Download Functional Analysis - Mathematics - Exam and more Exams Mathematics in PDF only on Docsity!
MATHEMATICAL TRIPOS Part II Alternative A
Monday 3 June 2002 1.30 to 4.
PAPER 1
Before you begin read these instructions carefully.
Each question is divided into Part (i) and Part (ii), which may or may not be related. Candidates may attempt either or both Parts of any question, but must not attempt Parts from more than SIX questions. If you submit answers to Parts of more than six questions, your lowest scoring attempt(s) will be rejected.
The number of marks for each question is the same, with Part (ii) of each question carrying twice as many marks as Part (i). Additional credit will be given for a substantially complete answer to either Part.
Begin each answer on a separate sheet.
Write legibly and on only one side of the paper.
At the end of the examination:
Tie your answers in separate bundles, marked C,D,E,... , M according to the letter affixed to each question. (For example, 2G, 19G should be in one bundle and 7J, 9J in another bundle.)
Attach a completed cover sheet to each bundle.
Complete a master cover sheet listing all Parts of all questions attempted.
It is essential that every cover sheet bear the candidate’s examination number and desk number.
1M Markov Chains
(i) We are given a finite set of airports. Assume that between any two airports, i and j, there are aij = aji flights in each direction on every day. A confused traveller takes one flight per day, choosing at random from all available flights. Starting from i, how many days on average will pass until the traveller returns again to i? Be careful to allow for the case where there may be no flights at all between two given airports.
(ii) Consider the infinite tree T with root R, where, for all m > 0, all vertices at distance 2m^ from R have degree 3, and where all other vertices (except R) have degree 2. Show that the random walk on T is recurrent.
2G Principles of Dynamics
(i) Derive Hamilton’s equations from Lagrange’s equations. Show that the Hamilto- nian H is constant if the Lagrangian L does not depend explicitly on time.
(ii) A particle of mass m is constrained to move under gravity, which acts in the negative z-direction, on the spheroidal surface −^2 (x^2 + y^2 ) + z^2 = l^2 , with 0 < 6 1. If θ, φ parametrize the surface so that
x = l sin θ cos φ, y = l sin θ sin φ, z = l cos θ,
find the Hamiltonian H(θ, φ, pθ , pφ).
Show that the energy
E =
p^2 θ 2 ml^2 (^2 cos^2 θ + sin^2 θ)
α sin^2 θ
is a constant of the motion, where α is a non-negative constant.
Rewrite this equation as 1 2
θ˙^2 + Veff (θ) = 0
and sketch Veff (θ) for = 1 and α > 0, identifying the maximal and minimal values of θ(t) for fixed α and E. If is now taken not to be unity, how do these values depend on ?
Paper 1
5D Electromagnetism
(i) Show that, in a region where there is no magnetic field and the charge density vanishes, the electric field can be expressed either as minus the gradient of a scalar potential φ or as the curl of a vector potential A. Verify that the electric field derived from
A =
4 π 0
p ∧ r r^3
is that of an electrostatic dipole with dipole moment p.
[You may assume the following identities:
∇(a · b) = a ∧ (∇ ∧ b) + b ∧ (∇ ∧ a) + (a · ∇)b + (b · ∇)a,
∇ ∧ (a ∧ b) = (b · ∇)a − (a · ∇)b + a∇ · b − b∇ · a.]
(ii) An infinite conducting cylinder of radius a is held at zero potential in the presence of a line charge parallel to the axis of the cylinder at distance s 0 > a, with charge density q per unit length. Show that the electric field outside the cylinder is equivalent to that produced by replacing the cylinder with suitably chosen image charges.
6F Dynamics of Differential Equations
(i) A system in R^2 obeys the equations:
x˙ = x − x^5 − 2 xy^4 − 2 y^3 (a − x^2 ) , y ˙ = y − x^4 y − 2 y^5 + x^3 (a − x^2 ) ,
where a is a positive constant.
