Part II, 2021, Paper 3
Part II, 2021, Paper 3
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Paper 3, Section II, I
commentIn this question, all varieties are over an algebraically closed field of characteristic zero.
What does it mean for a projective variety to be smooth? Give an example of a smooth affine variety whose projective closure is not smooth.
What is the genus of a smooth projective curve? Let be the hypersurface . Prove that contains a smooth curve of genus
Let be an irreducible curve of degree 2 . Prove that is isomorphic to .
We define a generalized conic in to be the vanishing locus of a non-zero homogeneous quadratic polynomial in 3 variables. Show that there is a bijection between the set of generalized conics in and the projective space , which maps the conic to the point whose coordinates are the coefficients of .
(i) Let be the subset of conics that consist of unions of two distinct lines. Prove that is not Zariski closed, and calculate its dimension.
(ii) Let be the homogeneous ideal of polynomials vanishing on . Determine generators for the ideal .
Paper 3, Section II, 20F
commentLet be a space. We define the cone of to be
where if and only if either or .
(a) Show that if is triangulable, so is . Calculate . [You may use any results proved in the course.]
(b) Let be a simplicial complex and a subcomplex. Let , and let be the space obtained by identifying with . Show that there is a long exact sequence
(c) In part (b), suppose that and for some . Calculate for all .
Paper 3, Section II, H
comment(a) State the Riemann-Lebesgue lemma. Show that the Fourier transform maps to itself continuously.
(b) For some , let . Consider the following system of equations for
Show that there exists a unique solving the equations with for . You need not find explicitly, but should give an expression for the Fourier transform of . Show that there exists a constant such that
For what values of can we conclude that ?
Paper 3, Section II, 34B
comment(a) In three dimensions, define a Bravais lattice and its reciprocal lattice .
A particle is subject to a potential with for and . State and prove Bloch's theorem and specify how the Brillouin zone is related to the reciprocal lattice.
(b) A body-centred cubic lattice consists of the union of the points of a cubic lattice and all the points at the centre of each cube:
where and are unit vectors parallel to the Cartesian coordinates in . Show that is a Bravais lattice and determine the primitive vectors and .
Find the reciprocal lattice Briefly explain what sort of lattice it is.
Hint: The matrix has inverse .
Paper 3, Section II,
comment(a) Customers arrive at a queue at the event times of a Poisson process of rate . The queue is served by two independent servers with exponential service times with parameter each. If the queue has length , an arriving customer joins with probability and leaves otherwise (where . For which and is there a stationary distribution?
(b) A supermarket allows a maximum of customers to shop at the same time. Customers arrive at the event times of a Poisson process of rate 1 , they enter the supermarket when possible, and they leave forever for another supermarket otherwise. Customers already in the supermarket pay and leave at the event times of an independent Poisson process of rate . When is there a unique stationary distribution for the number of customers in the supermarket? If it exists, find it.
(c) In the situation of part (b), started from equilibrium, show that the departure process is Poissonian.
Paper 3, Section II, 30A
comment(a) Carefully state Watson's lemma.
(b) Use the method of steepest descent and Watson's lemma to obtain an infinite asymptotic expansion of the function
Paper 3, Section I, F
commentDefine a regular expression and explain how this gives rise to a language .
Define a deterministic finite-state automaton and the language that it accepts.
State the relationship between languages obtained from regular expressions and languages accepted by deterministic finite-state automata.
Let and be regular languages. Is always regular? What about ?
Now suppose that are regular languages. Is the countable union always regular? What about the countable intersection ?
Paper 3, Section II,
commentSuppose that is a context-free grammar without -productions. Given a derivation of some word in the language of , describe a parse tree for this derivation.
State and prove the pumping lemma for . How would your proof differ if you did not assume that was in Chomsky normal form, but merely that has no - or unit productions?
For the alphabet of terminal symbols, state whether the following languages over are context free, giving reasons for your answer. (i) , (ii) , (iii) .
