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MATHEMATICAL TRIPOS Part II
Monday 6 June 2005 1.30 to 4.
PAPER 1
Before you begin read these instructions carefully.
The examination paper is divided into two sections. Each question in Section II carries twice the number of marks of each question in Section I. Candidates may attempt at most six questions from Section I and any number of questions from Section II.
Complete answers are preferred to fragments.
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 up your answers in bundles, marked A, B, C,.. .,J according to the code letter affixed to each question. Include in the same bundle all questions from Sections I and II with the same code letter.
Attach a gold cover sheet to each bundle; write the code letter in the box marked ‘EXAMINER LETTER’ on the cover sheet.
You must also complete a green master cover sheet listing all the questions you have attempted.
Every cover sheet must bear your examination number and desk number.
STATIONERY REQUIRMENTS
Gold cover sheet Green master cover sheet
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
SECTION I
1H Number Theory
Define the Legendre symbol
( (^) a p
. Prove that, if p is an odd prime, then
( (^2)
p
p^2 − 1 (^8).
Use the law of quadratic reciprocity to calculate
167
[You may use the Gauss Lemma without proof.]
2F Topics in Analysis
Prove that cosh(1/2) is irrational.
3G Geometry of Group Actions Let G be a subgroup of the group of isometries Isom(R^2 ) of the Euclidean plane. What does it mean to say that G is discrete?
Supposing that G is discrete, show that the subgroup GT of G consisting of all translations in G is generated by translations in at most two linearly independent vectors in R^2. Show that there is a homomorphism G → O(2) with kernel GT.
Draw, and briefly explain, pictures which illustrate two different possibilities for G when GT is isomorphic to the additive group Z.
4J Coding and Cryptography
Briefly describe the methods of Shannon-Fano and Huffman for economical coding. Illustrate both methods by finding decipherable binary codings in the case where messages μ 1 ,... , μ 5 are emitted with probabilities 0. 45 , 0. 25 , 0. 2 , 0. 05 , 0 .05. Compute the expected word length in each case.
Paper 1
7B Dynamical Systems
State Dulac’s Criterion and the Poincar´e–Bendixson Theorem regarding the exis- tence of periodic solutions to the dynamical system ˙x = f (x) in R^2. Hence show that
x˙ = y y ˙ = −x + y(μ − 2 x^2 − y^2 )
has no periodic solutions if μ < 0 and at least one periodic solution if μ > 0.
8A Further Complex Methods
Explain what is meant by the Papperitz symbol
P
z 1 z 2 z 3 α β γ z α′^ β′^ γ′
The hypergeometric function F (a, b; c; z) is defined as the solution of the equation determined by the Papperitz symbol
P
0 a 0 z 1 − c b c − a − b
that is analytic at z = 0 and satisfies F (a, b; c; 0) = 1.
Show, explaining each step, that
F (a, b; c; z) = (1 − z)c−a−bF (c − a, c − b; c; z).
9C Classical Dynamics
A particle of mass m 1 is constrained to move on a circle of radius r 1 , centre x = y = 0 in a horizontal plane z = 0. A second particle of mass m 2 moves on a circle of radius r 2 , centre x = y = 0 in a horizontal plane z = c. The two particles are connected by a spring whose potential energy is
V = 12 ω^2 d^2 ,
where d is the distance between the particles. How many degrees of freedom are there? Identify suitable generalized coordinates and write down the Lagrangian of the system in terms of them.
Paper 1
10D Cosmology
(a) Around t ≈ 1 s after the big bang (kT ≈ 1 MeV), neutrons and protons are kept in equilibrium by weak interactions such as
n + νe ↔ p + e−^. (∗)
Show that, in equilibrium, the neutron-to-proton ratio is given by
nn np
≈ e−Q/kT^ ,
where Q = (mn − mp)c^2 = 1.29 MeV corresponds to the mass difference between the neutron and the proton. Explain briefly why we can neglect the difference μn − μp in the chemical potentials.
(b) The ratio of the weak interaction rate ΓW ∝ T 5 which maintains (∗) to the Hubble expansion rate H ∝ T 2 is given by
ΓW
H
kT 0 .8 MeV
Explain why the neutron-to-proton ratio effectively “freezes out” once kT < 0 .8 MeV, except for some slow neutron decay. Also explain why almost all neutrons are subsequently captured in 4 He; estimate the value of the relative mass density Y (^4) He = ρ (^4) He/ρB (with ρB = ρn + ρp) given a final ratio nn/np ≈ 1 /8.
