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MATHEMATICAL TRIPOS Part II
Monday 4 June 2007 9 to 12
PAPER 1
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 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
1F Number Theory
State the prime number theorem, and Bertrand’s postulate. Let S be a finite set of prime numbers, and write fs(x) for the number of positive integers no larger than x, all of whose prime factors belong to S. Prove that
fs(x) 6 2 #(S)
x,
where #(S) denotes the number of elements in S. Deduce that, if x is a strictly positive integer, we have
π(x) >
log x 2 log 2
2F Topics in Analysis
Let n be an integer with n > 1. Are the following statements true or false? Give proofs.
(i) There exists a real polynomial Tn of degree n such that
Tn(cos t) = cos nt
for all real t. (ii) There exists a real polynomial Rn of degree n such that
Rn(cosh t) = cosh nt
for all real t. (iii) There exists a real polynomial Sn of degree n such that
Sn(cos t) = sin nt
for all real t.
3G Geometry of Group Actions Show that there are two ways to embed a regular tetrahedron in a cube C so that the vertices of the tetrahedron are also vertices of C. Show that the symmetry group of C permutes these tetrahedra and deduce that the symmetry group of C is isomorphic to the Cartesian product S 4 × C 2 of the symmetric group S 4 and the cyclic group C 2.
Paper 1
5I Statistical Modelling
According to the Independent newspaper (London, 8 March 1994) the Metropolitan Police in London reported 30475 people as missing in the year ending March 1993. For those aged 18 or less, 96 of 10527 missing males and 146 of 11363 missing females were still missing a year later. For those aged 19 and above, the values were 157 of 5065 males and 159 of 3520 females. This data is summarised in the table below.
age gender still total 1 Kid M 96 10527
2 Kid F 146 11363
3 Adult M 157 5065 4 Adult F 159 3520
Explain and interpret the R commands and (slightly abbreviated) output below. You should describe the model being fitted, explain how the standard errors are calculated, and comment on the hypothesis tests being described in the summary. In particular, what is the worst of the four categories for the probability of remaining missing a year later?
fit <- glm(still/total ~ age + gender, family = binomial,
summary(fit)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -3.06073 0.07216 -42.417 < 2e-16 *** ageKid -1.27079 0.08698 -14.610 < 2e-16 ***
genderM -0.37211 0.08671 -4.291 1.78e-05 ***
Residual deviance: 0.06514 on 1 degrees of freedom
For a person who was missing in the year ending in March 1993, find a formula, as a function of age and gender, for the estimated expected probability that they are still missing a year later.
Paper 1
6B Mathematical Biology
A chemostat is a well-mixed tank of given volume V 0 that contains water in which lives a population N (t) of bacteria that consume nutrient whose concentration is C(t) per unit volume. An inflow pipe supplies a solution of nutrient at concentration C 0 and at a constant flow rate of Q units of volume per unit time. The mixture flows out at the same rate through an outflow pipe. The bacteria consume the nutrient at a rate N K(C), where
K(C) =
KmaxC K 0 + C
and the bacterial population grows at a rate γN K(C), where 0 < γ < 1.
Write down the differential equations for N (t), C(t) and show that they can be rescaled into the following form:
dn dτ
= α
cn 1 + c
− n ,
dc dτ
cn 1 + c
− c + β ,
where α, β are positive constants, to be found.
Show that this system of equations has a non-trivial steady state if α > 1 and
β >
α − 1
, and that it is stable.
7E Dynamical Systems Given a non-autonomous kth-order differential equation
dky dtk^
= g
t, y,
dy dt
d^2 y dt^2
dk−^1 y dtk−^1
with y ∈ R, explain how it may be written in the autonomous first-order form x˙ = f (x) for suitably chosen vectors x and f.
Given an autonomous system ˙x = f (x) in Rn, define the corresponding flow φt(x). What is φs(φt(x)) equal to? Define the orbit O(x) through x and the limit set ω(x) of x. Define a homoclinic orbit.
Paper 1 [TURN OVER
10A Cosmology
Describe the motion of light rays in an expanding universe with scale factor a(t), and derive the redshift formula
1 + z =
a(t 0 ) a(te)
where the light is emitted at time te and observed at time t 0.
A galaxy at comoving position x is observed to have a redshift z. Given that the galaxy emits an amount of energy L per unit time, show that the total energy per unit time crossing a sphere centred on the galaxy and intercepting the earth is L/(1 + z)^2. Hence, show that the energy per unit time per unit area passing the earth is
L (1 + z)^2
4 π|x|^2 a^2 (t 0 )
Paper 1 [TURN OVER
SECTION II
11G Coding and Cryptography
Define the bar product C 1 |C 2 of linear codes C 1 and C 2 , where C 2 is a subcode of C 1. Relate the rank and minimum distance of C 1 |C 2 to those of C 1 and C 2. Show that if C⊥^ denotes the dual code of C, then
(C 1 |C 2 )⊥^ = C 2 ⊥ |C 1 ⊥.
