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MATHEMATICAL TRIPOS Part IB
Friday 10 June 2005 1.30 to 4.
PAPER 4
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 REQUIRMENTS 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
1B Linear Algebra
Define what it means for an n × n complex matrix to be unitary or Hermitian. Show that every eigenvalue of a Hermitian matrix is real. Show that every eigenvalue of a unitary matrix has absolute value 1.
Show that two eigenvectors of a Hermitian matrix that correspond to different eigenvalues are orthogonal, using the standard inner product on Cn.
2C Groups, Rings and Modules State Eisenstein’s irreducibility criterion. Let n be an integer > 1. Prove that 1 + x +... + xn−^1 is irreducible in Z[x] if and only if n is a prime number.
3B Analysis II
Let V be the vector space of continuous real-valued functions on [0, 1]. Show that the function
||f || =
0
|f (x)| dx
defines a norm on V.
For n = 1, 2 ,.. ., let fn(x) = e−nx. Is fn a convergent sequence in the space V with this norm? Justify your answer.
Paper 4
6G Quantum Mechanics
Define the commutator [A , B] of two operators, A and B. In three dimensions angular momentum is defined by a vector operator L with components
Lx = y pz − z py Ly = z px − x pz Lz = x py − y px.
Show that [Lx , Ly ] = i ℏ Lz and use this, together with permutations, to show that [L^2 , Lw] = 0, where w denotes any of the directions x, y, z.
At a given time the wave function of a particle is given by
ψ = (x + y + z) exp
x^2 + y^2 + z^2
Show that this is an eigenstate of L^2 with eigenvalue equal to 2ℏ^2.
7H Electromagnetism
For a static current density J(x) show that we may choose the vector potential A(x) so that −∇^2 A = μ 0 J.
For a loop L, centred at the origin, carrying a current I show that
A(x) =
μ 0 I 4 π
L
|x − r|
dr ∼ −
μ 0 I 4 π
|x|^3
L
1 2 x^ ×^ (r^ ×^ dr)^ as^ |x| → ∞^.
[You may assume
−∇^2
4 π|x|
= δ^3 (x) ,
and for fixed vectors a, b
∮
L
a · dr = 0,
L
(a · r b · dr + b · r a · dr) = 0.
]
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8F Numerical Analysis
Define Gaussian quadrature.
Evaluate the coefficients of the Gaussian quadrature of the integral ∫ (^1)
− 1
(1 − x^2 )f (x)dx
which uses two function evaluations.
9D Markov Chains Prove that the simple symmetric random walk in three dimensions is transient.
[You may wish to recall Stirling’s formula: n! ∼ (2π)
(^12) nn+^
(^12) e−n.]
Paper 4 [TURN OVER
12A Geometry
Given a parametrized smooth embedded surface σ : V → U ⊂ R^3 , where V is an open subset of R^2 with coordinates (u, v), and a point P ∈ U , define what is meant by the tangent space at P , the unit normal N at P , and the first fundamental form
Edu^2 + 2F du dv + Gdv^2.
[You need not show that your definitions are independent of the parametrization.]
The second fundamental form is defined to be
Ldu^2 + 2M du dv + N dv^2 ,
where L = σuu · N, M = σuv · N and N = σvv · N. Prove that the partial derivatives of N (considered as a vector-valued function of u, v) are of the form Nu = aσu + bσv , Nv = cσu + dσv , where
L M
M N
a b c d
E F
F G
Explain briefly the significance of the determinant ad − bc.
13B Analysis II
Let F : [−a, a] × [x 0 − r, x 0 + r] → R be a continuous function. Let C be the maximum value of |F (t, x)|. Suppose there is a constant K such that
|F (t, x) − F (t, y)| 6 K|x − y|
for all t ∈ [−a, a] and x, y ∈ [x 0 − r, x 0 + r]. Let b < min(a, r/C, 1 /K). Show that there is a unique C^1 function x : [−b, b] → [x 0 − r, x 0 + r] such that
x(0) = x 0
and dx dt
= F (t, x(t)).
[Hint: First show that the differential equation with its initial condition is equivalent to the integral equation
x(t) = x 0 +
∫ (^) t
0
F (s, x(s)) ds.
]
Paper 4 [TURN OVER
14A Metric and Topological Spaces
Let (M, d) be a metric space, and F a non-empty closed subset of M. For x ∈ M , set d(x, F ) = inf z∈F
d(x, z).