Paper 3 , Section I, D
commentThe Lagrangian of a particle of mass and charge in an electromagnetic field takes the form
Explain the meaning of and , and how they are related to the electric and magnetic fields.
Obtain the canonical momentum and the Hamiltonian .
Suppose that the electric and magnetic fields have Cartesian components and , respectively, where and are positive constants. Explain why the Hamiltonian of the particle can be taken to be
State three independent integrals of motion in this case.
Paper 3, Section I, K
commentLet . Define the Hamming code of length . Explain what it means to be a perfect code and show that is a perfect code.
Suppose you are using the Hamming code of length and you receive the message of length . How would you decode this message using minimum distance decoding? Explain why this leads to correct decoding if at most one channel error has occurred.
Paper 3, Section I, 9B
commentThe expansion of the universe during inflation is governed by the Friedmann equation
and the equation of motion for the inflaton field ,
Consider the potential
with and .
(a) Show that the inflationary equations have the exact solution
for arbitrary and appropriate choices of and . Determine the range of for which the solution exists. For what values of does inflation occur?
(b) Using the inflaton equation of motion and
together with the continuity equation
determine .
(c) What is the range of the pressure energy density ratio for which inflation occurs?
Paper 3, Section II, B
comment(a) Consider a closed universe endowed with cosmological constant and filled with radiation with pressure and energy density . Using the equation of state and the continuity equation
determine how depends on . Give the physical interpretation of the scaling of with
(b) For such a universe the Friedmann equation reads
What is the physical meaning of
(c) Making the substitution , determine and such that the Friedmann equation takes the form
Using the substitution and the boundary condition , deduce the boundary condition for .
Show that
and hence that
Express the constant in terms of and .
Sketch the graphs of for the cases and .
Paper 3, Section II, F
commentLet and be smooth boundaryless manifolds. Suppose is a smooth map. What does it mean for to be a regular value of ? State Sard's theorem and the stack-of-records theorem.
Suppose is another smooth map. What does it mean for and to be smoothly homotopic? Assume now that is compact, and has the same dimension as . Suppose that is a regular value for both and . Prove that
Let be a non-empty open subset of the sphere. Suppose that is a smooth map such that for all . Show that there must exist a pair of antipodal points on which is mapped to another pair of antipodal points by .
[You may assume results about compact 1-manifolds provided they are accurately stated.]
Paper 3, Section II, A
commentConsider the system
where and are constants with .
(a) Find the fixed points, and classify those on . State how the number of fixed points depends on and . Hence, or otherwise, deduce the values of at which stationary bifurcations occur for fixed .
(b) Sketch bifurcation diagrams in the -plane for the cases and , indicating the stability of the fixed points and the type of the bifurcations in each case. [You are not required to prove that the stabilities or bifurcation types are as you indicate.]
(c) For the case , analyse the bifurcation at using extended centre manifold theory and verify that the evolution equation on the centre manifold matches the behaviour you deduced from the bifurcation diagram in part (b).
(d) For , sketch the phase plane in the immediate neighbourhood of where the bifurcation of part (c) occurs.
Paper 3, Section II, 36C
comment(a) Derive the Larmor formula for the total power emitted through a large sphere of radius by a non-relativistic particle of mass and charge with trajectory . You may assume that the electric and magnetic fields describing radiation due to a source localised near the origin with electric dipole moment can be approximated as
Here, the radial distance is assumed to be much larger than the wavelength of emitted radiation which, in turn, is large compared to the spatial extent of the source.
(b) A non-relativistic particle of mass , moving at speed along the -axis in the positive direction, encounters a step potential of width and height described by
where is a monotonically increasing function with and . The particle carries charge and loses energy by emitting electromagnetic radiation. Assume that the total energy loss through emission is negligible compared with the particle's initial kinetic energy . For , show that the total energy lost is
Find the total energy lost also for the case .
