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MATHEMATICAL TRIPOS Part II
Friday 10 June 2005 9 to 12
PAPER 4
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
If n is an odd integer and b is an integer prime with n, state what it means for n to be a pseudoprime to the base b. What is a Carmichael number? State a criterion for n to be a Carmichael number and use the criterion to show that:
(i) Every Carmichael number is the product of at least three distinct primes.
(ii) 561 is a Carmichael number.
2F Topics in Analysis
(i) Let D ⊂ C be a domain, let f : D → C be an analytic function and let z 0 ∈ D. What does Taylor’s theorem say about z 0 , f and D?
(ii) Let K be the square consisting of all complex numbers z such that
− 1 6 Re(z) 6 1 and − 1 6 Im(z) 6 1 ,
and let w be a complex number not belonging to K. Prove that the function f (z) = (z − w)−^1 can be uniformly approximated on K by polynomials.
3G Geometry of Group Actions
Show that a set F ⊂ Rn^ with Hausdorff dimension strictly less than one is totally disconnected.
What does it mean for a M¨obius transformation to pair two discs? By considering a pair of disjoint discs and a pair of tangent discs, or otherwise, explain in words why there is a 2-generator Schottky group with limit set Λ ⊂ S^2 which has Hausdorff dimension at least 1 but which is not homeomorphic to a circle.
4J Coding and Cryptography What does it mean to transmit reliably at rate r through a binary symmetric channel (BSC) with error probability p? Assuming Shannon’s second coding theorem, compute the supremum of all possible reliable transmission rates of a BSC. What happens if (i) p is very small, (ii) p = 1/2, or (iii) p > 1 /2?
Paper 4
6E Mathematical Biology
The output of a linear perceptron is given by y = w · x, where w is a vector of weights connecting a fluctuating input vector x to an output unit. The weights are given random initial values and are then updated according to a learning rule that has a time-constant τ much greater than the fluctuation timescale of the inputs.
(a) Find the behaviour of |w| for each of the following two rules
(i) τ
dw dt
= yx
(ii) τ
dw dt
= yx − αy^2 w|w|^2 , where α is a positive constant.
(b) Consider a third learning rule
(iii) τ
dw dt
= yx − w|w|^2.
Show that in a steady state the vector of weights satisfies the eigenvalue equation
Cw = λw ,
where the matrix C and eigenvalue λ should be identified.
(c) Comment briefly on the biological implications of the three rules.
7B Dynamical Systems Find and classify the fixed points of the system
x˙ = x(1 − y) y˙ = −y + x^2.
Sketch the phase plane.
What is the ω-limit for the point (2, −1)? Which points have (0, 0) as their ω-limit?
Paper 4
8A Further Complex Methods
Write down necessary and sufficient conditions on the functions p(z) and q(z) for the point z = 0 to be (i) an ordinary point and (ii) a regular singular point of the equation
w′′^ + p(z)w′^ + q(z)w = 0. (∗)
Show that the point z = ∞ is an ordinary point if and only if
p(z) = 2z−^1 + z−^2 P (z−^1 ), q(z) = z−^4 Q(z−^1 ),
where P and Q are analytic in a neighbourhood of the origin.
Find the most general equation of the form (∗) that has a regular singular point at z = 0 but no other singular points.
9C Classical Dynamics Define a canonical transformation for a one-dimensional system with coordinates (q, p) → (Q, P ). Show that if the transformation is canonical then {Q, P } = 1.
Find the values of constants α and β such that the following transformations are canonical: (i) Q = pqβ^ , P = αq−^1. (ii) Q = qα^ cos(βp), P = qα^ sin(βp).
10D Cosmology The linearised equation for the growth of a density fluctuation δk in a homogeneous and isotropic universe is
d^2 δk dt^2
a˙ a
dδk dt
4 πGρm −
v s^2 k^2 a^2
δk = 0 , (∗)
where ρm is the non-relativistic matter density, k is the comoving wavenumber and vs is the sound speed (v^2 s ≡ dP/dρ).
(a) Define the Jeans length λJ and discuss its significance for perturbation growth.
(b) Consider an Einstein–de Sitter universe with a(t) = (t/t 0 )^2 /^3 filled with pressure-free matter (P = 0). Show that the perturbation equation (∗) can be re-expressed as ¨δk +^4 3 t
δ˙k − 2 3 t^2
δk = 0.
By seeking power law solutions, find the growing and decaying modes of this equation.
