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MATHEMATICAL TRIPOS Part IB
Wednesday 4 June 2008 1.30 to 4.
PAPER 2
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
Suppose that V and W are finite-dimensional vector spaces over R. What does it mean to say that ψ : V → W is a linear map? State the rank-nullity formula. Using it, or otherwise, prove that a linear map ψ : V → V is surjective if, and only if, it is injective.
Suppose that ψ : V → V is a linear map which has a right inverse, that is to say there is a linear map φ : V → V such that ψφ = idV , the identity map. Show that φψ = idV.
Suppose that A and B are two n × n matrices over R such that AB = I. Prove that BA = I.
2G Groups, Rings and Modules
What does it means to say that a complex number α is algebraic over Q? Define the minimal polynomial of α.
Suppose that α satisfies a nonconstant polynomial f ∈ Z[X] which is irreducible over Z. Show that there is an isomorphism Z[X]/(f ) ∼= Z[α].
[You may assume standard results about unique factorisation, including Gauss’s lemma.]
3F Analysis II Explain what is meant by the statement that a sequence (fn) of functions defined on an interval [a, b] converges uniformly to a function f. If (fn) converges uniformly to f , and each fn is continuous on [a, b], prove that f is continuous on [a, b].
Now suppose additionally that (xn) is a sequence of points of [a, b] converging to a limit x. Prove that fn(xn) → f (x).
4F Metric and Topological Spaces
Stating carefully any results on compactness which you use, show that if X is a compact space, Y is a Hausdorff space and f : X → Y is bijective and continuous, then f is a homeomorphism.
Hence or otherwise show that the unit circle S = {(x, y) ∈ R^2 : x^2 + y^2 = 1} is homeomorphic to the quotient space [0, 1]/ ∼, where ∼ is the equivalence relation defined by x ∼ y ⇔ either x = y or {x, y} = { 0 , 1 }.
Paper 2
9H Optimization
Goods from three warehouses have to be delivered to five shops, the cost of transporting one unit of good from warehouse i to shop j being cij , where
C =
The requirements of the five shops are respectively 9, 6, 12, 5 and 10 units of the good, and each warehouse holds a stock of 15 units. Find a minimal-cost allocation of goods from warehouses to shops and its associated cost.
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SECTION II
10E Linear Algebra
Define the determinant det(A) of an n × n square matrix A over the complex numbers. If A and B are two such matrices, show that det(AB) = det(A) det(B).
Write pM (λ) = det(M − λI) for the characteristic polynomial of a matrix M. Let A, B, C be n × n matrices and suppose that C is nonsingular. Show that pBC = pCB. Taking C = A + tI for appropriate values of t, or otherwise, deduce that pBA = pAB.
Show that if pA = pB then tr(A) = tr(B). Which of the following statements is true for all n × n matrices A, B, C? Justify your answers.
(i) pABC = pACB ;
(ii) pABC = pBCA.
11G Groups, Rings and Modules Let F be a field. Prove that every ideal of the ring F [X 1 ,... , Xn] is finitely generated.
Consider the set
R =
p(X, Y ) =
cij XiY j^ ∈ F [X, Y ]
∣ c 0 j =^ cj 0 = 0 whenever^ j >^0
Show that R is a subring of F [X, Y ] which is not Noetherian.
12G Geometry Show that the area of a spherical triangle with angles α, β, γ is α + β + γ − π. Hence derive the formula for the area of a convex spherical n-gon.
Deduce Euler’s formula F − E + V = 2 for a decomposition of a sphere into F convex polygons with a total of E edges and V vertices.
A sphere is decomposed into convex polygons, comprising m quadrilaterals, n pentagons and p hexagons, in such a way that at each vertex precisely three edges meet. Show that there are at most 7 possibilities for the pair (m, n), and that at least 3 of these do occur.
