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MATHEMATICAL TRIPOS Part II Alternative B
Wednesday 5 June 2002 9 to 12
PAPER 3
Before you begin read these instructions carefully.
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 C, D, E,... , M according to the letter affixed to each question. (For example, 9K, 10K should be in one bundle and 1M, 15M 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.
1M Markov Chains
(i) Consider the continuous-time Markov chain (Xt)t> 0 on { 1 , 2 , 3 , 4 , 5 , 6 , 7 } with generator matrix
Q =
Compute the probability, starting from state 3, that Xt hits state 2 eventually.
Deduce that lim t→∞
P(Xt = 2|X 0 = 3) =
[Justification of standard arguments is not expected.]
(ii) A colony of cells contains immature and mature cells. Each immature cell, after an exponential time of parameter 2, becomes a mature cell. Each mature cell, after an exponential time of parameter 3, divides into two immature cells. Suppose we begin with one immature cell and let n(t) denote the expected number of immature cells at time t. Show that n(t) = (4et^ + 3e−^6 t)/ 7.
Paper 3
3D Electromagnetism
(i) A plane electromagnetic wave in a vacuum has an electric field
E = (E 1 , E 2 , 0) cos(kz − ωt),
referred to cartesian axes (x, y, z). Show that this wave is plane polarized and find the orientation of the plane of polarization. Obtain the corresponding plane polarized magnetic field and calculate the rate at which energy is transported by the wave.
(ii) Suppose instead that
E = (E 1 cos(kz − ωt), E 2 cos(kz − ωt + φ), 0),
with φ a constant, 0 < φ < π. Show that, if the axes are now rotated through an angle ψ so as to obtain an elliptically polarized wave with an electric field
E′^ = (F 1 cos(kz − ωt + χ), F 2 sin(kz − ωt + χ), 0),
then
tan 2ψ =
2 E 1 E 2 cos φ E 12 − E 22
Show also that if E 1 = E 2 = E there is an elliptically polarized wave with
E′^ =
2 E
cos(kz − ωt + 12 φ) cos 12 φ, sin(kz − ωt + 12 φ) sin 12 φ, 0
Paper 3
4F Dynamics of Differential Equations
(i) Define the Floquet multiplier and Liapunov exponent for a periodic orbit ˆx(t) of a dynamical system ˙x = f (x) in R^2. Show that one multiplier is always unity, and that the other is given by
exp
(∫ T
0
∇· f (ˆx(t))dt
where T is the period of the orbit.
The Van der Pol oscillator ¨x + x˙(x^2 − 1) + x = 0 , 0 < 1 has a limit cycle xˆ(t) ≈ 2 sin t. Show using (∗) that this orbit is stable.
(ii) Show, by considering the normal form for a Hopf bifurcation from a fixed point x 0 (μ) of a dynamical system ˙x = f (x, μ), that in some neighbourhood of the bifurcation the periodic orbit is stable when it exists in the range of μ for which x 0 is unstable, and unstable in the opposite case.
Now consider the system
x˙ = x(1 − y) + μx y ˙ = y(x − 1) − μx
x > 0.
Show that the fixed point (1+μ , 1+μ) has a Hopf bifurcation when μ = 0, and is unstable (stable) when μ > 0 (μ < 0).
Suppose that a periodic orbit exists in μ > 0. Show without solving for the orbit that the result of part (i) shows that such an orbit is unstable. Define a similar result for μ < 0.
What do you conclude about the existence of periodic orbits when μ 6 = 0? Check your answer by applying Dulac’s criterion to the system, using the weighting ρ = e−(x+y).
5J Representation Theory Let G be a finite group acting on a finite set X. Define the permutation representation (ρ, C[X]) of G and compute its character πX. Prove that 〈πX , (^1) G〉G equals the number of orbits of G on X. If G acts also on the finite set Y , with character πY , show that 〈πX , πY 〉G equals the number of orbits of G on X × Y.
Now let G be the symmetric group Sn acting naturally on the set X = { 1 ,... , n}, and let Xr be the set of all r-element subsets of X. Let πr be the permutation character of G on Xr. Prove that
〈πk, π〉G = + 1 for 0 6 ` 6 k 6 n/ 2.
