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MATHEMATICAL TRIPOS Part II
Wednesday, 6 June, 2012 9:00 am to 12:00 pm
PAPER 2
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,.. ., K 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 completed gold cover sheet to each bundle.
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 REQUIREMENTS
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
1I Number Theory Define the Legendre symbol and the Jacobi symbol. State the law of quadratic reciprocity for the Jacobi symbol.
Compute the value of the Jacobi symbol
, stating clearly any results you use.
2F Topics in Analysis (a) Let γ : [0, 1] → C \ { 0 } be a continuous map such that γ(0) = γ(1). Define the winding number w(γ; 0) of γ about the origin. State precisely a theorem about homotopy invariance of the winding number.
(b) Let f : C → C be a continuous map such that z−^10 f (z) is bounded as |z| → ∞. Prove that there exists a complex number z 0 such that
f (z 0 ) = z^110.
3G Geometry and Groups Define the modular group acting on the upper half-plane. Explain briefly why it acts discontinuously and describe a fundamental domain. You should prove that the region which you describe is a fundamental domain.
4G Coding and Cryptography What is a (binary) linear code? What does it mean to say that a linear code has length n and minimum weight d? When is a linear code perfect? Show that, if n = 2r^ − 1, there exists a perfect linear code of length n and minimum weight 3.
Part II, Paper 2
6C Mathematical Biology Consider a birth-death process in which the birth rate per individual is λ and the death rate per individual in a population of size n is βn.
Let P (n, t) be the probability that the population has size n at time t. Write down the master equation for the system, giving an expression for ∂P (n, t)/∂t.
Show that d dt
〈n〉 = λ〈n〉 − β〈n^2 〉 ,
where 〈.〉 denotes the mean.
Deduce that in a steady state 〈n〉 6 λ/β.
Part II, Paper 2
7D Dynamical Systems Consider the dynamical system
x˙ = μx + x^3 − axy, y˙ = μ − x^2 − y ,
where a is a constant.
(a) Show that there is a bifurcation from the fixed point (0, μ) at μ = 0.
(b) Find the extended centre manifold at leading non-trivial order in x. Hence find the type of bifurcation, paying particular attention to the special values a = 1 and a = −1. [Hint. At leading order, the extended centre manifold is of the form y = μ + αx^2 + βμx^2 + γx^4 , where α, β, γ are constants to be determined.]
8E Further Complex Methods The hypergeometric function F (a, b; c; z) is defined as the particular solution of the second order linear ODE characterised by the Papperitz symbol
P
0 0 a z 1 − c c − a − b b
that is analytic at z = 0 and satisfies F (a, b; c; 0) = 1. Using the fact that a second solution w(z) of the above ODE is of the form
w(z) = z^1 −cu(z) ,
where u(z) is analytic in the neighbourhood of the origin, express w(z) in terms of F.
Part II, Paper 2 [TURN OVER
10E Cosmology The Friedmann equation for the scale factor a(t) of a homogeneous and isotropic universe of mass density ρ is
H^2 =
8 πGρ 3
kc^2 a^2
H =
a˙ a
where ˙a = da/dt and k is a constant. The mass conservation equation for a fluid of mass density ρ and pressure P is ρ˙ = − 3
ρ + P/c^2
H.
Conformal time τ is defined by dτ = a−^1 dt. Show that
H = aH ,
H =
a′ a
where a′^ = da/dτ. Hence show that the acceleration equation can be written as
H′^ = −
4 π 3
G (ρ + 3P/c^2 ) a^2.
Define the density parameter Ωm and show that in a matter-dominated era, in which P = 0, it satisfies the equation
Ω′ m = H Ωm(Ωm − 1).
Use this result to briefly explain the “flatness problem” of cosmology.
Part II, Paper 2 [TURN OVER
SECTION II
11F Topics in Analysis
(a) State Runge’s theorem about uniform approximability of analytic functions by com- plex polynomials.
(b) Let K be a compact subset of the complex plane.
