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MATHEMATICAL TRIPOS Part II
Thursday, 7 June, 2012 1:30 pm to 4:30 pm
PAPER 3
Before you begin read these instructions carefully.
The examination paper is divided into two sections. Each question in Section II carries twice the number of marks of each question in Section I. Candidates may attempt at most six questions from Section I and any number of 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 bundles, marked A, B, C,.. ., K according to the code letter affixed to each question. Include in the same bundle all questions from Sections I and II with the same code 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
Gold cover sheet 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
1I Number Theory Define the discriminant of the binary quadratic form f (x, y) = ax^2 + bxy + cy^2. Assuming that this form is positive definite, define what it means for f to be reduced. Show that there are precisely two reduced positive definite binary quadratic forms of discriminant −35.
2F Topics in Analysis State and prove Liouville’s theorem concerning approximation of algebraic numbers by rationals.
3G Geometry and Groups Let A be a M¨obius transformation acting on the Riemann sphere. Show that, if A is not loxodromic, then there is a disc ∆ in the Riemann sphere with A(∆) = ∆. Describe all such discs for each M¨obius transformation A.
Hence, or otherwise, show that the group G of M¨obius transformations generated by A : z 7 → iz and B : z 7 → 2 z
does not map any disc onto itself.
Describe the set of points of the Riemann sphere at which G acts discontinuously. What is the quotient of this set by the action of G?
4G Coding and Cryptography Describe the RSA system with public key (N, e) and private key d. Give a simple example of how the system is vulnerable to a homomorphism attack. Explain how a signature system prevents such an attack.
Part II, Paper 3
7D Dynamical Systems State without proof Lyapunov’s first theorem, carefully defining all the terms that you use.
Consider the dynamical system
x˙ = − 2 x + y − xy + 3y^2 − xy^2 + x^3 , y ˙ = − 2 y − x − y^2 − 3 xy + 2x^2 y.
By choosing a Lyapunov function V (x, y) = x^2 +y^2 , prove that the origin is asymptotically stable.
By factorising the expression for V˙ , or otherwise, show that the basin of attraction of the origin includes the set V < 7 /4.
8E Further Complex Methods The Beta function, denoted by B(z 1 , z 2 ), is defined by
B(z 1 , z 2 ) = Γ(z 1 )Γ(z 2 ) Γ(z 1 + z 2 )
, z 1 , z 2 ∈ C ,
where Γ(z) denotes the Gamma function. It can be shown that
B(z 1 , z 2 ) =
0
vz^2 −^1 dv (1 + v)z^1 +z^2 , Re z 1 > 0 , Re z 2 > 0.
By computing this integral for the particular case of z 1 +z 2 = 1, and by employing analytic continuation, deduce that Γ(z) satisfies the functional equation
Γ(z)Γ(1 − z) =
π sin πz , z ∈ C.
Part II, Paper 3
9A Classical Dynamics The motion of a particle of charge q and mass m in an electromagnetic field with scalar potential φ(r, t) and vector potential A(r, t) is characterized by the Lagrangian
L =
mr˙^2 2
− q(φ − r˙ · A).
(a) Show that the Euler–Lagrange equation is invariant under the gauge transformation
φ → φ −
∂t
, A → A + ∇Λ,
for an arbitrary function Λ(r, t).
(b) Derive the equations of motion in terms of the electric and magnetic fields E(r, t) and B(r, t). [Recall that B = ∇ × A and E = −∇φ − ∂ ∂tA .]
10E Cosmology For an ideal Fermi gas in equilibrium at temperature T and chemical potential μ, the average occupation number of the kth energy state, with energy Ek, is
¯nk =
e(Ek^ −μ)/kB^ T^ + 1
Discuss the limit T → 0. What is the Fermi energy ǫF? How is it related to the Fermi momentum pF? Explain why the density of states with momentum between p and p + dp is proportional to p^2 dp and use this fact to deduce that the fermion number density at zero temperature takes the form n ∝ p^3 F.
Consider an ideal Fermi gas that, at zero temperature, is either (i) non-relativistic or (ii) ultra-relativistic. In each case show that the fermion energy density ǫ takes the form ǫ ∝ nγ^ , for some constant γ which you should compute.
