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MATHEMATICAL TRIPOS Part II
Thursday, 9 June, 2011 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
(i) State Lagrange’s Theorem, and prove that, if p is an odd prime,
(p − 1)! ≡ − 1 mod p.
(ii) Still assuming p is an odd prime, prove that
32 · 52 · · · (p − 2)^2 ≡ (−1)
p+1 2 mod p.
2F Topics in Analysis Let Γ = {z ∈ C : z 6 = 1, |Re(z)| + |Im(z)| = 1}.
(i) Prove that, for any ζ ∈ C with |Re(ζ)| + |Im(ζ)| > 1 and any ǫ > 0, there exists a complex polynomial p such that
sup z∈Γ
|p(z) − (z − ζ)−^1 | < ǫ.
(ii) Does there exist a sequence of polynomials pn such that pn(z) → (z − 1)−^1 for every z ∈ Γ? Justify your answer.
3G Geometry and Groups Define a Kleinian group. Give an example of a Kleinian group that is a free group on two generators and explain why it has this property.
4G Coding and Cryptography What is the rank of a binary linear code C? What is the weight enumeration polynomial WC of C?
Show that WC (1, 1) = 2r^ where r is the rank of C. Show that WC (s, t) = WC (t, s) for all s and t if and only if WC (1, 0) = 1.
Find, with reasons, the weight enumeration polynomial of the repetition code of length n, and of the simple parity check code of length n.
Part II, Paper 3
7C Dynamical Systems For the map xn+1 = λxn(1 − x^2 n), with λ > 0, show the following:
(i) If λ < 1, then the origin is the only fixed point and is stable.
(ii) If λ > 1, then the origin is unstable. There are two further fixed points which are stable for 1 < λ < 2 and unstable for λ > 2.
(iii) If λ < 3
3 /2, then xn has the same sign as the starting value x 0 if |x 0 | < 1.
(iv) If λ < 3, then |xn+1| < 2
3 /3 when |xn| < 2
3 /3. Deduce that iterates starting sufficiently close to the origin remain bounded, though they may change sign.
[Hint: For (iii) and (iv) a graphical representation may be helpful.]
8E Further Complex Methods Explain the meaning of zj in the Weierstrass canonical product formula
f (z) = f (0) exp
[
f ′(0) f (0)
z
] ∏∞
j=
z zj
e zz j
Show that sin(πz) πz
∏^ ∞
n=
z^2 n^2
Deduce that
π cot(πz) =
z
∑^ ∞
n=
z z^2 − n^2
9C Classical Dynamics The Lagrangian for a heavy symmetric top is
L =
I 1
θ˙^2 + φ˙^2 sin^2 θ
I 3
ψ˙ + φ˙ cos θ
− M gl cos θ.
State Noether’s Theorem. Hence, or otherwise, find two conserved quantities linear in momenta, and a third conserved quantity quadratic in momenta.
Writing μ = cos θ, deduce that μ obeys an equation of the form
μ˙^2 = F (μ) ,
where F (μ) is cubic in μ. [You need not determine the explicit form of F (μ).]
Part II, Paper 3
10E Cosmology For an ideal gas of fermions of mass m in volume V , and at temperature T and chemical potential μ, the number density n and kinetic energy E are given by
n = 4 πgs h^3
0
¯n(p) p^2 dp , E = 4 πgs h^3
V
0
n ¯(p)ǫ(p)p^2 dp ,
where gs is the spin-degeneracy factor, h is Planck’s constant, ǫ(p) = c
p^2 + m^2 c^2 is the single-particle energy as a function of the momentum p, and
n¯(p) =
[
exp
ǫ(p) − μ kT
]− 1
where k is Boltzmann’s constant.
(i) Sketch the function ¯n(p) at zero temperature, explaining why ¯n(p) = 0 for p > pF (the Fermi momentum). Find an expression for n at zero temperature as a function of pF. Assuming that a typical fermion is ultra-relativistic (pc ≫ mc^2 ) even at zero temperature, obtain an estimate of the energy density E/V as a function of pF , and hence show that E ∼ hc n^4 /^3 V (∗) in the ultra-relativistic limit at zero temperature.
(ii) A white dwarf star of radius R has total mass M = 43 π mpnpR^3 , where mp is the proton mass and np the average proton number density. On the assumption that the star’s degenerate electrons are ultra-relativistic, so that (∗) applies with n replaced by the average electron number density ne, deduce the following estimate for the star’s internal kinetic energy:
Ekin ∼ hc
M
mp
R
By comparing this with the total gravitational potential energy, briefly discuss the consequences for white dwarf stability.
