Part II, 2002, Paper 3
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B3.10
commentLet be an irreducible polynomial of degree (over an algebraically closed field of characteristic zero) and the corresponding affine plane curve. Assume that is smooth (non-singular) and that the projectivization of intersects the line at infinity in distinct points. Show that is smooth and determine the divisor of the rational differential on . Deduce a formula for the genus of .
B3.7
commentFor a finite simplicial complex , let denote the rank of the finitely generated abelian group . Define the Euler characteristic by the formula
Let denote the number of -simplices in , for each . Show that
A3.10
comment(i) Consider the unconstrained geometric programme GP
subject to .
State the dual problem to GP. Give a careful statement of the AM-GM inequality, and use it to prove the primal-dual inequality for GP.
(ii) Define min-path and max-tension problems. State and outline the proof of the max-tension min-path theorem.
A company has branches in five cities and . The fares for direct flights between these cities are as follows:
\begin{tabular}{l|lllll} & & & & & \ \hline & & 50 & 40 & 25 & 10 \ & 50 & & 20 & 90 & 25 \ & 40 & 20 & & 10 & 25 \ & 25 & 90 & 10 & & 55 \ & 10 & 25 & 25 & 55 & \end{tabular}
Formulate this as a min-path problem. Illustrate the max-tension min-path algorithm by finding the cost of travelling by the cheapest routes between and each of the other cities.
B3.23
commentA periodic potential is expressed as , where are reciprocal lattice vectors and . In the nearly free electron model explain why it is appropriate, near the boundaries of energy bands, to consider a Bloch wave state
where is a free electron state for wave vector , and the sum is restricted to reciprocal lattice vectors such that . Obtain a determinantal formula for the possible energies corresponding to Bloch wave states of this form.
[You may take and assume for any .]
Suppose the sum is restricted to just and . Show that there is a gap between energy bands. Setting , show that there are two Bloch wave states with energies near the boundaries of the energy bands
What is meant by effective mass? Determine the value of the effective mass at the top and the bottom of the adjacent energy bands if is parallel to .
B3.13
comment(a) Define a renewal process and a discrete renewal process.
(b) State and prove the Discrete Renewal Theorem.
(c) The sequence satisfies
for some collection of non-negative numbers summing to 1 . Let . Show that
Give a probabilistic interpretation of the numbers and .
(d) Let the sequence be given by
How is this related to the simple symmetric random walk on the integers starting from the origin, and its subsequent returns to the origin? Determine in this case, either by calculating or by showing that satisfies the quadratic equation
B3.17
commentLet be a finite alphabet of letters and either the semi-infinite space or the doubly infinite space of sequences whose elements are drawn from . Define the natural topology on . If is a set of words, denote by the subspace of consisting of those sequences none of whose subsequences is in . Prove that is a closed subspace of ; and state and prove a necessary and sufficient condition for a closed subspace of to have the form for some .
what is the space ?
A3.6 B3.4
comment(i) Define the Floquet multiplier and Liapunov exponent for a periodic orbit of a dynamical system in . Show that one multiplier is always unity, and that the other is given by
where is the period of the orbit.
The Van der Pol oscillator has a limit cycle . Show using that this orbit is stable.
(ii) Show, by considering the normal form for a Hopf bifurcation from a fixed point of a dynamical system , that in some neighbourhood of the bifurcation the periodic orbit is stable when it exists in the range of for which is unstable, and unstable in the opposite case.
Now consider the system
Show that the fixed point has a Hopf bifurcation when , and is unstable (stable) when .
Suppose that a periodic orbit exists in . Show without solving for the orbit that the result of part (i) shows that such an orbit is unstable. Define a similar result for .
What do you conclude about the existence of periodic orbits when ? Check your answer by applying Dulac's criterion to the system, using the weighting .
A3.5 B3.3
comment(i) A plane electromagnetic wave in a vacuum has an electric field
referred to cartesian axes . Show that this wave is plane polarized and find the orientation of the plane of polarization. Obtain the corresponding plane polarized magnetic field and calculate the rate at which energy is transported by the wave.
(ii) Suppose instead that
with a constant, . Show that, if the axes are now rotated through an angle so as to obtain an elliptically polarized wave with an electric field
then
Show also that if there is an elliptically polarized wave with
B3.24
comment(i) Suppose that, with spherical polar coordinates, the Stokes streamfunction
represents a Stokes flow and thus satisfies the equation , where
Show that the possible values of are and . For which of these values is the corresponding flow irrotational? Sketch the streamlines of the flow for the case .