Part II, Paper 3 [TURN OVER
13C Mathematical Biology Consider the two-variable reaction-diffusion system ∂u ∂t = a − u + u^2 v + ∇^2 u , ∂v ∂t
= b − u^2 v + d∇^2 v ,
where a, b and d are positive constants. Show that there is one possible spatially homogeneous steady state with u > 0 and v > 0 and show that it is stable to small-amplitude spatially homogeneous disturbances provided that γ < β, where
γ =
b − a b + a and β = (a + b)^2.
Now assuming that the condition γ < β is satisfied, investigate the stability of the homogeneous steady state to spatially varying perturbations by considering the time- dependence of disturbances whose spatial form is such that ∇^2 u = −k^2 u and ∇^2 v = −k^2 v, with k constant. Show that such disturbances vary as ept, where p is one of the roots of
p^2 + (β − γ + dk^2 + k^2 )p + dk^4 + (β − dγ)k^2 + β.
By comparison with the stability condition for the homogeneous case above, give a simple argument as to why the system must be stable if d = 1. Show that the boundary between stability and instability (as some combination of β, γ and d is varied) must correspond to p = 0. Deduce that dγ > β is a necessary condition for instability and, furthermore, that instability will occur for some k if
d > β γ
γ
γ
γ^2
Deduce that the value of k^2 at which instability occurs as the stability boundary is crossed is given by k^2 =
β d
Part II, Paper 3 [TURN OVER
14D Dynamical Systems Consider the dynamical system
¨x − (a − bx) ˙x + x − x^2 = 0, a, b > 0. (1)
(a) Show that the fixed point at the origin is an unstable node or focus, and that the fixed point at x = 1 is a saddle point.
(b) By considering the phase plane (x, x˙), or otherwise, show graphically that the maximum value of x for any periodic orbit is less than one.
(c) By writing the system in terms of the variables x and z = ˙x − (ax − bx^2 /2), or otherwise, show that for any periodic orbit C ∮
C
(x − x^2 )(2ax − bx^2 ) dt = 0. (2)
Deduce that if a/b > 1 /2 there are no periodic orbits.
(d) If a = b = 0 the system (1) is Hamiltonian and has homoclinic orbit
X(t) =
3 tanh^2
t 2
which approaches X = 1 as t → ±∞. Now suppose that a, b are very small and that we seek the value of a/b corresponding to a periodic orbit very close to X(t). By using equation (3) in equation (2), find an approximation to the largest value of a/b for a periodic orbit when a, b are very small.
[Hint. You may use the fact that (1 − X) = 32 sech^2 ( 2 t ) = 3 (^) dtd (tanh( 2 t ))]
15E Cosmology In a flat expanding universe with scale factor a(t), average mass density ¯ρ and average pressure P¯ ≪ ρc¯^2 , the fractional density perturbations δk(t) at co-moving wavenumber k satisfy the equation
δ¨k = − 2
a˙ a
δ^ ˙k + 4πGρδ¯k − c
2 sk 2 a^2 δk. (∗)
Discuss briefly the meaning of each term on the right hand side of this equation. What is the Jeans length λJ , and what is its significance? How is it related to the Jeans mass?
How does the equation (∗) simplify at λ ≫ λJ in a flat universe? Use your result to show that density perturbations can grow. For a growing density perturbation, how does δ/δ˙ compare to the inverse Hubble time?
Explain qualitatively why structure only forms after decoupling, and why cold dark matter is needed for structure formation.
Part II, Paper 3
19H Representation Theory Show that every complex representation of a finite group G is equivalent to a unitary representation. Let χ be a character of some finite group G and let g ∈ G. Explain why there are roots of unity ω 1 ,... , ωd such that
χ(gi) = ωi 1 + · · · + ωdi
for all integers i.
For the rest of the question let G be the symmetric group on some finite set. Explain why χ(g) = χ(gi) whenever i is coprime to the order of g.
Prove that χ(g) ∈ Z. State without proof a formula for
g∈G χ(g) (^2) when χ is irreducible. Is there an
irreducible character χ of degree at least 2 with χ(g) 6 = 0 for all g ∈ G? Explain your answer.
[You may assume basic facts about the symmetric group, and about algebraic integers, without proof. You may also use without proof the fact that
16 i 6 n gcd(i,n)=
ωi^ ∈ Z
for any nth root of unity ω.]