By considering the quantity V = αx^4 + βy^4 , where α and β are appropriately chosen, show that if a > 1 then there is a unique fixed point and a unique limit cycle. How many fixed points are there when a < 1?
(ii) Consider the second order system
x¨ − (a − bx^2 ) ˙x + x − x^3 = 0 ,
where a, b are constants.
(a) Find the fixed points and determine their stability.
(b) Show that if the fixed point at the origin is unstable and 3a > b then there are no limit cycles.
[You may find it helpful to use the Li´enard coordinate z = ˙x − ax + 13 bx^3 .]
Paper 1
7J Logic, Computation and Set Theory
(i) State the Knaster-Tarski fixed point theorem. Use it to prove the Cantor-Bernstein Theorem; that is, if there exist injections A → B and B → A for two sets A and B then there exists a bijection A → B.
(ii) Let A be an arbitrary set and suppose given a subset R of P A × A. We define a subset B ⊆ A to be R-closed just if whenever (S, a) ∈ R and S ⊆ B then a ∈ B. Show that the set of all R-closed subsets of A is a complete poset in the inclusion ordering. Now assume that A is itself equipped with a partial ordering 6. (a) Suppose R satisfies the condition that if b > a ∈ A then ({b}, a) ∈ R. Show that if B is R-closed then c 6 b ∈ B implies c ∈ B. (b) Suppose that R satisfies the following condition. Whenever (S, a) ∈ R and b 6 a then there exists T ⊆ A such that (T, b) ∈ R, and for every t ∈ T we have (i) ({b}, t) ∈ R, and (ii) t 6 s for some s ∈ S. Let B and C be R-closed subsets of A. Show that the set [B → C] = {a ∈ A | ∀b 6 a (b ∈ B ⇒ b ∈ C)}
is R-closed.
8H Graph Theory
(i) State and prove a necessary and sufficient condition for a graph to be Eulerian (that is, to have an Eulerian circuit).
Prove that, given any connected non-Eulerian graph G, there is an Eulerian graph H and a vertex v ∈ H such that G = H − v.
(ii) Let G be a connected plane graph with n vertices, e edges and f faces. Prove that n − e + f = 2. Deduce that e ≤ g(n − 2)/(g − 2), where g is the smallest face size.
The crossing number c(G) of a non-planar graph G is the minimum number of edge- crossings needed when drawing the graph in the plane. (The crossing of three edges at the same point is not allowed.) Show that if G has n vertices and e edges then c(G) ≥ e− 3 n+6. Find c(K 6 ).
9J Number Theory
(i) Let p be a prime number. Prove that the multiplicative group of the field with p elements is cyclic.
(ii) Let p be an odd prime, and let k > 1 be an integer. Prove that we have x^2 ≡ 1 mod pk^ if and only if either x ≡ 1 mod pk^ or x ≡ −1 mod pk. Is this statement true when p = 2?
Let m be an odd positive integer, and let r be the number of distinct prime factors of m. Prove that there are precisely 2r^ different integers x satisfying x^2 ≡ 1 mod m and 0 < x < m.
Paper 1 [TURN OVER
11L Stochastic Financial Models
(i) The prices, Si, of a stock in a binomial model at times i = 0, 1 , 2 are represented by the following binomial tree.
The fixed interest rate per period is 1/5 and the probability that the stock price increases in a period is 1/3. Find the price at time 0 of a European call option with strike price 78 and expiry time 2.
Explain briefly the ideas underlying your calculations.
(ii) Consider an investor in a one-period model who may invest in s assets, all of which are risky, with a random return vector R having mean ER = r and positive- definite covariance matrix V ; assume that not all the assets have the same expected return. Show that any minimum-variance portfolio is equivalent to the investor dividing his wealth between two portfolios, the global minimum-variance portfolio and the diversified portfolio, both of which should be specified clearly in terms of r and V.
