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MATHEMATICAL TRIPOS Part IB
Friday, 4 June, 2010 1:30 pm to 4:30 pm
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.
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
1F Linear Algebra Define the notion of an inner product on a finite-dimensional real vector space V , and the notion of a self-adjoint linear map α : V → V.
Suppose that V is the space of real polynomials of degree at most n in a variable t. Show that
〈f, g〉 =
− 1
f (t)g(t) dt
is an inner product on V , and that the map α : V → V :
α(f )(t) = (1 − t^2 )f ′′(t) − 2 tf ′(t)
is self-adjoint.
2H Groups Rings and Modules Let M be a free Z-module generated by e 1 and e 2. Let a, b be two non-zero integers, and N be the submodule of M generated by ae 1 + be 2. Prove that the quotient module M/N is free if and only if a, b are coprime.
3G Analysis II Let S denote the set of continuous real-valued functions on the interval [0, 1]. For f, g ∈ S , set
d 1 (f, g) = sup {|f (x) − g(x)| : x ∈ [0, 1]} and d 2 (f, g) =
0
|f (x) − g(x)| dx.
Show that both d 1 and d 2 define metrics on S. Does the identity map on S define a continuous map of metric spaces (S, d 1 ) → (S, d 2 )? Does the identity map define a continuous map of metric spaces (S, d 2 ) → (S, d 1 )?
4G Complex Analysis State the principle of the argument for meromorphic functions and show how it follows from the Residue theorem.
Part IB, Paper 4
8C Numerical Analysis Suppose x 0 , x 1 ,... , xn ∈ [a, b] ⊂ R are pointwise distinct and f (x) is continuous on [a, b]. For k = 1, 2 ,... , n define
I 0 ,k(x) = f (x 0 )(xk − x) − f (xk)(x 0 − x) xk − x 0
and for k = 2, 3 ,... , n
I 0 , 1 , ... , k− 2 , k− 1 ,k(x) =
I 0 , 1 , ... , k− 2 , k− 1 (x)(xk − x) − I 0 , 1 , ... , k− 2 , k(x)(xk− 1 − x) xk − xk− 1
Show that I 0 , 1 , ... , k− 2 ,k− 1 ,k(x) is a polynomial of order k which interpolates f (x) at x 0 , x 1 ,... , xk.
Given xk = {− 1 , 0 , 2 , 5 } and f (xk) = { 33 , 5 , 9 , 1335 }, determine the interpolating polynomial.
9E Markov Chains Consider a Markov chain (Xn)n > 0 with state space {a, b, c, d} and transition probabilities given by the following table.
a b c d a 1 / 4 1 / 4 1 / 2 0 b 0 1 / 4 0 3 / 4 c 1 / 2 0 1 / 4 1 / 4 d 0 1 / 2 0 1 / 2
By drawing an appropriate diagram, determine the communicating classes of the chain, and classify them as either open or closed. Compute the following transition and hitting probabilities:
- P (Xn = b | X 0 = d) for a fixed n > 0 ,
- P (Xn = c for some n > 1 | X 0 = a).
Part IB, Paper 4
SECTION II
10F Linear Algebra (i) Show that the group On(R) of orthogonal n × n real matrices has a normal subgroup SOn(R) = {A ∈ On(R) | det A = 1}.
(ii) Show that On(R) = SOn(R) × {±In} if and only if n is odd.
(iii) Show that if n is even, then On(R) is not the direct product of SOn(R) with any normal subgroup.
[You may assume that the only elements of On(R) that commute with all elements of On(R) are ±In.]
11H Groups Rings and Modules Let V = (Z/ 3 Z)^2 , a 2-dimensional vector space over the field Z/ 3 Z, and let e 1 =
, e 2 =
∈ V.
(1) List all 1-dimensional subspaces of V in terms of e 1 , e 2. (For example, there is a subspace 〈e 1 〉 generated by e 1 .)
(2) Consider the action of the matrix group
G = GL 2 (Z/ 3 Z) =
{( (^) a b c d
∣ a, b, c, d ∈ Z/ 3 Z , ad − bc 6 = 0
on the finite set X of all 1-dimensional subspaces of V. Describe the stabiliser group K of 〈e 1 〉 ∈ X. What is the order of K? What is the order of G?
(3) Let H ⊂ G be the subgroup of all elements of G which act trivially on X. Describe H, and prove that G/H is isomorphic to S 4 , the symmetric group on four letters.