(c) Suppose instead that the weak interaction rate were very much weaker than that described by equation (†). Describe the effect on the relative helium density Y (^4) He. Briefly discuss the wider implications of this primordial helium-to-hydrogen ratio on stellar lifetimes and life on earth.
Paper 1 [TURN OVER
Here the column headings are, respectively: Three-course meal, Bottle of Beer, Suntan Lotion, Taxi (5km), Film (24 exp), Car Hire (per week). Interpret the R commands, and explain how to interpret the corresponding (slightly abbreviated) R output given below. Your solution should include a careful statement of the underlying statistical model, but you may quote without proof any distributional results required.
price = scan("dresorts") ; price Goods = gl(6,1,length=84); Resort=gl(14,6,length=84)
first.lm = lm(log(price) ~ Goods + Resort)
summary(first.lm) Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.8778 0.1629 11.527 < 2e- Goods2 -2.1084 0.1295 -16.286 < 2e-
Goods3 -0.6343 0.1295 -4.900 6.69e-
Goods4 -0.6284 0.1295 -4.854 7.92e- Goods5 -0.9679 0.1295 -7.476 2.49e-
Goods6 2.8016 0.1295 21.640 < 2e- Resort2 0.4463 0.1978 2.257 0.
Resort3 0.4105 0.1978 2.076 0.
Resort4 0.3067 0.1978 1.551 0. Resort5 0.4235 0.1978 2.142 0.
Resort6 0.2883 0.1978 1.458 0.
Resort7 0.3457 0.1978 1.748 0. Resort8 0.3787 0.1978 1.915 0.
Resort9 0.0943 0.1978 0.477 0.
Resort10 0.5981 0.1978 3.025 0. Resort11 0.3281 0.1978 1.659 0.
Resort12 0.2525 0.1978 1.277 0. Resort13 0.5508 0.1978 2.785 0.
Resort14 0.4590 0.1978 2.321 0.
Residual standard error: 0.3425 on 65 degrees of freedom
Multiple R-Squared: 0.
Paper 1 [TURN OVER
14B Dynamical Systems
Consider the equations
x˙ = (a − x^2 )(a^2 − y) y˙ = x − y
as a function of the parameter a. Find the fixed points and plot their location in the (a, x) plane. Hence, or otherwise, deduce that there are bifurcations at a = 0 and a = 1.
Investigate the bifurcation at a = 1 by making the substitutions u = x−1, v = y−x and μ = a − 1. Find the equation of the extended centre manifold to second order. Find the evolution equation on the centre manifold to second order, and determine the stability of its fixed points.
Show which branches of fixed points in the (a, x) plane are stable and which are unstable, and state, without calculation, the type of bifurcation at a = 0. Hence sketch the structure of the (x, y) phase plane very near the origin for |a| 1 in the cases (i) a < 0 and (ii) a > 0.
The system is perturbed to x˙ = (a − x^2 )(a^2 − y) + , where 0 < 1, with y˙ = x − y still. Sketch the possible changes to the bifurcation diagram near a = 0 and a = 1. [Calculation is not required.]
15C Classical Dynamics
(i) The action for a system with generalized coordinates (qa) is given by
S =
∫ (^) t 2
t 1
L(qa, q˙b) dt.
Derive Lagrange’s equations from the principle of least action by considering all paths with fixed endpoints, δqa(t 1 ) = δqa(t 2 ) = 0.
(ii) A pendulum consists of a point mass m at the end of a light rod of length l. The pivot of the pendulum is attached to a mass M which is free to slide without friction along a horizontal rail. Choose as generalized coordinates the position x of the pivot and the angle θ that the pendulum makes with the vertical.
Write down the Lagrangian and derive the equations of motion.
Find the frequency of small oscillations around the stable equilibrium.
Now suppose that a force acts on the pivot causing it to travel with constant acceleration in the x-direction. Find the equilibrium angle θ of the pendulum.
Paper 1
19G Representation Theory
Let the finite group G act on finite sets X and Y , and denote by C[X], C[Y ] the associated permutation representations on the spaces of complex functions on X and Y. Call their characters χX and χY.
