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MATHEMATICAL TRIPOS Part II Alternative A
Thursday 6 June 2002 1.30 to 4.
PAPER 4
Before you begin read these instructions carefully.
Candidates must not attempt more than FOUR questions. If you submit answers to more than four questions, your lowest scoring attempt(s) will be rejected.
The number of marks for each question is the same. Additional credit will be given for a substantially complete answer.
Write legibly and on only one side of the paper.
Begin each answer on a separate sheet.
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, 19C, 21C should be in one bundle and 12L, 14L 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’s examination number and desk number.
1M Markov Chains
Write an essay on the long-time behaviour of discrete-time Markov chains on a finite state space. Your essay should include discussion of the convergence of probabilities as well as almost-sure behaviour. You should also explain what happens when the chain is not irreducible.
2F Principles of Dynamics
Explain how the orientation of a rigid body can be specified by means of the three Eulerian angles, θ, φ and ψ.
An axisymmetric top of mass M has principal moments of inertia A, A and C, and is spinning with angular speed n about its axis of symmetry. Its centre of mass lies a distance h from the fixed point of support. Initially the axis of symmetry points vertically upwards. It then suffers a small disturbance. For what values of the spin is the initial configuration stable?
If the spin is such that the initial configuration is unstable, what is the lowest angle reached by the symmetry axis in the nutation of the top? Find the maximum and minimum values of the precessional angular velocity φ˙.
3K Functional Analysis Define the distribution function Φf of a non-negative measurable function f on the interval I = [0, 1]. Show that Φf is a decreasing non-negative function on [0, ∞] which is continuous on the right.
Define the Lebesgue integral
I f dm. Show that^
I f dm^ = 0 if and only if^ f^ = 0 almost everywhere.
Suppose that f is a non-negative Riemann integrable function on [0, 1]. Show that there are an increasing sequence (gn) and a decreasing sequence (hn) of non-negative step
functions with gn 6 f 6 hn such that
0 (hn(x)^ −^ gn(x))^ dx^ →^ 0. Show that the functions g = limn gn and h = limn hn are equal almost everywhere, that f is measurable and that the Lebesgue integral
I f dm^ is equal to the Riemann integral
0 f^ (x)^ dx. Suppose that j is a Riemann integrable function on [0, 1] and that j(x) > 0 for all
x. Show that
0 j(x)^ dx >^ 0.
Paper 4
6F Dynamics of Differential Equations
Define the terms homoclinic orbit, heteroclinic orbit and heteroclinic loop. In the case of a dynamical system that possesses a homoclinic orbit, explain, without detailed calculation, how to calculate its stability.
A second order dynamical system depends on two parameters μ 1 and μ 2. When μ 1 = μ 2 = 0 there is a heteroclinic loop between the points P 1 , P 2 as in the diagram.
When μ 1 , μ 2 are small there are trajectories that pass close to the fixed points P 1 , P 2 :
By adapting the method used above for trajectories near homoclinic orbits, show that the distances yn, yn+1 to the stable manifold at P 1 on successive returns are related to zn, zn+1, the corresponding distances near P 2 , by coupled equations of the form
zn = (yn)γ^1 + μ 1 , yn+1 = (zn)γ^2 + μ 2 ,
where any arbitrary constants have been removed by rescaling, and γ 1 , γ 2 depend on conditions near P 1 , P 2. Show from these equations that there is a stable heteroclinic orbit (μ 1 = μ 2 = 0) if γ 1 γ 2 > 1. Show also that in the marginal situation γ 1 = 2, γ 2 = 12 there can be a stable fixed point for small positive y, z if μ 2 < 0, μ^22 < μ 1. Explain carefully the form of the orbit of the original dynamical system represented by the solution of the above map when μ^22 = μ 1.
7K Geometry of Surfaces Write an essay on the Euler number of topological surfaces. Your essay should include a definition of subdivision, some examples of surfaces and their Euler numbers, and a discussion of the statement and significance of the Gauss–Bonnet theorem.
Paper 4
8J Logic, Computation and Set Theory
Let P be a set of primitive propositions. Let L(P ) denote the set of all compound propositions over P , and let S be a subset of L(P ). Consider the relation S on L(P ) defined by s S t if and only if S ∪ {s} ` t.
Prove that S is reflexive and transitive. Deduce that if we define ≈S by (s ≈S t if and only if s S t and t S s), then ≈S is an equivalence relation and the quotient BS = L(P )/ ≈S is partially ordered by the relation (^6) S induced by (^4) S (that is, [s] (^6) S [t] if and only if s (^4) S t, where square brackets denote equivalence classes).