20G Algebraic Topology State the Mayer–Vietoris Theorem for a simplicial complex K expressed as the union of two subcomplexes L and M. Explain briefly how the connecting homomorphism δ∗ : Hn(K) → Hn− 1 (L∩M ), which appears in the theorem, is defined. [You should include a proof that δ∗ is well-defined, but need not verify that it is a homomorphism.]
Now suppose that |K| ∼= S^3 , that |L| is a solid torus S^1 × B^2 , and that |L∩ M | is the boundary torus of |L|. Show that δ∗ : H 3 (K) → H 2 (L ∩ M ) is an isomorphism, and hence calculate the homology groups of M. [You may assume that a generator of H 3 (K) may be represented by a 3-cycle which is the sum of all the 3-simplices of K, with ‘matching’ orientations.]
Part II, Paper 3
21G Linear Analysis State the closed graph theorem. (i) Let X be a Banach space and Y a vector space. Suppose that Y is endowed with two norms ‖ · ‖ 1 and ‖ · ‖ 2 and that there is a constant c > 0 such that ‖y‖ 2 > c‖y‖ 1 for all y ∈ Y. Suppose that Y is a Banach space with respect to both norms. Suppose that T : X → Y is a linear operator, and that it is bounded when Y is endowed with the ‖ · ‖ 1 norm. Show that it is also bounded when Y is endowed with the ‖ · ‖ 2 norm.
∑ (ii) Suppose that^ X^ is a normed space and that (xn)∞^ n=1^ ⊆^ X^ is a sequence with ∞ n=1 |f^ (xn)|^ <^ ∞^ for all^ f^ in the dual space^ X
∗. Show that there is an M such that
∑^ ∞
n=
|f (xn)| 6 M ‖f ‖
for all f ∈ X∗. (iii) Suppose that X is the space of bounded continuous functions f : R → R with the sup norm, and that Y ⊆ X is the subspace of continuously differentiable functions with bounded derivative. Let T : Y → X be defined by T f = f ′. Show that the graph of T is closed, but that T is not bounded.
22I Riemann Surfaces Let Λ be the lattice Z + Zi, X the torus C/Λ, and ℘ the Weierstrass elliptic function with respect to Λ. (i) Let x ∈ X be the point given by 0 ∈ Λ. Determine the group
G = {f ∈ Aut(X) | f (x) = x}.
(ii) Show that ℘^2 defines a degree 4 holomorphic map h : X → C ∪ {∞}, which is invariant under the action of G, that is, h(f (y)) = h(y) for any y ∈ X and any f ∈ G. Identify a ramification point of h distinct from x which is fixed by every element of G. [If you use the Monodromy theorem, then you should state it correctly. You may use the fact that Aut(C) = {az + b | a ∈ C \ { 0 }, b ∈ C}, and may assume without proof standard facts about ℘.]
23I Algebraic Geometry Let X ⊂ P^2 (C) be the projective closure of the affine curve y^3 = x^4 + 1. Let ω denote the differential dx/y^2. Show that X is smooth, and compute vp(ω) for all p ∈ X. Calculate the genus of X.
Part II, Paper 3 [TURN OVER
26K Applied Probability We consider a system of two queues in tandem, as follows. Customers arrive in the first queue at rate λ. Each arriving customer is immediately served by one of infinitely many servers at rate μ 1. Immediately after service, customers join a single-server second queue which operates on a first-come, first-served basis, and has a service rate μ 2. After service in this second queue, each customer returns to the first queue with probability 0 < 1 − p < 1, and otherwise leaves the system forever. A schematic representation is given below:
λ
μ 1
μ 2
p
1 − p
.. .
1 2
(a) Let Mt and Nt denote the number of customers at time t in queues number 1 and 2 respectively, including those currently in service at time t. Give the transition rates of the Markov chain (Mt, Nt)t> 0. (b) Write down an equation satisfied by any invariant measure π for this Markov chain. Let α > 0 and β ∈ (0, 1). Define a measure π by
π(m, n) := e−α^
αm m! βn(1 − β), m, n ∈ { 0 , 1 ,.. .}.