(i) Show that the inner product 〈χX |χY 〉 is the number of orbits for the diagonal action of G on X × Y.
(ii) Assume that |X| > 1, and let S ⊂ C[X] be the subspace of those functions whose values sum to zero. By considering ‖χX ‖^2 , show that S is irreducible if and only if the G-action on X is doubly transitive: this means that for any two pairs (x 1 , x 2 ) and (x′ 1 , x′ 2 ) of points in X with x 1 6 = x 2 and x′ 1 6 = x′ 2 , there exists some g ∈ G with gx 1 = x′ 1 and gx 2 = x′ 2.
(iii) Let now G = Sn acting on the set X = { 1 , 2 ,... , n}. Call Y the set of 2- element subsets of X, with the natural action of Sn. If n > 4, show that C[Y ] decomposes under Sn into three irreducible representations, one of which is the trivial representation and another of which is S. What happens when n = 3?
[Hint: Consider 〈 1 |χY 〉, 〈χX |χY 〉 and ‖χY ‖^2 .]
20G Number Fields
Let K = Q(
p) where p is a prime with p ≡ 3 (mod 4). By computing the relative traces TrK/k(θ) where k runs through the three quadratic subfields of K, show that the algebraic integers θ in K have the form
θ = 12 (a + b
p) + 12 (c + d
p)
where a, b, c, d are rational integers. By further computing the relative norm NK/k(θ) where k = Q(
2), show that 4 divides
a^2 + pb^2 − 2
c^2 + pd^2
and 2
ab − 2 cd
Deduce that a and b are even and c ≡ d (mod 2). Hence verify that an integral basis for K is 1 ,
p, (^12)
p
Paper 1
21H Algebraic Topology
(i) Show that if E → T is a covering map for the torus T = S^1 × S^1 , then E is homeomorphic to one of the following: the plane R^2 , the cylinder R × S^1 , or the torus T.
(ii) Show that any continuous map from a sphere Sn^ (n > 2) to the torus T is homotopic to a constant map.
[General theorems from the course may be used without proof, provided that they are clearly stated.]
22F Linear Analysis Let K be a compact Hausdorff space, and let C(K) denote the Banach space of continuous, complex-valued functions on K, with the supremum norm. Define what it means for a set S ⊂ C(K) to be totally bounded, uniformly bounded, and equicontinuous.
Show that S is totally bounded if and only if it is both uniformly bounded and equicontinuous.
Give, with justification, an example of a Banach space X and a subset S ⊂ X such that S is bounded but not totally bounded.
23H Riemann Surfaces Let Λ be a lattice in C generated by 1 and τ , where τ is a fixed complex number with Imτ > 0. The Weierstrass ℘-function is defined as a Λ-periodic meromorphic function such that
(1) the only poles of ℘ are at points of Λ, and
(2) there exist positive constants ε and M such that for all |z| < ε, we have
|℘(z) − 1 /z^2 | < M |z|.
Show that ℘ is uniquely determined by the above properties and that ℘(−z) = ℘(z). By considering the valency of ℘ at z = 1/2, show that ℘′′(1/2) 6 = 0.
Show that ℘ satisfies the differential equation
℘′′(z) = 6℘^2 (z) + A,
for some complex constant A.
[Standard theorems about doubly-periodic meromorphic functions may be used without proof provided they are accurately stated, but any properties of the ℘-function that you use must be deduced from first principles.]
Paper 1 [TURN OVER
26I Applied Probability
A cell has been placed in a biological solution at time t = 0. After an exponential time of rate μ, it is divided, producing k cells with probability pk, k = 0, 1,.. ., with the mean value ρ =
k=1 kpk^ (k^ = 0 means that the cell dies). The same mechanism is applied to each of the living cells, independently.
(a) Let Mt be the number of living cells in the solution by time t > 0. Prove that EMt = exp
[
tμ(ρ − 1)
]
. [You may use without proof, if you wish, the fact that, if a positive function a(t) satisfies a(t + s) = a(t)a(s) for t, s > 0 and is differentiable at zero, then a(t) = eαt, t > 0 , for some α.]
Let φt(s) = E sMt^ be the probability generating function of Mt. Prove that it satisfies the following differential equation
d dt
φt(s) = μ
−φt(s) +
∑^ ∞
k=
pk
[
φt(s)
]k
, with φ 0 (s) = s.