(c) Take and explicitly evaluate the particle energy loss in each of the cases and . What is the maximum value attained by as is varied?
Paper 3, Section II, 38A
commentViscous fluid occupying is bounded by a rigid plane at and is extracted through a small hole at the origin at a constant flow rate . Assume that for sufficiently small values of the velocity is well-approximated by
except within a thin axisymmetric boundary layer near .
(a) Estimate the Reynolds number of the flow as a function of , and thus give an estimate for how small needs to be for such a solution to be applicable. Show that the radial pressure gradient is proportional to .
(b) In cylindrical polar coordinates , the steady axisymmetric boundary-layer equations for the velocity components can be written as
and is the Stokes streamfunction. Verify that the condition of incompressibility is satisfied by the use of .
Use scaling arguments to estimate the thickness of the boundary layer near and then to motivate seeking a similarity solution of the form
(c) Obtain the differential equation satisfied by , and state the conditions that would determine its solution. [You are not required to find this solution.]
By considering the flux in the boundary layer, explain why there should be a correction to the approximation of relative magnitude .
Paper 3 , Section I, 7E
commentThe Beta function is defined by
for and .
(a) Prove that and find .
(b) Show that .
(c) For each fixed with , use part (b) to obtain the analytic continuation of as an analytic function of , with the exception of the points
(d) Use part (c) to determine the type of singularity that the function has at , for fixed with .
Paper 3, Section II, 18I
commentDefine the elementary symmetric functions in the variables . State the fundamental theorem of symmetric functions.
Let , where is a field. Define the discriminant of , and explain why it is a polynomial in .
Compute the discriminant of .
Let . When does the discriminant of equal zero? Compute the discriminant of .
Paper 3, Section II, 37C
comment(a) Determine the signature of the metric tensor given by
Is it Riemannian, Lorentzian, or neither?
(b) Consider a stationary black hole with the Schwarzschild metric:
These coordinates break down at the horizon . By making a change of coordinates, show that this metric can be converted to infalling Eddington-Finkelstein coordinates.
(c) A spherically symmetric, narrow pulse of radiation with total energy falls radially inwards at the speed of light from infinity, towards the origin of a spherically symmetric spacetime that is otherwise empty. Assume that the radial width of the pulse is very small compared to the energy , and the pulse can therefore be treated as instantaneous.
(i) Write down a metric for the region outside the pulse, which is free from coordinate singularities. Briefly justify your answer. For what range of coordinates is this metric valid?
(ii) Write down a metric for the region inside the pulse. Briefly justify your answer. For what range of coordinates is this metric valid?
(iii) What is the final state of the system?
Paper 3, Section II, 17G
comment(a) Define the Ramsey number and show that .
Show that every 2-coloured complete graph with contains a monochromatic spanning tree. Is the same true if is coloured with 3 colours? Give a proof or counterexample.
(b) Let be a graph. Show that the number of paths of length 2 in is
Now consider a 2-coloured complete graph with . Show that the number of monochromatic triangles in is
where denotes the number of red edges incident with a vertex and denotes the number of blue edges incident with . [Hint: Count paths of length 2 in two different ways.]
Paper 3, Section II, 32D
comment(a) Consider the group of transformations of given by , where . Show that this acts as a group of Lie symmetries for the equation .
(b) Let and define . Show that the vector field generates the group of phase rotations .
(c) Show that the transformations of defined by
form a one-parameter group generated by the vector field
and find the second prolongation of the action of . Hence find the coefficients and in the second prolongation of ,
complex conjugate .
(d) Show that the group of transformations in part (c) acts as a group of Lie symmetries for the nonlinear Schrödinger equation . Given that solves the nonlinear Schrödinger equation for any , find a solution which describes a solitary wave travelling at arbitrary speed .
Paper 3, Section II, H
comment(a) State the Arzela-Ascoli theorem, including the definition of equicontinuity.