Part II, Paper 3 [TURN OVER
12F Topics in Analysis Let f : [0, 1] → R be continuous and let n be a positive integer. For g : [0, 1] → R a continuous function, write ‖f − g‖L∞ = supx∈[0,1] |f (x) − g(x)|.
(i) Let p be a polynomial of degree at most n with the property that there are (n + 2) distinct points x 1 , x 2 ,... , xn+2 ∈ [0, 1] with x 1 < x 2 <... < xn+2 such that
f (xj ) − p(xj ) = (−1)j^ ‖f − p‖L∞
for each j = 1, 2 ,... , n + 2. Prove that
‖f − p‖L∞^6 ‖f − q‖L∞
for every polynomial q of degree at most n.
(ii) Prove that there exists a polynomial p of degree at most n such that
‖f − p‖L∞ 6 ‖f − q‖L∞
for every polynomial q of degree at most n. [If you deduce this from a more general result about abstract normed spaces, you must prove that result.]
(iii) Let Y = {y 1 , y 2 ,... , yn+2} be any set of (n + 2) distinct points in [0, 1].
(a) For j = 1, 2 ,... , n + 2, let
rj (x) =
n∏+
k=1, k 6 =j
x − yk yj − yk
t(x) =
∑n+ j=1 f^ (yj^ )rj^ (x) and^ r(x) =^
∑n+ j=1 (−1) j (^) rj (x). Explain why there is a unique number λ ∈ R such that the degree of the polynomial t − λr is at most n. (b) Let ‖f − g‖L∞ (^) (Y ) = supx∈Y |f (x) − g(x)|. Deduce from part (a) that there exists a polynomial p of degree at most n such that
‖f − p‖L∞(Y ) 6 ‖f − q‖L∞(Y )
for every polynomial q of degree at most n.
Part II, Paper 3 [TURN OVER
13B Mathematical Biology The number density of a population of amoebae is n(x, t). The amoebae exhibit chemotaxis and are attracted to high concentrations of a chemical which has concentration a(x, t). The equations governing n and a are
∂n ∂t
= αn(n^20 − n^2 ) + ∇^2 n − ∇ · (χ(n)n∇a) , ∂a ∂t = βn − γa + D∇^2 a ,
where the constants n 0 , α, β, γ and D are all positive.
(i) Give a biological interpretation of each term in these equations and discuss the sign of χ(n).
(ii) Show that there is a non-trivial (i.e. a 6 = 0, n 6 = 0) steady-state solution for n and a, independent of x, and show further that it is stable to small disturbances that are also independent of x.
(iii) Consider small spatially varying disturbances to the steady state, with spatial structure such that ∇^2 ψ = −k^2 ψ, where ψ is any disturbance quantity. Show that if such disturbances also satisfy ∂ψ/∂t = pψ, where p is a constant, then p satisfies a quadratic equation, to be derived. By considering the conditions required for p = 0 to be a possible solution of this quadratic equation, or otherwise, deduce that instability is possible if
βχ 0 n 0 > 2 αn^20 D + γ + 2(2Dαn^20 γ)^1 /^2 ,
where χ 0 = χ(n 0 ).
(iv) Explain briefly how your conclusions might change if an additional geometric constraint implied that k^2 > k^20 , where k 0 is a given constant.
Part II, Paper 3
15E Cosmology An expanding universe with scale factor a(t) is filled with (pressure-free) cold dark matter (CDM) of average mass density ¯ρ(t). In the Zel’dovich approximation to gravitational clumping, the perturbed position r(q, t) of a CDM particle with unperturbed comoving position q is given by
r(q, t) = a(t)[q + ψ(q, t)] , (1)
where ψ is the comoving displacement.
(i) Explain why the conservation of CDM particles implies that
ρ(r, t) d^3 r = a^3 ρ¯(t) d^3 q ,
where ρ(r, t) is the CDM mass density. Use (1) to verify that d^3 q = a−^3 [1 − ∇q · ψ]d^3 r, and hence deduce that the fractional density perturbation is, to first order, δ ≡ ρ − ρ¯ ρ ¯ = −∇q · ψ.
