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MATHEMATICAL TRIPOS Part II
Friday, 5 June, 2009 9:00 am to 12:00 pm
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 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
1G Number Theory Let W denote the set of all positive definite binary quadratic forms, with integer coefficients, and having discriminant −67. Let SL 2 (Z) be the group of all 2 × 2 matrices with integer entries and determinant 1. Prove that W is infinite, but that all elements of W are equivalent under the action of the group SL 2 (Z)
2F Topics in Analysis State Liouville’s theorem on approximation of algebraic numbers by rationals, and use it to prove that the number ∑∞
n=
10 n!
is transcendental.
3F Geometry of Group Actions For every k ∈ R, show that there is a closed bounded totally disconnected subset X of some Euclidean space, such that X has Hausdorff dimension at least k. [Standard properties of Hausdorff dimension may be quoted without proof if carefully stated.]
4H Coding and Cryptography What is a general feedback register? What is a linear feedback register? Give an example of a general feedback register which is not a linear feedback register and prove that it has the stated property.
By giving proofs or counterexamples, establish which, if any, of the following statements are true and which, if any, are false.
(i) Given two linear feedback registers, there always exist non-zero initial fills for which the outputs are identical.
(ii) If two linear feedback registers have different lengths, there do not exist non-zero initial fills for which the outputs are identical.
(iii) If two linear feedback registers have different lengths, there exist non-zero initial fills for which the outputs are not identical.
(iv) There exist two linear feedback registers of different lengths and non-zero initial fills for which the outputs are identical.
Part II, Paper 4
6A Mathematical Biology The diffusion equation for a chemical concentration C(r, t) in three dimensions which depends only on the radial coordinate r is
Ct = D
r^2
r^2 Cr
r.^ (∗)
The general similarity solution of this equation takes the form
C(r, t) = tαF (ξ), ξ =
r tβ^
where α and β are to be determined. By direct substitution into (∗) and the requirement of a valid similarity solution, find one relation involving the exponents. Use the conservation of the total number of molecules to determine a second relation. Comment on the relationship between these exponents and the ones appropriate to the similarity solution of the one-dimensional diffusion equation. Show that F obeys
D
F ′′^ +
ξ
F ′
ξF ′^ +
F = 0,
and that the relevant solution describing the spreading of a delta-function initial condition is F (ξ) = A exp(−ξ^2 / 4 D), where A is a suitable normalisation that need not be found.
7E Dynamical Systems Consider the two-dimensional dynamical system ˙x = f (x) given in polar coordinates by r˙ = (r − r^2 )(r − g(θ)), θ^ ˙ = r, (∗) where g(θ) is continuously differentiable and 2π-periodic. Find a periodic orbit γ for (∗) and, using the hint or otherwise, compute the Floquet multipliers of γ in terms of g(θ). Explain why one of the Floquet multipliers is independent of g(θ). Give a sufficient condition for γ to be asymptotically stable. Investigate the stability of γ and the dynamics of (∗) in the case g(θ) = 2 sin θ. [Hint: The determinant of the fundamental matrix Φ(t) satisfies
d dt
det Φ = (∇ · f ) det Φ.
]
Part II, Paper 4
8D Further Complex Methods Show that
Γ(α)Γ(β) = Γ(α + β)
0
tα−^1 (1 − t)β−^1 dt, Re α > 0 , Re β > 0 ,
where Γ(z) denotes the Gamma function
Γ(z) =
0
xz−^1 e−xdx, Re z > 0.
9E Classical Dynamics (a) A Hamiltonian system with n degrees of freedom has the Hamiltonian H(p, q), where q = (q 1 , q 2 , q 3 ,... , qn) are the coordinates and p = (p 1 , p 2 , p 3 ,... , pn) are the momenta. A second Hamiltonian system has the Hamiltonian G = G(p, q). Neither H nor G contains the time explicitly. Show that the condition for H(p, q) to be invariant under the evolution of the coordinates and momenta generated by the Hamiltonian G(p, q) is that the Poisson bracket [H, G] vanishes. Deduce that G is a constant of the motion for evolution under H. Show that, when G = α
∑n k=1 pk,^ where^ α^ is constant, the motion it generates is a translation of each qk by an amount αt, while the corresponding pk remains fixed. What do you infer is conserved when H is invariant under this transformation? (b) When n = 3 and H is a function of p^21 + p^22 + p^23 and q^21 + q 22 + q 32 only, find [H, Li] when Li = ǫijkqj pk.