(b) Consider a sequence of continuous real-valued functions on such that for all is bounded and the sequence is equicontinuous at . Prove that there exists and a subsequence such that uniformly on any closed bounded interval.
(c) Let be a Hausdorff compact topological space, and the real-valued continuous functions on . Let be a compact subset of . Prove that the collection of functions is equicontinuous.
(d) We say that a Hausdorff topological space is locally compact if every point has a compact neighbourhood. Let be such a space, compact and open such that . Prove that there exists continuous with compact support contained in and equal to 1 on . [Hint: Construct an open set such that and is compact, and use Urysohn's lemma to construct a function in and then extend it by zero.]
Paper 3, Section II, 16G
comment(a) Let and be cardinals. What does it mean to say that ? Explain briefly why, assuming the Axiom of Choice, every infinite cardinal is of the form for some ordinal , and that for every ordinal we have .
(b) Henceforth, you should not assume the Axiom of Choice.
Show that, for any set , there is an injection from to its power set , but there is no bijection from to . Deduce that if is a cardinal then .
Let and be sets, and suppose that there exists a surjection . Show that there exists an injection .
Let be an ordinal. Prove that .
By considering as the set of relations on , or otherwise, show that there exists a surjection . Deduce that .
Paper 3, Section I, E
commentThe population density of individuals of age at time satisfies the partial differential equation
with the boundary condition
where and are, respectively, the per capita age-dependent birth and death rates.
(a) What is the biological interpretation of the boundary condition?
(b) Solve equation (1) assuming a separable form of solution, .
(c) Use equation (2) to obtain a necessary condition for the existence of a separable solution to the full problem.
(d) For a birth rate with and an age-independent death rate , show that a separable solution to the full problem exists and find the critical value of above which the population density grows with time.
Paper 3, Section II, 13E
commentConsider an epidemic spreading in a population that has been aggregated by age into groups numbered . The th age group has size and the numbers of susceptible, infective and recovered individuals in this group are, respectively, and . The spread of the infection is governed by the equations
where
and is a matrix satisfying , for .
(a) Describe the biological meaning of the terms in equations (1) and (2), of the matrix and the condition it satisfies, and of the lack of dependence of and on .
State the condition on the matrix that would ensure the absence of any transmission of infection between age groups.
(b) In the early stages of an epidemic, and . Use this information to linearise the dynamics appropriately, and show that the linearised system predicts
where is the vector of infectives at time is the identity matrix and is a matrix that should be determined.
(c) Deduce a condition on the eigenvalues of the matrix that allows the epidemic to grow.
Paper 3, Section I, I
commentDefine the continued fraction expansion of , and show that this expansion terminates if and only if .
Define the convergents of the continued fraction expansion of , and show that for all ,
Deduce that if , then for all , at least one of
must hold.
[You may assume that lies strictly between and for all ]
Paper 3, Section II, I
commentState what it means for two binary quadratic forms to be equivalent, and define the class number .
Let be a positive integer, and let be a binary quadratic form. Show that properly represents if and only if is equivalent to a binary quadratic form
for some integers and .
Let be an integer such that or . Show that is properly represented by some binary quadratic form of discriminant if and only if is a square modulo .
Fix a positive integer . Show that is composite for some integer such that if and only if is a square modulo for some prime .
Deduce that if and only if is prime for all .
Paper 3, Section II, 40E
commentConsider discretisation of the diffusion equation
by the Crank-Nicholson method:
where is the Courant number, is the step size in the space discretisation, is the step size in the time discretisation, and , where is the solution of . The initial condition is given.
(a) Consider the Cauchy problem for on the whole line, (thus ), and derive the formula for the amplification factor of the Crank-Nicholson method ( ). Use the amplification factor to show that the Crank-Nicholson method is stable for the Cauchy problem for all .
[You may quote basic properties of the Fourier transform mentioned in lectures, but not the theorem on sufficient and necessary conditions on the amplification factor to have stability.]
