Part II, 2021, Paper 4
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Paper 4, Section II, I
commentLet be a smooth irreducible projective algebraic curve over an algebraically closed field.
Let be an effective divisor on . Prove that the vector space of rational functions with poles bounded by is finite dimensional.
Let and be linearly equivalent divisors on . Exhibit an isomorphism between the vector spaces and .
What is a canonical divisor on ? State the Riemann-Roch theorem and use it to calculate the degree of a canonical divisor in terms of the genus of .
Prove that the canonical divisor on a smooth cubic plane curve is linearly equivalent to the zero divisor.
Paper 4, Section II, 21F
comment(a) Define the Euler characteristic of a triangulable space .
(b) Let be an orientable surface of genus . A is a doublebranched cover if there is a set of branch points, such that the restriction is a covering map of degree 2 , but for each , consists of one point. By carefully choosing a triangulation of , use the Euler characteristic to find a formula relating and .
Paper 4, Section II,
commentFix and let satisfy
(a) Let be a sequence of functions in . For , what is meant by (i) in and (ii) in ? Show that if , then
(b) Suppose that is a sequence with , and that there exists such that for all . Show that there exists and a subsequence , such that for any sequence with and , we have
Give an example to show that the result need not hold if the condition is replaced by in .
Paper 4, Section II, B
comment(a) Consider the nearly free electron model in one dimension with mass and periodic potential with and
Ignoring degeneracies, the energy spectrum of Bloch states with wavenumber is
where are normalized eigenstates of the free Hamiltonian with wavenumber . What is in this formula?
If we impose periodic boundary conditions on the wavefunctions, with and a positive integer, what are the allowed values of and ? Determine for these allowed values.
(b) State when the above expression for ceases to be a good approximation and explain why. Quoting any result you need from degenerate perturbation theory, calculate to the location and width of the band gaps.
(c) Determine the allowed energy bands for each of the potentials
(d) Briefly discuss a macroscopic physical consequence of the existence of energy bands.
Paper 4, Section II,
commentLet be a continuous-time Markov process with state space and generator satisfying for all . The local time up to time of is the random vector defined by
(a) Let be any function that is differentiable with respect to its second argument, and set
Show that
where
(b) For , write for the vector of squares of the components of . Let be a function such that whenever for some fixed . Using integration by parts, or otherwise, show that for all
where denotes .
(c) Let be a function with whenever for some fixed . Given , now let
in part (b) and deduce, using part (a), that
[You may exchange the order of integrals and derivatives without justification.]
Paper 4, Section II, A
comment(a) Classify the nature of the point at for the ordinary differential equation
(b) Find a transformation from to an equation of the form
and determine .
(c) Given satisfies ( , use the Liouville-Green method to find the first three terms in an asymptotic approximation as for , verifying the consistency of any approximations made.
(d) Hence obtain corresponding asymptotic approximations as of two linearly independent solutions of .
Paper 4, Section I,
commentState the pumping lemma for regular languages.
Which of the following languages over the alphabet are regular?
(i) .
(ii) where is the reverse of the word .
(iii) does not contain the subwords 01 or 10.
Paper 4, Section I, D
commentBriefly describe a physical object (a Lagrange top) whose Lagrangian is
Explain the meaning of the symbols in this equation.
Write down three independent integrals of motion for this system, and show that the nutation of the top is governed by the equation
where and is a certain cubic function that you need not determine.
Paper 4, Section II, 15D
comment(a) Let be a set of canonical phase-space variables for a Hamiltonian system with degrees of freedom. Define the Poisson bracket of two functions and . Write down the canonical commutation relations that imply that a second set of phase-space variables is also canonical.
(b) Consider the near-identity transformation
where and are small. Determine the approximate forms of the canonical commutation relations, accurate to first order in and . Show that these are satisfied when
where is a small parameter and is some function of the phase-space variables.
(c) In the limit this near-identity transformation is called the infinitesimal canonical transformation generated by . Let be an autonomous Hamiltonian. Show that the change in the Hamiltonian induced by the infinitesimal canonical transformation is
Explain why is an integral of motion if and only if the Hamiltonian is invariant under the infinitesimal canonical transformation generated by .
