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MATHEMATICAL TRIPOS Part II Alternative B
Thursday 7 June 2001 1.30 to 4.
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,... , L according to the letter affixed to each question. (For example, 2C, 5C should be in one bundle and 11D, 14D 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.
1A Combinatorics
Write an essay on extremal graph theory. You should give proofs of at least two major theorems and you should also include a description of alternative proofs or of further results.
2C Representation Theory
Let G be the Heisenberg group of order p^3. This is the subgroup
G =
1 a x 0 1 b 0 0 1
∣ a, b, x^ ∈^ Fp
of 3 × 3 matrices over the finite field Fp (p prime). Let H be the subgroup of G of such matrices with a = 0.
(i) Find all one dimensional representations of G.
[You may assume without proof that [G, G] is equal to the set of matrices in G with a = b = 0.]
(ii) Let ψ : Fp = Z/pZ −→ C∗^ be a non-trivial one dimensional representation of Fp, and define a one dimensional representation ρ of H by
ρ
1 0 x 0 1 b 0 0 1
(^) = ψ(x).
Show that Vψ = IndGH (ρ) is irreducible.
(iii) List all the irreducible representations of G and explain why your list is complete.
Paper 4
6B Number Fields
For a prime number p > 2, set ζ = e^2 πi/p, K = Q(ζ) and K+^ = Q(ζ + ζ−^1 ).
(a) Show that the (normalized) minimal polynomial of ζ − 1 over Q is equal to
f (x) =
(x + 1)p^ − 1 x
(b) Determine the degrees [K : Q] and [K+^ : Q].
(c) Show that p∏− 1
j=
(1 − ζj^ ) = p.
(d) Show that disc(f ) = (−1)
p− 2 1 pp−^2.
(e) Show that K contains Q(
p∗), where p∗^ = (−1)
p− 2 1 p.
(f) If j, k ∈ Z are not divisible by p, show that 1 −ζ
j 1 −ζk^ lies in^ O
∗ K. (g) Show that the ideal (p) = pOK is equal to (1 − ζ)p−^1.
7A Hilbert Spaces
Write an essay on the use of Hermite functions in the elementary theory of the Fourier transform on L^2 (R).
[You should assume, without proof, any results that you need concerning the approximation of functions by Hermite functions.]
Paper 4
8B Riemann Surfaces
Let λ and μ be fixed, non-zero complex numbers, with λ/μ 6 ∈ R, and let Λ = Zμ+Zλ be the lattice they generate in C. The series
℘(z) =
z^2
m,n
[ 1
(z − mλ − nμ)^2
(mλ + nμ)^2
]
with the sum taken over all pairs (m, n) ∈ Z × Z other than (0,0), is known to converge to an elliptic function, meaning a meromorphic function ℘ : C → C ∪ {∞} satisfying ℘(z) = ℘(z + μ) = ℘(z + λ) for all z ∈ C. (℘ is called the Weierstrass function.)
(a) Find the three zeros of ℘′^ modulo Λ, explaining why there are no others.
(b) Show that, for any number a ∈ C, other than the three values ℘(λ/2), ℘(μ/2) and ℘((λ + μ)/2), the equation ℘(z) = a has exactly two solutions, modulo Λ; whereas, for each of the specified values, it has a single solution.
[In (a) and (b), you may use, without proof, any known results about valencies and degrees of holomorphic maps between compact Riemann surfaces, provided you state them correctly.] (c) Prove that every even elliptic function φ(z) is a rational function of ℘(z); that is, there exists a rational function R for which φ(z) = R(℘(z)).
9B Algebraic Curves
Write an essay on curves of genus one (over an algebraically closed field k of characteristic zero). Legendre’s normal form should not be discussed.
10B Logic, Computation and Set Theory
What is a wellfounded relation, and how does wellfoundedness underpin wellfounded induction?
A formula φ(x, y) with two free variables defines an ∈-automorphism if for all x there is a unique y, the function f , defined by y = f (x) if and only if φ(x, y), is a permutation of the universe, and we have (∀xy)(x ∈ y ↔ f (x) ∈ f (y)).
Use wellfounded induction over ∈ to prove that all formulæ defining ∈-automorphisms are equivalent to x = y.