(ii) A spherical drop of liquid of viscosity , radius and centre at , is suspended in another liquid of viscosity which flows with streamfunction
far from the drop. The two liquids are of equal densities, surface tension is sufficiently strong to keep the drop spherical, and inertia is negligible. Show that
and obtain four equations determining the constants and . (You need not solve these equations.)
[You may assume, with standard notation, that
A3.13 B3.21
comment(i) Two particles with angular momenta and basis states are combined to give total angular momentum and basis states . State the possible values of and show how a state with can be constructed. Briefly describe, for a general allowed value of , what the Clebsch-Gordan coefficients are.
(ii) If the angular momenta and are both 1 show that the combined state is
Determine the corresponding expressions for the combined states and , assuming that they are respectively antisymmetric and symmetric under interchange of the two particles.
If the combined system is in state what is the probability that measurements of the -component of angular momentum for either constituent particle will give the value of 1 ?
Hint:
A3.3 B3.2
comment(i) Suppose that is a decreasing sequence of continuous real-valued functions on a compact metric space which converges pointwise to 0 . By considering sets of the form , for , or otherwise, show that converges uniformly to 0 .
Can the condition that is decreasing be dropped? Can the condition that is compact be dropped? Justify your answers.
(ii) Suppose that is a positive integer. Define polynomials recursively by
Show that , for , and show that converges to uniformly on .
[You may wish to use the identity .]
Suppose that is a closed subalgebra of the algebra of continuous real-valued functions on a compact metric space , equipped with the uniform norm, and suppose that has the property that for each there exists with . Show that there exists such that for all .
Show that for each positive integer , and show that contains the constant functions.
B3.6
commentShow that the polynomial has no rational roots. Show that the splitting field of over the finite field is an extension of degree 4 . Hence deduce that is irreducible over the rationals. Prove that has precisely two (non-multiple) roots over the finite field . Find the Galois group of over the rationals.
[You may assume any general results you need including the fact that is the only index 2 subgroup of .]
A3.7
comment(i) State what it means for surfaces and to be isometric.
Let be a surface, a diffeomorphism, and let
State a formula comparing the first fundamental forms of and .
(ii) Give a proof of the formula referred to at the end of part (i). Deduce that "isometry" is an equivalence relation.
The catenoid and the helicoid are the surfaces defined by
and
Show that the catenoid and the helicoid are isometric.
A3.4
comment(i) What does it mean for a ring to be Noetherian? State Hilbert's Basis Theorem. Give an example of a Noetherian ring which is not a principal ideal domain.
(ii) Prove Hilbert's Basis Theorem.
Is it true that if the ring is Noetherian, then so is
B3.8
commentLet be an infinite-dimensional, separable Hilbert space. Let be a compact linear operator on , and let be a non-zero, approximate eigenvalue of . Prove that is an eigenvalue, and that the corresponding eigenspace is finite-dimensional.
Let be a compact, self-adjoint operator on . Prove that there is an orthonormal basis of , and a sequence in , such that (i) and (ii) as .
Now let be compact, self-adjoint and injective. Let be a bounded self-adjoint operator on such that . Prove that has an orthonormal basis , where, for every is an eigenvector, both of and of .
[You may assume, without proof, results about self-adjoint operators on finite-dimensional spaces.]
A3.8 B3.11
comment(i) Explain briefly what is meant by the terms register machine and computable function.
Let be the universal computable function and a total computable function with . Here and are the unary and binary functions computed by the -th register machine program . Suppose is a total computable function. By considering the function
show that there is a number such that .
(ii) Let be the set of all partial functions . Consider the mapping defined by
(a) Show that any fixed point of is a total function . Deduce that has a unique fixed point.
[The Bourbaki- Witt Theorem may be assumed if stated precisely.]
(b) It follows from standard closure properties of the computable functions that there is a computable function such that
Assuming this, show that there is a total computable function such that
Deduce that the fixed point of is computable.
A3.1 B3.1
comment(i) Consider the continuous-time Markov chain on with generator matrix
Compute the probability, starting from state 3 , that hits state 2 eventually.
Deduce that
[Justification of standard arguments is not expected.]
(ii) A colony of cells contains immature and mature cells. Each immature cell, after an exponential time of parameter 2, becomes a mature cell. Each mature cell, after an exponential time of parameter 3, divides into two immature cells. Suppose we begin with one immature cell and let denote the expected number of immature cells at time . Show that
A3.17
comment(i) State the Fredholm alternative for Fredholm integral equations of the second kind.
Show that the integral equation
where is a continuous function, has a unique solution for if . Derive this solution.