Assuming the result that BS is a Boolean algebra with lattice operations induced by the logical operations on L(P ) (that is, [s] ∧ [t] = [s ∧ t], etc.), show that there is a bijection between the following two sets: (a) The set of lattice homomorphisms BS → { 0 , 1 }. (b) The set of models of the propositional theory S.
Deduce that the completeness theorem for propositional logic is equivalent to the assertion that, for any Boolean algebra B with more than one element, there exists a homomorphism B → { 0 , 1 }.
[You may assume the result that the completeness theorem implies the compactness theorem.]
9H Graph Theory Write an essay on connectivity in graphs.
Your essay should include proofs of at least two major theorems, along with a discussion of one or two significant corollaries.
10J Number Theory
Write an essay on quadratic reciprocity. Your essay should include (i) a proof of the law of quadratic reciprocity for the Legendre symbol, (ii) a proof of the law of quadratic reciprocity for the Jacobi symbol, and (iii) a comment on why this latter law is useful in primality testing.
11M Algorithms and Networks
Write an essay on Strong Lagrangian problems. You should give an account of dual- ity and how it relates to the Strong Lagrangian property. In particular, establish carefully the relationship between the Strong Lagrangian property and supporting hyperplanes.
Also, give an example of a class of problems that are Strong Lagrangian. [You should explain carefully why your example has the Strong Lagrangian property.]
Paper 4 [TURN OVER
Sulphur dioxide is one of the major air pollutants. A data-set presented by Sokal and Rohlf (1981) was collected on 41 US cities in 1969-71, corresponding to the following variables:
Y = sulphur dioxide content of air in micrograms per cubic metre
X1 = average annual temperature in degrees Fahrenheit X2 = number of manufacturing enterprises employing 20 or more workers
X3 = population size (1970 census) in thousands
X4 = average annual wind speed in miles per hour X5 = average annual precipitation in inches
X6 = average annual of days with precipitation per year.
Interpret the R output that follows below, quoting any standard theorems that you need to use.
next.lm lm(log(Y) ∼ X1 + X2 + X3 + X4 + X5 + X6)
summary(next.lm)
Call: lm(formula = log(Y) ∼ X1 + X2 + X3 + X4 + X5 + X6)
Residuals: Min 1Q Median 3Q Max -0.79548 -0.25538 -0.01968 0.28328 0.
Coefficients: Estimate Std. Error t value Pr(> |t|) (Intercept) 7.2532456 1.4483686 5.008 1.68e-05 *** X1 -0.0599017 0.0190138 -3.150 0.00339 ** X2 0.0012639 0.0004820 2.622 0.01298 * X3 -0.0007077 0.0004632 -1.528 0. X4 -0.1697171 0.0555563 -3.055 0.00436 ** X5 0.0173723 0.0111036 1.565 0. X6 0.0004347 0.0049591 0.088 0.
Signif. codes: 0 ‘’ 0.001 ‘’ 0.01 ‘’ 0.05 ‘.’
Residual standard error: 0.448 on 34 degrees of freedom
Multiple R-Squared: 0. F-statistic: 10.72 on 6 and 34 degrees of freedom, p-value: 1.126e-
Paper 4 [TURN OVER
15E Foundations 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 π 0 r
has normalized, spherically symmetric, real wave functions ψn(r) and energy eigenvalues En with E 0 < E 1 < E 2 < · · ·. What are the consequences of the Pauli exclusion principle for the ground state of the helium atom? Assuming that wavefunctions which are not spherically symmetric can be ignored, what are the states of the first excited energy level of the helium atom? [You may assume here that the electrons are non-interacting. ]
Show that, taking into account the interaction between the two electrons, the estimate for the energy of the ground state of the helium atom is
2 E 0 +
e^2 4 π 0
d^3 r 1 d^3 r 2 |r 1 − r 2 |
ψ^20 (r 1 )ψ^20 (r 2 ).
Paper 4
18D Statistical Physics and Cosmology
What is an ideal gas? Explain how the microstates of an ideal gas of indistinguish- able particles can be labelled by a set of integers. What range of values do these integers take for (a) a boson gas and (b) a Fermi gas?
Let Ei be the energy of the i-th one-particle energy eigenstate of an ideal gas in thermal equilibrium at temperature T and let pi(ni) be the probability that there are ni particles of the gas in this state. Given that
pi(ni) = e−βEini^ /Zi (β =
kT
determine the normalization factor Zi for (a) a boson gas and (b) a Fermi gas. Hence obtain an expression for ¯ni, the average number of particles in the i-th one-particle energy eigenstate for both cases (a) and (b).
