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MATHEMATICAL TRIPOS Part IB
Tuesday 3 June 2008 9.00 to 12.
PAPER 1
Before you begin read these instructions carefully.
Each question in Section II carries twice the number of marks of each question in Section I. Candidates may attempt at most four questions from Section I and at most six 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 separate bundles labelled A, B,... , H according to the examiner letter affixed to each question, including in the same bundle questions from Sections I and II with the same examiner letter.
Attach a completed gold cover sheet to each bundle; write the examiner 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 SPECIAL REQUIREMENTS Gold cover sheet None 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
1E Linear Algebra
Let A be an n × n matrix over C. What does it mean to say that λ is an eigenvalue of A? Show that A has at least one eigenvalue. For each of the following statements, provide a proof or a counterexample as appropriate.
(i) If A is Hermitian, all eigenvalues of A are real.
(ii) If all eigenvalues of A are real, A is Hermitian.
(iii) If all entries of A are real and positive, all eigenvalues of A have positive real part.
(iv) If A and B have the same trace and determinant then they have the same eigenvalues.
2G Geometry Show that any element of SO(3, R) is a rotation, and that it can be written as the product of two reflections.
3C Complex Analysis or Complex Methods
Given that f (z) is an analytic function, show that the mapping w = f (z)
(a) preserves angles between smooth curves intersecting at z if f ′(z) 6 = 0;
(b) has Jacobian given by |f ′(z)|^2.
4C Special Relativity
In an inertial frame S a photon of energy E is observed to travel at an angle θ relative to the x-axis. The inertial frame S′^ moves relative to S at velocity v in the x- direction and the x′-axis of S′^ is taken parallel to the x-axis of S. Observed in S′, the photon has energy E′^ and travels at an angle θ′^ relative to the x′-axis. Show that
E′^ =
E(1 − β cos θ) √ 1 − β^2
, cos θ′^ =
cos θ − β 1 − β cos θ
where β = v/c.
Paper 1
SECTION II
9E Linear Algebra
Let A be an m × n matrix of real numbers. Define the row rank and column rank of A and show that they are equal.
Show that if a matrix A′^ is obtained from A by elementary row and column operations then rank(A′) = rank(A).
Let P, Q and R be n × n matrices. Show that the 2n × 2 n matrices
P Q 0
Q QR
and
0 P QR
Q 0
have the same rank.
Hence, or otherwise, prove that
rank(P Q) + rank(QR) 6 rank(Q) + rank(P QR).
10G Groups, Rings and Modules (i) Show that A 4 is not simple.
(ii) Show that the group Rot(D) of rotational symmetries of a regular dodecahedron is a simple group of order 60.
(iii) Show that Rot(D) is isomorphic to A 5.
Paper 1
11F Analysis II
State and prove the Contraction Mapping Theorem.
Let (X, d) be a nonempty complete metric space and f : X → X a mapping such that, for some k > 0, the kth iterate f k^ of f (that is, f composed with itself k times) is a contraction mapping. Show that f has a unique fixed point.
Now let X be the space of all continuous real-valued functions on [0, 1], equipped with the uniform norm ‖h‖∞ = sup {|h(t)| : t ∈ [0, 1]}, and let φ : R × [0, 1] → R be a continuous function satisfying the Lipschitz condition
|φ(x, t) − φ(y, t)| 6 M |x − y|
for all t ∈ [0, 1] and all x, y ∈ R, where M is a constant. Let F : X → X be defined by
F (h)(t) = g(t) +
∫ (^) t
0
φ(h(s), s) ds ,
where g is a fixed continuous function on [0, 1]. Show by induction on n that
|F n(h)(t) − F n(k)(t)| 6
M ntn n!
‖h − k‖∞
for all h, k ∈ X and all t ∈ [0, 1]. Deduce that the integral equation
f (t) = g(t) +
∫ (^) t
0
φ(f (s), s) ds
has a unique continuous solution f on [0, 1].
