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MATHEMATICAL TRIPOS Part IB
Tuesday 1 June 2004 9 to 12
PAPER 1
Before you begin read these instructions carefully.
Each question in Section II carries twice the credit of each question in Section I. You should attempt at most four questions from Section I and at most six questions from Section II.
Additional credit will be given to substantially complete answers.
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 code letter affixed to each question, including in the same bundle questions from Sections I and II with the same code letter.
Attach a completed 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 candidate number and desk number.
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
SECTION I
1H Linear Algebra
Suppose that {e 1 ,... , er+1} is a linearly independent set of distinct elements of a vector space V and {e 1 ,... , er , fr+1,... , fm} spans V. Prove that fr+1,... , fm may be reordered, as necessary, so that {e 1 ,... er+1, fr+2,... , fm} spans V.
Suppose that {e 1 ,... , en} is a linearly independent set of distinct elements of V and that {f 1 ,... , fm} spans V. Show that n 6 m.
2F Groups, Rings and Modules Let G be a finite group of order n. Let H be a subgroup of G. Define the normalizer N (H) of H, and prove that the number of distinct conjugates of H is equal to the index of N (H) in G. If p is a prime dividing n, deduce that the number of Sylow p-subgroups of G must divide n.
[You may assume the existence and conjugacy of Sylow subgroups.]
Prove that any group of order 72 must have either 1 or 4 Sylow 3-subgroups.
3G Geometry Using the Riemannian metric
ds^2 =
dx^2 + dy^2 y^2
define the length of a curve and the area of a region in the upper half-plane H = {x + iy : y > 0 }.
Find the hyperbolic area of the region {(x, y) ∈ H : 0 < x < 1 , y > 1 }.
4G Analysis II
Define what it means for a sequence of functions Fn : (0, 1) → R, where n = 1, 2 ,... , to converge uniformly to a function F.
For each of the following sequences of functions on (0, 1), find the pointwise limit function. Which of these sequences converge uniformly? Justify your answers.
(i) Fn(x) = (^) n^1 ex
(ii) Fn(x) = e−nx
2
(iii) Fn(x) =
∑^ n
i=
xi
Paper 1
9C Fluid Dynamics
From the general mass-conservation equation, show that the velocity field u(x) of an incompressible fluid is solenoidal, i.e. that ∇ · u = 0.
Verify that the two-dimensional flow
u =
y x^2 + y^2
−x x^2 + y^2
is solenoidal and find a streamfunction ψ(x, y) such that u = (∂ψ/∂y, −∂ψ/∂x).
10H Statistics Use the generalized likelihood-ratio test to derive Student’s t-test for the equality of the means of two populations. You should explain carefully the assumptions underlying the test.
11H Markov Chains
Let P = (Pij ) be a transition matrix. What does it mean to say that P is (a) irreducible, (b) recurrent?
Suppose that P is irreducible and recurrent and that the state space contains at least two states. Define a new transition matrix P˜ by
P˜ij =
0 if i = j, (1 − Pii)−^1 Pij if i 6 = j.
Prove that P˜ is also irreducible and recurrent.
Paper 1
SECTION II
12H Linear Algebra
Let U and W be subspaces of the finite-dimensional vector space V. Prove that both the sum U + W and the intersection U ∩ W are subspaces of V. Prove further that
dim U + dim W = dim (U + W ) + dim (U ∩ W ).
Let U , W be the kernels of the maps A, B : R^4 → R^2 given by the matrices A and B respectively, where
A =
, B =
Find a basis for the intersection U ∩ W , and extend this first to a basis of U , and then to a basis of U + W.
13F Groups, Rings and Modules
State the structure theorem for finitely generated abelian groups. Prove that a finitely generated abelian group A is finite if and only if there exists a prime p such that A/pA = 0.
Show that there exist abelian groups A 6 = 0 such that A/pA = 0 for all primes p. Prove directly that your example of such an A is not finitely generated.
14G Geometry Show that for every hyperbolic line L in the hyperbolic plane H there is an isometry of H which is the identity on L but not on all of H. Call it the reflection RL.
Show that every isometry of H is a composition of reflections.
15G Analysis II
State the axioms for a norm on a vector space. Show that the usual Euclidean norm on Rn, ||x|| = (x^21 + x^22 +... + x^2 n)^1 /^2 ,
satisfies these axioms.
Let U be any bounded convex open subset of Rn^ that contains 0 and such that if x ∈ U then −x ∈ U. Show that there is a norm on Rn, satisfying the axioms, for which U is the set of points in Rn^ of norm less than 1.
Paper 1 [TURN OVER
18B Electromagnetism
Inside a volume D there is an electrostatic charge density ρ(r), which induces an electric field E(r) with associated electrostatic potential φ(r). The potential vanishes on the boundary of D. The electrostatic energy is
W =
D
ρφ d^3 r. (1)
Derive the alternative form
W =
D
E^2 d^3 r. (2)
A capacitor consists of three identical and parallel thin metal circular plates of area A positioned in the planes z = −H, z = a and z = H, with −H < a < H, with centres on the z axis, and at potentials 0, V and 0 respectively. Find the electrostatic energy stored, verifying that expressions (1) and (2) give the same results. Why is the energy minimal when a = 0?
19D Quantum Mechanics
The angular momentum operators are L = (L 1 , L 2 , L 3 ). Write down their commutation relations and show that [Li, L^2 ] = 0. Let
L± = L 1 ± iL 2 ,
and show that L^2 = L−L+ + L 32 + ℏL 3.
Verify that Lf (r) = 0, where r^2 = xixi, for any function f. Show that
L 3 (x 1 + ix 2 )nf (r) = nℏ(x 1 + ix 2 )nf (r) , L+(x 1 + ix 2 )nf (r) = 0 ,
for any integer n. Show that (x 1 + ix 2 )nf (r) is an eigenfunction of L^2 and determine its eigenvalue. Why must L−(x 1 + ix 2 )nf (r) be an eigenfunction of L^2? What is its eigenvalue?
Paper 1 [TURN OVER
20C Fluid Dynamics
A layer of water of depth h flows along a wide channel with uniform velocity (U, 0), in Cartesian coordinates (x, y), with x measured downstream. The bottom of the channel is at y = −h, and the free surface of the water is at y = 0. Waves are generated on the free surface so that it has the new position y = η(x, t) = a ei(ωt−kx).
Write down the equation and the full nonlinear boundary conditions for the velocity potential φ (for the perturbation velocity) and the motion of the free surface.
By linearizing these equations about the state of uniform flow, show that
∂η ∂t
+ U
∂η ∂x
∂φ ∂y
∂φ ∂t
+ U
∂φ ∂x
∂φ ∂y
= 0 on y = −h,
where g is the acceleration due to gravity.
Hence, determine the dispersion relation for small-amplitude surface waves
(ω − kU )^2 = gk tanh kh.
21H Statistics
State and prove the Rao–Blackwell Theorem. Suppose that X 1 , X 2 ,... , Xn are independent, identically-distributed random vari- ables with distribution
P (X 1 = r) = pr−^1 (1 − p), r = 1, 2 ,... ,
where p, 0 < p < 1, is an unknown parameter. Determine a one-dimensional sufficient statistic, T , for p.
By first finding a simple unbiased estimate for p, or otherwise, determine an unbiased estimate for p which is a function of T.
Paper 1