Paper 4 [TURN OVER
13E Information Theory
State the Kraft inequality. Prove that it gives a necessary and sufficient condition for the existence of a prefix-free code with given codeword lengths.
14D Optimization and Control
Consider the scalar system with plant equation xt+1 = xt + ut, t = 0, 1 ,... and cost
Cs(x 0 , u 0 , u 1 ,.. .) =
∑^ s
t=
[
u^2 t +
x^2 t
]
Show from first principles that minu 0 ,u 1 ,... Cs = Vsx^20 , where V 0 = 4/3 and for s = 0, 1 ,.. .,
Vs+1 = 4/3 + Vs/(1 + Vs).
Show that Vs → 2 as s → ∞. Prove that C∞ is minimized by the stationary control, ut = − 2 xt/3 for all t.
Consider the stationary policy π 0 that has ut = −xt for all t. What is the value of C∞ under this policy?
Consider the following algorithm, in which steps 1 and 2 are repeated as many times as desired.
- For a given stationary policy πn, for which ut = knxt for all t, determine the value of C∞ under this policy as V πn^ x^20 by solving for V πn^ in
V πn^ = k n^2 + 4/3 + (1 + kn)^2 V πn^.
- Now find kn+1 as the minimizer of
k^2 n+1 + 4/3 + (1 + kn+1)^2 V πn
and define πn+1 as the policy for which ut = kn+1xt for all t. Explain why πn+1 is guaranteed to be a better policy than πn.
Let π 0 be the stationary policy with ut = −xt. Determine π 1 and verify that it minimizes C∞ to within 0.2% of its optimum.
15E Principles of Statistics
Write an account, with appropriate examples, of one of the following: (a) Inference in multi-parameter exponential families;
(b) Asymptotic properties of maximum-likelihood estimators and their use in hypoth- esis testing;
(c) Bootstrap inference.
Paper 4 [TURN OVER
16D Stochastic Financial Models
Write an essay on the mean-variance approach to portfolio selection in a one-period model. Your essay should contrast the solution in the case when all the assets are risky with that for the case when there is a riskless asset.
17K Dynamical Systems
Define the rotation number ρ(f ) of an orientation-preserving circle map f and the rotation number ρ(F ) of a lift F of f. Prove that ρ(f ) and ρ(F ) are well-defined. Prove also that ρ(F ) is a continuous function of F.
State without proof the main consequence of ρ(f ) being rational.
18A Partial Differential Equations Write an essay on one of the following two topics:
(a) The notion of well-posedness for initial and boundary value problems for differential equations. Your answer should include a definition and give examples and state precise theorems for some specific problems.
(b) The concepts of distribution and tempered distribution and their use in the study of partial differential equations.
19L Methods of Mathematical Physics Show that
∫ (^) π 0 e
ix cos t (^) dt satisfies the differential equation
xy′′^ + y′^ + xy = 0,
and find a second solution, in the form of an integral, for x > 0.
Show, by finding the asymptotic behaviour as x → +∞, that your two solutions are linearly independent.
20K Numerical Analysis
Write an essay on the computation of eigenvalues and eigenvectors of matrices.
Paper 4
22F Foundations of Quantum Mechanics
(i) The two states of a spin- 12 particle corresponding to spin pointing along the z axis are denoted by | ↑〉 and | ↓〉. Explain why the states
| ↑, θ〉 = cos
θ 2
| ↑〉 + sin
θ 2
| ↓〉, | ↓, θ〉 = − sin
θ 2
| ↑〉 + cos
θ 2
correspond to the spins being aligned along a direction at an angle θ to the z direction.
The spin-0 state of two spin- 12 particles is
Show that this is independent of the direction chosen to define | ↑〉 1 , 2 , | ↓〉 1 , 2. If the spin of particle 1 along some direction is measured to be 12 ℏ show that the spin of particle 2 along the same direction is determined, giving its value.
[The Pauli matrices are given by
σ 1 =
, σ 2 =
0 −i i 0
, σ 3 =
]
(ii) Starting from the commutation relation for angular momentum in the form
[J 3 , J±] = ±ℏJ±, [J+, J−] = 2ℏJ 3 ,
obtain the possible values of j, m, where mℏ are the eigenvalues of J 3 and j(j + 1)ℏ^2 are the eigenvalues of J^2 = 12 (J+J− + J−J+) + J 32. Show that the corresponding normalized eigenvectors, |j, m〉, satisfy
J±|j, m〉 = ℏ ((j ∓ m)(j ± m + 1))^1 /^2 |j, m± 1 〉,
and that 1 n!