In the case of a Fermi gas, write down (without proof) the generalization of your formula for ¯ni to a gas at non-zero chemical potential μ. Show how it leads to the concept of a Fermi energy F for a gas at zero temperature. How is F related to the Fermi momentum pF for (a) a non-relativistic gas and (b) an ultra-relativistic gas?
In an approximation in which the discrete set of energies Ei is replaced with a continuous set with momentum p, the density of one-particle states with momentum in the range p to p + dp is g(p)dp. Explain briefly why
g(p) ∝ p^2 V, (∗)
where V is the volume of the gas. Using this formula, obtain an expression for the total energy E of an ultra-relativistic gas at zero chemical potential as an integral over p. Hence show that E V
∝ T α,
where α is a number that you should compute. Why does this result apply to a photon gas?
Using the formula (∗) for a non-relativistic Fermi gas at zero temperature, obtain an expression for the particle number density n in terms of the Fermi momentum and provide a physical interpretation of this formula in terms of the typical de Broglie wavelength. Obtain an analogous formula for the (internal) energy density and hence show that the pressure P behaves as P ∝ nγ
where γ is a number that you should compute. [You need not prove any relation between the pressure and the energy density you use.] What is the origin of this pressure given that T = 0 by assumption? Explain briefly and qualitatively how it is relevant to the stability of white dwarf stars.
Paper 4
19C Transport Processes
(a) A biological vessel is modelled two-dimensionally as a fluid-filled channel bounded by parallel plane walls y = ±a, embedded in an infinite region of fluid-saturated tissue. In the tissue a solute has concentration Cout(y, t), diffuses with diffusivity D and is consumed by biological activity at a rate kCout^ per unit volume, where D and k are constants. By considering the solute balance in a slice of tissue of infinitesimal thickness, show that
Ctout = DCyyout − kCout.
A steady concentration profile Cout(y) results from a flux β
Cin^ − Caout
, per unit area of wall, of solute from the channel into the tissue, where Cin^ is a constant concentration of solute that is maintained in the channel and Couta = Cout(a). Write down the boundary conditions satisfied by Cout(y). Solve for Cout(y) and show that
Caout =
γ γ + 1
Cin, (∗)
where γ = β/
kD.
(b) Now let the solute be supplied by steady flow down the channel from one end, x = 0, with the channel taken to be semi-infinite in the x-direction. The cross-sectionally averaged velocity in the channel u(x) varies due to a flux of fluid from the tissue to the channel (by osmosis) equal to λ
Cin^ − Couta
per unit area. Neglect both the variation of Cin(x) across the channel and diffusion in the x-direction.
By considering conservation of fluid, show that
aux = λ
Cin^ − Caout
and write down the corresponding equation derived from conservation of solute. Deduce that u(λCin^ + β) = u 0 (λC 0 in + β) ,
where u 0 = u(0) and C 0 in = Cin(0).
Assuming that equation (∗) still holds, even though Cout^ is now a function of x as well as y, show that u(x) satisfies the ordinary differential equation
(γ + 1)auux + βu = u 0
λCin 0 + β
Find scales ˆx and ˆu such that the dimensionless variables U = u/ˆu and X = x/ˆx satisfy U UX + U = 1.
Derive the solution (1 − U )eU^ = Ae−X^ and find the constant A.
To what values do u and Cin tend as x → ∞?
Paper 4 [TURN OVER
21C Mathematical Methods
State Watson’s lemma giving an asymptotic expansion as λ → ∞ for an integral of the form
I 1 =
∫ A
0
f (t)e−λtdt , A > 0.
Show how this result may be used to find an asymptotic expansion as λ → ∞ for an integral of the form
I 2 =
∫ B
−A
f (t)e−λt
2 dt , A > 0 , B > 0.
Hence derive Laplace’s method for obtaining an asymptotic expansion as λ → ∞ for an integral of the form
I 3 =
∫ (^) b
a
f (t)eλφ(t)dt ,
where φ(t) is differentiable, for the cases: (i) φ′(t) < 0 in a ≤ t ≤ b; and (ii) φ′(t) has a simple zero at t = c with a < c < b and φ′′(c) < 0.
Find the first two terms in the asymptotic expansion as x → ∞ of
I 4 =
−∞
log(1 + t^2 )e−xt
2 dt.
[You may leave your answer expressed in terms of Γ-functions.]
22F Numerical Analysis
Write an essay on the method of conjugate gradients. You should describe the algorithm, present an analysis of its properties and discuss its advantages.
[Any theorems quoted should be stated precisely but need not be proved.]
Paper 4