Deduce that the class functions χr = πr − πr− 1
are irreducible characters of Sn, for 1 6 r 6 n/2.
Paper 3 [TURN OVER
9K Riemann Surfaces
Let α 1 , α 2 be two non-zero complex numbers with α 1 /α 2 6 ∈ R. Let L be the lattice Zα 1 ⊕ Zα 2 ⊂ C. A meromorphic function f on C is elliptic if f (z + λ) = f (z), for all z ∈ C and λ ∈ L. The Weierstrass functions ℘(z), ζ(z), σ(z) are defined by the following properties:
- ℘(z) is elliptic, has double poles at the points of L and no other poles, and ℘(z) = 1 /z^2 + O(z^2 ) near 0;
- ζ′(z) = −℘(z), and ζ(z) = 1/z + O(z^3 ) near 0;
- σ(z) is odd, and σ′(z)/σ(z) = ζ(z), and σ(z)/z → 1 as z → 0.
Prove the following. (a) ℘, and hence ζ and σ, are uniquely determined by these properties. You are not expected to prove the existence of ℘, ζ, σ, and you may use Liouville’s theorem without proof. (b) ζ(z +αi) = ζ(z)+2ηi, and σ(z +αi) = kie^2 ηiz^ σ(z), for some constants ηi, ki (i = 1, 2). (c) σ is holomorphic, has simple zeroes at the points of L, and has no other zeroes. (d) Given a 1 ,... , an and b 1 ,... , bn in C with a 1 +... + an = b 1 +... + bn, the function
σ(z − a 1 ) · · · σ(z − an) σ(z − b 1 ) · · · σ(z − bn)
is elliptic.
(e) ℘(u) − ℘(v) = −
σ(u + v)σ(u − v) σ^2 (u)σ^2 (v)
(f) Deduce from (e), or otherwise, that
℘′(u) − ℘′(v) ℘(u) − ℘(v)
= ζ(u + v) − ζ(u) − ζ(v).
10K Algebraic Curves
Let f = f (x, y) be an irreducible polynomial of degree n ≥ 2 (over an algebraically closed field of characteristic zero) and V 0 = {f = 0} ⊂ A^2 the corresponding affine plane curve. Assume that V 0 is smooth (non-singular) and that the projectivization V ⊂ P^2 of V 0 intersects the line at infinity P^2 − A^2 in n distinct points. Show that V is smooth and
determine the divisor of the rational differential ω =
dx f (^) y′
on V. Deduce a formula for the
genus of V.
Paper 3 [TURN OVER
11J Logic, Computation and Set Theory
(i) Explain briefly what is meant by the terms register machine and computable function.
Let u be the universal computable function u(m, n) = fm(n) and s a total computable function with fs(m,n)(k) = fm(n, k). Here fm(n) and fm(n, k) are the unary and binary functions computed by the m-th register machine program Pm. Suppose h : N → N is a total computable function. By considering the function
g(m, n) = u(h(s(m, m)), n)
show that there is a number a such that fa = fh(a).
(ii) Let P be the set of all partial functions N × N → N. Consider the mapping Φ : P → P defined by
Φ(g)(m, n) =
n + 1 if m = 0, g(m − 1 , 1) if m > 0, n = 0 and g(m − 1 , 1) defined, g(m − 1 , g(m, n − 1)) if mn > 0 and g(m − 1 , g(m, n − 1)) defined, undefined otherwise.
(a) Show that any fixed point of Φ is a total function N × N → N. Deduce that Φ has a unique fixed point. [The Bourbaki-Witt Theorem may be assumed if stated precisely.]
(b) It follows from standard closure properties of the computable functions that there is a computable function ψ such that
ψ(p, m, n) = Φ(fp)(m, n).
Assuming this, show that there is a total computable function h such that
Φ(fp) = fh(p) for all p.
Deduce that the fixed point of Φ is computable.