Part IB, Paper 4 [TURN OVER
14A Complex Methods A linear system is described by the differential equation
y′′′(t) − y′′(t) − 2 y′(t) + 2 y(t) = f (t) ,
with initial conditions
y (0) = 0 , y′(0) = 1 , y′′(0) = 1.
The Laplace transform of f (t) is defined as
L[f (t)] = f˜ (s) =
0
e−stf (t) dt.
You may assume the following Laplace transforms,
L[y(t)] = ˜y(s), L[y′(t)] = sy˜(s) − y(0), L[y′′(t)] = s^2 ˜y(s) − sy(0) − y′(0), L[y′′′(t)] = s^3 ˜y(s) − s^2 y(0) − sy′(0) − y′′(0).
(a) Use Laplace transforms to determine the response, y 1 (t), of the system to the signal f (t) = − 2.
(b) Determine the response, y 2 (t), given that its Laplace transform is
y˜ 2 (s) =
s^2 (s − 1)^2
(c) Given that y′′′(t) − y′′(t) − 2 y′(t) + 2y(t) = g(t)
leads to the response with Laplace transform
y˜(s) =
s^2 (s − 1)^2
determine g(t).
Part IB, Paper 4 [TURN OVER
15F Geometry Suppose that D is the unit disc, with Riemannian metric
ds^2 = dx^2 + dy^2 1 − (x^2 + y^2 )
[Note that this is not a multiple of the Poincar´e metric.] Show that the diameters of D are, with appropriate parametrization, geodesics.
Show that distances between points in D are bounded, but areas of regions in D are unbounded.
Part IB, Paper 4
17B Methods Defining the function Gf 3 (r; r 0 ) = − 1 /(4π|r − r 0 |), prove Green’s third identity for functions u(r) satisfying Laplace’s equation in a volume V with surface S, namely
u(r 0 ) =
S
u
∂Gf 3 ∂n
∂u ∂n Gf 3
dS.
A solution is sought to the Neumann problem for ∇^2 u = 0 in the half plane z > 0:
u = O(|x|−a), ∂u ∂r
= O(|x|−a−^1 ) as |x| → ∞, ∂u ∂z
= p(x, y) on z = 0,
where a > 0. It is assumed that
−∞
−∞ p(x, y)^ dx dy^ = 0. Explain why this condition is necessary.
Construct an appropriate Green’s function G(r; r 0 ) satisfying ∂G/∂z = 0 at z = 0, using the method of images or otherwise. Hence find the solution in the form
u(x 0 , y 0 , z 0 ) =
−∞
−∞
p(x, y)f (x − x 0 , y − y 0 , z 0 ) dx dy,
where f is to be determined.
Now let p(x, y) =
x |x|, |y| < a, 0 otherwise.
By expanding f in inverse powers of z 0 , show that
u → − 2 a^4 x 0 3 πz^30
as z 0 → ∞.
Part IB, Paper 4
18B Fluid Dynamics Write down the velocity potential for a line source flow of strength m located at (r, θ) = (d, 0) in polar coordinates (r, θ) and derive the velocity components ur, uθ.
A two-dimensional flow field consists of such a source in the presence of a circular cylinder of radius a (a < d) centred at the origin. Show that the flow field outside the cylinder is the sum of the original source flow, together with that due to a source of the same strength at (a^2 /d, 0) and another at the origin, of a strength to be determined.
Use Bernoulli’s law to find the pressure distribution on the surface of the cylinder, and show that the total force exerted on it is in the x-direction and of magnitude
m^2 ρ 2 π^2
∫ (^2) π
0
ad^2 sin^2 θ cos θ (a^2 + d^2 − 2 ad cos θ)^2
dθ ,
where ρ is the density of the fluid. Without evaluating the integral, show that it is positive. Comment on the fact that the force on the cylinder is therefore towards the source.
19E Statistics Consider a collection X 1 ,... , Xn of independent random variables with common density function f (x; θ) depending on a real parameter θ. What does it mean to say T is a sufficient statistic for θ? Prove that if the joint density of X 1 ,... , Xn satisfies the factorisation criterion for a statistic T , then T is sufficient for θ.
Let each Xi be uniformly distributed on [−
θ,
θ ]. Find a two-dimensional sufficient statistic T = (T 1 , T 2 ). Using the fact that θˆ = 3X 12 is an unbiased estimator of θ , or otherwise, find an unbiased estimator of θ which is a function of T and has smaller variance than θˆ. Clearly state any results you use.
Part IB, Paper 4 [TURN OVER