Using the bar product construction, or otherwise, define the Reed–Muller code RM (d, r) for 0 6 r 6 d. Show that if 0 6 r 6 d−1, then the dual of RM (d, r) is again a Reed–Muller code.
12G Geometry of Group Actions Define the Hausdorff d-dimensional measure Hd(C) and the Hausdorff dimension of a subset C of R.
Set s = log 2/ log 3. Define the Cantor set C and show that its Hausdorff s-dimensional measure is at most 1.
Let (Xn) be independent Bernoulli random variables that take the values 0 and 2, each with probability 12. Define
ξ =
∑^ ∞
n=
Xn 3 n^
Show that ξ is a random variable that takes values in the Cantor set C.
Let U be a subset of R with 3−(k+1)^6 diam(U ) < 3 −k. Show that P(ξ ∈ U ) 6 2 −k and deduce that, for any set U ⊂ R, we have
P(ξ ∈ U ) 6 2(diam(U ))s^.
Hence, or otherwise, prove that Hs(C) > 12 and that the Cantor set has Hausdorff dimension s.
Paper 1
the summary from the fit of the one above.
fit2 <- glm(number ~ fuel + strain, family = poisson) fit3 <- glm(number ~ fuel, family = poisson)
Denote by H 1 , H 2 , H 3 the three hypotheses being fitted in sequence above.
Explain the hypothesis tests, including an approximate test of the fit of H 1 , that can be performed using the output from the following R code. Use these numbers to comment on the most appropriate model for the data.
c(fit1$dev, fit2$dev, fit3$dev)
[1] 84.59557 86.37646 118.
qchisq(0.95, df = 1) [1] 3.
14B Further Complex Methods
The function J(z) is defined by
J(z) =
P
tz−^1 (1 − t)b−^1 dt
where b is a constant (which is not an integer). The path of integration, P, is the Pochhammer contour, defined as follows. It starts at a point A on the axis between 0 and 1, then it circles the points 1 and 0 in the negative sense, then it circles the points 1 and 0 in the positive sense, returning to A. At the start of the path, arg(t) = arg(1−t) = 0 and the integrand is defined at other points on P by analytic continuation along P.
(i) For what values of z is J(z) analytic? Give brief reasons for your answer. (ii) Show that, in the case Re z > 0 and Re b > 0 ,
J(z) = − 4 e−πi(z+b)^ sin(πz) sin(πb) B(z, b) ,
where B(z, b) is the Beta function. (iii) Deduce that the only singularities of B(z, b) are simple poles. Explain carefully what happens if z is a positive integer.
Paper 1
15A Cosmology
In a homogeneous and isotropic universe, the scale factor a(t) obeys the Friedmann equation (^) ( a˙ a
kc^2 a^2
8 πG 3
ρ ,
where ρ is the matter density, which, together with the pressure P , satisfies
ρ˙ = − 3
a˙ a
ρ + P/c^2
Here, k is a constant curvature parameter. Use these equations to show that the rate of change of the Hubble parameter H = ˙a/a satisfies
H˙ + H^2 = − 4 πG 3
ρ + 3P/c^2
Suppose that an expanding Friedmann universe is filled with radiation (density ρR and pressure PR = ρRc^2 /3) as well as a “dark energy” component (density ρΛ and pressure PΛ = −ρΛc^2 ). Given that the energy densities of these two components are measured today (t = t 0 ) to be
ρR 0 = β
3 H^20
8 πG
and ρΛ0 =
3 H 02
8 πG
with constant β > 0 and a(t 0 ) = 1 ,
show that the curvature parameter must satisfy kc^2 = βH 02. Hence derive the following relations for the Hubble parameter and its time derivative:
H^2 =
H 02
a^4
β − βa^2 + a^4
H^ ˙ = −β H
2 0 a^4
2 − a^2
Show qualitatively that universes with β > 4 will recollapse to a Big Crunch in the future. [Hint: Sketch a^4 H^2 and a^4 H˙ versus a^2 for representative values of β.]
For β = 4, find an explicit solution for the scale factor a(t) satisfying a(0) = 0. Find the limiting behaviours of this solution for large and small t. Comment briefly on their significance.
Paper 1 [TURN OVER
19H Representation Theory
A finite group G has seven conjugacy classes C 1 = {e}, C 2 ,... , C 7 and the values of five of its irreducible characters are given in the following table.