12L Probability and Measure Derive the characteristic function of a real-valued random variable which is normally distributed with mean μ and variance σ^2. What does it mean to say that an Rn-valued random variable has a multivariate Gaussian distribution? Prove that the distribution of such a random variable is determined by its (Rn-valued) mean and its covariance matrix.
Let X and Y be random variables defined on the same probability space such that (X, Y ) has a Gaussian distribution. Show that X and Y are independent if and only if cov(X, Y ) = 0. Show that, even if they are not independent, one may always write X = aY + Z for some constant a and some random variable Z independent of Y.
[The inversion theorem for characteristic functions and standard results about indepen- dence may be assumed.]
Paper 3
14L Optimization and Control
Consider a scalar system with xt+1 = (xt + ut)ξt, where ξ 0 , ξ 1 ,... is a sequence of independent random variables, uniform on the interval [−a, a], with a 6 1. We wish to choose u 0 ,... , uh− 1 to minimize the expected value of
h∑− 1
t=
(c + x^2 t + u^2 t ) + 3x^2 h ,
where ut is chosen knowing xt but not ξt. Prove that the minimal expected cost can be written Vh(x 0 ) = hc + x^20 Πh and derive a recurrence for calculating Π 1 ,... , Πh.
How does your answer change if ut is constrained to lie in the set U(xt) = {u : |u + xt| < |xt|}?
Consider a stopping problem for which there are two options in state xt, t > 0:
(1) stop: paying a terminal cost 3x^2 t ; no further costs are incurred; (2) continue: choosing ut ∈ U(xt), paying c + u^2 t + x^2 t , and moving to state xt+1 = (xt + ut)ξt. Consider the problem of minimizing total expected cost subject to the constraint that no more than h continuation steps are allowed. Suppose a = 1. Show that an optimal policy stops if and only if either h continuation steps have already been taken or x^2 6 2 c/3.
[Hint: Use induction on h to show that a one-step-look-ahead rule is optimal. You should not need to find the optimal ut for the continuation steps.]
Paper 3
15M Principles of Statistics
(i) Describe in detail how to perform the Wald, score and likelihood ratio tests of a simple null hypothesis H 0 : θ = θ 0 given a random sample X 1 ,... , Xn from a regular one- parameter density f (x; θ). In each case you should specify the asymptotic null distribution of the test statistic.
(ii) Let X 1 ,... , Xn be an independent, identically distributed sample from a distribu- tion F , and let θˆ(X 1 ,... , Xn) be an estimator of a parameter θ of F.
Explain what is meant by: (a) the empirical distribution function of the sample; (b) the bootstrap estimator of the bias of θˆ, based on the empirical distribution function. Explain how a bootstrap estimator of the distribution function of θˆ − θ may be used to construct an approximate 1 − α confidence interval for θ.
Suppose the parameter of interest is θ = μ^2 , where μ is the mean of F , and the estimator is θˆ = X¯^2 , where X¯ = n−^1
∑n i=1 Xi^ is the sample mean. Derive an explicit expression for the bootstrap estimator of the bias of θˆ and show that it is biased as an estimator of the true bias of θˆ.
Let θˆi be the value of the estimator θˆ(X 1 ,... , Xi− 1 , Xi+1,... , Xn) computed from the sample of size n − 1 obtained by deleting Xi and let θˆ. = n−^1
∑n i=1 θˆi. The^ jackknife estimator of the bias of ˆθ is bJ = (n − 1) (θˆ. − ˆθ).
Derive the jackknife estimator bJ for the case θˆ = X¯^2 , and show that, as an estimator of the true bias of ˆθ, it is unbiased.
Paper 3 [TURN OVER
18G Partial Differential Equations
Define the Schwartz space S(Rn) and the space of tempered distributions S′(Rn). State the Fourier inversion theorem for the Fourier transform of a Schwartz function.
Consider the initial value problem:
∂^2 u ∂t^2
− ∆u + u = 0 , x ∈ Rn^ , 0 < t < ∞ ,
u(0, x) = f (x) ,
∂u ∂t
(0, x) = 0
for f in the Schwartz space S(Rn).