(b) Now consider the case where each cell is divided in two cells (p 2 = 1). Let Nt = Mt − 1 be the number of cells produced in the solution by time t.
Calculate the distribution of Nt. Is (Nt) an inhomogeneous Poisson process? If so, what is its rate λ(t)? Justify your answer.
27I Principles of Statistics
State Wilks’ Theorem on the asymptotic distribution of likelihood-ratio test statis- tics.
Suppose that X 1 ,... , Xn are independent with common N (μ, σ^2 ) distribution, where the parameters μ and σ are both unknown. Find the likelihood-ratio test statistic for testing H 0 : μ = 0 against H 1 : μ unrestricted, and state its (approximate) distribution.
What is the form of the t-test of H 0 against H 1? Explain why for large n the likelihood-ratio test and the t-test are nearly the same.
Paper 1 [TURN OVER
28J Stochastic Financial Models
Let X ≡ (X 0 , X 1 ,... , XJ )T^ be a zero-mean Gaussian vector, with covariance matrix V = (vjk). In a simple single-period economy with J agents, agent i will receive Xi at time 1 (i = 1,... , J). If Y is a contingent claim to be paid at time 1, define agent i’s reservation bid price for Y , assuming his preferences are given by E[Ui(Xi + Z)] for any contingent claim Z.
Assuming that Ui(x) ≡ − exp(−γix) for each i, where γi > 0, show that agent i’s reservation bid price for λ units of X 0 is
pi(λ) = −
γi(λ^2 v 00 + 2λv 0 i).
As λ → 0, find the limit of agent i’s per-unit reservation bid price for X 0 , and comment on the expression you obtain.
The agents bargain, and reach an equilibrium. Assuming that the contingent claim X 0 is in zero net supply, show that the equilibrium price of X 0 will be
p = −Γv 0 • ,
where Γ−^1 =
∑J
i=1 γ
− 1 i and^ v^0 •^ =^
∑J
i=1 v^0 i^. Show that at that price agent^ i^ will choose to buy θi = (Γv 0 • − γiv 0 i)/(γiv 00 )
units of X 0.
By computing the improvement in agent i’s expected utility, show that the value to agent i of access to this market is equal to a fixed payment of
(γiv 0 i − Γv 0 • )^2 2 γiv 00
Paper 1
31D Integrable Systems
Let φ(t) satisfy the linear singular integral equation
(t^2 + t − 1)φ(t) −
t^2 − t − 1 πi
L
φ(τ )dτ τ − t
πi
L
τ +
τ
φ(τ )dτ = t − 1 , t ∈ L,
where
denotes the principal value integral and L denotes a counterclockwise smooth closed contour, enclosing the origin but not the points ±1.
(a) Formulate the associated Riemann–Hilbert problem.
(b) For this Riemann–Hilbert problem, find the index, the homogeneous canonical solution and the solvability condition.
(c) Find φ(t).
32D Principles of Quantum Mechanics A one-dimensional harmonic oscillator has Hamiltonian
H =
2 m
pˆ^2 +
mω^2 xˆ^2 = ℏω
a†a +
where
a =
( (^) mω 2 ℏ
x ˆ +
i mω
pˆ
, a†^ =
( (^) mω 2 ℏ
x ˆ −
i mω
pˆ
obey [a, a†] = 1.
Assuming the existence of a normalised state | 0 〉 with a| 0 〉 = 0, verify that
|n〉 =
n!
a†^ n| 0 〉 , n = 0, 1 , 2 ,...
are eigenstates of H with energies En, to be determined, and that these states all have unit norm.
The Hamiltonian is now modified by a term
λV = λℏω( ar^ + a†^ r^ )
where r is a positive integer. Use perturbation theory to find the change in the lowest energy level to order λ^2 for any r. [You may quote any standard formula you need.]
Compute by perturbation theory, again to order λ^2 , the change in the first excited energy level when r = 1. Show that in this special case, r = 1, the exact change in all energy levels as a result of the perturbation is −λ^2 ℏω.
Paper 1
33B Applications of Quantum Mechanics
A beam of particles is incident on a central potential V (r) (r = |x|) that vanishes for r > R. Define the differential cross-section dσ/dΩ.