C 1 C 2 C 3 C 4 C 5 C 6 C 7
Calculate the number of elements in the various conjugacy classes and complete the character table.
[You may not identify G with any known group, unless you justify doing so.]
20H Number Fields
Let K = Q(
(a) Show that OK = Z[
−26] and that the discriminant dK is equal to −104. (b) Show that 2 ramifies in OK by showing that [2] = p^22 , and that p 2 is not a principal ideal. Show further that [3] = p 3 ¯p 3 with p 3 = [3, 1 −
−26]. Deduce that neither p 3 nor p^23 is a principal ideal, but p^33 = [1 −
−26].
(c) Show that 5 splits in OK by showing that [5] = p 5 ¯p 5 , and that
NK/Q(2 +
Deduce that p 2 p 3 p 5 has trivial class in the ideal class group of K. Conclude that the ideal class group of K is cyclic of order six.
[You may use the fact that (^2) π
104 ≈ 6. 492 .]
21H Algebraic Topology
(i) Compute the fundamental group of the Klein bottle. Show that this group is not abelian, for example by defining a suitable homomorphism to the symmetric group S 3.
(ii) Let X be the closed orientable surface of genus 2. How many (connected) double coverings does X have? Show that the fundamental group of X admits a homomorphism onto the free group on 2 generators.
Paper 1 [TURN OVER
22G Linear Analysis
Let X be a normed vector space over R. Define the dual space X∗^ and show directly that X∗^ is a Banach space. Show that the map φ : X → X∗∗^ defined by φ(x)v = v(x), for all x ∈ X, v ∈ X∗, is a linear map. Using the Hahn–Banach theorem, show that φ is injective and |φ(x)| = |x|.
Give an example of a Banach space X for which φ is not surjective. Justify your answer.
23F Riemann Surfaces
Define a complex structure on the unit sphere S^2 ⊂ R^3 using stereographic projection charts ϕ, ψ. Let U ⊂ C be an open set. Show that a continuous non-constant map F : U → S^2 is holomorphic if and only if ϕ ◦ F is a meromorphic function. Deduce that a non-constant rational function determines a holomorphic map S^2 → S^2. Define what is meant by a rational function taking the value a ∈ C ∪ {∞} with multiplicity m at infinity.
Define the degree of a rational function. Show that any rational function f satisfies (deg f ) − 1 6 deg f ′^6 2 deg f and give examples to show that the bounds are attained. Is it true that the product f.g satisfies deg(f.g) = deg f + deg g, for any non-constant rational functions f and g? Justify your answer.
24H Differential Geometry
Let f : X → Y be a smooth map between manifolds without boundary. Recall that f is a submersion if dfx : TxX → Tf (x)Y is surjective for all x ∈ X. The canonical submersion is the standard projection of Rk^ onto Rl^ for k > l, given by
(x 1 ,... , xk) 7 → (x 1 ,... , xl).
(i) Let f be a submersion, x ∈ X and y = f (x). Show that there exist local coordinates around x and y such that f , in these coordinates, is the canonical submersion. [You may assume the inverse function theorem.]
(ii) Show that submersions map open sets to open sets. (iii) If X is compact and Y connected, show that every submersion is surjective. Are there submersions of compact manifolds into Euclidean spaces Rk^ with k > 1?
Paper 1
27I Principles of Statistics
Suppose that X has density f (·|θ) where θ ∈ Θ. What does it mean to say that statistic T ≡ T (X) is sufficient for θ?
Suppose that θ = (ψ, λ), where ψ is the parameter of interest, and λ is a nuisance parameter, and that the sufficient statistic T has the form T = (C, S). What does it mean to say that the statistic S is ancillary? If it is, how (according to the conditionality principle) do we test hypotheses on ψ? Assuming that the set of possible values for X is discrete, show that S is ancillary if and only if the density (probability mass function) f (x|ψ, λ) factorises as
f (x|ψ, λ) = ϕ 0 (x) ϕC (C(x), S(x), ψ) ϕS (S(x), λ) (∗)
for some functions ϕ 0 , ϕC , and ϕS with the properties
∑
x∈C−^1 (c)∩S−^1 (s)
ϕ 0 (x) = 1 =
s
ϕS (s, λ) =
s
c
ϕC (c, s, ψ)
for all c, s, ψ, and λ.
Suppose now that X 1 ,... , Xn are independent observations from a Γ(a, b) distri- bution, with density f (x|a, b) = (bx)a−^1 e−bxbI{x> 0 }/Γ(a).
Assuming that the criterion (∗) holds also for observations which are not discrete, show that it is not possible to find (C(X), S(X)) sufficient for (a, b) such that S is ancillary when b is regarded as a nuisance parameter, and a is the parameter of interest.