(d) The Hamiltonian of the gravitational -body problem in three-dimensional space is
where and are the mass, position and momentum of body . Determine the form of and the infinitesimal canonical transformation that correspond to the translational symmetry of the system.
Paper 4 , Section I,
commentDescribe the Rabin scheme for coding a message as modulo a certain integer .
Describe the RSA encryption scheme with public key and private key .
[In both cases you should explain how you encrypt and decrypt.]
Give an advantage and a disadvantage that the Rabin scheme has over the RSA scheme.
Paper 4, Section I, B
commentA collection of particles, with masses and positions , interact through a gravitational potential
Assume that the system is gravitationally bound, and that the positions and velocities are bounded for all time. Further, define the time average of a quantity by
(a) Assuming that the time average of the kinetic energy and potential energy are well defined, show that
[You should consider the quantity , with all measured relative to the centre of mass.]
(b) Explain how part (a) can be used, together with observations, to provide evidence in favour of dark matter. [You may assume that time averaging may be replaced by an average over particles.]
Paper 4, Section II, F
commentLet be an interval, and be a surface. Assume that is a regular curve parametrised by arc-length. Define the geodesic curvature of . What does it mean for to be a geodesic curve?
State the global Gauss-Bonnet theorem including boundary terms.
Suppose that is a surface diffeomorphic to a cylinder. How large can the number of simple closed geodesics on be in each of the following cases?
(i) has Gaussian curvature everywhere zero;
(ii) has Gaussian curvature everywhere positive;
(iii) has Gaussian curvature everywhere negative.
In cases where there can be two or more simple closed geodesics, must they always be disjoint? Justify your answer.
[A formula for the Gaussian curvature of a surface of revolution may be used without proof if clearly stated. You may also use the fact that a piecewise smooth curve on a cylinder without self-intersections either bounds a domain homeomorphic to a disc or is homotopic to the waist-curve of the cylinder.]
Paper 4, Section II, A
comment(a) A continuous map of an interval into itself has a periodic orbit of period 3 . Prove that also has periodic orbits of period for all positive integers .
(b) What is the minimum number of distinct orbits of of periods 2,4 and 5 ? Explain your reasoning with a directed graph. [Formal proof is not required.]
(c) Consider the piecewise linear map defined by linear segments between and . Calculate the orbits of periods 2,4 and 5 that are obtained from the directed graph in part (b).
[In part (a) you may assume without proof:
(i) If and are non-empty closed bounded intervals such that then there is a closed bounded interval such that .
(ii) The Intermediate Value Theorem.]
Paper 4, Section II, 36C
comment(a) Define the electric displacement for a medium which exhibits a linear response with polarisation constant to an applied electric field with polarisation constant . Write down the effective Maxwell equation obeyed by in the timeindependent case and in the absence of any additional mobile charges in the medium. Describe appropriate boundary conditions for the electric field at an interface between two regions with differing values of the polarisation constant. [You should discuss separately the components of the field normal to and tangential to the interface.]
(b) Consider a sphere of radius , centred at the origin, composed of dielectric material with polarisation constant placed in a vacuum and subjected to a constant, asymptotically homogeneous, electric field, with as . Using the ansatz
with constants and to be determined, find a solution to Maxwell's equations with appropriate boundary conditions at .
(c) By comparing your solution with the long-range electric field due to a dipole consisting of electric charges located at displacements find the induced electric dipole moment of the dielectric sphere.
Paper 4, Section II, A
commentConsider a steady axisymmetric flow with components in cylindrical polar coordinates , where is a positive constant. The fluid has density and kinematic viscosity .
(a) Briefly describe the flow and confirm that it is incompressible.
(b) Show that the vorticity has one component , in the direction. Write down the corresponding vorticity equation and derive the solution
Hence find and show that it has a maximum at some finite radius , indicating how scales with and .