(c) Qualitatively describe the evolution of non-relativistic matter perturbations (k > aH) in the radiation era, a(t) ∝ t^1 /^2 , when ρr ρm. What feature in the power spectrum is associated with the matter–radiation transition?
Paper 4 [TURN OVER
13I Statistical Modelling
(i) Suppose that Y 1 ,... , Yn are independent random variables, and that Yi has probability density function
f (yi|β, ν) =
νyi μi
)ν e−yiν/μi^
Γ(ν)
yi
for yi > 0
where 1 /μi = βT^ xi , for 1 6 i 6 n,
and x 1 ,... , xn are given p-dimensional vectors, and ν is known.
Show that E(Yi) = μi and that var (Yi) = μ^2 i /ν.
(ii) Find the equation for βˆ, the maximum likelihood estimator of β, and suggest an iterative scheme for its solution.
(iii) If p = 2, and xi =
zi
, find the large-sample distribution of βˆ 2. Write your
answer in terms of a, b, c and ν, where a, b, c are defined by
a =
μ^2 i , b =
ziμ^2 i , c =
z i^2 μ^2 i.
14A Further Complex Methods Two representations of the zeta function are
ζ(z) =
Γ(1 − z) 2 πi
−∞
tz−^1 e−t^ − 1
dt and ζ(z) =
∑^ ∞
1
n−z^ ,
where, in the integral representation, the path is the Hankel contour and the principal branch of tz−^1 , for which | arg z| < π, is to be used. State the range of z for which each representation is valid.
Evaluate the integral (^) ∫
γ
tz−^1 e−t^ − 1
dt,
where γ is a closed path consisting of the straight line z = πi + x, with |x| < 2 N π, and the semicircle |z − πi| = 2N π, with Im z > π, where N is a positive integer.
Making use of this result and assuming, when necessary, that the integral along the curved part of γ is negligible when N is large, derive the functional equation
ζ(z) = 2z^ πz−^1 sin(πz/2)Γ(1 − z)ζ(1 − z)
for z 6 = 1.
Paper 4 [TURN OVER
15D Cosmology
For an ideal gas of bosons, the average occupation number can be expressed as
n¯k =
gk e(Ek^ −μ)/kT^ − 1
where gk has been included to account for the degeneracy of the energy level Ek. In the approximation in which a discrete set of energies Ek is replaced with a continuous set with momentum p, the density of one-particle states with momentum in the range p to p + dp is g(p)dp. Explain briefly why g(p) ∝ p^2 V ,
where V is the volume of the gas. Using this formula with equation (∗), obtain an expression for the total energy density = E/V of an ultra-relativistic gas of bosons at zero chemical potential as an integral over p. Hence show that
∝ T α^ ,
where α is a number you should find. Why does this formula apply to photons?
Prior to a time t ∼ 100 , 000 years, the universe was filled with a gas of photons and non-relativistic free electrons and protons. Subsequently, at around t ∼ 400 , 000 years, the protons and electrons began combining to form neutral hydrogen,
p + e−^ ↔ H + γ.
Deduce Saha’s equation for this recombination process stating clearly the steps required:
n^2 e nH
2 πmekT h^2
exp(−I/kT ) ,
where I is the ionization energy of hydrogen. [Note that the equilibrium number density of
a non-relativistic species (kT mc^2 ) is given by n = gs
( (^2) πmkT h^2
exp
[
(μ − mc^2 )/kT
]
while the photon number density is nγ = 16πζ(3)
( (^) kT hc
, where ζ(3) ≈ 1. 20 .... ] Consider now the fractional ionization Xe = ne/nB, where nB = np + nH = ηnγ is the baryon number of the universe and η is the baryon-to-photon ratio. Find an expression for the ratio (1 − Xe)/Xe^2
in terms only of kT and constants such as η and I. One might expect neutral hydrogen to form at a temperature given by kT ≈ I ≈ 13 eV, but instead in our universe it forms at the much lower temperature kT ≈ 0 .3 eV. Briefly explain why.
Paper 4
20G Number Fields
State Dedekind’s theorem on the factorisation of rational primes into prime ideals.
A rational prime is said to ramify totally in a field with degree n if it is the n-th power of a prime ideal in the field. Show that, in the quadratic field Q(
d) with d a square- free integer, a rational prime ramifies totally if and only if it divides the discriminant of the field.