Part II, Paper 4 [TURN OVER
SECTION II
11G Number Theory Let s = σ + it , where σ and t are real, and for σ > 1 let
ζ(s) =
∑^ ∞
n=
ns^
Prove that ζ(s) has no zeros in the half plane σ > 1. Show also that for σ > 1 ,
1 ζ(s)
∑^ ∞
n=
μ(n) ns^
where μ denotes the M¨obius function. Assuming that ζ(s) −
s − 1 has an analytic continuation to the half plane σ > 0 , show that if s = 1 + it , with t 6 = 0 , and ζ(s) = 0 then s is at most a simple zero of ζ.
12F Geometry of Group Actions Define three-dimensional hyperbolic space, the translation length of an isometry of hyperbolic 3-space, and the axis of a hyperbolic isometry. Briefly explain how and why the latter two concepts are related. Find the translation length of the isometries defined by (i) z 7 → kz, k ∈ C \ { 0 } and (ii) z 7 → 3 z + 2 7 z + 5
13I Statistical Modelling Consider the linear model Y = Xβ + ε, where ε ∼ Nn(0, σ^2 I) and X is an n × p matrix of full rank p < n. Find the form of the maximum likelihood estimator ˆβ of β, and derive its distribution assuming that σ^2 is known. Assuming the prior π(β, σ^2 ) ∝ σ−^2 find the joint posterior of (β, σ^2 ) up to a normalising constant. Derive the posterior conditional distribution π(β|σ^2 , X, Y ). Comment on the distribution of βˆ found above and the posterior conditional π(β|σ^2 , X, Y ). Comment further on the predictive distribution of y∗^ at input x∗^ under both the maximum likelihood and Bayesian approaches.
Part II, Paper 4 [TURN OVER
14E Dynamical Systems Let I, J be closed bounded intervals in R, and let F : R → R be a continuous map. Explain what is meant by the statement that ‘I F -covers J’ (written I → J). For a collection of intervals I 0 ,... , Ik define the associated directed graph Γ and transition matrix A. Derive an expression for the number of (not necessarily least) period-n points of F in terms of A.
Let F have a 5-cycle
x 0 < x 1 < x 2 < x 3 < x 4
such that xi+1 = F (xi) for i = 0,... , 4 where indices are taken modulo 5. Write down the directed graph Γ and transition matrix A for the F -covering relations between the intervals [xi, xi+1]. Compute the number of n-cycles which are guaranteed to exist for F , for each integer 1 6 n 6 4, and the intervals the points move between.
Explain carefully whether or not F is guaranteed to have a horseshoe. Must F be chaotic? Could F be a unimodal map? Justify your answers.
Similarly, a continuous map G : R → R has a 5-cycle
x 3 < x 1 < x 0 < x 2 < x 4.
For what integer values of n, 1 6 n 6 4, is G guaranteed to have an n-cycle?
Is G guaranteed to have a horseshoe? Must G be chaotic? Justify your answers.
Part II, Paper 4
16G Logic and Set Theory What is a transitive class? What is the significance of this notion for models of set theory?
Prove that for any set x there is a least transitive set TC(x), the transitive closure of x, with x ⊆ TC(x). Show that for any set x, one has TC(x) = x ∪ TC(
x), and deduce that TC({x}) = {x} ∪ TC(x).
A set x is hereditarily countable when every member of TC({x}) is countable. Let (HC, ∈) be the collection of hereditarily countable sets with the usual membership relation. Is HC transitive? Show that (HC, ∈) satisfies the axiom of unions. Show that (HC, ∈) satisfies the axiom of separation. What other axioms of ZF set theory are satisfied in (HC, ∈)?