(c) Find an expression for the net advection of angular momentum, prv, into the finite cylinder defined by and . Show that this is always positive and asymptotes to the value
as
(d) Show that the torque exerted on the cylinder of part (c) by the exterior flow is always negative and demonstrate that it exactly balances the net advection of angular momentum. Comment on why this has to be so.
[You may assume that for a flow in cylindrical polar coordinates
Paper 4 , Section I, 7E
comment(a) Explain in general terms the meaning of the Papperitz symbol
State a condition satisfied by and . [You need not write down any differential equations explicitly, but should provide explicit explanation of the meaning of and
(b) The Papperitz symbol
where are constants, can be transformed into
(i) Provide an explicit description of the transformations required to obtain ( from .
(ii) One of the solutions to the -equation that corresponds to is a hypergeometric function . Express and in terms of and .
Paper 4 , Section II, 18I
commentLet be a field, and a group which acts on by field automorphisms.
(a) Explain the meaning of the phrase in italics in the previous sentence.
Show that the set of fixed points is a subfield of .
(b) Suppose that is finite, and set . Let . Show that is algebraic and separable over , and that the degree of over divides the order of .
Assume that is a primitive element for the extension , and that is a subgroup of . What is the degree of over ? Justify your answer.
(c) Let , and let be a primitive th root of unity in for some integer . Show that the -automorphisms of defined by
generate a group isomorphic to the dihedral group of order .
Find an element for which .
Paper 4 , Section II,
comment(a) A flat , isotropic and homogeneous universe has metric given by
(i) Show that the non-vanishing Christoffel symbols and Ricci tensor components are
where dots are time derivatives and (no summation assumed).
(ii) Derive the first-order Friedmann equation from the Einstein equations,
(b) Consider a flat universe described by ( ) with in which late-time acceleration is driven by "phantom" dark energy obeying an equation of state with pressure , where and the energy density . The remaining matter is dust, so we have with each component separately obeying .
(i) Calculate an approximate solution for the scale factor that is valid at late times. Show that the asymptotic behaviour is given by a Big Rip, that is, a singularity in which at some finite time .
(ii) Sketch a diagram of the scale factor as a function of for a convenient choice of , ensuring that it includes (1) the Big Bang, (2) matter domination, (3) phantom-energy domination, and (4) the Big Rip. Label these epochs and mark them on the axes.
(iii) Most reasonable classical matter fields obey the null energy condition, which states that the energy-momentum tensor everywhere satisfies for any null vector . Determine if this applies to phantom energy.
[The energy-momentum tensor for a perfect fluid is
Paper 4, Section II, 17G
commentState and prove Hall's theorem, giving any definitions required by the proof (e.g. of an -alternating path).
Let be a (not necessarily bipartite) graph, and let be the size of the largest matching in . Let be the smallest for which there exist vertices such that every edge in is incident with at least one of . Show that and that . For each positive integer , find a graph with and . Determine and when is the Turan graph on 30 vertices.
By using Hall's theorem, or otherwise, show that if is a bipartite graph then
Define the chromatic index of a graph . Prove that if with then .
Paper 4, Section II, H
comment(a) Let be two Hilbert spaces, and be a bounded linear operator. Show that there exists a unique bounded linear operator such that
(b) Let be a separable Hilbert space. We say that a sequence is a frame of if there exists such that
State briefly why such a frame exists. From now on, let be a frame of . Show that is dense in .
(c) Show that the linear map given by is bounded and compute its adjoint .
(d) Assume now that is a Hilbertian (orthonormal) basis of and let . Show that the Hilbert cube such that is a compact subset of .
Paper 4, Section II, 16G
commentWrite down the Axiom of Foundation.
What is the transitive closure of a set ? Prove carefully that every set has a transitive closure. State and prove the principle of -induction.
Let be a model of . Let be a surjective function class such that for all we have if and only if . Show, by -induction or otherwise, that is the identity.