Given that each incoming particle has momentum ℏk, explain the relevance of solutions to the time-independent Schr¨odinger equation with the asymptotic form
ψ (x) ∼ eik·x^ + f (ˆx)
eikr r
as r → ∞, where k = |k| and ˆx = x/r. Write down a formula that determines dσ/dΩ in this case.
Write down the time-independent Schr¨odinger equation for a particle of mass m
and energy E =
ℏ^2 k^2 2 m
in a central potential V (r), and show that it allows a solution of
the form
ψ (x) = eik·x^ −
m 2 πℏ^2
d^3 x′^
eik|x−x
′|
|x − x′|
V (r′)ψ (x′).
Show that this is consistent with (∗) and deduce an expression for f (ˆx). Obtain the Born approximation for f (ˆx), and show that f (ˆx) = F (kxˆ − k), where
F (q) = −
m 2 πℏ^2
d^3 x e−iq·x^ V (r).
Under what conditions is the Born approximation valid?
Obtain a formula for f (ˆx) in terms of the scattering angle θ in the case that
V (r) = K
e−μr r
for constants K and μ. Hence show that f (ˆx) is independent of ℏ in the limit μ → 0, when expressed in terms of θ and the energy E.
[You may assume that (∇^2 + k^2 )
eikr r
= − 4 πδ^3 (x).]
Paper 1 [TURN OVER
35C General Relativity
Suppose (x(τ ), t(τ )) is a timelike geodesic of the metric
ds^2 =
dx^2 1 + x^2
− (1 + x^2 ) dt^2 ,
where τ is proper time along the world line. Show that dt/dτ = E/(1 + x^2 ), where E > 1 is a constant whose physical significance should be stated. Setting a^2 = E^2 − 1, show that
dτ =
dx √ a^2 − x^2
, dt =
E dx (1 + x^2 )
a^2 − x^2
Deduce that x is a periodic function of proper time τ with period 2π. Sketch dx/dτ as a function of x and superpose on this a sketch of dx/dt as a function of x. Given the identity
∫ (^) a
−a
E dx (1 + x^2 )
a^2 − x^2
= π ,
deduce that x is also a periodic function of t with period 2π.
Next consider the family of metrics
ds^2 =
[1 + f (x)]^2 dx^2 1 + x^2
− (1 + x^2 ) dt^2 ,
where f is an odd function of x, f (−x) = −f (x), and |f (x)| < 1 for all x. Derive expressions analogous to (∗) above. Deduce that x is a periodic function of τ and also that x is a periodic function of t. What are the periods?
Paper 1 [TURN OVER
36E Fluid Dynamics II
Consider a unidirectional flow with dynamic viscosity μ along a straight rigid-walled channel of uniform cross-sectional shape D driven by a uniform applied pressure gradient G. Write down the differential equation and boundary conditions governing the velocity w along the channel.
Consider the situation when the boundary includes a sharp corner of angle 2α. Explain why one might expect that, sufficiently close to the corner, the solution should be of the form w = (G/μ)r^2 f (θ) ,
where r and θ are polar co-ordinates with origin at the vertex and θ = ±α describing the two planes emanating from the corner. Determine f (θ).
If D is the sector bounded by the lines θ = ±α and the circular arc r = a, show that the flow is given by
w = (G/μ)r^2 f (θ) +
∑^ ∞
n=
Anrλn^ cos λnθ,
where λn and An are to be determined.
[Note that
cos(ax) cos(bx) dx = {a sin(ax) cos(bx) − b sin(bx) cos(ax)}/(a^2 − b^2 ).]
Considering the values of λ 0 and λ 1 , comment briefly on the cases: (i) 2α < 12 π; (ii) 12 π < 2 α < 32 π; and (iii) 32 π < 2 α < 2 π.
37E Waves
An elastic solid with density ρ has Lam´e moduli λ and μ. Write down equations satisfied by the dilational and shear potentials φ and ψ.
For a two-dimensional disturbance give expressions for the displacement field u = (ux, uy , 0) in terms of φ(x, y; t) and ψ = (0, 0 , ψ(x, y; t)).
Suppose the solid occupies the region y < 0 and that the surface y = 0 is free of traction. Find a combination of solutions for φ and ψ that represent a propagating surface wave (a Rayleigh wave) near y = 0. Show that the wave is non-dispersive and obtain an equation for the speed c. [You may assume without proof that this equation has a unique positive root.]
Paper 1