Use this result to integrate the Poisson equation ∇^2 Φ = 4πGρ¯ for the gravitational potential Φ. Then use the particle equation of motion ¨r = −∇Φ to deduce a second-order differential equation for ψ, and hence that
δ¨ + 2
a˙ a
δ^ ˙ − 4 πGρ δ¯ = 0. (2)
[You may assume that ∇^2 Φ = 4πGρ¯ implies ∇Φ = (4πG/3)¯ρ r and that the pressure-free acceleration equation is ¨a = −(4πG/3)¯ρa.]
(ii) A flat matter-dominated universe with background density ¯ρ = (6πGt^2 )−^1 has scale factor a(t) = (t/t 0 )^2 /^3. The universe is filled with a pressure-free homogeneous (non-clumping) fluid of mass density ρH (t), as well as cold dark matter of mass density ρC (r, t). Assuming that the Zel’dovich perturbation equation in this case is as in (2) but with ¯ρ replaced by ¯ρC , i.e. that
¨δ + 2
a˙ a
δ^ ˙ − 4 πG¯ρC δ = 0 ,
seek power-law solutions δ ∝ tα^ to find growing and decaying modes with
α =
25 − 24 ΩH
where ΩH = ρH /ρ¯. Given that matter domination starts (t = teq) at a redshift z ≈ 105 , and given an initial perturbation δ(teq) ≈ 10 −^5 , show that ΩH = 2/3 yields a model that is not compatible with the large-scale structure observed today.
Part II, Paper 3
16H Logic and Set Theory State and prove the Upward L¨owenheim–Skolem Theorem. [You may assume the Compactness Theorem, provided that you state it clearly.] A total ordering (X, <) is called dense if for any x < y there exists z with x < z < y. Show that a dense total ordering (on more than one point) cannot be a well-ordering. For each of the following theories, either give axioms, in the language of posets, for the theory or prove carefully that the theory is not axiomatisable in the language of posets. (i) The theory of dense total orderings. (ii) The theory of countable dense total orderings. (iii) The theory of uncountable dense total orderings. (iv) The theory of well-orderings.
17F Graph Theory Define the Tur´an graph Tr (n). State and prove Tur´an’s theorem. Hence, or otherwise, find ex(K 3 ; n). Let G be a bipartite graph with n vertices in each class. Let k be an integer, 1 6 k 6 n, and assume e(G) > (k − 1)n. Show that G contains a set of k independent edges. [Hint: Suppose G contains a set D of a independent edges but no set of a + 1 independent edges. Let U be the set of vertices of the edges in D and let F be the set of edges in G with precisely one vertex in U ; consider |F |.] Hence, or otherwise, show that if H is a triangle-free tripartite graph with n vertices in each class then e(H) 6 2 n^2.
18H Galois Theory Let n > 1 and K = Q(μn) be the cyclotomic field generated by the nth roots of unity. Let a ∈ Q with a 6 = 0, and consider F = K( n
a).
(i) State, without proof, the theorem which determines Gal(K/Q).
(ii) Show that F/Q is a Galois extension and that Gal(F/Q) is soluble. [When using facts about general Galois extensions and their generators, you should state them clearly.]
(iii) When n = p is prime, list all possible degrees [F : Q], with justification.
Part II, Paper 3 [TURN OVER
21G Linear Analysis Let H be a complex Hilbert space with orthonormal basis (en)∞ n=−∞. Let T : H → H be a bounded linear operator. What is meant by the spectrum σ(T ) of T? Define T by setting T (en) = en− 1 + en+1 for n ∈ Z. Show that T has a unique extension to a bounded, self-adjoint linear operator on H. Determine the norm ‖T ‖. Exhibit, with proof, an element of σ(T ). Show that T has no eigenvectors. Is T compact? [General results from spectral theory may be used without proof. You may also use the fact that if a sequence (xn) satisfies a linear recurrence λxn = xn− 1 + xn+1 with λ ∈ R, |λ| 6 2, λ 6 = 0, then it has the form xn = Aαn^ sin(θ 1 n + θ 2 ) or xn = (A + nB)αn, where A, B, α ∈ R and 0 6 θ 1 < π, |θ 2 | 6 π/2.]
22G Riemann Surfaces State the Classical Monodromy Theorem for analytic continuations in subdomains of the plane. Let n, r be positive integers with r > 1 and set h(z) = zn^ − 1. By removing n semi-infinite rays from C, find a subdomain U ⊂ C on which an analytic function h^1 /r may be defined, justifying this assertion. Describe briefly a gluing procedure which will produce the Riemann surface R for the complete analytic function h^1 /r^. Let Z denote the set of nth roots of unity and assume that the natural analytic covering map π : R → C \ Z extends to an analytic map of Riemann surfaces ˜π : R˜ → C∞, where R˜ is a compactification of R and C∞ denotes the extended complex plane. Show that ˜π has precisely n branch points if and only if r divides n.