Show that the solution can be written as
u(t, x) = (2π)−n/^2
Rn
eix·ξ^ uˆ(t, ξ)dξ ,
where uˆ(t, ξ) = cos
t
1 + |ξ|^2
fˆ (ξ)
and
fˆ (ξ) = (2π)−n/^2
Rn
e−ix·ξ^ f (x)dx.
State the Plancherel-Parseval theorem and hence deduce that ∫
Rn
|u(t, x)|^2 dx ≤
Rn
|f (x)|^2 dx.
Paper 3 [TURN OVER
19G Methods of Mathematical Physics
Show that the equation
zw′′^ + w′^ + (λ − z)w = 0
has solutions of the form
w(z) =
γ
(t − 1)(λ−1)/^2 (t + 1)−(λ+1)/^2 eztdt.
Give examples of possible choices of γ to provide two independent solutions, assuming Re(z) > 0. Distinguish between the cases Re λ > −1 and Re λ < 1. Comment on the case − 1 < Re λ < 1 and on the case that λ is an odd integer.
Show that, if Re λ < 1 , there is a solution w 1 (z) that is bounded as z → +∞, and that, in this limit,
w 1 (z) ∼ A e−z^ z(λ−1)/^2
(1 − λ)^2 8 z
where A is a constant.
20F Numerical Analysis (i) Determine the order of the multistep method
yn+2 − (1 + α)yn+1 + αyn = h[ 121 (5 + α)fn+2 + 23 (1 − α)fn+1 − 121 (1 + 5α)fn]
for the solution of ordinary differential equations for different choices of α in the range − 1 6 α 6 1.
(ii) Prove that no such choice of α results in a method whose linear stability domain includes the interval (−∞, 0).
Paper 3
22D Statistical Physics
A system consisting of non-interacting bosons has single-particle levels uniquely labelled by r with energies r , r ≥ 0. Show that the free energy in the grand canonical ensemble is F = kT
r
log(1 − e−β(r^ −μ)).
What is the maximum value for μ?
A system of N bosons in a large volume V has one energy level of energy zero and a large number M 1 of energy levels of the same energy , where M takes the form M = AV with A a positive constant. What are the dimensions of A?
Show that the free energy is
F = kT
log(1 − eβμ) + AV log(1 − e−β(−μ))
The numbers of particles with energies 0, are respectively N 0 , N. Write down expressions for N 0 , N in terms of μ.
At temperature T what is the maximum number of bosons N (^) max in the normal phase (the state with energy )? Explain what happens when N > N (^) max.
Given N and T calculate the transition temperature TB at which Bose condensation occurs.
For T > TB show that μ = (TB − T )/TB. What is the value of μ for T < TB?
Calculate the mean energy E for (a) T > TB (b) T < TB , and show that the heat capacity of the system at constant volume is
CV =
kT 2
AV ^2
(eβ^ − 1)^2
T < TB
0 T > TB.
Paper 3
23E Applications of Quantum Mechanics
A periodic potential is expressed as V (x) =
g ag^ e
ig·x, where {g} are reciprocal
lattice vectors and ag∗^ = a−g, a 0 = 0. In the nearly free electron model explain why it is appropriate, near the boundaries of energy bands, to consider a Bloch wave state
|ψk〉 =
r
αr |kr 〉 , kr = k + gr ,
where |k〉 is a free electron state for wave vector k, 〈k′|k〉 = δk′k, and the sum is restricted to reciprocal lattice vectors gr such that |kr | ≈ |k|. Obtain a determinantal formula for the possible energies E(k) corresponding to Bloch wave states of this form.
[You may take g 1 = 0 and assume eib·x|k〉 = |k + b〉 for any b.]
Suppose the sum is restricted to just k and k + g. Show that there is a gap 2|ag| between energy bands. Setting k = − 12 g + q, show that there are two Bloch wave states with energies near the boundaries of the energy bands
E±(k) ≈
ℏ^2 |g|^2 8 m
± |ag| +
ℏ^2 |q|^2 2 m
ℏ^4
8 m^2 |ag|
(q·g)^2.
What is meant by effective mass? Determine the value of the effective mass at the top and the bottom of the adjacent energy bands if q is parallel to g.
Paper 3 [TURN OVER