J (^) −n |j, j〉 = ℏn
(2j)! n!(2j − n)!
|j, j−n〉, n ≤ 2 j.
The state |w〉 is defined by
|w〉 = ewJ−/ℏ|j, j〉,
for any complex w. By expanding the exponential show that 〈w|w〉 = (1 + |w|^2 )^2 j^. Verify that e−wJ−/ℏJ 3 ewJ−/ℏ^ = J 3 − wJ−,
and hence show that
J 3 |w〉 = ℏ
j − w
∂w
|w〉.
If H = αJ 3 verify that |eiαt〉 e−ijαt^ is a solution of the time-dependent Schr¨odinger equation.
Paper 4
23F Statistical Physics
Given that the free energy F can be written in terms of the partition function Z as F = −kT log Z show that the entropy S and internal energy E are given by
S = k
∂(T log Z) ∂T
, E = kT 2
∂ log Z ∂T
A system of particles has Hamiltonian H(p, q) where p is the set of particle momenta and q the set of particle coordinates. Write down the expression for the classical partition function ZC for this system in equilibrium at temperature T. In a particular case H is given by H(p, q) = pαAαβ (q)pβ + qαBαβ (q)qβ.
Let H be a homogeneous function in all the pα, 1 ≤ α ≤ N , and in a subset of the qα, 1 ≤ α ≤ M (M ≤ N ). Derive the principle of equipartition for this system.
A system consists of N weakly interacting harmonic oscillators each with Hamilto- nian
h(p, q) =
p^2 +
ω^2 q^2.
Using equipartition calculate the classical specific heat of the system, CC (T ). Also calculate the classical entropy SC (T ).
Write down the expression for the quantum partition function of the system and derive expressions for the specific heat C(T ) and the entropy S(T ) in terms of N and the parameter θ = ℏω/kT. Show for θ 1 that
C(T ) = CC (T ) + O (θ) , S(T ) = SC (T ) + S 0 + O (θ) ,
where S 0 should be calculated. Comment briefly on the physical significance of the constant S 0 and why it is non-zero.
Paper 4 [TURN OVER
26H Fluid Dynamics II
Starting from the steady planar vorticity equation u .∇ω = ν∇^2 ω,
outline briefly the derivation of the boundary layer equation
uux + υuy = U dU/dx + νuyy ,
explaining the significance of the symbols used.
Viscous fluid occupies the region y > 0 with rigid stationary walls along y = 0 for x > 0 and x < 0. There is a line sink at the origin of strength πQ, Q > 0, with Q/ν 1. Assuming that vorticity is confined to boundary layers along the rigid walls:
(a) Find the flow outside the boundary layers.
(b) Explain why the boundary layer thickness δ along the wall x > 0 is proportional to x, and deduce that
δ =
ν Q
x.
(c) Show that the boundary layer equation admits a solution having stream function ψ = (νQ)^1 /^2 f (η) with η = y/δ. Find the equation and boundary conditions satisfied by f. (d) Verify that a solution is
f ′^ =
1 + cosh(η
2 + c)
provided that c has one of two values to be determined. Should the positive or negative value be chosen?
27L Waves in Fluid and Solid Media
Derive the ray-tracing equations governing the evolution of a wave packet φ(x, t) = A(x, t) exp{iψ(x, t)} in a slowly varying medium, stating the conditions under which the equations are valid.
Consider now a stationary obstacle in a steadily moving homogeneous two-dimen- sional medium which has the dispersion relation
ω(k 1 , k 2 ) = α
k 12 + k^22
− V k 1 ,
where (V, 0) is the velocity of the medium. The obstacle generates a steady wave system. Writing (k 1 , k 2 ) = κ(cos φ, sin φ), show that the wave satisfies
κ =
α^2 V 2 cos^2 φ
Show that the group velocity of these waves can be expressed as
cg = V ( 12 cos^2 φ − 1 , 12 cos φ sin φ).
Deduce that the waves occupy a wedge of semi-angle sin−1 1 3 about the negative x 1 -axis.
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