Download Part II Paper 1: Mathematical Models and Systems and more Exams Mathematics in PDF only on Docsity!
MATHEMATICAL TRIPOS Part II
Monday, 31 May, 2010 9:00 am to 12:00 pm
PAPER 1
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,.. ., J 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
1G Number Theory (i) Let N be an integer > 2. Define the addition and multiplication on the set of congruence classes modulo N.
(ii) Let an integer M > 1 have expansion to the base 10 given by a (^) s... a 0. Prove that 11 divides M if and only if
∑ (^) s i=0 (−1) ia (^) i is divisible by 11.
2F Topics in Analysis Let (X, d) be a non-empty complete metric space with no isolated points, G an open dense subset of X and E a countable dense subset of X.
(i) Stating clearly any standard theorem you use, prove that G \ E is a dense subset of X.
(ii) If G is only assumed to be uncountable and dense in X, does it still follow that G \ E is dense in X? Justify your answer.
3F Geometry of Group Actions Explain what it means to say that G is a crystallographic group of isometries of the Euclidean plane and that G is its point group. Prove the crystallographic restriction: a rotation in such a point group G must have order 1, 2 , 3 , 4 or 6.
4H Coding and Cryptography Explain what is meant by saying that a binary code C is a decodable code with words Cj of length lj for 1 6 j 6 n. Prove the MacMillan inequality which states that, for such a code, ∑n
j=
2 −lj^6 .
Part II, Paper 1
8E Further Complex Methods Let the complex-valued function f (z) be analytic in the neighbourhood of the point z 0 and let u(x, y) be the real part of f (z). Show that
f (z) = 2 u
z + ¯z 0 2
z − z¯ 0 2 i
− f (z 0 ) , z = x + iy.
Hence find the analytic function whose real part is
e−y^ [ x cos x − y sin x ].
9D Classical Dynamics A system with coordinates qi, i = 1,... , n, has the Lagrangian L(qi, q˙i). Define the energy E.
Consider a charged particle, of mass m and charge e, moving with velocity v in the presence of a magnetic field B = ∇ × A. The usual vector equation of motion can be derived from the Lagrangian
L =
m v^2 + e v · A ,
where A is the vector potential.
The particle moves in the presence of a field such that
A = (0, r g(z), 0) , g(z) > 0 ,
referred to cylindrical polar coordinates (r, φ, z). Obtain two constants of the motion, and write down the Lagrangian equations of motion obtained by variation of r, φ and z.
Show that, if the particle is projected from the point (r 0 , φ 0 , z 0 ) with velocity (0, −2 (e/m) r 0 g(z 0 ), 0), it will describe a circular orbit provided that g′(z 0 ) = 0.
Part II, Paper 1
10D Cosmology What is meant by the expression ‘Hubble time’?
For a(t) the scale factor of the universe and assuming a(0) = 0 and a(t 0 ) = 1, where t 0 is the time now, obtain a formula for the size of the particle horizon R 0 of the universe. Taking a(t) = (t/t 0 )α^ , show that R 0 is finite for certain values of α. What might be the physically relevant values of α? Show that the age of the universe is less than the Hubble time for these values of α.
Part II, Paper 1 [TURN OVER
14E Further Complex Methods Consider the partial differential equation for u(x, t),
∂u ∂t
∂^2 u ∂x^2
, β > 0 , 0 < x < ∞ , t > 0 , (∗)
where u(x, t) is required to vanish rapidly for all t as x → ∞. (i) Verify that the PDE (∗) can be written in the following form ( e−ikx+(k (^2) −iβk)t u
t
e−ikx+(k (^2) −iβk)t[ (ik + β)u + ux
])
x
(ii) Define ˆu(k, t) =
0 e
−ikx (^) u(x, t) dx, which is analytic for Im k 6 0. Determine uˆ(k, t) in terms of ˆu(k, 0) and also the functions f 0 , f 1 defined by
f 0 (ω, t) =
∫ (^) t
0
e−ω(t−t
′ (^) ) u(0, t′) dt′^ , f 1 (ω, t) =
∫ (^) t
0
e−ω(t−t
′ (^) ) ux(0, t′) dt′^.