Show that it is possible to find α > 0 , β ∈ (0, 1) so that π is an invariant measure of (Mt, Nt)t> 0 , if and only if λ < μ 2 p. Give the values of α and β in this case. (c) Assume now that λp > μ 2. Show that the number of customers is not positive recurrent. [Hint. One way to solve the problem is as follows. Assume it is positive recurrent. Observe that Mt is greater than a M/M/∞ queue with arrival rate λ. Deduce that Nt is greater than a M/M/ 1 queue with arrival rate λp and service rate μ 2. You may use without proof the fact that the departure process from the first queue is then, at equilibrium, a Poisson process with rate λ, and you may use without proof properties of thinned Poisson processes.]
Part II, Paper 3 [TURN OVER
27K Principles of Statistics The parameter vector is Θ ≡ (Θ 1 , Θ 2 , Θ 3 ), with Θi > 0, Θ 1 + Θ 2 + Θ 3 = 1. Given Θ = θ ≡ (θ 1 , θ 2 , θ 3 ), the integer random vector X = (X 1 , X 2 , X 3 ) has a trinomial distribution, with probability mass function
p(x | θ) =
n! x 1! x 2! x 3! θx 1 1 θ 2 x 2 θx 3 3 ,
xi > 0 ,
∑^3
i=
xi = n
Compute the score vector for the parameter Θ∗^ := (Θ 1 , Θ 2 ), and, quoting any relevant general result, use this to determine E(Xi) (i = 1, 2 , 3).
Considering (1) as an exponential family with mean-value parameter Θ∗, what is the corresponding natural parameter Φ ≡ (Φ 1 , Φ 2 )?
Compute the information matrix I for Θ∗, which has (i, j)-entry
Iij = −E
∂^2 l ∂θi∂θj
(i, j = 1, 2) ,
where l denotes the log-likelihood function, based on X, expressed in terms of (θ 1 , θ 2 ).
Show that the variance of log(X 1 /X 3 ) is asymptotic to n−^1 (θ− 1 1 + θ 3 − 1 ) as n → ∞. [Hint. The information matrix IΦ for Φ is I−^1 and the dispersion matrix of the maximum likelihood estimator Φ̂ behaves, asymptotically (for n → ∞) as IΦ− 1 .]
28J Optimization and Control A state variable x = (x 1 , x 2 ) ∈ R^2 is subject to dynamics
x˙ 1 (t) = x 2 (t) x˙ 2 (t) = u(t),
where u = u(t) is a scalar control variable constrained to the interval [− 1 , 1]. Given an initial value x(0) = (x 1 , x 2 ), let F (x 1 , x 2 ) denote the minimal time required to bring the state to (0, 0). Prove that
max u∈[− 1 ,1]
−x 2
∂F
∂x 1
− u
∂F
∂x 2
Explain how this equation figures in Pontryagin’s maximum principle.
Use Pontryagin’s maximum principle to show that, on an optimal trajectory, u(t) only takes the values 1 and −1, and that it makes at most one switch between them.
Show that u(t) = 1, 0 6 t 6 2 is optimal when x(0) = (2, −2). Find the optimal control when x(0) = (7, −2).
Part II, Paper 3
31B Asymptotic Methods Find the two leading terms in the asymptotic expansion of the Laplace integral
I(x) =
0
f (t)ext 4 dt
as x → ∞, where f (t) is smooth and positive on [0, 1].
32D Integrable Systems Consider a one-parameter group of transformations acting on R^4
(x, y, t, u) −→ (exp (ǫα)x, exp (ǫβ)y, exp (ǫγ)t, exp (ǫδ)u) , (1)
where ǫ is a group parameter and (α, β, γ, δ) are constants.
(a) Find a vector field W which generates this group.
(b) Find two independent Lie point symmetries S 1 and S 2 of the PDE
(ut − uux)x = uyy, u = u(x, y, t) , (2)
which are of the form (1).
(c) Find three functionally-independent invariants of S 1 , and do the same for S 2. Find a non-constant function G = G(x, y, t, u) which is invariant under both S 1 and S 2.
(d) Explain why all the solutions of (2) that are invariant under a two-parameter group of transformations generated by vector fields
W = u
∂u
∂x
y
∂y
, V =
∂y
are of the form u = xF (t), where F is a function of one variable. Find an ODE for F characterising these group-invariant solutions.