17F Graph Theory Let X denote the number of triangles in a random graph G chosen from G(n, p). Find the mean and variance of X. Hence show that p = n−^1 is a threshold for the existence of a triangle, in the sense that if pn → 0 then almost surely G does not contain a triangle, while if pn → ∞ then almost surely G does contain a triangle.
Now let p = n−^1 /^2 , and let Y denote the number of edges of G (chosen as before from G(n, p)). By considering the mean of Y − X, show that for each n > 3 there exists a graph on n vertices with at least 16 n^3 /^2 edges that is triangle-free. Is this within a constant factor of the best-possible answer (meaning the greatest number of edges that a triangle-free graph on n vertices can have)?
18H Galois Theory (a) Let K be a field. State what it means for ξn ∈ K to be a primitive nth root of unity. Show that if ξn is a primitive nth root of unity, then the characteristic of K does not divide n. Prove any theorems you use. (b) Determine the minimum polynomial of a primitive 10th root of unity ξ 10 over Q. Show that
5 ∈ Q(ξ 10 ). (c) Determine F 3 (ξ 10 ), F 11 (ξ 10 ), F 19 (ξ 10 ). [Hint: Write a necessary and sufficient condition on q for a finite field Fq to contain a primitive 10 th root of unity.]
Part II, Paper 4
19F Representation Theory Let H 6 G be finite groups. (a) Let ρ be a representation of G affording the character χ. Define the restriction, ResGH ρ of ρ to H. Suppose χ is irreducible and suppose ResGH ρ affords the character χH. Let ψ 1 ,... , ψr be the irreducible characters of H. Prove that χH = d 1 ψ 1 + · · · + dr ψr, where the non- negative integers d 1 ,... , dr satisfy the inequality
∑^ r
i=
d^2 i 6 |G : H|. (1)
Prove that there is equality in (1) if and only if χ(g) = 0 for all elements g of G which lie outside H. (b) Let ψ be a class function of H. Define the induced class function, IndGH ψ. State the Frobenius reciprocity theorem for class functions and deduce that if ψ is a character of H then IndGH ψ is a character of G. Assuming ψ is a character, identify a G-space affording the character IndGH ψ. Briefly justify your answer. (c) Let χ 1 ,... , χk be the irreducible characters of G and let ψ be an irreducible character of H. Show that the integers e 1 ,... , ek , which are given by IndGH (ψ) = e 1 χ 1 + · · · + ekχk, satisfy ∑k
i=
e^2 i 6 |G : H|.
20H Number Fields Suppose that K is a number field of degree n = r + 2s, where K has exactly r real embeddings. Show that the group of units in OK is a finitely generated abelian group of rank at most r + s − 1. Identify the torsion subgroup in terms of roots of unity. [General results about discrete subgroups of a Euclidean real vector space may be used without proof, provided that they are stated clearly.] Find all the roots of unity in Q(
Part II, Paper 4 [TURN OVER
23G Algebraic Geometry State the Riemann–Roch theorem for a smooth projective curve V , and use it to outline a proof of the Riemann–Hurwitz formula for a non-constant morphism between projective nonsingular curves in characteristic zero. Let V ⊂ P^2 be a smooth projective plane cubic over an algebraically closed field k of characteristic zero, written in normal form X 0 X 22 = F (X 0 , X 1 ) for a homogeneous cubic polynomial F , and let P 0 = (0 : 0 : 1) be the point at infinity. Taking the group law on V for which P 0 is the identity element, let P ∈ V be a point of order 3. Show that there exists a linear form H ∈ k[X 0 , X 1 , X 2 ] such that V ∩ V (H) = {P }. Let H 1 , H 2 ∈ k[X 0 , X 1 , X 2 ] be nonzero linear forms. Suppose the lines {Hi = 0} are distinct, do not meet at a point of V , and are nowhere tangent to V. Let W ⊂ P^3 be given by the vanishing of the polynomials
X 0 X 22 − F (X 0 , X 1 ), X 32 − H 1 (X 0 , X 1 , X 2 )H 2 (X 0 , X 1 , X 2 ).
