Part II, 2018, Paper 3
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Paper 3, Section II, I
comment(a) State the Riemann-Roch theorem.
(b) Let be a smooth projective curve of genus 1 over an algebraically closed field , with char . Show that there exists an isomorphism from to the plane cubic in defined by the equation
for some distinct .
(c) Let be the point at infinity on . Show that the map is an isomorphism.
Describe how this defines a group structure on . Denote addition by . Determine all the points with in terms of the equation of the plane curve in part (b).
Paper 3, Section II, H
comment(a) State a version of the Seifert-van Kampen theorem for a cell complex written as the union of two subcomplexes .
(b) Let
for , and take any . Write down a presentation for .
(c) By computing a homology group of a suitable four-sheeted covering space of , prove that is not homotopy equivalent to a compact, connected surface whenever .
Paper 3, Section II,
comment(a) Let be a measure space. Define the spaces for . Prove that if then for all .
(b) Now let endowed with Borel sets and Lebesgue measure. Describe the dual spaces of for . Define reflexivity and say which are reflexive. Prove that is not the dual space of
(c) Now let be a Borel subset and consider the measure space induced from Borel sets and Lebesgue measure on .
(i) Given any , prove that any sequence in converging in to some admits a subsequence converging almost everywhere to .
(ii) Prove that if for then . [Hint: You might want to prove first that the inclusion is continuous with the help of one of the corollaries of Baire's category theorem.]
Paper 3, Section II, A
commentA beam of particles of mass and momentum is incident along the -axis. The beam scatters off a spherically symmetric potential . Write down the asymptotic form of the wavefunction in terms of the scattering amplitude .
The incoming plane wave and the scattering amplitude can be expanded in partial waves as,
where are Legendre polynomials. Define the -matrix. Assuming that the S-matrix is unitary, explain why we can write
for some real phase shifts . Obtain an expression for the total cross-section in terms of the phase shifts .
[Hint: You may use the orthogonality of Legendre polynomials:
Consider the repulsive, spherical potential
where . By considering the s-wave solution to the Schrödinger equation, show that
For low momenta, , compute the s-wave contribution to the total cross-section. Comment on the physical interpretation of your result in the limit .
Paper 3, Section II, J
commentIndividuals arrive in a shop in the manner of a Poisson process with intensity , and they await service in the order of their arrival. Their service times are independent, identically distributed random variables . For , let be the number remaining in the shop immediately after the th departure. Show that
where is the number of arrivals during the th service period, and .
Show that
where is a typical service period, and is the traffic intensity of the queue.
Suppose , and the queue is in equilibrium in the sense that and have the same distribution for all . Express in terms of . Deduce that the mean waiting time (prior to service) of a typical individual is .
Paper 3, Section II, B
comment(a) Find the curves of steepest descent emanating from for the integral
for and determine the angles at which they meet at , and their asymptotes at infinity.
(b) An integral representation for the Bessel function for real is
Show that, as , with fixed,
Paper 3, Section I, G
comment(a) Define what it means for a context-free grammar (CFG) to be in Chomsky normal form ( .
(b) Give an algorithm for converting a CFG into a corresponding CFG in CNF satisfying . [You need only outline the steps, without proof.]
(c) Convert the following :
into a grammar in CNF.
Paper 3, Section II, G
comment(a) State and prove the pumping lemma for regular languages.
(b) Let be a minimal deterministic finite-state automaton whose language is finite. Let be the transition diagram of , and suppose there exists a non-empty closed path in starting and ending at state .
(i) Show that there is no path in from to any accept state of .
(ii) Show that there is no path in from to any other state of .
Paper 3, Section I, B
commentThree particles of unit mass move along a line in a potential
where is the coordinate of the th particle, .
Write the Lagrangian in the form
and specify the matrices and .
Find the normal frequencies and normal modes for this system.
Paper 3, Section , H
commentCompute the rank and minimum distance of the cyclic code with generator polynomial and parity check polynomial . Now let be a root of in the field with 8 elements. We receive the word . Verify that , and hence decode using minimum-distance decoding.