Paper 4, Section I, E
commentA marine population grows logistically and disperses by diffusion. It is moderately predated on up to a distance from a straight coast. Beyond that distance, predation is sufficiently excessive to eliminate the population. The density of the population at a distance from the coast satisfies
subject to the boundary conditions
(a) Interpret the terms on the right-hand side of , commenting on their dependence on . Interpret the boundary conditions.
(b) Show that a non-zero population is viable if and
Interpret these conditions.
Paper 4, Section II, E
commentThe spatial density of a population at location and time satisfies
where and .
(a) Give a biological example of the sort of phenomenon that this equation describes.
(b) Show that there are three spatially homogeneous and stationary solutions to , of which two are linearly stable to homogeneous perturbations and one is linearly unstable.
(c) For , find the stationary solution to subject to the conditions
(d) Write down the differential equation that is satisfied by a travelling-wave solution to of the form . Let be the solution from part (c). Verify that satisfies this differential equation for , provided the speed is chosen appropriately. [Hint: Consider the change to the equation from part (c).]
(e) State how the sign of depends on , and give a brief qualitative explanation for why this should be the case.
Paper 4, Section II, J
commentLet be a dataset of input-output pairs lying in for . Describe the random-forest algorithm as applied to using decision trees to produce a fitted regression function . [You need not explain in detail the construction of decision trees, but should describe any modifications specific to the random-forest algorithm.]
Briefly explain why for each and , we have .
State the bounded-differences inequality.
Treating as deterministic, show that with probability at least ,
where .
Hint: Treat each as a random variable taking values in an appropriate space (of functions), and consider a function satisfying
Paper 4 , Section II, 20G
comment(a) Compute the class group of . Find also the fundamental unit of , stating clearly any general results you use.
[The Minkowski bound for a real quadratic field is ]
(b) Let be real quadratic, with embeddings . An element is totally positive if and . Show that the totally positive elements of form a subgroup of the multiplicative group of index 4 .
Let be non-zero ideals. We say that is narrowly equivalent to if there exists a totally positive element of such that . Show that this is an equivalence relation, and that the equivalence classes form a group under multiplication. Show also that the order of this group equals
Paper 4, Section I, I
commentLet be a prime, and let for some positive integer .
Show that if a prime power divides for some , then .
Given a positive real , define , where is the von Mangoldt function, taking the value if for some prime and integer , and 0 otherwise. Show that
Deduce that for all integers .
Paper 4, Section II, I
comment(a) Let be an odd integer and an integer with . What does it mean to say that is a (Fermat) pseudoprime to base b?
Let be integers. Show that if is an odd composite integer dividing and satisfying , then is a pseudoprime to base .
(b) Fix . Let be an odd prime not dividing , and let
Use the conclusion of part (a) to show that is a pseudoprime to base . Deduce that there are infinitely many pseudoprimes to base .
(c) Let be integers, and let , where are distinct primes not dividing . For each , let . Show that is a pseudoprime to base if and only if for all , the order of modulo divides .
(d) By considering products of prime factors of and for primes , deduce that there are infinitely many pseudoprimes to base 2 with two prime factors.
[Hint: You may assume that for implies , and that for is not a power of 3.]
Paper 4 , Section II, 40E
comment(a) Show that if and are real matrices such that both and are symmetric positive definite, then the spectral radius of is strictly less than
(b) Consider the Poisson equation (with zero Dirichlet boundary condition) on the unit square, where is some smooth function. Given and an equidistant grid on the unit square with stepsize , the standard five-point method is given by
where and . Equation can be written as a linear system , where and both depend on the chosen ordering of the grid points.
Use the result in part (a) to show that the Gauss-Seidel method converges for the linear system described above, regardless of the choice of ordering of the grid points.
[You may quote convergence results - based on the spectral radius of the iteration matrix - mentioned in the lecture notes.]
Paper 4, Section II, 33B
comment(a) A quantum system has Hamiltonian . Let be an orthonormal basis of eigenstates, with corresponding energies . For , and the system is in state . Calculate the probability that it is found to be in state at time , correct to lowest non-trivial order in .