Verify that the same holds in the cyclotomic field Q(ζ), where ζ = e^2 πi/q^ with q an odd prime, and also in the cubic field Q( 3
[The cases d ≡ 2 , 3 (mod 4) and d ≡ 1 (mod 4) for the quadratic field should be carefully distinguished. It can be assumed that Q(ζ) has a basis 1 , ζ,... , ζq−^2 and discriminant (−1)(q−1)/^2 qq−^1 , and that Q( 3
- has a basis 1 , 3
2)^2 and discriminant − 108 .]
21H Algebraic Topology
Let X be a simplicial complex. Suppose X = B ∪C for subcomplexes B and C, and let A = B ∩ C. Show that the inclusion of A in B induces an isomorphism H∗A → H∗B if and only if the inclusion of C in X induces an isomorphism H∗C → H∗X.
22F Linear Analysis
Let X and Y be normed vector spaces. Show that a linear map T : X → Y is continuous if and only if it is bounded.
Now let X, Y , Z be Banach spaces. We say that a map F : X × Y → Z is bilinear if F (αx + βy, z) = αF (x, z) + βF (y, z), for all scalars α, β and x, y ∈ X, z ∈ Y
F (x, αy + βz) = αF (x, y) + βF (x, z), for all scalars α, β and x ∈ X, y, z ∈ Y.
Suppose that F is bilinear and is continuous in each variable separately. Show that there exists a constant M > 0 such that
||F (x, y)|| 6 M ||x|| ||y||
for all x ∈ X, y ∈ Y.
[Hint: For each fixed x ∈ X, consider the map y 7 → F (x, y) from Y to Z.]
Paper 4
23H Riemann Surfaces
Define what is meant by the degree of a non-constant holomorphic map between compact connected Riemann surfaces, and state the Riemann–Hurwitz formula.
Let EΛ = C/Λ be an elliptic curve defined by some lattice Λ. Show that the map
ψ : z + Λ ∈ EΛ → −z + Λ ∈ EΛ
is biholomorphic, and that there are four points in EΛ fixed by ψ.
Let S = EΛ/ ∼ be the quotient surface (the topological surface obtained by identifying z + Λ and ψ(z + Λ), for each z) and let π : EΛ → S be the corresponding projection map. Denote by EΛ^0 ⊂ EΛ the complement of the four points fixed by ψ, and let S^0 = π(EΛ^0 ). Describe briefly a family of charts making S^0 into a Riemann surface, so that π : EΛ^0 → S^0 is a holomorphic map.
Now assume that the complex structure of S^0 extends to S, so that S is a Riemann surface, and that the map π is in fact holomorphic on all of EΛ. Calculate the genus of S.
Paper 4 [TURN OVER
25J Probability and Measure
Let f : R^2 → R be Borel-measurable. State Fubini’s theorem for the double integral ∫
y∈R
x∈R
f (x, y) dx dy.
Let 0 < a < b. Show that the function
f (x, y) =
e−xy^ if x ∈ (0, ∞), y ∈ [a, b] 0 otherwise
is measurable and integrable on R^2.
Evaluate (^) ∞ ∫
0
e−ax^ − e−bx x
dx
by Fubini’s theorem or otherwise.
Paper 4 [TURN OVER
26I Applied Probability
A particle performs a continuous-time nearest neighbour random walk on a regular triangular lattice inside an angle π/3, starting from the corner. See the diagram below. The jump rates are 1/3 from the corner and 1/6 in each of the six directions if the particle is inside the angle. However, if the particle is on the edge of the angle, the rate is 1/ along the edge away from the corner and 1/6 to each of three other neighbouring sites in the angle. See the diagram below, where a typical trajectory is also shown.
The particle position at time t > 0 is determined by its vertical level Vt and its horizontal position Gt. For k > 0, if Vt = k then Gt = 0,... , k. Here 1,... , k − 1 are positions inside, and 0 and k positions on the edge of the angle, at vertical level k.
Let J 1 V , J 2 V , ... be the times of subsequent jumps of process (Vt) and consider the embedded discrete-time Markov chains
Y (^) nin =
Gin n , V̂n
and Y (^) nout =
Gout n , V̂n
where V̂n is the vertical level immediately after time JnV , Ĝ in n is the horizontal position
immediately after time JnV , and Ĝ out n is the horizontal position immediately before time JnV+1.
Paper 4
27I Principles of Statistics
A group of n hospitals is to be ‘appraised’; the ‘performance’ θi of hospital i has a N (0, 1 /τ ) prior distribution, different hospitals being independent. The ‘performance’ cannot be measured directly, so an expensive firm of management consultants has been hired to arrive at each hospital’s Standardised Index of Quality [SIQ], this being a number Xi for hospital i related to θi by the commercially-sensitive formula
Xi = θi + εi,
where the εi are independent with common N (0, 1 /τε) distribution.
