation when μ = 14. By finding the first approximation to the extended centre manifold, construct the normal form at the bifurcation point in each case, and determine the respective bifurcation types. Deduce that for μ just greater than 14 , and for μ just less than 1, there is a stable pair of “mixed-mode” solutions with x^2 > 0, y > 0.
7H Geometry of Surfaces
Write an essay on the Theorema Egregium for surfaces in R^3.
8H Logic, Computation and Set Theory
Write an essay on propositional logic. You should include all relevant definitions, and should cover the Completeness Theorem, as well as the Compactness Theorem and the Decidability Theorem.
[You may assume that the set of primitive propositions is countable. You do not need to give proofs of simple examples of syntactic implication, such as the fact that p ⇒ p is a theorem or that p ⇒ q and q ⇒ r syntactically imply p ⇒ r.]
9F Graph Theory Write an essay on the vertex-colouring of graphs drawn on compact surfaces other than the sphere. You should include a proof of Heawood’s bound, and an example of a surface for which this bound is not attained.
10G Number Theory Write an essay describing the factor base method for factorising a large odd positive integer n. Your essay should include a detailed explanation of how the continued fraction of
n can be used to find a suitable factor base.
Paper 4 [TURN OVER
13I Principles of Statistics
Write an account, with appropriate examples, of inference in multiparameter exponential families. Your account should include a discussion of natural statistics and their properties and of various conditional tests on natural parameters.
14I Computational Statistics and Statistical Modelling
The nave height x, and the nave length y for 16 Gothic-style cathedrals and 9 Romanesque-style cathedrals, all in England, have been recorded, and the corresponding R output (slightly edited) is given below.
first.lm _ lm(y ~ x + Style); summary(first.lm)
Call: lm(formula = y ~ x + Style)
Residuals: Min 1Q Median 3Q Max
-172.67 -30.44 20.38 55.02 96.
Coefficients:
Estimate Std. Error t value Pr(>|t|) (Intercept) 44.298 81.648 0.543 0.
x 4.712 1.058 4.452 0.
Style2 80.393 32.306 2.488 0.
Residual standard error: 77.53 on 22 degrees of freedom
Multiple R-Squared: 0.
You may assume that x, y are in suitable units, and that “style” has been set up as a factor with levels 1,2 corresponding to Gothic, Romanesque respectively.
(a) Explain carefully, with suitable graph(s) if necessary, the results of this analysis.
(b) Using the general model Y = Xβ + (in the conventional notation) explain carefully the theory needed for (a).
[Standard theorems need not be proved.]
Paper 4 [TURN OVER
15C Foundations of Quantum Mechanics
Discuss the quantum mechanics of the one-dimensional harmonic oscillator using creation and annihilation operators, showing how the energy levels are calculated.
A quantum mechanical system consists of two interacting harmonic oscillators and has the Hamiltonian
H =
pˆ^21 +
xˆ^21 +
pˆ^22 +
xˆ^22 + λˆx 1 ˆx 2.
For λ = 0, what are the degeneracies of the three lowest energy levels? For λ 6 = 0 compute, to lowest non-trivial order in perturbation theory, the energies of the ground state and first excited state.
[Standard results for perturbation theory may be stated without proof.]
16C Quantum Physics Describe the energy band structure available to electrons moving in crystalline materials. How can it be used to explain the properties of crystalline materials that are conductors, insulators and semiconductors?
Where does the Fermi energy lie in an intrinsic semiconductor? Describe the process of doping of semiconductors and explain the difference between n-type and p-type doping. What is the effect of the doping on the position of the Fermi energy in the two cases?
Why is there a potential difference across a junction of n-type and p-type semicon- ductors?
Derive the relation I = I 0
1 − e−qV /kT^
between the current, I, and the voltage, V , across an np junction, where I 0 is the total minority current in the semiconductor and q is the charge on the electron, T is the temperature and k is Boltzmann’s constant. Your derivation should include an explanation of the terms majority current and minority current.
Why can the np junction act as a rectifier?
Paper 4
18A Statistical Physics and Cosmology
Let g(p) be the density of states of a particle in volume V as a function of the magnitude p of the particle’s momentum. Explain why g(p) ∝ V p^2 /h^3 , where h is Planck’s constant. Write down the Bose–Einstein and Fermi–Dirac distributions for the (average) number ¯n(p) of particles of an ideal gas with momentum p. Hence write down integrals for the (average) total number N of particles and the (average) total energy E as functions of temperature T and chemical potential μ. Why do N and E also depend on the volume V?
Electromagnetic radiation in thermal equilibrium can be regarded as a gas of photons. Why are photons “ultra-relativistic” and how is photon momentum p related to the frequency ν of the radiation? Why does a photon gas have zero chemical potential? Use your formula for ¯n(p) to express the energy density εγ of electromagnetic radiation in the form
εγ =
0
(ν)dν
where (ν) is a function of ν that you should determine up to a dimensionless multiplicative constant. Show that (ν) is independent of h when kT hν, where k is Boltzmann’s constant. Let νpeak be the value of ν at the maximum of the function (ν); how does νpeak depend on T?
Let nγ be the photon number density at temperature T. Show that nγ ∝ T q for some power q, which you should determine. Why is nγ unchanged as the volume V is increased quasi-statically? How does T depend on V under these circumstances? Applying your result to the Cosmic Microwave Background Radiation (CMBR), deduce how the temperature Tγ of the CMBR depends on the scale factor a of the Universe. At a time when Tγ ∼ 3000 K, the Universe underwent a transition from an earlier time at which it was opaque to a later time at which it was transparent. Explain briefly the reason for this transition and its relevance to the CMBR.