(iii) Show that in the inverse transform expression for u(x, t) the integrals involving f 0 , f 1 may be transformed to the contour
L =
k ∈ C : Re (k^2 − iβk) = 0, Im k > β
By considering ˆu(k′, t) where k′^ = −k + iβ and k ∈ L, show that it is possible to obtain an equation which allows f 1 to be eliminated. (iv) Obtain an integral expression for the solution of (∗) subject to the the initial- boundary value conditions of given u(x, 0), u(0, t). [You need to show that (^) ∫
L
eikx^ ˆu(k′, t) dk = 0 , x > 0 ,
by an appropriate closure of the contour which should be justified.]
Part II, Paper 1 [TURN OVER
15D Cosmology A star has pressure P (r) and mass density ρ(r), where r is the distance from the centre of the star. These quantities are related by the pressure support equation
P ′^ = −
Gmρ r 2
where P ′^ = dP/dr and m(r) is the mass within radius r. Use this to derive the virial theorem Egrav = − 3 〈P 〉 V ,
where Egrav is the total gravitational potential energy and 〈P 〉 the average pressure.
The total kinetic energy of a spherically symmetric star is related to 〈P 〉 by
Ekin = α 〈P 〉 V ,
where α is a constant. Use the virial theorem to determine the condition on α for gravitational binding. By considering the relation between pressure and ‘internal energy’ U for an ideal gas, determine α for the cases of a) an ideal gas of non-relativistic particles, b) an ideal gas of ultra-relativistic particles.
Why does your result imply a maximum mass for any star? Briefly explain what is meant by the Chandrasekhar limit.
A white dwarf is in orbit with a companion star. It slowly accretes matter from the other star until its mass exceeds the Chandrasekhar limit. Briefly explain its subsequent evolution.
16G Logic and Set Theory Show that ℵ (^) α^2 = ℵ (^) α for all α.
An infinite cardinal m is called regular if it cannot be written as a sum of fewer than m cardinals each of which is smaller than m. Prove that ℵ 0 and ℵ 1 are regular.
Is ℵ 2 regular? Is ℵ (^) ω regular? Justify your answers.
Part II, Paper 1
19F Representation Theory (i) Let N be a normal subgroup of the finite group G. Without giving detailed proofs, define the process of lifting characters from G/N to G. State also the orthogonality relations for G.
(ii) Let a, b be the following two permutations in S 12 ,
a = (1 2 3 4 5 6)(7 8 9 10 11 12) ,
b = (1 7 4 10)(2 12 5 9)(3 11 6 8) ,
and let G = 〈a, b〉 , a subgroup of S 12. Prove that G is a group of order 12 and list the conjugacy classes of G. By identifying a normal subgroup of G of index 4 and lifting irreducible characters, calculate all the linear characters of G. Calculate the complete character table of G. By considering 6th roots of unity, find explicit matrix representations affording the non-linear characters of G.
20G Number Fields Suppose that m is a square-free positive integer, m > 5 , m 6 ≡ 1 (mod 4). Show that, if the class number of K = Q(
−m ) is prime to 3 , then x^3 = y^2 + m has at most two solutions in integers. Assume the m is even.
21H Algebraic Topology State the path lifting and homotopy lifting lemmas for covering maps. Suppose that X is path connected and locally path connected, that p 1 : Y 1 → X and p 2 : Y 2 → X are cov- ering maps, and that Y 1 and Y 2 are simply connected. Using the lemmas you have stated, but without assuming the correspondence between covering spaces and subgroups of π 1 , prove that Y 1 is homeomorphic to Y 2.
Part II, Paper 1
22H Linear Analysis a) State and prove the Banach–Steinhaus Theorem.
[You may use the Baire Category Theorem without proving it.]
b) Let X be a (complex) normed space and S ⊂ X. Prove that if {f (x) : x ∈ S} is a bounded set in C for every linear functional f ∈ X∗^ then there exists K > 0 such that ‖x‖ 6 K for all x ∈ S.
[You may use here the following consequence of the Hahn–Banach Theorem without proving it: for a given x ∈ X, there exists f ∈ X∗^ with ‖f ‖ = 1 and |f (x)| = ‖x‖.]
c) Conclude that if two norms ‖.‖ 1 and ‖.‖ 2 on a (complex) vector space V are not equivalent, there exists a linear functional f : V → C which is continuous with respect to one of the two norms, and discontinuous with respect to the other.