(i) Let Σ be an unbounded, connected subset of C \ K. Prove that for each ζ ∈ Σ, the function f (z) = (z − ζ)−^1 is uniformly approximable on K by a sequence of complex polynomials. [You may not use Runge’s theorem without proof.] (ii) Let Γ be a bounded, connected component of C \ K. Prove that there is no point ζ ∈ Γ such that the function f (z) = (z − ζ)−^1 is uniformly approximable on K by a sequence of complex polynomials.
12G Coding and Cryptography What does it mean to say that f : Fd 2 → Fd 2 is a linear feedback shift register? Let (xn)n> 0 be a stream produced by such a register. Show that there exist N, M with N + M 6 2 d^ − 1 such that xr+N = xr for all r > M.
Describe and justify the Berlekamp–Massey method for ‘breaking’ a cipher stream arising from a linear feedback register of unknown length.
Let xn, yn, zn be three streams produced by linear feedback registers. Set
kn = xn if yn = zn
kn = yn if yn 6 = zn.
Show that kn is also a stream produced by a linear feedback register. Sketch proofs of any theorems you use.
Part II, Paper 2
14E Further Complex Methods Let the complex function q(x, t) satisfy
i
∂q(x, t) ∂t
∂^2 q(x, t) ∂x^2 = 0 , 0 < x < ∞ , 0 < t < T ,
where T is a positive constant. The unified transform method implies that the solution of any well-posed problem for the above equation is given by
q(x, t) =
2 π
−∞
eikx−ik
(^2) t qˆ 0 (k)dk
2 π
L
eikx−ik (^2) t [ k˜g 0 (ik^2 , t) − i˜g 1 (ik^2 , t)
]
dk , (1)
where L is the union of the rays (i∞, 0) and (0, ∞), ˆq 0 (k) denotes the Fourier transform of the initial condition q 0 (x), and ˜g 0 , ˜g 1 denote the t-transforms of the boundary values q(0, t), qx(0, t):
qˆ 0 (k) =
0
e−ikxq 0 (x)dx, Im k 6 0 ,
g ˜ 0 (k, t) =
∫ (^) t
0
eksq(0, s)ds , ˜g 1 (k, t) =
∫ (^) t
0
eksqx(0, s)ds , k ∈ C , 0 < t < T.
Furthermore, q 0 (x), q(0, t) and qx(0, t) are related via the so-called global relation
eik (^2) t qˆ(k, t) = ˆq 0 (k) + kg˜ 0 (ik^2 , t) − ig˜ 1 (ik^2 , t) , Im k 6 0 , (2)
where ˆq(k, t) denotes the Fourier transform of q(x, t).
(a) Assuming the validity of (1) and (2), use the global relation to eliminate ˜g 1 from equation (1).
(b) For the particular case that
q 0 (x) = e−a (^2) x , 0 < x < ∞ ; q(0, t) = cos bt , 0 < t < T ,
where a and b are real numbers, use the representation obtained in (a) to express the solution in terms of an integral along the real axis and an integral along L (you should not attempt to evaluate these integrals). Show that it is possible to deform these two integrals to a single integral along a new contour L˜, which you should sketch.
[You may assume the validity of Jordan’s lemma.]
Part II, Paper 2
15A Classical Dynamics Consider a rigid body with principal moments of inertia I 1 , I 2 , I 3.
(a) Derive Euler’s equations of torque-free motion
I 1 ω˙ 1 = (I 2 − I 3 )ω 2 ω 3 , I 2 ω˙ 2 = (I 3 − I 1 )ω 3 ω 1 , I 3 ω˙ 3 = (I 1 − I 2 )ω 1 ω 2 ,
with components of the angular velocity ω = (ω 1 , ω 2 , ω 3 ) given in the body frame.
(b) Show that rotation about the second principal axis is unstable if (I 2 − I 3 )(I 1 − I 2 ) > 0.
(c) The principal moments of inertia of a uniform cylinder of radius R, height h and mass M about its centre of mass are
I 1 = I 2 =
M R^2
M h^2 12
; I 3 =
M R^2
The cylinder has two identical cylindrical holes of radius r drilled along its length. The axes of symmetry of the holes are at a distance a from the axis of symmetry of the cylinder such that r < R/2 and r < a < R − r. All three axes lie in a single plane. Compute the principal moments of inertia of the body.