(ii) Describe the WKB method for finding approximate solutions of the equation
where is an arbitrary non-zero, differentiable function and is a small parameter. Obtain these solutions in terms of an exponential with slowly varying exponent and slowly varying amplitude.
Hence, by means of a suitable change of independent variable, find approximate solutions of the equation
in , where is a large parameter.
B3.19
commentShow that the equation
has solutions of the form
Give examples of possible choices of to provide two independent solutions, assuming . Distinguish between the cases and . Comment on the case and on the case that is an odd integer.
Show that, if , there is a solution that is bounded as , and that, in this limit,
where is a constant.
A3.18
comment(i) Show that the equation
has two solutions which are independent of both and . Show that one of these is linearly stable. Show that the other solution is linearly unstable, and find the range of wavenumbers that exhibit the instability.
Sketch the nonlinear evolution of the unstable solution after it receives a small, smooth, localized perturbation in the direction towards the stable solution.
(ii) Show that the equations
are a Bäcklund pair for the equations
By choosing to be a suitable constant, and using the Bäcklund pair, find a solution of the equation
which is non-singular in the region of the plane and has the value at .
A3.9
comment(i) Let denote the number of primes , where is a positive real number. State and prove Legendre's formula relating to . Use this formula to compute
(ii) Let , where is a real number greater than 1 . Prove the following two assertions rigorously, assuming always that . (a) , where the product is taken over all primes ; (b) .
Explain why (b) enables us to define for . Deduce from (b) that .
A3.19 B3.20
comment(i) Determine the order of the multistep method
for the solution of ordinary differential equations for different choices of in the range .
(ii) Prove that no such choice of results in a method whose linear stability domain includes the interval .
B3.14
commentConsider a scalar system with , where is a sequence of independent random variables, uniform on the interval , with . We wish to choose to minimize the expected value of
where is chosen knowing but not . Prove that the minimal expected cost can be written and derive a recurrence for calculating .
How does your answer change if is constrained to lie in the set
Consider a stopping problem for which there are two options in state :
(1) stop: paying a terminal cost ; no further costs are incurred;
(2) continue: choosing , paying , and moving to state
Consider the problem of minimizing total expected cost subject to the constraint that no more than continuation steps are allowed. Suppose . Show that an optimal policy stops if and only if either continuation steps have already been taken or .
[Hint: Use induction on to show that a one-step-look-ahead rule is optimal. You should not need to find the optimal for the continuation steps.]
B3.18
commentDefine the Schwartz space and the space of tempered distributions . State the Fourier inversion theorem for the Fourier transform of a Schwartz function.
Consider the initial value problem:
for in the Schwartz space .
Show that the solution can be written as
where
and
State the Plancherel-Parseval theorem and hence deduce that
A3.2
comment(i) Show that Hamilton's equations follow from the variational principle
under the restrictions . Comment on the difference from the variational principle for Lagrange's equations.
(ii) Suppose we transform from and to and , with
where is the new Hamiltonian. Show that and obey Hamilton's equations with Hamiltonian .
Show that the time independent generating function takes the Hamiltonian
to harmonic oscillator form. Show that and obey the Poisson bracket relation
A3.12 B3.15
comment(i) Describe in detail how to perform the Wald, score and likelihood ratio tests of a simple null hypothesis given a random sample from a regular oneparameter density . In each case you should specify the asymptotic null distribution of the test statistic.
(ii) Let be an independent, identically distributed sample from a distribution , and let be an estimator of a parameter of .
Explain what is meant by: (a) the empirical distribution function of the sample; (b) the bootstrap estimator of the bias of , based on the empirical distribution function. Explain how a bootstrap estimator of the distribution function of may be used to construct an approximate confidence interval for .
Suppose the parameter of interest is , where is the mean of , and the estimator is , where is the sample mean.
Derive an explicit expression for the bootstrap estimator of the bias of and show that it is biased as an estimator of the true bias of .
Let be the value of the estimator computed from the sample of size obtained by deleting and let . The jackknife estimator of the bias of is
Derive the jackknife estimator for the case , and show that, as an estimator of the true bias of , it is unbiased.
B3.12
commentDerive the characteristic function of a real-valued random variable which is normally distributed with mean and variance . What does it mean to say that an -valued random variable has a multivariate Gaussian distribution? Prove that the distribution of such a random variable is determined by its -valued) mean and its covariance matrix.
Let and be random variables defined on the same probability space such that has a Gaussian distribution. Show that and are independent if and only if . Show that, even if they are not independent, one may always write for some constant and some random variable independent of .
[The inversion theorem for characteristic functions and standard results about independence may be assumed.]