23H Algebraic Geometry Let X be a smooth projective curve over an algebraically closed field k of charac- teristic 0. (i) Let D be a divisor on X. Define L(D), and show dim L(D) 6 deg D + 1. (ii) Define the space of rational differentials Ω^1 k(X)/k.
If p is a point on X, and t a local parameter at p, show that Ω^1 k(X)/k = k(X)dt.
Use that equality to give a definition of vp(ω) ∈ Z, for ω ∈ Ω^1 k(X)/k, p ∈ X. [You need not show that your definition is independent of the choice of local parameter.]
Part II, Paper 3 [TURN OVER
24I Differential Geometry For an oriented surface S in R^3 , define the Gauss map, the second fundamental form and the normal curvature in the direction w ∈ TpS at a point p ∈ S.
Let k˜ 1 ,... , ˜km be normal curvatures at p in the directions v 1 ,... , vm, such that the angle between vi and vi+1 is π/m for each i = 1,... , m − 1 (m > 2). Show that
k˜ 1 +... + ˜km = mH ,
where H is the mean curvature of S at p.
What is a minimal surface? Show that if S is a minimal surface, then its Gauss map N at each point p ∈ S satisfies
〈dNp(w 1 ), dNp(w 2 )〉 = μ(p)〈w 1 , w 2 〉 , for all w 1 , w 2 ∈ TpS , (∗)
where μ(p) ∈ R depends only on p. Conversely, if the identity (∗) holds at each point in S, must S be minimal? Justify your answer.
25K Probability and Measure
(i) State and prove Kolmogorov’s zero-one law.
(ii) Let (E, E, μ) be a finite measure space and suppose that (Bn)n> 1 is a sequence of events such that Bn+1 ⊂ Bn for all n > 1. Show carefully that μ(Bn) → μ(B), where B = ∩∞ n=1Bn.
(iii) Let (Xi)i> 1 be a sequence of independent and identically distributed random variables such that E(X 12 ) = σ^2 < ∞ and E(X 1 ) = 0. Let K > 0 and consider the event An defined by
An =
Sn √ n
> K
, where Sn =
∑^ n
i=
Xi.
Prove that there exists c > 0 such that for all n large enough, P(An) > c. Any result used in the proof must be stated clearly.
(iv) Prove using the results above that An occurs infinitely often, almost surely. Deduce that lim sup n→∞
Sn √ n
almost surely.
Part II, Paper 3
27K Principles of Statistics Random variables X 1 , X 2 ,... are independent and identically distributed from the exponential distribution E(θ), with density function
pX (x | θ) = θe−θx^ (x > 0) ,
when the parameter Θ takes value θ > 0. The following experiment is performed. First X 1 is observed. Thereafter, if X 1 = x 1 ,... , Xi = xi have been observed (i > 1), a coin having probability α(xi) of landing heads is tossed, where α : R → (0, 1) is a known function and the coin toss is independent of the X’s and previous tosses. If it lands heads, no further observations are made; if tails, Xi+1 is observed.
Let N be the total number of X’s observed, and X := (X 1 ,... , XN ). Write down the likelihood function for Θ based on data X = (x 1 ,... , xn), and identify a minimal sufficient statistic. What does the likelihood principle have to say about inference from this experiment?
Now consider the experiment that only records Y := XN. Show that the density function of Y has the form
pY (y | θ) = exp{a(y) − k(θ) − θy}.
Assuming the function a(·) is twice differentiable and that both pY (y | θ) and ∂pY (y | θ)/∂y vanish at 0 and ∞, show that a′(Y ) is an unbiased estimator of Θ, and find its variance.
Stating clearly any general results you use, deduce that
−k′′(θ) Eθ{a′′(Y )} > 1.
Part II, Paper 3
28K Optimization and Control An observable scalar state variable evolves as xt+1 = xt + ut, t = 0, 1 ,.... Let controls u 0 , u 1 ,... be determined by a policy π and define
Cs(π, x 0 ) =
∑^ s−^1
t=
(x^2 t + 2xtut + 7u^2 t ) and Cs(x 0 ) = inf π Cs(π, x 0 ).