23G Riemann Surfaces Given a lattice Λ ⊂ C, we may define the corresponding Weierstrass ℘-function to be the unique even Λ-periodic elliptic function ℘ with poles only on Λ and for which ℘(z) − 1 /z^2 → 0 as z → 0. For w 6 ∈ Λ , we set
f (z) = det
℘ (z) ℘ (w) ℘ (−z − w) ℘′(z) ℘′(w) ℘′(−z − w)
an elliptic function with periods Λ. By considering the poles of f , show that f has valency at most 4 (i.e. is at most 4 to 1 on a period parallelogram).
If w 6 ∈ 13 Λ , show that f has at least six distinct zeros. If w ∈ 13 Λ , show that f has at least four distinct zeros, at least one of which is a multiple zero. Deduce that the meromorphic function f is identically zero.
If z 1 , z 2 , z 3 are distinct non-lattice points in a period parallelogram such that z 1 + z 2 + z 3 ∈ Λ , what can be said about the points (℘(zi), ℘′(zi)) ∈ C^2 (i = 1, 2 , 3)?
Part II, Paper 1 [TURN OVER
26I Probability and Measure State Carath´eodory’s extension theorem. Define all terms used in the statement.
Let A be the ring of finite unions of disjoint bounded intervals of the form
A =
⋃^ m
i=
(ai, bi]
where m ∈ Z+^ and a 1 < b 1 <... < am < bm. Consider the set function μ defined on A by μ(A) =
∑^ m
i=
(bi − ai).
You may assume that μ is additive. Show that for any decreasing sequence (Bn : n ∈ N) in A with empty intersection we have μ(Bn) → 0 as n → ∞.
Explain how this fact can be used in conjunction with Carath´eodory’s extension theorem to prove the existence of Lebesgue measure.
Part II, Paper 1 [TURN OVER
27I Applied Probability (a) Define what it means to say that π is an equilibrium distribution for a Markov chain on a countable state space with Q-matrix Q = (q (^) ij ), and give an equation which is satisfied by any equilibrium distribution. Comment on the possible non-uniqueness of equilibrium distributions.
(b) State a theorem on convergence to an equilibrium distribution for a continuous- time Markov chain.
A continuous-time Markov chain (Xt, t > 0) has three states 1, 2, 3 and the Q- matrix Q = (q (^) ij ) is of the form
Q =
−λ 1 λ 1 / 2 λ 1 / 2 λ 2 / 2 −λ 2 λ 2 / 2 λ 3 / 2 λ 3 / 2 −λ 3
where the rates λ 1 , λ 2 , λ 3 ∈ [ 0, ∞) are not all zero.
[Note that some of the λi may be zero, and those cases may need special treatment.]
(c) Find the equilibrium distributions of the Markov chain in question. Specify the cases of uniqueness and non-uniqueness.
(d) Find the limit of the transition matrix P (t) = exp(tQ) when t → ∞.
(e) Describe the jump chain (Yn) and its equilibrium distributions. If P̂ is the jump probability matrix, find the limit of P̂ n^ as n → ∞.
28J Principles of Statistics The distribution of a random variable X is obtained from the binomial distribution B(n; Π) by conditioning on X > 0; here Π ∈ (0, 1) is an unknown probability parameter and n is known. Show that the distributions of X form an exponential family and identify the natural sufficient statistic T , natural parameter Φ, and cumulant function k(φ). Using general properties of the cumulant function, compute the mean and variance of X when Π = π. Write down an equation for the maximum likelihood estimate Π of Π and explain̂ why, when Π = π, the distribution of Π is approximately normal̂ N (π, π(1 − π)/n) for large n.
Suppose we observe X = 1. It is suggested that, since the condition X > 0 is then automatically satisfied, general principles of inference require that the inference to be drawn should be the same as if the distribution of X had been B(n; Π) and we had observed X = 1. Comment briefly on this suggestion.
Part II, Paper 1
30E Partial Differential Equations (a) Solve by using the method of characteristics
x 1
∂x 1 u + 2 x 2
∂x 2 u = 5 u , u(x 1 , 1) = g(x 1 ) ,
where g : R → R is continuous. What is the maximal domain in R^2 in which u is a solution of the Cauchy problem?
(b) Prove that the function
u(x, t) =
0 , x < 0 , t > 0 , x/t , 0 < x < t , t > 0 , 1 , x > t > 0 ,
is a weak solution of the Burgers equation
∂ ∂t
u +
∂x
u^2 = 0 , x ∈ R, t > 0 , (∗)
with initial data
u(x, 0) =
0 , x < 0 , 1 , x > 0.