(b) Consider on the interval (thus and ) with Dirichlet boundary conditions and , for some sufficiently smooth functions and . Show directly (without using the Lax equivalence theorem) that, given sufficient smoothness of , the Crank-Nicholson method is convergent, for any , in the norm defined by for .
[You may assume that the Trapezoidal method has local order 3 , and that the standard three-point centred discretisation of the second derivative (as used in the CrankNicholson method) has local order 2.]
Paper 3, Section II, B
comment(a) A quantum system with total angular momentum is combined with another of total angular momentum . What are the possible values of the total angular momentum of the combined system? For given , what are the possible values of the angular momentum along any axis?
(b) Consider the case . Explain why all the states with are antisymmetric under exchange of the angular momenta of the two subsystems, while all the states with are symmetric.
(c) An exotic particle of spin 0 and negative intrinsic parity decays into a pair of indistinguishable particles . Assume each particle has spin 1 and that the decay process conserves parity. Find the probability that the direction of travel of the particles is observed to lie at an angle from some axis along which their total spin is observed to be ?
Paper 3, Section II, J
commentLet iid for some known and some unknown . [The gamma distribution has probability density function
and its mean and variance are and , respectively.]
(a) Find the maximum likelihood estimator for and derive the distributional limit of . [You may not use the asymptotic normality of the maximum likelihood estimator proved in the course.]
(b) Construct an asymptotic -level confidence interval for and show that it has the correct (asymptotic) coverage.
(c) Write down all the steps needed to construct a candidate to an asymptotic -level confidence interval for using the nonparametric bootstrap.
Paper 3, Section II,
commentShow that random variables defined on some probability space are independent if and only if
for all bounded measurable functions .
Now let be an infinite sequence of independent Gaussian random variables with zero means, , and finite variances, . Show that the series converges in if and only if .
[You may use without proof that for .]
Paper 3 , Section I, D
commentLet be the joint state of a bipartite system with subsystems and separated in space. Suppose that Alice and Bob have access only to subsystems and respectively, on which they can perform local quantum operations.
Alice performs a unitary operation on and then a (generally incomplete) measurement on , with projectors labelled by her possible measurement outcomes . Then Bob performs a complete measurement on relative to the orthonormal basis labelled by his possible outcomes .
Show that the probability distribution of Bob's measurement outcomes is unaffected by whether or not Alice actually performs the local operations on described above.
Paper 3, Section II, D
commentLet denote the set of all -bit strings and let denote the space of qubits.
(a) Suppose has the property that for a unique and suppose we have a quantum oracle .
(i) Let and introduce the operators
on , where is the identity operator. Give a geometrical description of the actions of and on the 2-dimensional subspace of given by the real span of and . [You may assume without proof that the product of two reflections in is a rotation through twice the angle between the mirror lines.]
(ii) Using the results of part (i), or otherwise, show how we may determine with certainty, starting with a supply of qubits each in state and using only once, together with other quantum operations that are independent of .
(b) Suppose , where is a fixed linear subspace with orthogonal complement . Let denote the projection operator onto and let , where is the identity operator on .
(i) Show that any can be written as , where , and and are normalised.
(ii) Let and . Show that .
(iii) Now assume, in addition, that and that for some unitary operation . Suppose we can implement the operators as well as the operation . In the case , show how the -qubit state may be made exactly from by a process that succeeds with certainty.
Paper 3, Section II, I
commentIn this question we work over .
(a) (i) Let be a subgroup of a finite group . Given an -space , define the complex vector space . Define, with justification, the -action on .
(ii) Write for the conjugacy class of . Suppose that breaks up into conjugacy classes of with representatives . If is a character of , write down, without proof, a formula for the induced character as a certain sum of character values .
(b) Define permutations by and let be the subgroup of . It is given that the elements of are all of the form for and that has order 21 .
