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MATHEMATICAL TRIPOS Part IB
Friday 8 June 2007 1.30 to 4.
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; 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 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
1G Linear Algebra
Suppose that α : V → W is a linear map of finite-dimensional complex vector spaces. What is the dual map α∗^ of the dual vector spaces?
Suppose that we choose bases of V, W and take the corresponding dual bases of the dual vector spaces. What is the relation between the matrices that represent α and α∗ with respect to these bases? Justify your answer.
2G Groups, Rings and Modules If p is a prime, how many abelian groups of order p^4 are there, up to isomorphism?
3H Analysis II
Define uniform convergence for a sequence f 1 , f 2 ,... of real-valued functions on the interval (0, 1).
For each of the following sequences of functions on (0, 1), find the pointwise limit function. Which of these sequences converge uniformly on (0, 1)?
(i) fn(x) = log (x + (^) n^1 ),
(ii) fn(x) = cos ( xn ).
Justify your answers.
4H Complex Analysis State the argument principle.
Show that if f is an analytic function on an open set U ⊂ C which is one-to-one, then f ′(z) 6 = 0 for all z ∈ U.
Paper 4
8F Numerical Analysis
Given f ∈ C^3 [0, 2], we approximate f ′(0) by the linear combination
μ(f ) = −
f (0) + 2f (1) −
f (2).
Using the Peano kernel theorem, determine the least constant c in the inequality
|f ′(0) − μ(f )| ≤ c ‖f ′′′‖∞ ,
and give an example of f for which the inequality turns into equality.
9C Markov Chains
For a Markov chain with state space S, define what is meant by the following: (i) states i, j ∈ S communicate;
(ii) state i ∈ S is recurrent.
Prove that communication is an equivalence relation on S and that if two states i, j communicate and i is recurrent then j is recurrent.
Paper 4
SECTION II
10G Linear Algebra
(i) State and prove the Cayley–Hamilton theorem for square complex matrices. (ii) A square matrix A is of order n for a strictly positive integer n if An^ = I and no smaller positive power of A is equal to I.
Determine the order of a complex 2 × 2 matrix A of trace zero and determinant 1.
11G Groups, Rings and Modules
A regular icosahedron has 20 faces, 12 vertices and 30 edges. The group G of its rotations acts transitively on the set of faces, on the set of vertices and on the set of edges.
(i) List the conjugacy classes in G and give the size of each. (ii) Find the order of G and list its normal subgroups.
[A normal subgroup of G is a union of conjugacy classes in G.]
12A Geometry Write down the Riemannian metric for the upper half-plane model H of the hyperbolic plane. Describe, without proof, the group of isometries of H and the hyperbolic lines (i.e. the geodesics) on H.
Show that for any two hyperbolic lines 1 , 2 , there is an isometry of H which maps 1 onto 2.
Suppose that g is a composition of two reflections in hyperbolic lines which are ultraparallel (i.e. do not meet either in the hyperbolic plane or at its boundary). Show that g cannot be an element of finite order in the group of isometries of H.
[Existence of a common perpendicular to two ultraparallel hyperbolic lines may be assumed. You might like to choose carefully which hyperbolic line to consider as a common perpendicular.]
Paper 4 [TURN OVER
15F Complex Methods
(i) Use the definition of the Laplace transform of f (t):
L{f (t)} = F (s) =
0
e−stf (t) dt ,
to show that, for f (t) = tn,
L{f (t)} = F (s) =
n! sn+^
, L{eatf (t)} = F (s − a) =
n! (s − a)n+^
(ii) Use contour integration to find the inverse Laplace transform of
F (s) =
s^2 (s + 1)^2
(iii) Verify the result in (ii) by using the results in (i) and the convolution theorem.
(iv) Use Laplace transforms to solve the differential equation
f (iv)(t) + 2f ′′′(t) + f ′′(t) = 0,
subject to the initial conditions
f (0) = f ′(0) = f ′′(0) = 0, f ′′′(0) = 1.
Paper 4 [TURN OVER
16E Methods
Write down the Euler-Lagrange equation for extrema of the functional
I =
∫ (^) b
a
F (y, y′) dx.
Show that a first integral of this equation is given by
F − y′^
∂F
∂y′^
= C.
A road is built between two points A and B in the plane z = 0 whose polar coordinates are r = a, θ = 0 and r = a, θ = π/2 respectively. Owing to congestion, the traffic speed at points along the road is kr^2 with k a positive constant. If the equation describing the road is r = r(θ), obtain an integral expression for the total travel time T from A to B.
[Arc length in polar coordinates is given by ds^2 = dr^2 + r^2 dθ^2 .]
Calculate T for the circular road r = a.
Find the equation for the road that minimises T and determine this minimum value.
17B Special Relativity (a) A moving π^0 particle of rest-mass mπ decays into two photons of zero rest-mass,
π^0 → γ + γ.
Show that
sin
θ 2
mπ c^2 2
E 1 E 2
where θ is the angle between the three-momenta of the two photons and E 1 , E 2 are their energies.
(b) The π−^ particle of rest-mass mπ decays into an electron of rest-mass me and a neutrino of zero rest mass, π−^ → e−^ + ν.
Show that v, the speed of the electron in the rest frame of the π−, is
v = c
[
1 − (me/mπ )^2 1 + (me/mπ )^2
]
Paper 4
20C Optimization
Consider the linear programming problem
minimize 2 x 1 − 3 x 2 − 2 x 3 subject to − 2 x 1 + 2 x 2 + 4 x 3 6 5 4 x 1 + 2 x 2 − 5 x 3 6 8 5 x 1 − 4 x 2 + 12 x 3 6 5 , xi > 0 , i = 1, 2 , 3.
(i) After adding slack variables z 1 , z 2 and z 3 and performing one iteration of the simplex algorithm, the following tableau is obtained.
x 1 x 2 x 3 z 1 z 2 z 3
x 2 − 1 1 2 1 / 2 0 0 5 / 2 z 2 6 0 − 9 − 1 1 0 3 z 3 1 0 17 / 2 2 0 1 15
Payoff − 1 0 4 3 / 2 0 0 15 / 2
Complete the solution of the problem.
(ii) Now suppose that the problem is amended so that the objective function becomes
2 x 1 − 3 x 2 − 5 x 3.
Find the solution of this new problem.
(iii) Formulate the dual of the problem in (ii) and identify the optimal solution to the dual.
END OF PAPER
Paper 4