Show that W has genus 4. [You may assume without proof that W is an irreducible smooth curve.]
24H Differential Geometry (a) Let X be a compact surface (without boundary) in R^3. State the global Gauss– Bonnet formula for X, identifying all terms in the formula. (b) Let X ⊂ R^3 be a surface. Define what it means for a curve γ : I → X to be a geodesic. State a theorem concerning the existence of geodesics and define the exponential map. (c) Let ψ : X → Y be an isometry and let γ be a geodesic. Show that ψ ◦ γ is a geodesic. If KX denotes the Gaussian curvature of X, and KY denotes the Gaussian curvature of Y , show that KY ◦ ψ = KX. Now suppose ψ : X → Y is a smooth map such that ψ ◦ γ is a geodesic for all γ a geodesic. Is ψ necessarily an isometry? Give a proof or counterexample. Similarly, suppose ψ : X → Y is a smooth map such that KY ◦ ψ = KX. Is ψ necessarily an isometry? Give a proof or counterexample.
Part II, Paper 4 [TURN OVER
25J Probability and Measure Let (Ω, F, P) be a probability space and let G be a sub-σ-algebra of F. Show that, for any random variable X ∈ L^2 (P), there exists a G-measurable random variable Y ∈ L^2 (P) such that E((X − Y )Z) = 0 for all G-measurable random variables Z ∈ L^2 (P).
[You may assume without proof the completeness of L^2 (P).]
Let (G, X) be a Gaussian random variable in R^2 , with mean (μ, ν) and covariance
matrix
u v v w
. Assume that F = σ(G, X) and G = σ(G). Find the random variable Y
explicitly in this case.
26J Applied Probability A flea jumps on the vertices of a triangle ABC; its position is described by a continuous time Markov chain with a Q-matrix
Q =
A
B
C
(a) Draw a diagram representing the possible transitions of the flea together with the rates of each of these transitions. Find the eigenvalues of Q and express the transition probabilities pxy(t), x, y = A, B, C, in terms of these eigenvalues. [Hint: det (Q − μI) = (− 1 − μ)^3 + 1. Specifying the equilibrium distribution may help.] Hence specify the probabilities P
Nt = i mod 3
where (Nt, t > 0) is a Poisson process of rate 1. (b) A second flea jumps on the vertices of the triangle ABC as a Markov chain with Q-matrix
Q′^ =
−ρ 0 ρ ρ −ρ 0 0 ρ −ρ
A
B
C
where ρ > 0 is a given real number. Let the position of the second flea at time t be denoted by Yt. We assume that (Yt, t > 0) is independent of (Xt, t > 0). Let p(t) = P(Xt = Yt). Show that limt→∞ p(t) exists and is independent of the starting points of X and Y. Compute this limit.
Part II, Paper 4
29J Stochastic Financial Models An agent with utility U (x) = − exp(−γx), where γ > 0 is a constant, may select at time 0 a portfolio of n assets, which he then holds to time 1. The values X = (X 1 ,... , Xn)T of the assets at time 1 have a multivariate normal distribution with mean μ and nonsingular covariance matrix V. Prove that the agent will prefer portfolio ψ ∈ Rn^ to portfolio θ ∈ Rn if and only if q(ψ) > q(θ), where
q(x) = x · μ −
γ 2 x · V x.
Determine his optimal portfolio.
The agent initially holds portfolio θ, which he may change to portfolio θ + z at cost ε
∑n i=1 |zi|, where^ ε^ is some positive transaction cost.^ By considering the function t 7 → q(θ + tz) for 0 6 t 6 1, or otherwise, prove that the agent will have no reason to change his initial portfolio θ if and only if, for every i = 1,... , n,
|μi − γ (V θ)i| 6 ε.
