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MATHEMATICAL TRIPOS Part IB
Wednesday, 6 June, 2012 1:30 pm to 4:30 pm
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.
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 determinant det A of an n × n real matrix A. Suppose that X is a matrix with block form
X =
A B
0 C
where A, B and C are matrices of dimensions n × n, n × m and m × m respectively. Show that det X = (det A)(det C).
2G Groups, Rings and Modules What does it mean to say that the finite group G acts on the set Ω? By considering an action of the symmetry group of a regular tetrahedron on a set of pairs of edges, show there is a surjective homomorphism S 4 → S 3.
[You may assume that the symmetric group Sn is generated by transpositions.]
3E Analysis II Let f : R^2 → R be a function. What does it mean to say that f is differentiable at a point (x, y) ∈ R^2? Prove directly from this definition, that if f is differentiable at (x, y), then f is continuous at (x, y).
Let f : R^2 → R be the function:
f (x, y) =
x^2 + y^2 if x and y are rational
0 otherwise.
For which points (x, y) ∈ R^2 is f differentiable? Justify your answer.
Part IB, Paper 2
7A Fluid Dynamics Starting from Euler’s equation for the motion of an inviscid fluid, derive the vorticity equation in the form
Dω Dt
= ω · ∇u.
Deduce that an initially irrotational flow remains irrotational.
Consider a plane flow that at time t = 0 is described by the streamfunction
ψ = x^2 + y^2.
Calculate the vorticity everywhere at times t > 0.
8H Statistics Let the sample x = (x 1 ,... , xn) have likelihood function f (x; θ). What does it mean to say T (x) is a sufficient statistic for θ?
Show that if a certain factorization criterion is satisfied then T is sufficient for θ. Suppose that T is sufficient for θ and there exist two samples, x and y, for which T (x) 6 = T (y) and f (x; θ)/f (y; θ) does not depend on θ. Let
T 1 (z) =
T (z) z 6 = y T (x) z = y.
Show that T 1 is also sufficient for θ.
Explain why T is not minimally sufficient for θ.
9H Optimization Consider the two-player zero-sum game with payoff matrix
A =
Express the problem of finding the column player’s optimal strategy as a linear program- ming problem in which x 1 + x 2 + x 3 is to be maximized subject to some constraints.
Solve this problem using the simplex algorithm and find the optimal strategy for the column player.
Find also, from the final tableau you obtain, both the value of the game and the row player’s optimal strategy.
Part IB, Paper 2
SECTION II
10F Linear Algebra (i) Define the transpose of a matrix. If V and W are finite-dimensional real vector spaces, define the dual of a linear map T : V → W. How are these two notions related? Now suppose V and W are finite-dimensional inner product spaces. Use the inner product on V to define a linear map V → V ∗^ and show that it is an isomorphism. Define the adjoint of a linear map T : V → W. How are the adjoint of T and its dual related? If A is a matrix representing T , under what conditions is the adjoint of T represented by the transpose of A? (ii) Let V = C[0, 1] be the vector space of continuous real-valued functions on [0, 1], equipped with the inner product
〈f, g〉 =
0
f (t)g(t) dt.
Let T : V → V be the linear map
T f (t) =
∫ (^) t
0
f (s) ds.
What is the adjoint of T?
11G Groups, Rings and Modules State Gauss’s Lemma. State Eisenstein’s irreducibility criterion.
(i) By considering a suitable substitution, show that the polynomial 1 + X^3 + X^6 is irreducible over Q.
(ii) By working in Z 2 [X], show that the polynomial 1 − X^2 + X^5 is irreducible over Q.
Part IB, Paper 2 [TURN OVER
15B Variational Principles (i) A two-dimensional oscillator has action
S =
∫ (^) t 1
t 0
x˙^2 +
y˙^2 −
ω^2 x^2 −
ω^2 y^2
dt.
Find the equations of motion as the Euler-Lagrange equations associated to S, and use them to show that J = ˙xy − yx˙ is conserved. Write down the general solution of the equations of motion in terms of sin ωt and cos ωt, and evaluate J in terms of the coefficients which arise in the general solution. (ii) Another kind of oscillator has action
S^ ˜ =
∫ (^) t 1
t 0
x˙^2 +
y˙^2 −
αx^4 −
βx^2 y^2 −
αy^4
dt ,
where α and β are real constants. Find the equations of motion and use these to show that in general J = ˙xy − yx˙ is not conserved. Find the special value of the ratio β/α for which J is conserved. Explain what is special about the action S˜ in this case, and state the interpretation of J.
16C Methods Consider the linear differential operator L defined by
Ly := − d^2 y dx^2
on the interval 0 6 x < ∞. Given the boundary conditions y(0) = 0 and limx→∞ y(x) = 0, find the Green’s function G(x, ξ) for L with these boundary conditions. Hence, or otherwise, obtain the solution of
Ly =
1 , 0 6 x 6 μ 0 , μ < x < ∞
subject to the above boundary conditions, where μ is a positive constant. Show that your piecewise solution is continuous at x = μ and has the value
y(μ) =
(1 + e−^2 μ^ − 2 e−μ).
Part IB, Paper 2 [TURN OVER
17C Quantum Mechanics Consider a quantum mechanical particle in a one-dimensional potential V (x), for which V (x) = V (−x). Prove that when the energy eigenvalue E is non-degenerate, the energy eigenfunction χ(x) has definite parity.
Now assume the particle is in the double potential well
V (x) =
U , 0 6 |x| 6 l 1 0 , l 1 < |x| 6 l 2 ∞ , l 2 < |x| ,
where 0 < l 1 < l 2 and 0 < E < U (U being large and positive). Obtain general expressions for the even parity energy eigenfunctions χ+(x) in terms of trigonometric and hyperbolic functions. Show that
− tan[k(l 2 − l 1 )] = k κ
coth(κl 1 ) ,
where k^2 =
2 mE ℏ^2 and κ^2 =
2 m(U − E) ℏ^2
18B Electromagnetism A straight wire has n mobile, charged particles per unit length, each of charge q. Assuming the charges all move with velocity v along the wire, show that the current is I = nqv.
Using the Lorentz force law, show that if such a current-carrying wire is placed in a uniform magnetic field of strength B perpendicular to the wire, then the force on the wire, per unit length, is BI.
Consider two infinite parallel wires, with separation L, carrying (in the same sense of direction) positive currents I 1 and I 2 , respectively. Find the force per unit length on each wire, determining both its magnitude and direction.
Part IB, Paper 2
20H Markov Chains Let (Xn)n> 0 be the symmetric random walk on vertices of a connected graph. At each step this walk jumps from the current vertex to a neighbouring vertex, choosing uniformly amongst them. Let Ti = inf{n > 1 : Xn = i}. For each i 6 = j let qij = P (Tj < Ti | X 0 = i) and mij = E(Tj | X 0 = i). Stating any theorems that you use:
(i) Prove that the invariant distribution π satisfies detailed balance.
(ii) Use reversibility to explain why πiqij = πj qji for all i, j.
Consider a symmetric random walk on the graph shown below.
(iii) Find m 33.
(iv) The removal of any edge (i, j) leaves two disjoint components, one which includes i and one which includes j. Prove that mij = 1 + 2eij (i), where eij (i) is the number of edges in the component that contains i.
(v) Show that mij + mji ∈ { 18 , 36 , 54 , 72 } for all i 6 = j.
END OF PAPER
Part IB, Paper 2
