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MATHEMATICAL TRIPOS Part IB
Friday, 8 June, 2012 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 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 Let V be a complex vector space with basis {e 1 ,... , en}. Define T : V → V by T (ei) = ei − ei+1 for i < n and T (en) = en − e 1. Show that T is diagonalizable and find its eigenvalues. [You may use any theorems you wish, as long as you state them clearly.]
2G Groups, Rings and Modules An idempotent element of a ring R is an element e satisfying e^2 = e. A nilpotent element is an element e satisfying eN^ = 0 for some N > 0.
Let r ∈ R be non-zero. In the ring R[X], can the polynomial 1 + rX be (i) an idempotent, (ii) a nilpotent? Can 1 + rX satisfy the equation (1 + rX)^3 = (1 + rX)? Justify your answers.
3E Analysis II Let f : Rn^ × Rm^ → R be a bilinear function. Show that f is differentiable at any point in Rn^ × Rm^ and find its derivative.
4E Complex Analysis Let h : C → C be a holomorphic function with h(i) 6 = h(−i). Does there exist a
holomorphic function f defined in |z| < 1 for which f ′(z) =
h(z) 1 + z^2 ? Does there exist a
holomorphic function f defined in |z| > 1 for which f ′(z) =
h(z) 1 + z^2 ? Justify your answers.
5D Methods Show that the general solution of the wave equation
1 c^2
∂^2 y ∂t^2
∂^2 y ∂x^2
can be written in the form
y(x, t) = f (x − ct) + g(x + ct).
Hence derive the solution y(x, t) subject to the initial conditions
y(x, 0) = 0, ∂y ∂t
(x, 0) = ψ(x).
Part IB, Paper 4
9H Markov Chains Let (Xn)n> 0 be an irreducible Markov chain with p( ijn ) = P (Xn = j | X 0 = i). Define the meaning of the statements:
(i) state i is transient,
(ii) state i is aperiodic.
Give a criterion for transience that can be expressed in terms of the probabilities (p( iin ), n = 0, 1 ,... ).
Prove that if a state i is transient then all states are transient. Prove that if a state i is aperiodic then all states are aperiodic. Suppose that p( iin )= 0 unless n is divisible by 3. Given any other state j, prove that
p( jjn )= 0 unless n is divisible by 3.
Part IB, Paper 4
SECTION II
10F Linear Algebra Let V be a finite-dimensional real vector space of dimension n. A bilinear form B : V × V → R is nondegenerate if for all v 6 = 0 in V , there is some w ∈ V with B(v, w) 6 = 0. For v ∈ V , define 〈v〉⊥^ = {w ∈ V | B(v, w) = 0}. Assuming B is nondegenerate, show that V = 〈v〉 ⊕ 〈v〉⊥^ whenever B(v, v) 6 = 0. Suppose that B is a nondegenerate, symmetric bilinear form on V. Prove that there is a basis {v 1 ,... , vn} of V with B(vi, vj ) = 0 for i 6 = j. [If you use the fact that symmetric matrices are diagonalizable, you must prove it.] Define the signature of a quadratic form. Explain how to determine the signature of the quadratic form associated to B from the basis you constructed above. A linear subspace V ′^ ⊂ V is said to be isotropic if B(v, w) = 0 for all v, w ∈ V ′. Show that if B is nondegenerate, the maximal dimension of an isotropic subspace of V is (n − |σ|)/2, where σ is the signature of the quadratic form associated to B.
11G Groups, Rings and Modules Let R be a commutative ring with unit 1. Prove that an R-module is finitely generated if and only if it is a quotient of a free module Rn, for some n > 0. Let M be a finitely generated R-module. Suppose now I is an ideal of R, and φ is an R-homomorphism from M to M with the property that
φ(M ) ⊂ I · M = {m ∈ M | m = rm′^ with r ∈ I , m′^ ∈ M }.
Prove that φ satisfies an equation
φn^ + an− 1 φn−^1 + · · · + a 1 φ + a 0 = 0
where each aj ∈ I. [You may assume that if T is a matrix over R, then adj(T ) T = det T (id), with id the identity matrix.] Deduce that if M satisfies I · M = M , then there is some a ∈ R satisfying
a − 1 ∈ I and aM = 0.
Give an example of a finitely generated Z-module M and a proper ideal I of Z satisfying the hypothesis I · M = M , and for your example, give an explicit such element a.
Part IB, Paper 4 [TURN OVER
15G Geometry Let Σ ⊂ R^3 be a smooth closed surface. Define the principal curvatures κmax and κmin at a point p ∈ Σ. Prove that the Gauss curvature at p is the product of the two principal curvatures. A point p ∈ Σ is called a parabolic point if at least one of the two principal curvatures vanishes. Suppose Π ⊂ R^3 is a plane and Σ is tangent to Π along a smooth closed curve C = Π ∩ Σ ⊂ Σ. Show that C is composed of parabolic points. Can both principal curvatures vanish at a point of C? Briefly justify your answer.