Part II, Paper 3
33A Principles of Quantum Mechanics Discuss the consequences of indistinguishability for a quantum mechanical state consisting of two identical, non-interacting particles when the particles have (a) spin zero, (b) spin 1/2. The stationary Schr¨odinger equation for one particle in the potential
− 2 e^2 4 πǫ 0 r
has normalised, spherically-symmetric real wavefunctions ψn(r) and energy eigenvalues En with E 0 < E 1 < E 2 < · · ·. The helium atom can be modelled by considering two non-interacting spin 1/2 particles in the above potential. What are the consequences of the Pauli exclusion principle for the ground state? Write down the two-electron state for this model in the form of a spatial wavefunction times a spin state. Assuming that wavefunctions are spherically-symmetric, find the states of the first excited energy level of the helium atom. What combined angular momentum quantum numbers J, M does each state have? Assuming standard perturbation theory results, arrive at a multi-dimensional integral in terms of the one-particle wavefunctions for the first-order correction to the helium ground state energy, arising from the electron-electron interaction.
Part II, Paper 3 [TURN OVER
35C Statistical Physics A ferromagnet has magnetization order parameter m and is at temperature T. The free energy is given by
F (T ; m) = F 0 (T ) + a 2
(T − Tc) m^2 + b 4
m^4 ,
where a, b and Tc are positive constants. Find the equilibrium value of the magnetization at both high and low temperatures. Evaluate the free energy of the ground state as a function of temperature. Hence compute the entropy and heat capacity. Determine the jump in the heat capacity and identify the order of the phase transition. After imposing a background magnetic field B, the free energy becomes
F (T ; m) = F 0 (T ) + Bm + a 2 (T − Tc) m^2 + b 4 m^4.
Explain graphically why the system undergoes a first-order phase transition at low temperatures as B changes sign. The spinodal point occurs when the meta-stable vacuum ceases to exist. Determine the temperature T of the spinodal point as a function of Tc, a, b and B.
36B Electrodynamics The non-relativistic Larmor formula for the power, P , radiated by a particle of charge q and mass m that is being accelerated with an acceleration a is
P =
μ 0 6 π q^2 |a|^2.
Starting from the Li´enard–Wiechert potentials, sketch a derivation of this result. Explain briefly why the relativistic generalization of this formula is
P =
μ 0 6 π
q^2 m^2
dpμ dτ
dpν dτ ημν
where pμ^ is the relativistic momentum of the particle and τ is the proper time along the worldline of the particle. A particle of mass m and charge q moves in a plane perpendicular to a constant magnetic field B. At time t = 0 as seen by an observer O at rest, the particle has energy E = γm. At what rate is electromagnetic energy radiated by this particle? At time t according to the observer O, the particle has energy E′^ = γ′m. Find an expression for γ′^ in terms of γ and t.
Part II, Paper 3 [TURN OVER
37B General Relativity (i) The Schwarzschild metric is given by
ds^2 = −
2 M
r
dt^2 +
2 M
r
dr^2 + r^2 (dθ^2 + sin^2 θ dφ^2 ).
Consider a time-like geodesic xa(τ ), where τ is the proper time, lying in the plane θ = π/2. Use the Lagrangian L = gab x˙a^ x˙b^ to derive the equations governing the geodesic, showing that r^2 φ˙ = h ,
with h constant, and hence demonstrate that
d^2 u dφ^2
M
h^2
where u = 1/r. State which term in this equation makes it different from an analogous equation in Newtonian theory.
(ii) Now consider Kruskal coordinates, in which the Schwarzschild t and r are replaced by U and V , defined for r > 2 M by
U ≡
( (^) r 2 M
er/(4M^ )^ cosh
t 4 M
V ≡
( (^) r 2 M
er/(4M^ )^ sinh
t 4 M
and for r < 2 M by
U ≡
r 2 M
er/(4M^ )^ sinh
t 4 M
V ≡
r 2 M
er/(4M^ )^ cosh
t 4 M
Given that the metric in these coordinates is
ds^2 =
32 M 3
r
e−r/(2M^ )(−dV 2 + dU 2 ) + r^2 (dθ^2 + sin^2 θdφ^2 ) ,
where r = r(U, V ) is defined implicitly by ( (^) r 2 M
er/(2M^ )^ = U 2 − V 2 ,
sketch the Kruskal diagram, indicating the positions of the singularity at r = 0, the event horizon at r = 2M , and general lines of constant r and of constant t.
Part II, Paper 3