Paper 3, Section I, B
commentThe energy density of a particle species is defined by
where is the energy, and the distribution function, of a particle with momentum . Here is the speed of light and is the rest mass of the particle. If the particle species is in thermal equilibrium then the distribution function takes the form
where is the number of degrees of freedom of the particle, is the temperature, and are constants and is for bosons and is for fermions.
(a) Stating any assumptions you require, show that in the very early universe the energy density of a given particle species is
(b) Show that the total energy density in the very early universe is
where is defined by
[Hint: You may use the fact that
Paper 3, Section II, B
commentThe pressure support equation for stars is
where is the density, is the pressure, is the radial distance, and is Newton's constant.
(a) What two boundary conditions should we impose on the above equation for it to describe a star?
(b) By assuming a polytropic equation of state,
where is a constant, derive the Lane-Emden equation
where , with the density at the centre of the star, and , for some that you should determine.
(c) Show that the mass of a polytropic star is
where and is the value of at the surface of the star.
(d) Derive the following relation between the mass, , and radius, , of a polytropic star
where you should determine the constant . What type of star does the polytrope represent and what is the significance of the mass being constant for this star?
Paper 3, Section II, I
commentLet be a surface.
(a) Define the Gaussian curvature of in terms of the coefficients of the first and second fundamental forms, computed with respect to a local parametrization of .
Prove the Theorema Egregium, i.e. show that the Gaussian curvature can be expressed entirely in terms of the coefficients of the first fundamental form and their first and second derivatives with respect to and .
(b) State the global Gauss-Bonnet theorem for a compact orientable surface .
(c) Now assume that is non-compact and diffeomorphic to but that there is a point such that is a compact subset of . Is it necessarily the case that Justify your answer.
Paper 3, Section II, 32E
commentConsider the system
where and are parameters.
By considering a function of the form , show that when the origin is globally asymptotically stable. Sketch the phase plane for this case.
Find the fixed points for the general case. Find the values of and for which the fixed points have (i) a stationary bifurcation and (ii) oscillatory (Hopf) bifurcations. Sketch these bifurcation values in the -plane.
For the case , find the leading-order approximation to the extended centre manifold of the bifurcation as varies, assuming that . Find also the evolution equation on the extended centre manifold to leading order. Deduce the type of bifurcation, and sketch the bifurcation diagram in the -plane.
Paper 3, Section II, D
commentStarting from the covariant form of the Maxwell equations and making a suitable choice of gauge which you should specify, show that the 4-vector potential due to an arbitrary 4-current obeys the wave equation,
where .
Use the method of Green's functions to show that, for a localised current distribution, this equation is solved by
for some that you should specify.
A point particle, of charge , moving along a worldline parameterised by proper time , produces a 4 -vector potential
where . Define and draw a spacetime diagram to illustrate its physical significance.
Suppose the particle follows a circular trajectory,
(with ), in some inertial frame with coordinates . Evaluate the resulting 4 -vector potential at a point on the -axis as a function of and .
Paper 3, Section II, C
commentFor two Stokes flows and inside the same volume with different boundary conditions on its boundary , prove the reciprocal theorem
where and are the stress tensors associated with the flows.
Stating clearly any properties of Stokes flow that you require, use the reciprocal theorem to prove that the drag on a body translating with uniform velocity is given by
where is a symmetric second-rank tensor that depends only on the geometry of the body.
A slender rod falls slowly through very viscous fluid with its axis inclined to the vertical. Explain why the rod does not rotate, stating any properties of Stokes flow that you use.
When the axis of the rod is inclined at an angle to the vertical, the centre of mass of the rod travels at an angle to the vertical. Given that the rod falls twice as quickly when its axis is vertical as when its axis is horizontal, show that
Paper 3, Section I, B
commentUsing a suitable branch cut, show that
where .
Paper 3, Section II, I
commentLet be a finite field extension of a field , and let be a finite group of automorphisms of . Denote by the field of elements of fixed by the action of .
(a) Prove that the degree of over is equal to the order of the group .
(b) For any write .
(i) Suppose that . Prove that the coefficients of generate over .
(ii) Suppose that . Prove that the coefficients of and lie in . By considering the case with and in , or otherwise, show that they need not generate over .
Paper 3, Section II, E
commentThe Schwarzschild metric in isotropic coordinates , is given by:
where
and is the mass of the black hole.
