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MATHEMATICAL TRIPOS Part IB
Thursday 7 June 2001 9 to 12
PAPER 2
Before you begin read these instructions carefully.
Each question in Section II carries twice the credit of each question in Section I. Candidates may attempt at most four questions in 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:
Answers must be tied up in separate bundles, marked A, B,... , G according to the letter affixed to each question, and a blue cover sheet must be attached to each bundle.
A green master cover sheet listing all the questions attempted must be completed.
It is essential that every cover sheet bear the candidate’s examination number and desk number.
SECTION I
1A Analysis II
State and prove the contraction mapping theorem. Let A = {x, y, z}, let d be the discrete metric on A, and let d ′^ be the metric given by: d ′^ is symmetric and
d ′(x, y) = 2, d ′(x, z) = 2, d ′(y, z) = 1,
d ′(x, x) = d ′(y, y) = d ′(z, z) = 0.
Verify that d ′^ is a metric, and that it is Lipschitz equivalent to d.
Define an appropriate function f : A → A such that f is a contraction in the d ′ metric, but not in the d metric.
2G Methods
Show that the symmetric and antisymmetric parts of a second-rank tensor are them- selves tensors, and that the decomposition of a tensor into symmetric and antisymmetric parts is unique.
For the tensor A having components
A =
find the scalar a, vector p and symmetric traceless tensor B such that
Ax = ax + p ∧ x + Bx
for every vector x.
3D Statistics
Suppose the single random variable X has a uniform distribution on the interval [0, θ] and it is required to estimate θ with the loss function
L(θ, a) = c(θ − a)^2 ,
where c > 0.
Find the posterior distribution for θ and the optimal Bayes point estimate with respect to the prior distribution with density p(θ) = θe−θ^ , θ > 0.
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7E Complex Methods
A complex function is defined for every z ∈ V , where V is a non-empty open subset of C, and it possesses a derivative at every z ∈ V. Commencing from a formal definition of derivative, deduce the Cauchy–Riemann equations.
8B Quadratic Mathematics
Let V be a finite-dimensional vector space over a field k. Describe a bijective correspondence between the set of bilinear forms on V , and the set of linear maps of V to its dual space V ∗. If φ 1 , φ 2 are non-degenerate bilinear forms on V , prove that there exists an isomorphism α : V → V such that φ 2 (u, v) = φ 1 (u, αv) for all u, v ∈ V. If furthermore both φ 1 , φ 2 are symmetric, show that α is self-adjoint (i.e. equals its adjoint) with respect to φ 1.
9F Quantum Mechanics
Consider a solution ψ(x, t) of the time-dependent Schr¨odinger equation for a particle of mass m in a potential V (x). The expectation value of an operator O is defined as
〈 O 〉 =
dx ψ∗(x, t) O ψ(x, t).
Show that d dt
〈x〉 =
〈p〉 m
where
p =
i
∂x
and that d dt
〈p〉 =
∂V
∂x
(x)
[You may assume that ψ(x, t) vanishes as x → ±∞.]
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SECTION II
10A Analysis II
Define total boundedness for metric spaces. Prove that a metric space has the Bolzano–Weierstrass property if and only if it is complete and totally bounded.
11G Methods
Explain what is meant by an isotropic tensor.
Show that the fourth-rank tensor
Aijkl = αδij δkl + βδikδjl + γδilδjk (∗)
is isotropic for arbitrary scalars α, β and γ.
Assuming that the most general isotropic tensor of rank 4 has the form (∗), or otherwise, evaluate
Bijkl =
r<a
xixj
∂^2
∂xk∂xl
r
dV,
where x is the position vector and r = |x|.
12D Statistics What is meant by a generalized likelihood ratio test? Explain in detail how to perform such a test.
Let X 1 ,... , Xn be independent random variables, and let Xi have a Poisson distribution with unknown mean λi, i = 1,... , n.
Find the form of the generalized likelihood ratio statistic for testing H 0 : λ 1 =... = λn, and show that it may be approximated by
X¯
∑^ n
i=
(Xi − X¯)^2 ,
where X¯ = n−^1
∑n i=1 Xi. If, for n = 7, you found that the value of this statistic was 27.3, would you accept H 0? Justify your answer.
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16E Complex Methods
Let R be a rational function such that limz→∞{zR(z)} = 0. Assuming that R has no real poles, use the residue calculus to evaluate ∫ (^) ∞
−∞
R(x)dx.
Given that n > 1 is an integer, evaluate ∫ (^) ∞
0
dx 1 + x^2 n^
17B Quadratic Mathematics
Suppose p is an odd prime and a an integer coprime to p. Define the Legendre symbol ( ap ), and state (without proof) Euler’s criterion for its calculation.
For j any positive integer, we denote by rj the (unique) integer with |rj | ≤ (p−1)/ 2 and rj ≡ aj mod p. Let l be the number of integers 1 ≤ j ≤ (p − 1)/2 for which rj is negative. Prove that (^) ( a p
= (−1)l.
Hence determine the odd primes for which 2 is a quadratic residue.
Suppose that p 1 ,... , pm are primes congruent to 7 modulo 8, and let N = 8(p 1... pm)^2 − 1.
Show that 2 is a quadratic residue for any prime dividing N. Prove that N is divisible by some prime p ≡ 7 mod 8. Hence deduce that there are infinitely many primes congruent to 7 modulo 8.
18F Quantum Mechanics
(a) Write down the angular momentum operators L 1 , L 2 , L 3 in terms of xi and
pi = −iℏ
∂xi
, i = 1, 2 , 3.
Verify the commutation relation
[L 1 , L 2 ] = iℏL 3.
Show that this result and its cyclic permutations imply
[L 3 , L 1 ± iL 2 ] = ±ℏ (L 1 ± iL 2 ), [L^2 , L 1 ± iL 2 ] = 0.
(b) Consider a wavefunction of the form ψ = (x^23 + ar^2 )f (r), where r^2 = x^21 + x^22 + x^23. Show that for a particular value of a, ψ is an eigenfunction of both L^2 and L 3. What are the corresponding eigenvalues?
END OF PAPER
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