Prove that d(x, F ) is a continuous function of x, and that it is strictly positive for x 6 ∈ F.
A topological space is called normal if for any pair of disjoint closed subsets F 1 , F 2 , there exist disjoint open subsets U 1 ⊃ F 1 , U 2 ⊃ F 2. By considering the function
d(x, F 1 ) − d(x, F 2 ),
or otherwise, deduce that any metric space is normal.
Suppose now that X is a normal topological space, and that F 1 , F 2 are disjoint closed subsets in X. Prove that there exist open subsets W 1 ⊃ F 1 , W 2 ⊃ F 2 , whose closures are disjoint. In the case when X = R^2 with the standard metric topology, F 1 = {(x, − 1 /x) : x < 0 } and F 2 = {(x, 1 /x) : x > 0 }, find explicit open subsets W 1 , W 2 with the above property.
15F Complex Methods Determine the Fourier expansion of the function f (x) = sin λx, where −π 6 x 6 π, in the two cases where λ is an integer and λ is a real non-integer.
Using the Parseval identity in the case λ = 12 , find an explicit expression for the sum (^) ∞ ∑
n=
n^2 (4n^2 − 1)^2
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17G Special Relativity
Obtain the Lorentz transformations that relate the coordinates of an event mea- sured in one inertial frame (t, x, y, z) to those in another inertial frame moving with velocity v along the x axis. Take care to state the assumptions that lead to your result.
A star is at rest in a three-dimensional coordinate frame S′^ that is moving at constant velocity v along the x axis of a second coordinate frame S. The star emits light of frequency ν′, which may considered to be a beam of photons. A light ray from the star to the origin in S′^ is a straight line that makes an angle θ′^ with the x′^ axis. Write down the components of the four-momentum of a photon in this light ray.
The star is seen by an observer at rest at the origin of S at time t = t′^ = 0, when the origins of the coordinate frames S and S′^ coincide. The light that reaches the observer moves along a line through the origin that makes an angle θ to the x axis and has frequency ν. Make use of the Lorentz transformations between the four-momenta of a photon in these two frames to determine the relation
λ = λ′
v^2 c^2
v c
cos θ
where λ is the observed wavelength of the photon and λ′^ is the wavelength in the star’s rest frame.
Comment on the form of this result in the special cases with cos θ = 1, cos θ = − 1 and cos θ = 0.
[You may assume that the energy of a photon of frequency ν is hν and its three- momentum is a three-vector of magnitude hν/c.]
18E Fluid Dynamics
A fluid of density ρ 1 occupies the region z > 0 and a second fluid of density ρ 2 occupies the region z < 0. State the equations and boundary conditions that are satisfied by the corresponding velocity potentials φ 1 and φ 2 and pressures p 1 and p 2 when the system is perturbed so that the interface is at z = ζ(x, t) and the motion is irrotational.
Seek a set of linearised equations and boundary conditions when the disturbances are proportional to ei(kx−ωt), and derive the dispersion relation
ω^2 =
ρ 2 − ρ 1 ρ 2 + ρ 1
gk,
where g is the gravitational acceleration.
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19D Statistics
Let Y 1 ,... , Yn be observations satisfying
Yi = βxi + i, 1 6 i 6 n,
where 1 ,... , n are independent random variables each with the N (0, σ^2 ) distribution. Here x 1 ,... , xn are known but β and σ^2 are unknown.
(i) Determine the maximum-likelihood estimates (β,̂ ̂σ^2 ) of (β, σ^2 ).
(ii) Find the distribution of β̂.
(iii) By showing that Yi − βx̂ i and β̂ are independent, or otherwise, determine the joint distribution of β̂ and ̂σ^2.
(iv) Explain carefully how you would test the hypothesis H 0 : β = β 0 against H 1 : β 6 = β 0.
20D Optimization Describe the Ford–Fulkerson algorithm for finding a maximal flow from a source to a sink in a directed network with capacity constraints on the arcs. Explain why the algorithm terminates at an optimal flow when the initial flow and the capacity constraints are rational.
Illustrate the algorithm by applying it to the problem of finding a maximal flow from S to T in the network below.
H
F G
E
K J
T
A B
S
D C
END OF PAPER
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