An ideal gas of fermions f of mass m is in equilibrium at temperature T and chemical potential μf with a gas of its own anti-particles f¯ and photons (γ). Assuming that chemical equilibrium is maintained by the reaction
f + f¯ ↔ γ
determine the chemical potential μ (^) f¯ of the antiparticles. Let nf and n (^) f¯ be the number densities of f and f¯ , respectively. What will their values be for kT mc^2 if μf = 0? Given that μf > 0, but μf kT , show that
nf ≈ n 0 (T )
[
μf kT
F
mc^2 /kT
)]
where n 0 (T ) is the fermion number density at zero chemical potential and F is a positive function of the dimensionless ratio mc^2 /kT. What is F when kT mc^2?
Given that μf kT , obtain an expression for the ratio (nf − n (^) f¯ )/n 0 in terms of μ, T and the function F. Supposing that f is either a proton or neutron, why should you expect the ratio (nf − n (^) f¯ )/nγ to remain constant as the Universe expands?
Paper 4
19E Transport Processes
A shallow layer of fluid of viscosity μ, density ρ and depth h(x, t) lies on a rigid horizontal plane y = 0 and is bounded by impermeable barriers at x = −L and x = L (L h). Gravity acts vertically and a wind above the layer causes a shear stress τ (x) to be exerted on the upper surface in the +x direction. Surface tension is negligible compared to gravity.
(a) Assuming that the steady flow in the layer can be analysed using lubrication theory, show that the horizontal pressure gradient px is given by px = ρghx and hence that
hhx =
τ ρg
Show also that the fluid velocity at the surface y = h is equal to τ h/ 4 μ, and sketch the velocity profile for 0 6 y 6 h.
(b) In the case in which τ is a constant, τ 0 , and assuming that the difference between h and its average value h 0 remains small compared with h 0 , show that
h ≈ h 0
3 τ 0 x 2 ρgh^20
provided that τ 0 L ρgh^20
(c) Surfactant at surface concentration Γ(x) is added to the surface, so that now
τ = τ 0 − AΓx, (2)
where A is a positive constant. The surfactant is advected by the surface fluid velocity and also experiences a surface diffusion with diffusivity D. Write down the equation for conservation of surfactant, and hence show that
(τ 0 − AΓx) hΓ = 4μDΓx. (3)
From equations (1), (2) and (3) deduce that
Γ Γ 0
= exp
[
ρg 18 μD
h^3 − h^30
]
where Γ 0 is a constant. Assuming once more that h 1 ≡ h − h 0 h 0 , and that h = h 0 at x = 0, show further that
h 1 ≈
3 τ 0 x 2 ρgh 0
[
AΓ 0 h 0 4 μD
]− 1
provided that τ 0 h 0 L μD
1 as well as
τ 0 L ρgh^20
Paper 4 [TURN OVER
21B Mathematical Methods
Let y(x, λ) denote the solution for 0 6 x < ∞ of
d^2 y dx^2
− (x + λ^2 )y = 0,
subject to the conditions that y(0, λ) = a and y(x, λ)→0 as x→∞, where a > 0; it may be assumed that y(x, λ) > 0 for x > 0. Write y(x, λ) in the form
y(x, λ) = exp(z(x, λ)) ,
and consider an asymptotic expansion of the form
z(x, λ) ∼
∑^ ∞
n=
λ^1 −nφn(x) ,
valid in the limit λ→∞ with x = O(1). Find φ 0 (x), φ 1 (x), φ 2 (x) and φ 3 (x).
It is known that the solution y(x, λ) is of the form
y(x, λ) = c Y (X)
where X = x + λ^2
and the constant factor c depends on λ. By letting Y (X) = exp(Z(X)), show that the expression
Z(X) = −
X^3 /^2 −
lnX
satisfies the relevant differential equation with an error of O(1/X^3 /^2 ) as X → ∞. Comment on the relationship between your answers for z(x, λ) and Z(X).
Paper 4 [TURN OVER
22B Nonlinear Waves and Integrable Systems
Let Φ+(t), Φ−(t) denote the boundary values of functions which are analytic inside and outside a disc of radius 12 centred at the origin. Let C denote the boundary of this disc.
Suppose that Φ+, Φ−^ satisfy the jump condition
Φ+(t) =
t t^2 − 1
Φ−(t) +
t^3 − t^2 + 1 t^2 − t
, t ∈ C.
(a) Show that the associated index is 1. (b) Find the canonical solution of the homogeneous problem, i.e. the solution satisfying
X(z) ∼ z−^1 , z → ∞.
(c) Find the general solution of the Riemann–Hilbert problem satisfying the above jump condition as well as
Φ(z) = O(z−^1 ), z → ∞.
(d) Use the above result to solve the linear singular integral problem
(t^2 + t − 1)φ(t) +
t^2 − t − 1 πi
C
φ(τ ) τ − t
dτ =
2(t^3 − t^2 + 1)(t + 1) t
, t ∈ C.
23E Numerical Analysis
Write an essay on the conjugate gradient method. Your essay should include: (a) a statement of the method and a sketch of its derivation;
(b) discussion, without detailed proofs, but with precise statements of relevant theorems, of the conjugacy of the search directions;
(c) a description of the standard form of the algorithm; (d) discussion of the connection of the method with Krylov subspaces.
Paper 4