(i) Assume that τ and τε are known. What is the posterior distribution of θ given X? Suppose that hospital j was the hospital with the lowest SIQ, with a value Xj = x; conditional on X, what is the distribution of θj?
(ii) Now, instead of assuming τ and τε known, suppose that τ has a Gamma prior with parameters (α, β), density
f (t) = (βt)α−^1 βe−βt/Γ(α)
for known α and β, and that τε = κτ , where κ is a known constant. Find the posterior distribution of (θ, τ ) given X. Comment briefly on the form of the distribution.
Paper 4
28J Stochastic Financial Models
(a) In the context of the Black–Scholes formula, let S 0 be spot price, K be strike price, T be time to maturity, and assume constant interest rate r, volatility σ and absence of dividends. Write down explicitly the prices of a European call and put,
EC (S 0 , K, σ, r, T ) and EP (S 0 , K, σ, r, T ).
Use Φ for the normal distribution function. [No proof is required.]
(b) From here on assume r = 0. Keeping T, σ fixed, we shorten the notation to EC (S 0 , K) and similarly for EP. Show that put-call symmetry holds:
EC (S 0 , K) = EP (K, S 0 ).
Check homogeneity: for every real α > 0
EC (αS 0 , αK) = αEC (S 0 , K).
(c) Show that the price of a down-and-out European call with strike K < S 0 and barrier B 6 K is given by
EC (S 0 , K) −
S 0
B
EC
B^2
S 0
, K
(d)
(i) Specialize the last expression to B = K and simplify.
(ii) Answer a popular interview question in investment banks: What is the fair value of a down-and-out call given that S 0 = 100, B = K = 80, σ = 20%, r = 0, T = 1? Identify the corresponding hedge. [It may be helpful to compute Delta first.]
(iii) Does this hedge work beyond the Black–Scholes model? When does it fail?
Paper 4 [TURN OVER
31A Asymptotic Methods
Consider the differential equation d^2 w dx^2
= q(x)w ,
where q(x) > 0 in an interval (a, ∞). Given a solution w(x) and a further smooth function ξ(x), define W (x) = [ξ′(x)]^1 /^2 w(x).
Show that, when ξ is regarded as the independent variable, the function W (ξ) obeys the differential equation
d^2 W dξ^2
x ˙^2 q(x) + ˙x^1 /^2
d^2 dξ^2
[ ˙x−^1 /^2 ]
W, (∗)
where ˙x denotes dx/dξ.
Taking the choice ξ(x) =
q^1 /^2 (x)dx ,
show that equation (∗) becomes
d^2 W dξ^2
= (1 + φ)W ,
where
φ = −
q^3 /^4
d^2 dx^2
q^1 /^4
In the case that φ is negligible, deduce the Liouville–Green approximate solutions
w± = q−^1 /^4 exp
q^1 /^2 dx
Consider the Whittaker equation
d^2 w dx^2
[
s(s − 1) x^2
]
w ,
where s is a real constant. Show that the Liouville–Green approximation suggests the existence of solutions wA,B (x) with asymptotic behaviour of the form
wA ∼ exp(x/2)
∑^ ∞
n=
anx−n
, wB ∼ exp(−x/2)
∑^ ∞
n=
bnx−n
as x → ∞.
Given that these asymptotic series may be differentiated term-by-term, show that
an =
(−1)n n!
(s − n)(s − n + 1)... (s + n − 1).
Paper 4 [TURN OVER
32D Principles of Quantum Mechanics
The Hamiltonian for a quantum system in the Schr¨odinger picture is
H 0 + λV (t) ,
where H 0 is independent of time and the parameter λ is small. Define the interaction picture corresponding to this Hamiltonian and derive a time evolution equation for interaction picture states.
Let |a〉 and |b〉 be eigenstates of H 0 with distinct eigenvalues Ea and Eb respectively. Show that if the system is initially in state |a〉 then the probability of measuring it to be in state |b〉 after a time t is
λ^2 ℏ^2
∫ (^) t
0
dt′〈b|V (t′)|a〉ei(Eb−Ea)t
′/ℏ
2
Deduce that if V (t) = e−μt/ℏW , where W is a time-independent operator and μ is a positive constant, then the probability for such a transition to have occurred after a very long time is approximately
λ^2 μ^2 + (Eb − Ea)^2
|〈b|W |a〉|^2.
Paper 4