30B Partial Differential Equations Consider the two-dimensional domain
G = {(x, y) | R^21 < x^2 + y^2 < R^22 },
where 0 < R 1 < R 2 < ∞. Solve the Dirichlet boundary value problem for the Laplace equation ∆u = 0 in G, u = u 1 (ϕ), r = R 1 , u = u 2 (ϕ), r = R 2 ,
where (r, ϕ) are polar coordinates. Assume that u 1 , u 2 are 2π-periodic functions on the real line and u 1 , u 2 ∈ L^2 loc(R).
[Hint: Use separation of variables in polar coordinates, u = R(r)Φ(ϕ), with periodic boundary conditions for the function Φ of the angle variable. Use an ansatz of the form R(r) = rα^ for the radial function.]
Part II, Paper 4
31A Asymptotic Methods The differential equation f ′′^ = Q(x)f (∗) has a singular point at x = ∞. Assuming that Q(x) > 0, write down the Liouville–Green lowest approximations f±(x) for x → ∞, with f−(x) → 0. The Airy function Ai(x) satisfies (∗) with
Q(x) = x,
and Ai(x) → 0 as x → ∞. Writing
Ai(x) = w(x)f−(x),
show that w(x) obeys
x^2 w′′^ −
2 x^5 /^2 +
x
w′^ +
w = 0.
Derive the expansion w ∼ c
x−^3 /^2
as x → ∞,
where c is a constant.
Part II, Paper 4 [TURN OVER
34D Statistical Physics Briefly state the ergodic hypothesis and explain its importance. Consider an ideal, classical, monatomic gas in the presence of a uniform gravitational field in the negative z-direction. For convenience, assume the gas is in an arbitrarily large cubic box. (i) Compute the internal energy E of the gas. (ii) Explain your result for E in relation to the equipartition theorem. (iii) What is the probability that an atom is located at a height between z and z + dz? (iv) What is the most probable speed of an atom of this gas?
35C Electrodynamics In a superconductor, the charge carriers have a charge q, mass m and number density n. Describe how to construct the superconducting current in terms of the vector potential A and the wavefunction of the charge carriers. Show that the current is gauge invariant. Derive the Helmholtz equation
∇^2 B = B/ℓ^2
for a time-independent magnetic field and evaluate the length scale ℓ in terms of n, q and m. Why does this imply that magnetic flux cannot exist in a superconductor?
Part II, Paper 4 [TURN OVER
36D General Relativity The Schwarzschild metric is given by
ds^2 = −
2 M
r
dt^2 +
2 M
r
dr^2 + r^2
dθ^2 + sin^2 θdφ^2
where M is the mass in gravitational units. By using the radial component of the geodesic equations, or otherwise, show for a particle moving on a geodesic in the equatorial plane θ = π/2 with r constant that (^) ( dφ dt
M
r^3
Show that such an orbit is stable for r > 6 M.
An astronaut circles the Earth freely for a long time on a circular orbit of radius R, while the astronaut’s twin remains motionless on Earth, which is assumed to be spherical, with radius R 0 , and non-rotating. Show that, on returning to Earth, the astronaut will be younger than the twin only if 2R < 3 R 0.
37E Fluid Dynamics II Two regions of inviscid fluid with the same density are separated by a thin membrane at y = 0. The fluid in y > 0 has the uniform velocity (U, 0 , 0) in Cartesian coordinates, while the fluid in y < 0 is at rest.
The membrane is now slightly perturbed to y = η(x, t). The dynamical effect of the membrane is to induce a pressure difference across it equal to β∂^4 η/∂x^4 , where β is a constant and the sign is such that the pressure is higher below the interface when ∂^4 η/∂x^4 > 0.
On the assumption that the flow remains irrotational and all perturbations are small, derive the relation between σ and k for disturbances of the form η(x, t) = Re(Ceikx+σt), where k is real but σ may be complex. Show that there is instability only for |k| < kmax, where kmax is to be determined. Find the maximum growth rate and the value of |k| for which this is obtained.
Part II, Paper 4