16B Variational Principles Consider a functional I =
∫ (^) b
a
F (x, y, y′)dx
where F is smooth in all its arguments, y(x) is a C^1 function and y′^ = (^) dxdy. Consider the function y(x) + h(x) where h(x) is a small C^1 function which vanishes at a and b. Obtain formulae for the first and second variations of I about the function y(x). Derive the Euler-Lagrange equation from the first variation, and state its variational interpretation. Suppose now that I =
0
(y′^2 − 1)^2 dx
where y(0) = 0 and y(1) = β. Find the Euler-Lagrange equation and the formula for the second variation of I. Show that the function y(x) = βx makes I stationary, and that it is a (local) minimizer if β > √^13. Show that when β = 0, the function y(x) = 0 is not a minimizer of I.
Part IB, Paper 4 [TURN OVER
17D Methods Let D ⊂ R^2 be a two-dimensional domain with boundary S = ∂D, and let
G 2 = G 2 (r, r 0 ) =
2 π
log |r − r 0 | ,
where r 0 is a point in the interior of D. From Green’s second identity, ∫
S
φ ∂ψ ∂n
− ψ ∂φ ∂n
dℓ =
D
(φ∇^2 ψ − ψ∇^2 φ) da ,
derive Green’s third identity
u(r 0 ) =
D
G 2 ∇^2 u da +
S
u
∂G 2
∂n
− G 2
∂u ∂n
dℓ.
[Here (^) ∂n∂ denotes the normal derivative on S.]
Consider the Dirichlet problem on the unit disc D 1 = {r ∈ R^2 : |r| 6 1 }:
∇^2 u = 0, r ∈ D 1 , u(r) = f (r), r ∈ S 1 = ∂D 1.
Show that, with an appropriate function G(r, r 0 ), the solution can be obtained by the formula
u(r 0 ) =
S 1
f (r)
∂n G(r , r 0 ) dℓ.
State the boundary conditions on G and explain how G is related to G 2.
For r, r 0 ∈ R^2 , prove the identity ∣∣ ∣∣^ r |r|
− r 0 |r|
∣∣^ r^0 |r 0 |
− r|r 0 |
and deduce that if the point r lies on the unit circle, then
|r − r 0 | = |r 0 ||r − r∗ 0 | , where r∗ 0 =
r 0 |r 0 |^2
Hence, using the method of images, or otherwise, find an expression for the function G(r , r 0 ). [An expression for (^) ∂n∂ G is not required.]
Part IB, Paper 4
19H Statistics From each of 3 populations, n data points are sampled and these are believed to obey yij = αi + βi(xij − x¯i) + ǫij , j ∈ { 1 ,... , n}, i ∈ { 1 , 2 , 3 },
where ¯xi = (1/n)
j xij^ , the^ ǫij^ are independent and identically distributed as^ N^ (0, σ
and σ^2 is unknown. Let ¯yi = (1/n)
j yij^. (i) Find expressions for ˆαi and βˆi, the least squares estimates of αi and βi. (ii) What are the distributions of ˆαi and βˆi? (iii) Show that the residual sum of squares, R 1 , is given by
R 1 =
∑^3
i=
∑^ n
j=
(yij − y¯i)^2 − βˆ i^2
∑n
j=
(xij − x¯i)^2
Calculate R 1 when n = 9, {αˆi}^3 i=1 = { 1. 6 , 0. 6 , 0. 8 }, { βˆi}^3 i=1 = { 2 , 1 , 1 },
∑^9
j=
(yij − y¯i)^2
3
i=
∑^9
j=
(xij − x¯i)^2
3
i=
(iv) H 0 is the hypothesis that α 1 = α 2 = α 3. Find an expression for the maximum likelihood estimator of α 1 under the assumption that H 0 is true. Calculate its value for the above data.
(v) Explain (stating without proof any relevant theory) the rationale for a statistic which can be referred to an F distribution to test H 0 against the alternative that it is not true. What should be the degrees of freedom of this F distribution? What would be the outcome of a size 0.05 test of H 0 with the above data?
Part IB, Paper 4
20H Optimization Describe the Ford-Fulkerson algorithm. State conditions under which the algorithm is guaranteed to terminate in a finite number of steps. Explain why it does so, and show that it finds a maximum flow. [You may assume that the value of a flow never exceeds the value of any cut.] In a football league of n teams the season is partly finished. Team i has already won wi matches. Teams i and j are to meet in mij further matches. Thus the total number of remaining matches is M =
i<j mij^.^ Assume there will be no drawn matches.^ We wish to determine whether it is possible for the outcomes of the remaining matches to occur in such a way that at the end of the season the numbers of wins by the teams are (x 1 ,... , xn). Invent a network flow problem in which the maximum flow from source to sink equals M if and only if (x 1 ,... , xn) is a feasible vector of final wins. Illustrate your idea by answering the question of whether or not x = (7, 5 , 6 , 6) is a possible profile of total end-of-season wins when n = 4, w = (1, 2 , 3 , 4), and M = 14 with
(mij ) =
END OF PAPER
Part IB, Paper 4