16H Logic and Set Theory Explain what is meant by a substructure of a Σ-structure A, where Σ is a first-order signature (possibly including both predicate symbols and function symbols). Show that if B is a substructure of A, and φ is a first-order formula over Σ with n free variables, then [φ]B = [φ]A ∩ Bn^ if φ is quantifier-free. Show also that [φ]B ⊆ [φ]A ∩ Bn^ if φ is an existential formula (that is, one of the form (∃x 1 ,... , xm)ψ where ψ is quantifier-free), and [φ]B ⊇ [φ]A ∩ Bn^ if φ is a universal formula. Give examples to show that the two latter inclusions can be strict. Show also that (a) if T is a first-order theory whose axioms are all universal sentences, then any substructure of a T -model is a T -model; (b) if T is a first-order theory such that every first-order formula φ is T -provably equivalent to a universal formula (that is, T ⊢ (φ ⇔ ψ) for some universal ψ), and B is a sub-T -model of a T -model A, then [φ]B = [φ]A ∩ Bn^ for every first-order formula φ with n free variables.
Part II, Paper 2 [TURN OVER
20F Number Fields Let K = Q(α) where α is a root of X^2 − X + 12 = 0. Factor the elements 2, 3 , α and α + 2 as products of prime ideals in OK. Hence compute the class group of K. Show that the equation y^2 + y = 3(x^5 − 4) has no integer solutions.
21G Algebraic Topology State the Seifert–Van Kampen Theorem. Deduce that if f : S^1 → X is a continuous map, where X is path-connected, and Y = X ∪f B^2 is the space obtained by adjoining a disc to X via f , then Π 1 (Y ) is isomorphic to the quotient of Π 1 (X) by the smallest normal subgroup containing the image of f∗ : Π 1 (S^1 ) → Π 1 (X). State the classification theorem for connected triangulable 2-manifolds. Use the result of the previous paragraph to obtain a presentation of Π 1 (Mg ), where Mg denotes the compact orientable 2-manifold of genus g > 0.
22G Linear Analysis What is meant by a normal topological space? State and prove Urysohn’s lemma. Let X be a normal topological space and let S ⊆ X be closed. Show that there is a continuous function f : X → [0, 1] with f −^1 (0) = S if, and only if, S is a countable intersection of open sets. [Hint. If S =
n=1 Un^ then consider^
n=1 2
−nfn, where the functions fn : X → [0, 1] are supplied by an appropriate application of Urysohn’s lemma.]
23I Riemann Surfaces Let X be the algebraic curve in C^2 defined by the polynomial p(z, w) = zd^ + wd^ + 1 where d is a natural number. Using the implicit function theorem, or otherwise, show that there is a natural complex structure on X. Let f : X → C be the function defined by f (a, b) = b. Show that f is holomorphic. Find the ramification points and the corresponding branching orders of f. Assume that f extends to a holomorphic map g : Y → C ∪ {∞} from a compact Riemann surface Y to the Riemann sphere so that g−^1 (∞) = Y \ X and that g has no ramification points in g−^1 (∞). State the Riemann–Hurwitz formula and apply it to g to calculate the Euler characteristic and the genus of Y.
Part II, Paper 2 [TURN OVER
24I Algebraic Geometry Let k be a field, J an ideal of k[x 1 ,... , xn], and let R = k[x 1 ,... , xn]/J. Define the radical
J of J and show that it is also an ideal. The Nullstellensatz says that if J is a maximal ideal, then the inclusion k ⊆ R is an algebraic extension of fields. Suppose from now on that k is algebraically closed. Assuming the above statement of the Nullstellensatz, prove the following.
(i) If J is a maximal ideal, then J = (x 1 − a 1 ,... , xn − an), for some (a 1 ,... , an) ∈ kn. (ii) If J 6 = k[x 1 ,... , xn], then Z(J) 6 = ∅, where
Z(J) = {a ∈ kn^ | f (a) = 0 for all f ∈ J}.
(iii) For V an affine subvariety of kn, we set
I(V ) = {f ∈ k[x 1 ,... , xn] | f (a) = 0 for all a ∈ V }.