12F Metric and Topological Spaces Write down the definition of a topology on a set X.
For each of the following families T of subsets of Z, determine whether T is a topology on Z. In the cases where the answer is ‘yes’, determine also whether (Z, T ) is a Hausdorff space and whether it is compact.
(a) T = {U ⊆ Z : either U is finite or 0 ∈ U }. (b) T = {U ⊆ Z : either Z \ U is finite or 0 6 ∈ U }.
(c) T = {U ⊆ Z : there exists k > 0 such that, for all n, n ∈ U ⇔ n + k ∈ U }.
(d) T = {U ⊆ Z : for all n ∈ U , there exists k > 0 such that {n+km : m ∈ Z} ⊆ U }.
Paper 1 [TURN OVER
15A Quantum Mechanics
The radial wavefunction g(r) for the hydrogen atom satisfies the equation
ℏ^2
2 mr^2
d dr
r^2
dg(r) dr
e^2 g(r) 4 π 0 r
+ ℏ^2
( + 1)
2 mr^2
g(r) = Eg(r). (∗)
With reference to the general form for the time-independent Schr¨odinger equation, explain the origin of each term. What are the allowed values of `?
The lowest-energy bound-state solution of (∗), for given , has the form rαe−βr^. Find α and β and the corresponding energy E in terms of.
A hydrogen atom makes a transition between two such states corresponding to + and. What is the frequency of the emitted photon?
Paper 1 [TURN OVER
16B Electromagnetism
Suppose that the current density J(r) is constant in time but the charge density ρ(r, t) is not.
(i) Show that ρ is a linear function of time:
ρ(r, t) = ρ(r, 0) + ˙ρ(r, 0)t,
where ˙ρ(r, 0) is the time derivative of ρ at time t = 0.
(ii) The magnetic induction due to a current density J(r) can be written as
B(r) =
μ 0 4 π
J(r′) × (r − r′) |r − r′|^3
dV ′^.
Show that this can also be written as
B(r) =
μ 0 4 π
∇ ×
J(r′) |r − r′|
dV ′. (1)
(iii) Assuming that J vanishes at infinity, show that the curl of the magnetic field in (1) can be written as
∇ × B(r) = μ 0 J(r) +
μ 0 4 π
∇′^ · J(r′) |r − r′|
dV ′^. (2)
[You may find useful the identities ∇ × (∇ × A) = ∇(∇ · A) − ∇^2 A and also ∇^2 (1/|r − r′|) = − 4 πδ(r − r′).]
(iv) Show that the second term on the right hand side of (2) can be expressed in terms of the time derivative of the electric field in such a way that B itself obeys Amp`ere’s law with Maxwell’s displacement current term, i.e. ∇ × B = μ 0 J + μ 0 0 ∂E/∂t.
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18H Statistics
Suppose that X 1 ,... , Xn is a sample of size n with common N (μX , 1) distribution, and Y 1 ,... , Yn is an independent sample of size n from a N (μY , 1) distribution.
(i) Find (with careful justification) the form of the size-α likelihood–ratio test of the null hypothesis H 0 : μY = 0 against alternative H 1 : (μX , μY ) unrestricted.
(ii) Find the form of the size-α likelihood–ratio test of the hypothesis
H 0 : μX > A, μY = 0 ,
against H 1 : (μX , μY ) unrestricted, where A is a given constant.
Compare the critical regions you obtain in (i) and (ii) and comment briefly.
19H Markov Chains
The village green is ringed by a fence with N fenceposts, labelled 0, 1 ,... , N −1. The village idiot is given a pot of paint and a brush, and started at post 0 with instructions to paint all the posts. He paints post 0, and then chooses one of the two nearest neighbours, 1 or N −1 , with equal probability, moving to the chosen post and painting it. After painting a post, he chooses with equal probability one of the two nearest neighbours, moves there and paints it (regardless of whether it is already painted). Find the distribution of the last post unpainted.
END OF PAPER
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