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MATHEMATICAL TRIPOS Part IB
Tuesday, 5 June, 2012 9:00 am to 12:00 pm
PAPER 1
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 sheets 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 notions of basis and dimension of a vector space. Prove that two finite- dimensional real vector spaces with the same dimension are isomorphic.
In each case below, determine whether the set S is a basis of the real vector space V :
(i) V = C is the complex numbers; S = { 1 , i}.
(ii) V = R[x] is the vector space of all polynomials in x with real coefficients; S = { 1 , (x − 1), (x − 1)(x − 2), (x − 1)(x − 2)(x − 3),.. .}.
(iii) V = {f : [0, 1] → R}; S = {χp | p ∈ [0, 1]}, where
χp(x) =
1 x = p 0 x 6 = p.
2A Complex Analysis or Complex Methods Find a conformal transformation ζ = ζ(z) that maps the domain D, 0 < arg z < 32 π , on to the strip 0 < Im(ζ) < 1. Hence find a bounded harmonic function φ on D subject to the boundary conditions φ = 0, A on arg z = 0, 32 π , respectively, where A is a real constant.
3G Geometry Describe a collection of charts which cover a circular cylinder of radius R. Compute the first fundamental form, and deduce that the cylinder is locally isometric to the plane.
4B Variational Principles State how to find the stationary points of a C^1 function f (x, y) restricted to the circle x^2 + y^2 = 1, using the method of Lagrange multipliers. Explain why, in general, the method of Lagrange multipliers works, in the case where there is just one constraint.
Find the stationary points of x^4 + 2y^3 restricted to the circle x^2 + y^2 = 1.
Part IB, Paper 1
8H Optimization State the Lagrangian sufficiency theorem. Use Lagrange multipliers to find the optimal values of x 1 and x 2 in the problem:
maximize x^21 + x 2 subject to x^21 + 12 x^22 6 b 1 , x 1 > b 2 and x 1 , x 2 > 0 ,
for all values of b 1 , b 2 such that b 1 − b^22 > 0.
Part IB, Paper 1
SECTION II
9F Linear Algebra Define what it means for two n × n matrices to be similar to each other. Show that if two n × n matrices are similar, then the linear transformations they define have isomorphic kernels and images. If A and B are n × n real matrices, we define [A, B] = AB − BA. Let
KA = {X ∈ Mn×n(R) | [A, X] = 0} LA = {[A, X] | X ∈ Mn×n(R)}.
Show that KA and LA are linear subspaces of Mn×n(R). If A and B are similar, show that KA ∼= KB and LA ∼= LB. Suppose that A is diagonalizable and has characteristic polynomial
(x − λ 1 )m^1 (x − λ 2 )m^2 ,
where λ 1 6 = λ 2. What are dim KA and dim LA?
10G Groups, Rings and Modules Let G be a finite group. What is a Sylow p-subgroup of G? Assuming that a Sylow p-subgroup H exists, and that the number of conjugates of H is congruent to 1 mod p, prove that all Sylow p-subgroups are conjugate. If np denotes the number of Sylow p-subgroups, deduce that
np ≡ 1 mod p and np
|G|.
If furthermore G is simple prove that either G = H or
|G|
np!
Deduce that a group of order 1, 000 , 000 cannot be simple.
Part IB, Paper 1 [TURN OVER
13A Complex Analysis or Complex Methods Using Cauchy’s integral theorem, write down the value of a holomorphic function f (z) where |z| < 1 in terms of a contour integral around the unit circle, ζ = eiθ^. By considering the point 1/z, or otherwise, show that
f (z) =
2 π
∫ (^2) π
0
f (ζ) 1 − |z|^2 |ζ − z|^2
dθ.