28J Stochastic Financial Models
(i) What does it mean to say that a process (Mt)t> 0 is a martingale? What does the martingale convergence theorem tell us when applied to positive martingales?
(ii) What does it mean to say that a process (Bt)t> 0 is a Brownian motion? Show that supt> 0 Bt = ∞ with probability one. (iii) Suppose that (Bt)t> 0 is a Brownian motion. Find μ such that
St = exp (x 0 + σBt + μt)
is a martingale. Discuss the limiting behaviour of St and E (St) for this μ as t → ∞.
Paper 1
29A Partial Differential Equations
(i) Consider the problem of solving the equation
∑^ n
j=
aj (x)
∂u ∂xj
= b(x, u)
for a C^1 function u = u(x) = u(x 1 ,... , xn), with data specified on a C^1 hypersurface S ⊂ Rn u(x) = φ(x), ∀x ∈ S.
Assume that a 1 ,... , an, φ, b are C^1 functions. Define the characteristic curves and explain what it means for the non-characteristic condition to hold at a point on S. State a local existence and uniqueness theorem for the problem. (ii) Consider the case n = 2 and the equation
∂u ∂x 1
∂u ∂x 2
= x 2 u
with data u(x 1 , 0) = φ(x 1 , 0) = f (x 1 ) specified on the axis {x ∈ R^2 : x 2 = 0}. Obtain a formula for the solution.
(iii) Consider next the case n = 2 and the equation
∂u ∂x 1
∂u ∂x 2
with data u(g(s)) = φ(g(s)) = f (s) specified on the hypersurface S, which is given parametrically as S ≡ {x ∈ R^2 : x = g(s)} where g : R → R^2 is defined by
g(s) = (s, 0), s < 0 ,
g(s) = (s, s^2 ), s > 0.
Find the solution u and show that it is a global solution. (Here “global” means u is C^1 on all of R^2 .) (iv) Consider next the equation ∂u ∂x 1
∂u ∂x 2
to be solved with the same data given on the same hypersurface as in (iii). Explain, with reference to the characteristic curves, why there is generally no global C^1 solution. Discuss the existence of local solutions defined in some neighbourhood of a given point y ∈ S for various y. [You need not give formulae for the solutions.]
Paper 1 [TURN OVER
31E Integrable Systems
(i) Using the Cole–Hopf transformation
u = −
2 ν φ
∂φ ∂x
map the Burgers equation
∂u ∂t
∂u ∂x
= ν
∂^2 u ∂x^2
to the heat equation ∂φ ∂t
= ν
∂^2 φ ∂x^2
(ii) Given that the solution of the heat equation on the infinite line R with initial condition φ(x, 0) = Φ(x) is given by
φ(x, t) =
4 πνt
−∞
Φ(ξ) e−^
(x 4 −νtξ) 2 dξ ,
show that the solution of the analogous problem for the Burgers equation with initial condition u(x, 0) = U (x) is given by
u =
−∞
x − ξ t
e−^ 21 ν G(x,ξ,t)^ dξ ∫ (^) ∞
−∞
e−^ 21 ν G(x,ξ,t)^ dξ
where the function G is to be determined in terms of U.
(iii) Determine the ODE characterising the scaling reduction of the spherical modified Korteweg – de Vries equation
∂u ∂t
∂u ∂x
∂^3 u ∂x^3
u t
Paper 1 [TURN OVER
32D Principles of Quantum Mechanics
A particle in one dimension has position and momentum operators ˆx and ˆp whose eigenstates obey
〈x|x′〉 = δ(x−x′) , 〈p|p′〉 = δ(p−p′) , 〈x|p〉 = (2πℏ)−^1 /^2 eixp/ℏ^.
Given a state |ψ〉, define the corresponding position-space and momentum-space wave- functions ψ(x) and ψ˜(p) and show how each of these can be expressed in terms of the other. Derive the form taken in momentum space by the time-independent Schr¨odinger equation ( (^) pˆ 2
2 m
|ψ〉 = E|ψ〉
for a general potential V.
Now let V (x) = −(ℏ^2 λ/m)δ(x) with λ a positive constant. Show that the Schr¨odinger equation can be written
( (^) p 2
2 m
− E
ψ˜(p) = ℏλ 2 πm
−∞
dp′^ ψ˜(p′)
and verify that it has a solution ψ˜(p) = N/(p^2 + α^2 ) for unique choices of α and E, to be determined (you need not find the normalisation constant, N ). Check that this momentum space wavefunction can also be obtained from the position space solution ψ(x) =
λe−λ|x|.
Paper 1