Now suppose that R = (R 1 , R 2 ,... , Rs)
where R 1 , R 2 ,... , Rs are independent random variables with Ri having the exponential distribution with probability density function λie−λix, x > 0, where λi > 0, 1 6 i 6 s. Determine the global minimum-variance portfolio and the diversified portfolio explicitly.
Consider further the situation when the investor has the utility function u(x) = 1 − e−x, where x denotes his wealth. Suppose that he acts to maximize the expected utility of his final wealth, and that his initial wealth is w > 0. Show that he now divides his wealth between the diversified portfolio and the uniform portfolio, in which wealth is apportioned equally between the assets, and determine the amounts that he invests in each.
Paper 1 [TURN OVER
12M Principles of Statistics
(i) Explain in detail the minimax and Bayes principles of decision theory.
Show that if d(X) is a Bayes decision rule for a prior density π(θ) and has constant risk function, then d(X) is minimax.
(ii) Let X 1 ,... , Xp be independent random variables, with Xi ∼ N (μi, 1), i = 1,... , p.
Consider estimating μ = (μ 1 ,... , μp)T^ by d = (d 1 ,... , dp)T^ , with loss function
L(μ, d) =
∑^ p
i=
(μi − di)^2.
What is the risk function of X = (X 1 ,... , Xp)T^?
Consider the class of estimators of μ of the form
da(X) =
a XT^ X
X ,
indexed by a > 0. Find the risk function of da(X) in terms of E
1 /XT^ X
, which you should not attempt to evaluate, and deduce that X is inadmissible. What is the optimal value of a?
[You may assume Stein’s Lemma, that for suitably behaved real-valued functions h,
E {(Xi − μi)h(X)} = E
∂h(X) ∂Xi
. ]
Paper 1
14E Quantum Physics
(i) A system of N identical non-interacting bosons has energy levels Ei with degen- eracy gi, 1 ≤ i < ∞, for each particle. Show that in thermal equilibrium the number of particles Ni with energy Ei is given by
Ni =
gi eβ(Ei−μ)^ − 1
where β and μ are parameters whose physical significance should be briefly explained.
(ii) A photon moves in a cubical box of side L. Assuming periodic boundary conditions, show that, for large L, the number of photon states lying in the frequency range ω → ω+dω is ρ(ω)dω where
ρ(ω) = L^3
ω^2 π^2 c^3
If the box is filled with thermal radiation at temperature T , show that the number of photons per unit volume in the frequency range ω → ω + dω is n(ω)dω where
n(ω) =
ω^2 π^2 c^3
eℏω/kT^ − 1
Calculate the energy density W of the thermal radiation. Show that the pressure P exerted on the surface of the box satisfies
P =
W.
[You may use the result
0
x^3 dx ex− 1 =^
π^4 15 .]
Paper 1
15D General Relativity
(i) Given a covariant vector field Va, define the curvature tensor Rabcd by
Va;bc − Va;cb = VeReabc. (∗)
Express Reabc in terms of the Christoffel symbols and their derivatives. Show that
Reabc = −Reacb.
Further, by setting Va = ∂φ/∂xa, deduce that
Reabc + Recab + Rebca = 0.
(ii) Write down an expression similar to (∗) given in Part (i) for the quantity
gab;cd − gab;dc
and hence show that Reabc = −Raebc.
Define the Ricci tensor, show that it is symmetric and write down the contracted Bianchi identities.
In certain spacetimes of dimension n ≥ 2, Rabcd takes the form
Rabcd = K(xe)[gacgbd − gadgbc].
Obtain the Ricci tensor and Ricci scalar. Deduce that K is a constant in such spacetimes if the dimension n is greater than 2.
Paper 1 [TURN OVER
17E Symmetries and Groups in Physics
(i) Let H be a normal subgroup of the group G. Let G/H denote the group of cosets ˜g = gH for g ∈ G. If D : G → GL(Cn) is a representation of G with D(h 1 ) = D(h 2 ) for all h 1 , h 2 ∈ H show that D˜(˜g) = D(g) is well-defined and that it is a representation of G/H. Show further that D˜(˜g) is irreducible if and only if D(g) is irreducible.