B3.5
commentLet be a finite group acting on a finite set . Define the permutation representation of and compute its character . Prove that equals the number of orbits of on . If acts also on the finite set , with character , show that equals the number of orbits of on .
Now let be the symmetric group acting naturally on the set , and let be the set of all -element subsets of . Let be the permutation character of on . Prove that
Deduce that the class functions
are irreducible characters of , for .
B3.9
commentLet be two non-zero complex numbers with . Let be the lattice . A meromorphic function on is elliptic if , for all and . The Weierstrass functions are defined by the following properties:
is elliptic, has double poles at the points of and no other poles, and near 0
, and near 0 ;
is odd, and , and as .
Prove the following
(a) , and hence and , are uniquely determined by these properties. You are not expected to prove the existence of , and you may use Liouville's theorem without proof.
(b) , and , for some constants .
(c) is holomorphic, has simple zeroes at the points of , and has no other zeroes.
(d) Given and in with , the function
is elliptic.
(e) .
(f) Deduce from (e), or otherwise, that .
B3.22
commentA system consisting of non-interacting bosons has single-particle levels uniquely labelled by with energies . Show that the free energy in the grand canonical ensemble is
What is the maximum value for ?
A system of bosons in a large volume has one energy level of energy zero and a large number of energy levels of the same energy , where takes the form with a positive constant. What are the dimensions of
Show that the free energy is
The numbers of particles with energies are respectively . Write down expressions for in terms of .
At temperature what is the maximum number of bosons in the normal phase (the state with energy )? Explain what happens when .
Given and calculate the transition temperature at which Bose condensation occurs.
For show that . What is the value of for ?
Calculate the mean energy for (a) (b) , and show that the heat capacity of the system at constant volume is
A3.14
comment(i) Write down the first law of thermodynamics for the change in the internal energy of a gas of particles in a volume with entropy .
Given that
where is the pressure, use the first law to show that is constant at constant and
Write down the Boyle-Charles law for a non-relativistic ideal gas and hence deduce that the temperature is proportional to at constant and .
State the principle of equipartition of energy and use it to deduce that
Hence deduce the value of . Show that this value of is such that the ratio is unchanged by a change of volume at constant and , where is the energy of the -th one particle eigenstate of a non-relativistic ideal gas.
(ii) A classical gas of non-relativistic particles of mass at absolute temperature and number density has a chemical potential
where is the particle's spin degeneracy factor. What condition on is needed for the validity of this formula and why?
Thermal and chemical equilibrium between two species of non-relativistic particles and is maintained by the reaction
where and are massless particles with zero chemical potential. Given that particles and have masses and respectively, but equal spin degeneracy factors, find the number density ratio as a function of and . Given that but show that
for some function which you should determine.
Explain how a reaction of the above type is relevant to a determination of the neutron to proton ratio in the early universe and why this ratio does not fall rapidly to zero as the universe cools. Explain briefly the process of primordial nucleosynthesis by which neutrons are converted into stable helium nuclei. Let
be the fraction of the universe that ends up in helium. Compute as a function of the ratio at the time of nucleosynthesis.
A3.11 B3.16
comment(i) Explain briefly what it means to say that a stochastic process is a standard Brownian motion.
Let be a standard Brownian motion and let be real numbers. What condition must and satisfy to ensure that the process is a martingale? Justify your answer carefully.
(ii) At the beginning of each of the years an investor has income , of which he invests a proportion , and consumes the rest during the year. His income at the beginning of the next year is
where are independent positive random variables with finite means and is a constant. He decides on after he has observed both and at the beginning of year , but at that time he does not have any knowledge of the value of , for any . The investor retires in year and consumes his entire income during that year. He wishes to determine the investment policy that maximizes his expected total consumption
Prove that the optimal policy may be expressed in terms of the numbers , where , for , and determine the optimal expected total consumption.
A3.15
(i) Let denote the symmetry group of rotations and reflections of a regular hexagon. The elements of are given by with and . The conjugacy classes of are and .
Show that the character table of is
\begin{tabular}{l|rrrrrr} & & & & & & \ \hline & 1 & 1 & 1 & 1 & 1 & 1 \ & 1 & 1 & 1 & 1 & & \ & 1 & & 1 & & 1 & \ & 1 & & 1 & & & 1 \ & 2 & 1 & & & 0 & 0 \ & 2 & & & 2 & 0 & 0 \end{tabular}
(ii) Show that the character of an rotation with angle in the dimensional irreducible representation of is given by
For a hexagonal crystal of atoms find how the degeneracy of the D-wave orbital states in the atomic central potential is split by the crystal potential with symmetry and give the new degeneracies.
By using the fact that