Prove that J = I(V ) for some affine subvariety V ⊆ kn, if and only if J =
J.
[Hint. Given f ∈ J, you may wish to consider the ideal in k[x 1 ,... , xn, y] generated by J and yf − 1 .] (iv) If A is a finitely generated algebra over k, and A does not contain nilpotent elements, then there is an affine variety V ⊆ kn, for some n, with A = k[x 1 ,... , xn]/I(V ).
Assuming char(k) 6 = 2, find
J when J is the ideal (x(x − y)^2 , y(x + y)^2 ) in k[x, y].
25I Differential Geometry Define the Gauss map N for an oriented surface S ⊂ R^3. Show that at each p ∈ S the derivative of the Gauss map
dNp : TpS → TN (p)S^2 = TpS
is self-adjoint. Define the principal curvatures k 1 , k 2 of S. Now suppose that S is compact (and without boundary). By considering the square of the distance to the origin, or otherwise, prove that S has a point p with k 1 (p)k 2 (p) > 0. [You may assume that the intersection of S with a plane through the normal direction at p ∈ S contains a regular curve through p.]
Part II, Paper 2
27K Applied Probability (a) A colony of bacteria evolves as follows. Let X be a random variable with values in the positive integers. Each bacterium splits into X copies of itself after an exponentially distributed time of parameter λ > 0. Each of the X daughters then splits in the same way but independently of everything else. This process keeps going forever. Let Zt denote the number of bacteria at time t. Specify the Q-matrix of the Markov chain Z = (Zt, t > 0). [It will be helpful to introduce pn = P(X = n), and you may assume for simplicity that p 0 = p 1 = 0.] (b) Using the Kolmogorov forward equation, or otherwise, show that if u(t) = E(Zt|Z 0 = 1), then u′(t) = αu(t) for some α to be explicitly determined in terms of X. Assuming that E(X) < ∞, deduce the value of u(t) for all t > 0, and show that Z does not explode. [You may differentiate series term by term and exchange the order of summation without justification.] (c) We now assume that X = 2 with probability 1. Fix 0 < q < 1 and let φ(t) = E(qZt^ |Z 0 = 1). Show that φ satisfies
φ(t) = qe−λt^ +
∫ (^) t
0
λe−λsφ(t − s)^2 ds.
By making the change of variables u = t − s, show that dφ/dt = λφ(φ − 1). Deduce that for all n > 1, P(Zt = n|Z 0 = 1) = βn−^1 (1 − β) where β = 1 − e−λt.
28K Principles of Statistics Carefully defining all italicised terms, show that, if a sufficiently general method of inference respects both the Weak Sufficiency Principle and the Conditionality Principle, then it respects the Likelihood Principle.
The position Xt of a particle at time t > 0 has the Normal distribution N (0, φt), where φ is the value of an unknown parameter Φ; and the time, Tx, at which the particle first reaches position x 6 = 0 has probability density function
px(t) = |x| √ 2 πφt^3
exp
x^2 2 φt
(t > 0).
Experimenter E 1 observes Xτ , and experimenter E 2 observes Tξ , where τ > 0, ξ 6 = 0 are fixed in advance. It turns out that Tξ = τ. What does the Likelihood Principle say about the inferences about Φ to be made by the two experimenters?
E 1 bases his inference about Φ on the distribution and observed value of X τ^2 /τ , while E 2 bases her inference on the distribution and observed value of ξ^2 /Tξ. Show that these choices respect the Likelihood Principle.
Part II, Paper 2
29J Optimization and Control Describe the elements of a generic stochastic dynamic programming equation for the problem of maximizing the expected sum of discounted rewards accrued at times 0, 1 ,.... What is meant by the positive case? What is specially true in this case that is not true in general? An investor owns a single asset which he may sell once, on any of the days t = 0, 1 ,.... On day t he will be offered a price Xt. This value is unknown until day t, is independent of all other offers, and a priori it is uniformly distributed on [0, 1]. Offers remain open, so that on day t he may sell the asset for the best of the offers made on days 0,... , t. If he sells for x on day t then the reward is xβt. Show from first principles that if 0 < β < 1 then there exists ¯x such that the expected reward is maximized by selling the first day the offer is at least ¯x. For β = 4/5, find both ¯x and the expected reward under the optimal policy. Explain what is special about the case β = 1.
