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MATHEMATICAL TRIPOS Part II Alternative A
Friday 4 June 2003 9 to 12
PAPER 4
Before you begin read these instructions carefully.
Candidates must not attempt more than FOUR questions.
The number of marks for each question is the same.
Additional credit will be given for a substantially complete answer.
Write on one side 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 your answers in separate bundles, marked A, B, C,... , J according to the letter affixed to each question. (For example, 3F, 8F should be in one bundle and 11J, 13J in another bundle.)
Attach a completed cover sheet to each bundle.
Complete a master cover sheet listing all questions attempted.
It is essential that every cover sheet bear the candidate number and desk number.
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
1I Markov Chains
(a) Give three definitions of a continuous-time Markov chain with a given Q-matrix on a finite state space: (i) in terms of holding times and jump probabilities, (ii) in terms of transition probabilities over small time intervals, and (iii) in terms of finite-dimensional distributions.
(b) A flea jumps clockwise on the vertices of a triangle; the holding times are independent exponential random variables of rate one. Find the eigenvalues of the corresponding Q-matrix and express transition probabilities pxy (t), t ≥ 0 , x, y = A, B, C, in terms of these roots. Deduce the formulas for the sums
S 0 (t) =
∑^ ∞
n=
t^3 n (3n)!
, S 1 (t) =
∑^ ∞
n=
t^3 n+ (3n + 1)!
, S 2 (t) =
∑^ ∞
n=
t^3 n+ (3n + 2)!
in terms of the functions et, e−t/^2 , cos(
3 t/2) and sin(
3 t/2).
Find the limits lim t→∞
e−tSj (t), j = 0, 1 , 2.
What is the connection between the decompositions et^ = S 0 (t) + S 1 (t) + S 2 (t) and et^ = cosh t + sinh t?
Paper 4
4G Groups, Rings and Fields
(a) Let t be the maximal power of the prime p dividing the order of the finite group G, and let N (pt) denote the number of subgroups of G of order pt. State clearly the numerical restrictions on N (pt) given by the Sylow theorems.
If H and K are subgroups of G of orders r and s respectively, and their intersection H ∩ K has order t, show the set HK = {hk : h ∈ H, k ∈ K} contains rs/t elements.
(b) The finite group G has 48 elements. By computing the possible values of N (16), show that G cannot be simple.
5C Electromagnetism Consider a frame S′^ moving with velocity v relative to the laboratory frame S where |v|^2 c^2. The electric and magnetic fields in S are E and B, while those measured in S′^ are E′^ and B′. Given that B′^ = B, show that
∮
Γ
E′^ · dl =
Γ
(E + v ∧ B) · dl,
for any closed circuit Γ and hence that E′^ = E + v ∧ B.
Now consider a fluid with electrical conductivity σ and moving with velocity v(r). Use Ohm’s law in the moving frame to relate the current density j to the electric field E in the laboratory frame, and show that if j remains finite in the limit σ → ∞ then
∂B ∂t
= ∇ ∧ (v ∧ B).
The magnetic helicity H in a volume V is given by
V A^ ·^ B^ dτ^ where^ A^ is the vector potential. Show that if the normal components of v and B both vanish on the surface bounding V then dH/dt = 0.
Paper 4
6B Nonlinear Dynamical Systems
(a) Consider the map G 1 (x) = f (x+a), defined on 0 6 x < 1, where f (x) = x [mod 1], 0 6 f < 1, and the constant a satisfies 0 6 a < 1. Give, with reasons, the values of a (if any) for which the map has (i) a fixed point, (ii) a cycle of least period n, (iii) an aperiodic orbit. Does the map exhibit sensitive dependence on initial conditions?
Show (graphically if you wish) that if the map has an n-cycle then it has an infinite number of such cycles. Is this still true if G 1 is replaced by f (cx + a), 0 < c < 1?
(b) Consider the map
G 2 (x) = f (x + a +
b 2 π
sin 2πx),
where f (x) and a are defined as in Part (a), and b > 0 is a parameter.