(ii) For a matrix U ∈ SU (2) define the linear map ΦU : R^3 → R^3 by ΦU (x).σ =
U x.σU †^ with σ = (σ 1 , σ 2 , σ 3 )T^ as the vector of the Pauli spin matrices
σ 1 =
, σ 2 =
0 −i i 0
, σ 3 =
Show that ‖ΦU (x)‖ = ‖x‖. Because of the linearity of ΦU there exists a matrix R(U ) such that ΦU (x) = R(U )x. Given that any SU (2) matrix can be written as
U = cos α I − i sin α n.σ ,
where α ∈ [0, π] and n is a unit vector, deduce that R(U ) ∈ SO(3) for all U ∈ SU (2). Compute R(U )n and R(U )x in the case that x.n = 0 and deduce that R(U ) is the matrix of a rotation about n with angle 2α.
[Hint: m.σ n.σ = m.n I + i(m × n).σ .]
Show that R(U ) defines a surjective homomorphism Θ : SU (2) → SO(3) and find the kernel of Θ.
Paper 1 [TURN OVER
18C Transport Processes
(i) Material of thermal diffusivity D occupies the semi-infinite region x > 0 and is initially at uniform temperature T 0. For time t > 0 the temperature at x = 0 is held at a constant value T 1 > T 0. Given that the temperature T (x, t) in x > 0 satisfies the diffusion equation Tt = DTxx, write down the equation and the boundary and initial conditions satisfied by the dimensionless temperature θ = (T − T 0 ) / (T 1 − T 0 ).
Use dimensional analysis to show that the lengthscale of the region in which T is significantly different from T 0 is proportional to (Dt)^1 /^2. Hence show that this problem has a similarity solution
θ = erfc (ξ/2) ≡
π
ξ/ 2
e−u
2 du ,
where ξ = x/(Dt)^1 /^2.
What is the rate of heat input, −DTx, across the plane x = 0?
(ii) Consider the same problem as in Part (i) except that the boundary condition at x = 0 is replaced by one of constant rate of heat input Q. Show that θ(ξ, t) satisfies the partial differential equation
θξξ +
ξ 2
θξ = tθt
and write down the boundary conditions on θ(ξ, t). Deduce that the problem has a similarity solution of the form
θ =
Q(t/D)^1 /^2 T 1 − T 0
f (ξ).
Derive the ordinary differential equation and boundary conditions satisfied by f (ξ). Differentiate this equation once to obtain
f ′′′^ +
ξ 2
f ′′^ = 0
and solve for f ′(ξ). Hence show that
f (ξ) =
π
e−ξ
(^2) / 4 − ξ erfc (ξ/2).
Sketch the temperature distribution T (x, t) for various times t, and calculate T (0, t) explicitly.
Paper 1
20F Numerical Analysis
(i) Let A be an n×n symmetric real matrix with distinct eigenvalues λ 1 , λ 2 ,... , λn and corresponding eigenvectors v 1 , v 2 , ...., vn, where ‖vl‖ = 1. Given x(0)^ ∈ Rn, ‖x(0)‖ = 1, the sequence x(k)^ is generated in the following manner. We set
μ = x(k)^ T^ Ax(k),
y = (A − μI)−^1 x(k),
x(k+1)^ =
y ‖y‖
Show that if
x(k)^ = c−^1
v 1 + α
∑^ n
l=
dlvl
where α is a real scalar and c is chosen so that ‖x(k)‖ = 1, then
μ = c−^2
λ 1 + α^2
∑^ n
j=
λj d^2 j
Give an explicit expression for c.
(ii) Use the above result to prove that, if |α| is small,
x(k+1)^ = ˜c−^1
v 1 + α^3
∑^ n
l=
d^ ˜lvl
and obtain the numbers ˜c and d˜ 2 ,... , d˜n.
Paper 1