(i) Find the orders of the centralisers and . Hence show that there are five conjugacy classes of .
(ii) Find all characters of degree 1 of by lifting from a suitable quotient group.
(iii) Let . By first inducing linear characters of using the formula stated in part (a)(ii), find the remaining irreducible characters of .
Paper 3, Section II, F
comment(a) Let be a polynomial of degree , and let be the multiplicities of the ramification points of . Prove that
Show that, for any list of integers satisfying , there is a polynomial of degree such that the are the multiplicities of the ramification points of .
(b) Let be an analytic map, and let be the set of branch points. Prove that the restriction is a regular covering map. Given , explain how a closed loop in gives rise to a permutation of . Show that the group of all such permutations is transitive, and that the permutation only depends on up to homotopy.
(c) Prove that there is no meromorphic function of degree 4 with branch points such that every preimage of 0 and 1 has ramification index 2 , while some preimage of has ramification index equal to 3. [Hint: You may use the fact that every non-trivial product of -cycles in the symmetric group is a -cycle.]
Paper 3, Section I, J
commentConsider the normal linear model , where is a design matrix, is a vector of responses, is the identity matrix, and are unknown parameters.
Derive the maximum likelihood estimator of the pair and . What is the distribution of the estimator of ? Use it to construct a -level confidence interval of . [You may use without proof the fact that the "hat matrix" is a projection matrix.]
Paper 3, Section II, C
comment(a) A gas of non-interacting particles with spin degeneracy has the energymomentum relationship , for constants . Show that the density of states, , in a -dimensional volume with is given by
where is a constant that you should determine. [You may denote the surface area of a unit -dimensional sphere by .]
(b) Write down the Bose-Einstein distribution for the average number of identical bosons in a state with energy in terms of and the chemical potential . Explain why .
(c) Show that an ideal quantum Bose gas in a -dimensional volume , with , as above, has
where is the pressure and is a constant that you should determine.
(d) For such a Bose gas, write down an expression for the number of particles that do not occupy the ground state. Use this to determine the values of for which there exists a Bose-Einstein condensate at sufficiently low temperatures.
Paper 3, Section II, 29K
comment(a) Let be a martingale and a supermartingale. If , show that for any bounded stopping time . [If you use a general result about supermartingales, you must prove it.]
(b) Consider a market with one stock with time- price and constant interest rate . Explain why a self-financing investor's wealth process satisfies
where is the number of shares of the stock held during the th period.
(c) Given an initial wealth , an investor seeks to maximize , where is a given utility function. Suppose the stock price is such that , where is a sequence of independent copies of a random variable . Let be defined inductively by
with terminal condition for all .
Show that the process is a supermartingale for any trading strategy . Suppose that the trading strategy with corresponding wealth process are such that the process is a martingale. Show that is optimal.
Paper 3 , Section I,
commentState Runge's theorem on the approximation of analytic functions by polynomials.
Let . Establish whether the following statements are true or false by giving a proof or a counterexample in each case.
(i) If is the uniform limit of a sequence of polynomials , then is a polynomial.
(ii) If is analytic, then there exists a sequence of polynomials such that for each integer and each we have .
Paper 3, Section II, 39A
commentConsider a two-dimensional stratified fluid of sufficiently slowly varying background density that small-amplitude vertical-velocity perturbations can be assumed to satisfy the linear equation
and is a constant. The background density profile is such that is piecewise constant with for and with in a layer of uniform density .
A monochromatic internal wave of amplitude is incident on the intermediate layer from , and produces velocity perturbations of the form
where and .
(a) Show that the vertical variations have the form
where and are (in general) complex amplitudes and
In particular, you should justify the choice of signs for the coefficients involving .
(b) What are the appropriate boundary conditions to impose on at to determine the unknown amplitudes?
(c) Apply these boundary conditions to show that
where .
(d) Hence show that
where is the angle between the incident wavevector and the downward vertical.