(a) Let , denote a coordinate system related to by
where and . Write down the transformation matrix , briefly explain its physical meaning and show that the inverse transformation is of the same form, but with .
(b) Using the coordinate transformation matrix of part (a), or otherwise, show that the components of the metric in coordinates are given by
where and are functions of that you should determine. You should also express in terms of the coordinates .
(c) Consider the limit with held constant. Show that for points the function , while tends to a finite value, which you should determine. Hence determine the metric components at points in this limit.
Paper 3, Section II, I
commentWhat does it mean to say that a graph has a -colouring? What are the chromatic number and the independence number of a graph ? For each , give an example of a graph such that but .
Let . Show that there exists a graph containing no cycle of length with .
Show also that if is sufficiently large then there is a triangle-free of order with .
Paper 3, Section II, A
commentSuppose is a smooth one-parameter group of transformations acting on .
(a) Define the generator of the transformation,
where you should specify and in terms of .
(b) Define the prolongation of and explicitly compute in terms of .
Recall that if is a Lie point symmetry of the ordinary differential equation:
then it follows that whenever .
(c) Consider the ordinary differential equation:
for a smooth function. Show that if generates a Lie point symmetry of this equation, then:
(d) Find all the Lie point symmetries of the equation:
where is an arbitrary smooth function.
Paper 3, Section II, F
comment(a) Let be a normed vector space and let be a Banach space. Show that the space of bounded linear operators is a Banach space.
(b) Let and be Banach spaces, and let be a dense linear subspace. Prove that a bounded linear map can be extended uniquely to a bounded linear map with the same operator norm. Is the claim also true if one of and is not complete?
(c) Let be a normed vector space. Let be a sequence in such that
Prove that there is a constant such that
Paper 3, Section II, G
commentState and prove the Compactness Theorem for first-order predicate logic. State and prove the Upward Löwenheim-Skolem Theorem.
[You may assume the Completeness Theorem for first-order predicate logic.]
For each of the following theories, either give axioms (in the specified language) for the theory or prove that the theory is not axiomatisable.
(i) The theory of finite groups (in the language of groups).
(ii) The theory of groups in which every non-identity element has infinite order (in the language of groups).
(iii) The theory of total orders (in the language of posets).
(iv) The theory of well-orderings (in the language of posets).
If a theory is axiomatisable by a set of sentences, and also by a finite set of sentences, does it follow that the theory is axiomatisable by some finite subset of ? Justify your answer.
Paper 3, Section I,
commentConsider a nonlinear model for the axisymmetric dispersal of a population in two spatial dimensions whose density, , obeys
where is a positive constant, is a radial polar coordinate, and is time.
Show that
is constant. Interpret this condition.
Show that a similarity solution of the form
is valid for provided that the scaling function satisfies
Show that there exists a value (which need not be evaluated) such that for but for . Determine the area within which at time in terms of .
[Hint: The gradient and divergence operators in cylindrical polar coordinates act on radial functions and as
Paper 3, Section II, C
commentConsider fluctuations of a population described by the vector . The probability of the state at time , obeys the multivariate Fokker-Planck equation
where is a drift vector and is a symmetric positive-definite diffusion matrix, and the summation convention is used throughout.
(a) Show that the Fokker-Planck equation can be expressed as a continuity equation
for some choice of probability flux which you should determine explicitly. Here, denotes the gradient operator.
(b) Show that the above implies that an initially normalised probability distribution remains normalised,
at all times, where the volume element .
(c) Show that the first two moments of the probability distribution obey
(d) Now consider small fluctuations with zero mean, and assume that it is possible to linearise the drift vector and the diffusion matrix as and where has real negative eigenvalues and is a symmetric positive-definite matrix. Express the probability flux in terms of the matrices and and assume that it vanishes in the stationary state.
(e) Hence show that the multivariate normal distribution,
where is a normalisation and is symmetric, is a solution of the linearised FokkerPlanck equation in the stationary state, and obtain an equation that relates to the matrices and .
(f) Show that the inverse of the matrix is the matrix of covariances and obtain an equation relating to the matrices and .