Paper 2 [TURN OVER
15D Methods
(a) Legendre’s equation may be written in the form
d dx
(1 − x^2 )
dy dx
Show that there is a series solution for y of the form
y =
∑^ ∞
k=
akxk,
where the ak satisfy the recurrence relation
ak+ ak
(λ − k(k + 1)) (k + 1)(k + 2)
Hence deduce that there are solutions for y(x) = Pn(x) that are polynomials of degree n, provided that λ = n(n + 1). Given that a 0 is then chosen so that Pn(1) = 1, find the explicit form for P 2 (x).
(b) Laplace’s equation for Φ(r, θ) in spherical polar coordinates (r, θ, φ) may be written in the axisymmetric case as
∂^2 Φ ∂r^2
r
∂r
r^2
∂x
(1 − x^2 )
∂x
where x = cos θ.
Write down without proof the general form of the solution obtained by the method of separation of variables. Use it to find the form of Φ exterior to the sphere r = a that satisfies the boundary conditions, Φ(a, x) = 1 + x^2 , and limr→∞ Φ(r, x) = 0.
Paper 2 [TURN OVER
16A Quantum Mechanics
Give the physical interpretation of the expression
〈A〉ψ =
ψ(x)∗^ Aψˆ (x)dx
for an observable A, where Aˆ is a Hermitian operator and ψ is normalised. By considering the norm of the state (A + iλB)ψ for two observables A and B, and real values of λ, show that
〈A^2 〉ψ 〈B^2 〉ψ >
|〈[A, B]〉ψ |^2.
Deduce the uncertainty relation
∆A∆B >
|〈[A, B]〉ψ | ,
where ∆A is the uncertainty of A.
A particle of mass m moves in one dimension under the influence of potential 1 2 mω
(^2) x (^2). By considering the commutator [x, p], show that the expectation value of the
Hamiltonian satisfies
〈H〉ψ >
ℏω.
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18D Numerical Analysis
(a) A Householder transformation (reflection) is given by
H = I −
2 uuT ‖u‖^2
where H ∈ Rm×m, u ∈ Rm, and I is the m × m unit matrix and u is a non-zero vector which has norm ‖u‖ = (
∑m i=1 u
(^2) i ) 1 / (^2). Show that H is orthogonal.
(b) Suppose that A ∈ Rm×n, x ∈ Rn^ and b ∈ Rm^ with n < m. Show that if x minimises ‖Ax − b‖^2 then it also minimises ‖QAx − Qb‖^2 , where Q is an arbitrary m × m orthogonal matrix.
(c) Using Householder reflection, find the x that minimises ‖Ax − b‖^2 when
A =
b^ =
19H Statistics Suppose that the joint distribution of random variables X, Y taking values in Z+^ = { 0 , 1 , 2 ,... } is given by the joint probability generating function
ϕ(s, t) ≡ E [sX^ tY^ ] =
1 − α − β 1 − αs − βt
where the unknown parameters α and β are positive, and satisfy the inequality α + β < 1. Find E(X). Prove that the probability mass function of (X, Y ) is
f (x, y | α, β) = (1 − α − β)
x + y x
αxβy^ (x, y ∈ Z+) ,
and prove that the maximum-likelihood estimators of α and β based on a sample of size n drawn from the distribution are
αˆ =
X
1 + X + Y
, βˆ =
Y
1 + X + Y
where X (respectively, Y ) is the sample mean of X 1 ,... , Xn (respectively, Y 1 ,... , Yn).
By considering ˆα + βˆ or otherwise, prove that the maximum-likelihood estimator is biased. Stating clearly any results to which you appeal, prove that as n → ∞, ˆα → α, making clear the sense in which this convergence happens.
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20H Markov Chains
A Markov chain with state–space I = Z+^ has non-zero transition probabilities p 00 = q 0 and pi,i+1 = pi , pi+1,i = qi+1 (i ∈ I).
Prove that this chain is recurrent if and only if
∑
n> 1
∏^ n
r=
qr pr
Prove that this chain is positive-recurrent if and only if
∑
n> 1
∏^ n
r=
pr− 1 qr
END OF PAPER
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