Find the regions of the (a, b) plane for which the map has (i) no fixed points, (ii) exactly two fixed points.
Now consider the possible existence of a 2-cycle of the map G 2 when b 1, and suppose the elements of the cycle are X, Y with X < 12. By expanding X, Y, a in powers of b, so that X = X 0 + bX 1 + b^2 X 2 + O(b^3 ), and similarly for Y and a, show that
a =
b^2 8 π
sin 4πX 0 + O(b^3 ).
Use this result to sketch the region of the (a, b) plane in which 2-cycles exist. How many distinct cycles are there for each value of a in this region?
7G Geometry of Surfaces
Write an essay on the Gauss–Bonnet theorem and its proof.
8F Logic, Computation and Set Theory Write an essay on recursive functions. Your essay should include a sketch of why every computable function is recursive, and an explanation of the existence of a universal recursive function, as well as brief discussions of the Halting Problem and of the relationship between recursive sets and recursively enumerable sets.
[You may assume that every recursive function is computable. You do not need to give proofs that particular functions to do with prime-power decompositions are recursive.]
Paper 4 [TURN OVER
12J Stochastic Financial Models
What is Brownian motion (Bt)t> 0? Briefly explain how Brownian motion can be considered as a limit of simple random walks. State the Reflection Principle for Brownian motion, and use it to derive the distribution of the first passage time τa ≡ inf{t : Bt = a} to some level a > 0.
Suppose that Xt = Bt + ct, where c > 0 is constant. Stating clearly any results to which you appeal, derive the distribution of the first-passage time τ (^) a(c )≡ inf{t : Xt = a} to a > 0.
Now let σa ≡ sup{t : Xt = a}, where a > 0. Find the density of σa.
13J Principles of Statistics
Suppose that θ ∈ Rd^ is the parameter of a non-degenerate exponential family. Derive the asymptotic distribution of the maximum-likelihood estimator ˆθn of θ based on a sample of size n. [You may assume that the density is infinitely differentiable with respect to the parameter, and that differentiation with respect to the parameter commutes with integration.]
Paper 4 [TURN OVER
14I Computational Statistics and Statistical Modelling
Suppose that Y 1 ,... , Yn are independent observations, with Yi having probability density function of the following form
f (yi|θi, φ) = exp
[
yiθi − b(θi) φ
]
where E(Yi) = μi and g(μi) = βT^ xi. You should assume that g( ) is a known function, and β, φ are unknown parameters, with φ > 0, and also x 1 ,... , xn are given linearly independent covariate vectors. Show that
∂` ∂β
∑ (^) (yi − βi) g′(μi)Vi
xi,
where ` is the log-likelihood and Vi = var (Yi) = φb′′(θi).
Discuss carefully the (slightly edited) R output given below, and briefly suggest another possible method of analysis using the function glm ( ).
s <- scan()
1: 33 63 157 38 108 159
7: Read 6 items
r <- scan()
1: 3271 7256 5065 2486 8877 3520 7:
Read 6 items
gender <- scan(,"")
1: b b b g g g
7: Read 6 items
age <- scan(,"")
1: 13&under 14-18 19&over 4: 13&under 14-18 19&over
7:
Read 6 items
gender <- factor(gender) ; age <- factor(age)
summary(glm(s/r ~ gender + age,binomial, weights=r))
Coefficients: Question continues on next page.
Paper 4
15E Foundations of Quantum Mechanics
The states of the hydrogen atom are denoted by |nlm〉 with l < n, −l ≤ m ≤ l and associated energy eigenvalue En, where
En = −
e^2 8 π 0 a 0 n^2
A hydrogen atom is placed in a weak electric field with interaction Hamiltonian
H 1 = −eEz.
a) Derive the necessary perturbation theory to show that to O(E^2 ) the change in the energy associated with the state | 100 〉 is given by
∆E 1 = e^2 E^2
∑^ ∞
n=
n∑− 1
l=
∑^ l
m=−l
〈 100 |z|nlm〉
^2
E 1 − En
The wavefunction of the ground state | 100 〉 is
ψn=1(r) =
(πa^30 )^1 /^2
e−r/a^0.
