Download Examination Instructions and Problems in Mathematics, Physics, and Engineering and more Exams Mathematics in PDF only on Docsity!
MATHEMATICAL TRIPOS Part IB
Wednesday 8 June 2005 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 REQUIRMENTS 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
1C Linear Algebra
Let Ω be the set of all 2 × 2 matrices of the form α = aI + bJ + cK + dL, where a, b, c, d are in R, and
I =
, J =
i 0 0 −i
, K =
, L =
0 i i 0
(i^2 = −1).
Prove that Ω is closed under multiplication and determine its dimension as a vector space over R. Prove that
(aI + bJ + cK + dL) (aI − bJ − cK − dL) = (a^2 + b^2 + c^2 + d^2 )I ,
and deduce that each non-zero element of Ω is invertible.
2C Groups, Rings and Modules Define an automorphism of a group G, and the natural group law on the set Aut(G) of all automorphisms of G. For each fixed h in G, put ψ(h)(g) = hgh−^1 for all g in G. Prove that ψ(h) is an automorphism of G, and that ψ defines a homomorphism from G into Aut(G).
3B Analysis II
Define uniform continuity for a real-valued function defined on an interval in R.
Is a uniformly continuous function on the interval (0, 1) necessarily bounded? Is 1/x uniformly continuous on (0, 1)?
Is sin(1/x) uniformly continuous on (0, 1)?
Justify your answers.
4A Metric and Topological Spaces Let X be a topological space. Suppose that U 1 , U 2 ,... are connected subsets of X with Uj ∩ U 1 non-empty for all j > 0. Prove that
W =
j> 0
Uj
is connected. If each Uj is path-connected, prove that W is path-connected.
Paper 2
8E Fluid Dynamics
For a steady flow of an incompressible fluid of density ρ, show that
u × ω = ∇H ,
where ω = ∇ × u is the vorticity and H is to be found. Deduce that H is constant along streamlines.
Now consider a flow in the xy-plane described by a streamfunction ψ(x, y). Evaluate u × ω and deduce from H = H(ψ) that
dH dψ
9D Optimization
Explain what is meant by a two-person zero-sum game with payoff matrix A = (aij ). Show that the problems of the two players may be expressed as a dual pair of linear programming problems. State without proof a set of sufficient conditions for a pair of strategies for the two players to be optimal.
Paper 2
SECTION II
10C Linear Algebra
(i) Let A = (aij ) be an n × n matrix with entries in C. Define the determinant of A, the cofactor of each aij , and the adjugate matrix adj(A). Assuming the expansion of the determinant of a matrix in terms of its cofactors, prove that
adj(A) A = det(A)In ,
where In is the n × n identity matrix.
(ii) Let
A =
Show the eigenvalues of A are ± 1 , ±i, where i^2 = −1, and determine the diagonal matrix to which A is similar. For each eigenvalue, determine a non-zero eigenvector.
11C Groups, Rings and Modules
Let A be the abelian group generated by two elements x, y, subject to the relation 6 x+9y = 0. Give a rigorous explanation of this statement by defining A as an appropriate quotient of a free abelian group of rank 2. Prove that A itself is not a free abelian group, and determine the exact structure of A.
12A Geometry
Let U be an open subset of R^2 equipped with a Riemannian metric. For γ : [0, 1] → U a smooth curve, define what is meant by its length and energy. Prove that length(γ)^2 ≤ energy(γ), with equality if and only if ˙γ has constant norm with respect to the metric.
Suppose now U is the upper half plane model of the hyperbolic plane, and P, Q are points on the positive imaginary axis. Show that a smooth curve γ joining P and Q represents an absolute minimum of the length of such curves if and only if γ(t) = i v(t), with v a smooth monotonic real function.
Suppose that a smooth curve γ joining the above points P and Q represents a stationary point for the energy under proper variations; deduce from an appropriate form of the Euler–Lagrange equations that γ must be of the above form, with ˙v/v constant.
Paper 2 [TURN OVER
15E Methods
Write down the Euler–Lagrange equation for the variational problem for r(z)
δ
∫ (^) h
−h
F (z, r, r′) dz = 0,
with boundary conditions r(−h) = r(h) = R, where R is a given positive constant. Show that if F does not depend explicitly on z, i.e. F = F (r, r′), then the equation has a first integral
F − r′^
∂F
∂r′^
k
where k is a constant.
An axisymmetric soap film r(z) is formed between two circular rings r = R at z = ±H. Find the equation governing the shape which minimizes the surface area. Show that the shape takes the form r(z) = k−^1 cosh kz.
Show that there exist no solution if R/H < sinh A, where A is the unique positive solution of A = coth A.
Paper 2 [TURN OVER
16G Quantum Mechanics
A particle of mass m moving in a one-dimensional harmonic oscillator potential satisfies the Schr¨odinger equation
H Ψ(x, t) = iℏ
∂t
Ψ(x, t) ,
where the Hamiltonian is given by
H = −
ℏ^2
2 m
d^2 dx^2
m ω^2 x^2.
The operators a and a†^ are defined by
a =
βx +
i βℏ
p
, a†^ =
βx −
i βℏ
p
where β =
mω/ℏ and p = −iℏ∂/∂x is the usual momentum operator. Show that [a, a†] = 1.
Express x and p in terms of a and a†^ and, hence or otherwise, show that H can be written in the form H =
a†a + (^12)
ℏω.
Show, for an arbitrary wave function Ψ, that
dx Ψ∗^ H Ψ ≥ 12 ℏω and hence that the energy of any state satisfies the bound
E ≥ 12 ℏω.
Hence, or otherwise, show that the ground state wave function satisfies aΨ 0 = 0 and that its energy is given by E 0 = 12 ℏω.
By considering H acting on a†^ Ψ 0 , (a†)^2 Ψ 0 , and so on, show that states of the form
(a†)n^ Ψ 0
(n > 0) are also eigenstates and that their energies are given by En =
n + (^12)
ℏω.
Paper 2
19D Statistics
Let X 1 ,... , Xn be a random sample from a probability density function f (x | θ), where θ is an unknown real-valued parameter which is assumed to have a prior density π(θ). Determine the optimal Bayes point estimate a(X 1 ,... , Xn) of θ, in terms of the posterior distribution of θ given X 1 ,... , Xn, when the loss function is
L(θ, a) =
γ(θ − a) when θ > a, δ(a − θ) when θ 6 a,
where γ and δ are given positive constants.
Calculate the estimate explicitly in the case when f (x | θ) is the density of the uniform distribution on (0, θ) and π(θ) = e−θ^ θn/n!, θ > 0.
20D Markov Chains
Consider a Markov chain (Xn)n> 0 with state space { 0 , 1 , 2 ,.. .} and transition probabilities given by
Pi,j = pqi−j+1, 0 < j 6 i + 1, and Pi, 0 = qi+1^ for i > 0 ,
with Pi,j = 0, otherwise, where 0 < p < 1 and q = 1 − p.
For each i > 1, let
hi = P (Xn = 0, for some n > 0 | X 0 = i) ,
that is, the probability that the chain ever hits the state 0 given that it starts in state i. Write down the equations satisfied by the probabilities {hi, i > 1 } and hence, or otherwise, show that they satisfy a second-order recurrence relation with constant coefficients. Calculate hi for each i > 1.
Determine for each value of p, 0 < p < 1, whether the chain is transient, null recurrent or positive recurrent and in the last case calculate the stationary distribution.
[Hint: When the chain is positive recurrent, the stationary distribution is geometric.]
END OF PAPER
Paper 2