(c) Let u = u(x, t), x ∈ R, t > 0 be a piecewise C^1 -function with a jump discontinuity along the curve
Γ : x = s(t)
and let u solve the Burgers equation (∗) on both sides of Γ. Prove that u is a weak solution of (1) if and only if
s˙(t) =
(ul(t) + ur(t))
holds, where ul(t), ur(t) are the one-sided limits
ul(t) = lim xրs(t)−^ u(x, t) , ur(t) = lim xցs(t)+^ u(x, t).
[Hint: Multiply the equation by a test function φ ∈ C 0 ∞ (R × [0, ∞)), split the integral appropriately and integrate by parts. Consider how the unit normal vector along Γ can be expressed in terms of s˙.]
Part II, Paper 1
31C Asymptotic Methods For λ > 0 let
I(λ) =
∫ (^) b
0
f (x) e−λx^ dx , with 0 < b < ∞.
Assume that the function f (x) is continuous on 0 < x 6 b, and that
f (x) ∼ xα
∑^ ∞
n=
an xnβ^ ,
as x → (^0) + , where α > −1 and β > 0.
(a) Explain briefly why in this case straightforward partial integrations in general cannot be applied for determining the asymptotic behaviour of I(λ) as λ → ∞.
(b) Derive with proof an asymptotic expansion for I(λ) as λ → ∞.
(c) For the function
B(s, t) =
0
us−^1 (1 − u)t−^1 du , s, t > 0 ,
obtain, using the substitution u = e−x, the first two terms in an asymptotic expansion as s → ∞. What happens as t → ∞?
[Hint: The following formula may be useful
Γ(y) =
0
xy−^1 e−x^ dt , for x > 0. ]
Part II, Paper 1 [TURN OVER
33C Principles of Quantum Mechanics Two states |j 1 m 1 〉 1 , |j 2 m 2 〉 2 , with angular momenta j 1 , j 2 , are combined to form states |J M 〉 with total angular momentum
J = |j 1 − j 2 |, |j 1 − j 2 | + 1 ,... , j 1 + j 2.
Write down the state with J = M = j 1 + j 2 in terms of the original angular momentum states. Briefly describe how the other combined angular momentum states may be found in terms of the original angular momentum states.
If j 1 = j 2 = j, explain why the state with J = 0 must be of the form
∑^ j
m=−j
αm|j m〉 1 |j −m〉 2.
By considering J+|0 0〉, determine a relation between αm+1 and αm, hence find αm.
If the system is in the state |j j〉 1 |j −j〉 2 what is the probability, written in terms of j, of measuring the combined total angular momentum to be zero?
[Standard angular momentum states |j m〉 are joint eigenstates of J^2 and J 3 , obeying
J±|j m〉 =
(j ∓ m)(j ± m + 1) |j m± 1 〉.
Units in which ℏ = 1 have been used throughout.]
Part II, Paper 1 [TURN OVER
34B Applications of Quantum Mechanics Give an account of the variational principle for establishing an upper bound on the ground-state energy, E 0 , of a particle moving in a potential V (x) in one dimension.
Explain how an upper bound on the energy of the first excited state can be found in the case that V (x) is a symmetric function.
A particle of mass 2m = ℏ^2 moves in the potential
V (x) = −V 0 e −x 2 , V 0 > 0.
Use the trial wavefunction ψ(x) = e −^
1 2 ax^2 ,
where a is a positive real parameter, to establish the upper bound E 0 6 E(a) for the energy of the ground state, where
E(a) =
a − V 0
a √ 1 + a
Show that, for a > 0, E(a) has one zero and find its position.
Show that the turning points of E(a) are given by
(1 + a)^3 =
V 02
a
and deduce that there is one turning point in a > 0 for all V 0 > 0.
Sketch E(a) for a > 0 and hence deduce that V (x) has at least one bound state for all V 0 > 0.
For 0 < V 0 ≪ 1 show that
−V 0 < E 0 6 ǫ(V 0 ) ,
where ǫ(V 0 ) = − 12 V 02 + O(V 04 ).
[You may use the result that
−∞ e^ −bx^2 dx = √^ π b for^ b >^ 0.]
Part II, Paper 1