By replacing En, ∀ n > 1, in the denominator of (∗) by E 2 show that
|∆E 1 | <
32 π 3
0 E^2 a^30.
b) Find a matrix whose eigenvalues are the perturbed energies to O(E) for the states | 200 〉 and | 210 〉. Hence, determine these perturbed energies to O(E) in terms of the matrix elements of z between these states. [Hint: 〈nlm|z|nlm〉 = 0 ∀ n, l, m 〈nlm|z|nl′m′〉 = 0 ∀ n, l, l′, m, m′, m 6 = m′
]
Paper 4
16E Quantum Physics
Explain the operation of the np junction. Your account should include a discussion of the following topics:
(a) the rˆole of doping and the fermi-energy; (b) the rˆole of majority and minority carriers;
(c) the contact potential;
(d) the relationship I(V ) between the current I flowing through the junction and the external voltage V applied across the junction; (e) the property of rectification.
Paper 4 [TURN OVER
18C Statistical Physics and Cosmology
(a) Consider an ideal gas of Fermi particles obeying the Pauli exclusion principle with a set of one-particle energy eigenstates Ei. Given the probability pi(ni) at tempera- ture T that there are ni particles in the eigenstate Ei:
pi(ni) =
e(μ−Ei)ni/kT Zi
determine the appropriate normalization factor Zi. Use this to find the average number ¯ni of Fermi particles in the eigenstate Ei.
Explain briefly why in generalizing these discrete eigenstates to a continuum in momentum space (in the range p to p + dp) we must multiply by the density of states
g(p)dp =
4 πgsV h^3
p^2 dp ,
where gs is the degeneracy of the eigenstates and V is the volume.
(b) With the energy expressed as a momentum integral
E =
0
E(p)¯n(p)dp ,
consider the effect of changing the volume V so slowly that the occupation numbers do not change (i.e. particle number N and entropy S remain fixed). Show that the momentum varies as dp/dV = −p/ 3 V and so deduce from the first law expression ( ∂E ∂V
N,S
= −P
that the pressure is given by
P =
3 V
0
pE′(p)¯n(p)dp.
Show that in the non-relativistic limit P = 23 U/V where U is the internal energy, while for ultrarelativistic particles P = 13 E/V.
(c) Now consider a Fermi gas in the limit T → 0 with all momentum eigenstates filled up to the Fermi momentum pF. Explain why the number density can be written as
n =
4 πgs h^3
∫ (^) pF
0
p^2 dp ∝ p^3 F.
From similar expressions for the energy, deduce in both the non-relativistic and ultra- relativistic limits that the pressure may be written as
P ∝ nγ^ ,
where γ should be specified in each case.
(d) Examine the stability of an object of radius R consisting of such a Fermi degenerate gas by comparing the gravitational binding energy with the total kinetic energy. Briefly point out the relevance of these results to white dwarfs and neutron stars.
Paper 4 [TURN OVER
19A Transport Processes
(a) Solute diffuses and is advected in a moving fluid. Derive the transport equation and deduce that the solute concentration C(x, t) satisfies the advection–diffusion equation
Ct + ∇ · (uC) = ∇ · (D∇C),
where u is the velocity field and D the diffusivity. Write down the form this equation takes when ∇ · u = 0, both u and ∇C are unidirectional, in the x-direction, and D is a constant.
(b) A solution occupies the region x > 0, bounded by a semi-permeable membrane at x = 0 across which fluid passes (by osmosis) with velocity
u = −k (C 1 − C(0, t)) ,
where k is a positive constant, C 1 is a fixed uniform solute concentration in the region x < 0, and C(x, t) is the solute concentration in the fluid. The membrane does not allow solute to pass across x = 0, and the concentration at x = L is a fixed value CL (where C 1 > CL > 0).