By setting z = reiα, show that for any harmonic function u(r, α),
u(r, α) =
2 π
∫ (^2) π
0
u(1, θ) 1 − r^2 1 − 2 r cos(α − θ) + r^2
dθ
if r < 1. Assuming that the function v(r, α), which is the conjugate harmonic function to u(r, α), can be written as
v(r, α) = v(0) +
π
∫ (^2) π
0
u(1, θ) r sin(α − θ) 1 − 2 r cos(α − θ) + r^2
dθ ,
deduce that
f (z) = iv(0) +
2 π
∫ (^2) π
0
u(1, θ) ζ + z ζ − z dθ.
[You may use the fact that on the unit circle, ζ = 1/ζ, and hence
ζ ζ − 1 /z
z ζ − z
. ]
Part IB, Paper 1 [TURN OVER
14C Methods Consider the regular Sturm-Liouville (S-L) system
(Ly)(x) − λω(x)y(x) = 0 , a 6 x 6 b ,
where
(Ly)(x) := −[p(x)y′(x)]′^ + q(x)y(x)
with ω(x) > 0 and p(x) > 0 for all x in [a, b], and the boundary conditions on y are { A 1 y(a) + A 2 y′(a) = 0 B 1 y(b) + B 2 y′(b) = 0.
Show that with these boundary conditions, L is self-adjoint. By considering yLy, or otherwise, show that the eigenvalue λ can be written as
λ =
∫ (^) b a (py
′ (^2) + qy (^2) ) dx − [pyy′]b a ∫ (^) b a ωy (^2) dx
Now suppose that a = 0 and b = ℓ, that p(x) = 1, q(x) > 0 and ω(x) = 1 for all x ∈ [0, ℓ], and that A 1 = 1, A 2 = 0, B 1 = k ∈ R+^ and B 2 = 1. Show that the eigenvalues of this regular S-L system are strictly positive. Assuming further that q(x) = 0, solve the system explicitly, and with the aid of a graph, show that there exist infinitely many eigenvalues λ 1 < λ 2 < · · · < λn < · · ·. Describe the behaviour of λn as n → ∞.
Part IB, Paper 1
17A Fluid Dynamics Consider inviscid, incompressible fluid flow confined to the (x, y) plane. The fluid has density ρ, and gravity can be neglected. Using the conservation of volume flux, determine the velocity potential φ(r) of a point source of strength m, in terms of the distance r from the source. Two point sources each of strength m are located at x+ = (0, a) and x− = (0, −a). Find the velocity potential of the flow. Show that the flow in the region y > 0 is equivalent to the flow due to a source at x+ and a fixed boundary at y = 0. Find the pressure on the boundary y = 0 and hence determine the force on the boundary. [Hint: you may find the substitution x = a tan θ useful for the calculation of the pressure.]
18D Numerical Analysis For a numerical method for solving y′^ = f (t, y), define the linear stability domain, and state when such a method is A-stable.
Determine all values of the real parameter a for which the Runge-Kutta method
k 1 = f
tn + ( 12 − a)h, yn +
4 hk^1 + (^
1 4 −^ a)hk^2
k 2 = f
tn + ( 12 + a)h, yn +
( 14 + a)hk 1 + 14 hk 2
yn+1 = yn + 12 h(k 1 + k 2 )
is A-stable.
Part IB, Paper 1
19H Statistics State and prove the Neyman-Pearson lemma. A sample of two independent observations, (x 1 , x 2 ), is taken from a distribution with density f (x; θ) = θxθ−^1 , 0 6 x 6 1. It is desired to test H 0 : θ = 1 against H 1 : θ = 2. Show that the best test of size α can be expressed using the number c such that
1 − c + c log c = α.
Is this the uniformly most powerful test of size α for testing H 0 against H 1 : θ > 1? Suppose that the prior distribution of θ is P (θ = 1) = 4γ/(1 + 4γ), P (θ = 2) = 1 /(1+4γ), where 1 > γ > 0. Find the test of H 0 against H 1 that minimizes the probability of error. Let w(θ) denote the power function of this test at θ (> 1). Show that
w(θ) = 1 − γθ^ + γθ^ log γθ.
Part IB, Paper 1 [TURN OVER