(b) Now suppose form a basis of the Hilbert space, with respect to which
where is the Heaviside step function and is a real constant. Calculate the exact probability that the system is in state at time . For which frequency is this probability maximized?
Paper 4, Section II, J
commentSuppose that , and suppose the prior on is a gamma distribution with parameters and . [Recall that has probability density function
and that its mean and variance are and , respectively. ]
(a) Find the -Bayes estimator for for the quadratic loss, and derive its quadratic risk function.
(b) Suppose we wish to estimate . Find the -Bayes estimator for for the quadratic loss, and derive its quadratic risk function. [Hint: The moment generating function of a Poisson distribution is for , and that of a Gamma distribution is for .]
(c) State a sufficient condition for an admissible estimator to be minimax, and give a proof of this fact.
(d) For each of the estimators in parts (a) and (b), is it possible to deduce using the condition in (c) that the estimator is minimax for some value of and ? Justify your answer.
Paper 4, Section II, 26H
commentLet be a probability space. Show that for any sequence satisfying one necessarily has
Let and be random variables defined on . Show that almost surely as implies that in probability as .
Show that in probability as if and only if for every subsequence there exists a further subsequence such that almost surely as .
Paper 4, Section I,
commentLet be a state space of dimension with standard orthonormal basis labelled by . Let QFT denote the quantum Fourier transform and let denote the operation defined by .
(a) Introduce the basis defined by . Show that each is an eigenstate of and determine the corresponding eigenvalue.
(b) By expressing a generic state in the basis, show that QFT and QFT have the same output distribution if measured in the standard basis.
(c) Let be positive integers with , and let be an integer with . Suppose that we are given the state
where and are unknown to us. Using part (b) or otherwise, show that a standard basis measurement on QFT has an output distribution that is independent of .
Paper 4, Section II, I
comment(a) Define the group . Sketch a proof of the classification of the irreducible continuous representations of . Show directly that the characters obey an orthogonality relation.
(b) Define the group .
(i) Show that there is a bijection between the conjugacy classes in and the subset of the real line. [If you use facts about a maximal torus , you should prove them.]
(ii) Write for the conjugacy class indexed by an element , where . Show that is homeomorphic to . [Hint: First show that is in bijection with .
(iii) Let be the parametrisation of conjugacy classes from part (i). Determine the representation of whose character is the function .
Paper 4 , Section I, J
commentThe data frame data contains the daily number of new avian influenza cases in a large poultry farm.
Write down the model being fitted by the code below. Does the model seem to provide a satisfactory fit to the data? Justify your answer.
The owner of the farm estimated that the size of the epidemic was initially doubling every 7 days. Is that estimate supported by the analysis below? [You may need .]
Paper 4, Section II, J
commentLet be an non-random design matrix and be a -vector of random responses. Suppose , where is an unknown vector and is known.
(a) Let be a constant. Consider the ridge regression problem
Let be the fitted values. Show that , where
(b) Show that
(c) Let , where is independent of . Show that is an unbiased estimator of .
(d) Describe the behaviour (monotonicity and limits) of as a function of when and . What is the minimum value of ?
Paper 4, Section II, 35C
comment(a) Explain what is meant by a first-order phase transition and a second-order phase transition.
(b) Explain why the (Helmholtz) free energy is the appropriate thermodynamic potential to consider at fixed and .
(c) Consider a ferromagnet with free energy
where is the temperature, is the magnetization, and are constants.
Find the equilibrium value of at high and low temperatures. Hence, evaluate the equilibrium thermodynamic free energy as a function of and compute the entropy and heat capacity. Determine the jump in the heat capacity and identify the order of the phase transition.
(d) Now consider a ferromagnet with free energy
where are constants with , but .
Find the equilibrium value of at high and low temperatures. What is the order of the phase transition?
For determine the behaviour of the heat capacity at high and low temperatures.
Paper 4, Section II, 29K
(a) What does it mean to say that a stochastic process is a Brownian motion? Show that, if is a continuous Gaussian process such that and for all , then is a Brownian motion.
For the rest of the question, let be a Brownian motion.
(b) Let