Write down the differential equation and boundary conditions to be satisfied by C in a steady state. Make the equations non-dimensional by using the substitutions
X =
xkC 1 D
, θ(X) =
C(x) C 1
, θL =
CL
C 1
and show that the concentration distribution is given by
θ(X) = θL exp [(1 − θ 0 )(Λ − X)] ,
where Λ and θ 0 should be defined, and θ 0 is given by the transcendental equation
θ 0 = θLeΛ−Λθ^0. (∗)
What is the dimensional fluid velocity u, in terms of θ 0?
(c) Show that if, instead of taking a finite value of L, you had tried to take L infinite, then you would have been unable to solve for θ unless θL = 0, but in that case there would be no way of determining θ 0.
(d) Find asymptotic expansions for θ 0 from equation (∗) in the following limits:
(i) For θL → 0, Λ fixed, expand θ 0 as a power series in θL, and equate coefficients to show that θ 0 ∼ eΛθL − Λe2Λθ^2 L + O
θ^3 L
(ii) For Λ → ∞, θL fixed, take logarithms, expand θ 0 as a power series in 1/Λ, and show that θ 0 ∼ 1 +
log θL Λ
+ O
Λ^2
What is the limiting value of θ 0 in the limits (i) and (ii)? Question continues on next page.
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21A Mathematical Methods
State Watson’s lemma, describing the asymptotic behaviour of the integral
I(λ) =
∫ A
0
e−λtf (t) dt, A > 0 ,
as λ → ∞, given that f (t) has the asymptotic expansion
f (t) ∼
∑^ ∞
n=
antnβ
as t → (^0) +, where β > 0.
Consider the integral
J(λ) =
∫ (^) b
a
eλφ(t)F (t)dt,
where λ 1 and φ(t) has a unique maximum in the interval [a, b] at c, with a < c < b, such that φ′(c) = 0, φ′′(c) < 0.
By using the change of variable from t to ζ, defined by
φ(t) − φ(c) = −ζ^2 ,
deduce an asymptotic expansion for J(λ) as λ → ∞. Show that the leading-order term gives
J(λ) ∼ eλφ(c)F (c)
( (^2) π λ|φ′′(c)|
The gamma function Γ(x) is defined for x > 0 by
Γ(x) =
0
e(x−1) log^ t−t^ dt.
By means of the substitution t = (x − 1)s, or otherwise, deduce that
Γ(x + 1) ∼ x(x+^
(^12) ) e−x
2 π
12 x
as x → ∞.
22D Numerical Analysis Write an essay on the method of conjugate gradients. You should define the method, list its main properties and sketch the relevant proof. You should also prove that (in exact arithmetic) the method terminates in a finite number of steps, briefly mention the connection with Krylov subspaces, and describe the approach of preconditioned conjugate gradients.
Paper 4
23B Nonlinear Waves
Let ψ(k; x, t) satisfy the linear integral equation
ψ(k; x, t) + iei(kx+k
(^3) t)
L
ψ(l; x, t) l + k
dλ(l) = ei(kx+k
(^3) t) ,
where the measure dλ(k) and the contour L are such that ψ(k; x, t) exists and is unique. Let q(x, t) be defined in terms of ψ(k; x, t) by
q(x, t) = −
∂x
L
ψ(k; x, t)dλ(k).
(a) Show that
(M ψ) + iei(kx+k
(^3) t)
L
(M ψ)(l; x, t) l + k
dλ(l) = 0,
where
M ψ ≡
∂^2 ψ ∂x^2
− ik
∂ψ ∂x
(b) Show that
(N ψ) + iei(kx+k
(^3) t)
L
(N ψ)(l; x, t) l + k
dλ(l) = 3kei(kx+k
(^3) t)
L
(M ψ)(l; x, t) l + k
dλ(l),
where
N ψ ≡
∂ψ ∂t
∂^3 ψ ∂x^3
∂ψ ∂x
(c) By recalling that the KdV equation
∂q ∂t
∂^3 q ∂x^3
∂q ∂x
admits the Lax pair M ψ = 0, N ψ = 0,
write down an expression for dλ(l) which gives rise to the one-soliton solution of the KdV equation. Write down an expression for ψ(k; x, t) and for q(x, t).
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