Show that it is possible to express Cs(x 0 ) in terms of Πs, which satisfies the recurrence
Πs = 6(1 + Πs− 1 ) 7 + Πs− 1
, s = 1, 2 ,... ,
with Π 0 = 0. Deduce that C∞(x 0 ) > 2 x^20. [C∞(x 0 ) is defined as (^) slim→∞ Cs(x 0 ).]
By considering the policy π∗^ which takes ut = −(1/3)(2/3)t^ x 0 , t = 0, 1 ,... , show that C∞(x 0 ) = 2x^20. Give an alternative description of π∗^ in closed-loop form.
29J Stochastic Financial Models
First, what is a Brownian motion?
(i) The price St of an asset evolving in continuous time is represented as
St = S 0 exp(σWt + μt) ,
where W is a standard Brownian motion, and σ and μ are constants. If riskless investment in a bank account returns a continuously-compounded rate of interest r, derive a formula for the time-0 price of a European call option on the asset S with strike K and expiry T. You may use any general results, but should state them clearly.
(ii) In the same financial market, consider now a derivative which pays
Y =
exp
T −^1
∫ T
0
log(Su) du
− K
at time T. Find the time-0 price for this derivative. Show that it is less than the price of the European call option which you derived in (i).
Part II, Paper 3 [TURN OVER
32A Integrable Systems Let U (ρ, τ, λ) and V (ρ, τ, λ) be matrix-valued functions. Consider the following system of overdetermined linear partial differential equations:
∂ ∂ρ
ψ = U ψ ,
∂τ
ψ = V ψ ,
where ψ is a column vector whose components depend on (ρ, τ, λ). Using the consistency condition of this system, derive the associated zero curvature representation (ZCR)
∂ ∂τ
U −
∂ρ
V + [U, V ] = 0 , (∗)
where [ · , · ] denotes the usual matrix commutator.
(i) Let U = i 2
2 λ ∂ρφ ∂ρφ − 2 λ
, V =
4 iλ
cos φ −i sin φ i sin φ − cos φ
Find a partial differential equation for φ = φ(ρ, τ ) which is equivalent to the ZCR (∗).
(ii) Assuming that U and V in (∗) do not depend on t := ρ − τ , show that the trace of (U − V )p^ does not depend on x := ρ + τ , where p is any positive integer. Use this fact to construct a first integral of the ordinary differential equation
φ′′^ = sin φ , where φ = φ(x).
Part II, Paper 3 [TURN OVER
33D Principles of Quantum Mechanics The Pauli matrices σ = (σx, σy , σz ) = (σ 1 , σ 2 , σ 3 ), with
σ 1 =
, σ 2 =
0 −i i 0
, σ 3 =
are used to represent angular momentum operators with respect to basis states | ↑〉 and | ↓〉 corresponding to spin up and spin down along the z-axis. They satisfy
σiσj = δij + iǫijkσk.
(i) How are | ↑〉 and | ↓〉 represented? How is the spin operator s related to σ and ℏ? Check that the commutation relations between the spin operators are as desired. Check that s^2 acting on a spin one-half state has the correct eigenvalue. What are the states obtained by applying sx, sy to the eigenstates | ↑〉 and | ↓〉 of sz?
(ii) Let V be the space of states for a spin one-half system. Consider a combination of three such systems with states belonging to V (1)^ ⊗ V (2)^ ⊗ V (3)^ and spin operators acting on each subsystem denoted by s( xi ), s( yi )with i = 1, 2 , 3. Find the eigenvalues of the operators
s(1) x s(2) y s(3) y , s(1) y s(2) x s(3) y , s(1) y s(2) y s(3) x and s(1) x s(2) x s(3) x
of the state |Ψ〉 =
[
]
(iii) Consider now whether these outcomes for measurements of particular combinations of the operators s( xi ), s( yi ) in the state |Ψ〉 could be reproduced by replacing the spin operators with classical variables ˜s( xi ), ˜s( yi ) which take values ±ℏ/2 according to some probabilities. Assume that these variables are identical to the quantum measurements of s(1) x s(2) y s(3) y , s(1) y s(2) x s(3) y , s(1) y s(2) y s(3) x on |Ψ〉. Show that classically this implies a unique possibility for
˜s(1) x ˜s(2) x s˜(3) x ,
and find its value. State briefly how this result could be used to experimentally test quantum mechan- ics.
Part II, Paper 3