30J Stochastic Financial Models (i) Give the definition of Brownian motion.
(ii) The price St of an asset evolving in continuous time is represented as
St = S 0 exp (σWt + μt) ,
where (Wt)t> 0 is a standard Brownian motion and σ and μ are constants. If riskless investment in a bank account returns a continuously compounded rate of interest r, derive the Black–Scholes formula for the time-0 price of a European call option on asset S with strike price K and expiry T. [Standard results from the course may be used without proof but must be stated clearly.]
(iii) In the same financial market, a certain contingent claim C pays (ST )n^ at time T , where n > 1. Find the closed-form expression for the time-0 value of this contingent claim. Show that for every s > 0 and n > 1,
sn^ = n(n − 1)
∫ (^) s
0
kn−^2 (s − k)dk.
Using this identity, how would you replicate (at least approximately) the contingent claim C with a portfolio consisting only of European calls?
Part II, Paper 2 [TURN OVER
33A Principles of Quantum Mechanics (a) Define the Heisenberg picture of quantum mechanics in relation to the Schr¨odinger picture. Explain how the two pictures provide equivalent descriptions of physical results. (b) Derive the equation of motion for an operator in the Heisenberg picture. For a particle of mass m moving in one dimension, the Hamiltonian is
Hˆ = pˆ
2 2 m
where ˆx and ˆp are the position and momentum operators, and the state vector is |Ψ〉. The eigenstates of ˆx and ˆp satisfy
〈x|p〉 =
2 πℏ
eipx/ℏ^ , 〈x|x′〉 = δ(x − x′) , 〈p|p′〉 = δ(p − p′).
Use standard methods in the Dirac formalism to show that
〈x|pˆ|x′〉 = −iℏ
∂x δ(x − x′) ,
〈p|xˆ|p′〉 = iℏ
∂p δ(p − p′).
Calculate 〈x|H ˆ|x′〉 and express 〈x|pˆ|Ψ〉, 〈x| Hˆ|Ψ〉 in terms of the position space wavefunction Ψ(x). Write down the momentum space Hamiltonian for the potential
V (ˆx) = mω^2 xˆ^4 / 2.
Part II, Paper 2 [TURN OVER
34E Applications of Quantum Mechanics A solution of the S-wave Schr¨odinger equation at large distances for a particle of mass m with momentum ℏk and energy E = ℏ^2 k^2 / 2 m, has the form
ψ 0 (r) ∼
A
r [sin kr + g(k) cos kr].
Define the phase shift δ 0 and verify that tan δ 0 (k) = g(k). Write down a formula for the cross-section σ, for a particle of momentum ℏk scattering on a radially symmetric potential of finite range, as a function of the phase shifts δl for the partial waves with quantum number l.
(i) Suppose that g(k) = −k/K for K > 0. Show that there is a bound state of energy EB = −ℏ^2 K^2 / 2 m. Neglecting the contribution from partial waves with l > 0 show that the cross section is σ =
4 π K^2 + k^2
(ii) Suppose now that g(k) = γ/(K 0 − k) with K 0 > 0 , γ > 0 and γ ≪ K 0. Neglecting the contribution from partial waves with l > 0, derive an expression for the cross section σ, and show that it has a local maximum when E ≈ ℏ^2 K^20 / 2 m. Discuss the interpretation of this phenomenon in terms of resonant behaviour and derive an expression for the decay width of the resonant state.
35C Statistical Physics Explain what is meant by an isothermal expansion and an adiabatic expansion of a gas.
By first establishing a suitable Maxwell relation, show that
∂E ∂V
T
= T
∂p ∂T
V
− p
and ∂CV ∂V
T
= T
∂^2 p ∂T 2
V
The energy in a gas of blackbody radiation is given by E = aV T 4 , where a is a constant. Derive an expression for the pressure p(V, T ).
Show that if the radiation expands adiabatically, V T 3 is constant.
Part II, Paper 2