Paper 3, Section I, G
commentWhat is a multiplicative function? Show that if is a multiplicative function, then so is .
Define the Möbius function , and show that it is multiplicative. Deduce that
and that
What is if What is if
Paper 3, Section II, G
commentWhat does it mean to say that a positive definite binary quadratic form is reduced? What does it mean to say that two binary quadratic forms are equivalent? Show that every positive definite binary quadratic form is equivalent to some reduced form.
Show that the reduced positive definite binary quadratic forms of discriminant are and . Show also that a prime is represented by if and only if
Paper 3, Section II, E
commentThe diffusion equation for :
is solved numerically by the difference scheme
Here is the Courant number, with , and .
(a) Prove that, as with constant , the local error of the method is .
(b) Applying the Fourier stability analysis, show that the method is stable if and only if . [Hint: If a polynomial has real roots, then those roots lie in if and only if and .]
(c) Prove that, for the same equation, the leapfrog scheme
is unstable for any choice of .
Paper 3, Section II, K
commentThe scalars are related by the equations
where the initial state is normally distributed with mean and variance 1 and is a sequence of independent random variables each normally distributed with mean 0 and variance 1 . The control variable is to be chosen at time on the basis of information , where and
(a) Let be the Kalman filter estimates of , i.e.
where is chosen to minimise . Calculate and show that, conditional on is normally distributed with mean and variance .
(b) Define
Show that , where and .
(c) Show that the minimising control can be expressed in the form and find . How would the expression for be altered if or had variances other than 1?
Paper 3, Section II, D
commentA quantum system is prepared in the ground state at time . It is subjected to a time-varying Hamiltonian . Show that, to first order in , the system evolves as
where and
A large number of hydrogen atoms, each in the ground state, are subjected to an electric field
where is a constant. Show that the fraction of atoms found in the state is, after a long time and to lowest non-trivial order in ,
where is the energy difference between the and states, and is the electron charge and the Bohr radius. What fraction of atoms lie in the state?
[Hint: You may assume the hydrogenic wavefunctions
and the integral
for a positive integer.]
Paper 3, Section II, K
commentIn the model of a Gaussian distribution in dimension , with unknown mean and known identity covariance matrix , we estimate based on a sample of i.i.d. observations drawn from .
(a) Define the Fisher information , and compute it in this model.
(b) We recall that the observed Fisher information is given by
Find the limit of , where is the maximum likelihood estimator of in this model.
(c) Define the Wald statistic and compute it. Give the limiting distribution of and explain how it can be used to design a confidence interval for .
[You may use results from the course provided that you state them clearly.]
Paper 3, Section II, J
commentLet be the Lebesgue measure on the real line. Recall that if is a Borel subset, then
where the infimum is taken over all covers of by countably many intervals, and denotes the length of an interval .
(a) State the definition of a Borel subset of .
(b) State a definition of a Lebesgue measurable subset of .
(c) Explain why the following sets are Borel and compute their Lebesgue measure:
(d) State the definition of a Borel measurable function .
(e) Let be a Borel measurable function . Is it true that the subset of all where is continuous at is a Borel subset? Justify your answer.
(f) Let be a Borel subset with . Show that
contains the interval .
(g) Let be a Borel subset such that . Show that for every , there exists in such that
Deduce that contains an open interval around 0 .
Paper 3, Section I, 10D
commentLet denote the set of all -bit strings. For any Boolean function on 2 bits consider the linear operation on 3 qubits defined by
for all and denoting addition of bits modulo 2 . Here the first register is a 2-qubit register and the second is a 1-qubit register. We are able to apply only the 1-qubit Pauli and Hadamard gates to any desired qubits, as well as the 3 -qubit gate to any three qubits. We can also perform measurements in the computational basis.
(a) Describe how we can construct the state
starting from the standard 3-qubit state .
(b) Suppose now that the gate is given to us but is not specified. However is promised to be one of two following cases:
(i) is a constant function (i.e. for all , or for all ),
(ii) for any 2-bit string we have (with as above).
Show how we may determine with certainty which of the two cases (i) or (ii) applies, using only a single application of .
Paper 3, Section II,
In this question you may assume the following fact about the quantum Fourier transform if and , where , then