MATHEMATICAL TRIPOS Part II Alternative B Friday 4 June 2004 9 to 12 PAPER 4 Before you begin read these instructions carefully. Candidates must not attempt more than FOUR questions. The number of marks for each question is the same. Additional credit will be given for a substantially complete answer. Write on one side of the paper only and begin each answer on a separate sheet. Write legibly; otherwise, you place yourself at a grave disadvantage. At the end of the examination: Tie your answers in separate bundles, marked A, B, C, . . . , J according to the letter affixed to each question. (For example, 1F, 10F should be in one bundle and 3H, 4H in another bundle.) Attach a completed cover sheet to each bundle. Complete a master cover sheet listing all questions attempted. It is essential that every cover sheet bear the candidate number and desk number. You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.2 1F Combinatorics Write an essay on Ramsey’s theorem. You should include the finite and infinite versions, together with some discussion of bounds in the finite case, and give at least one application. 2G Representation Theory Write an essay on the finite-dimensional representations of SU2, including a proof of their complete reducibility, and a description of the irreducible representations and the decomposition of their tensor products. 3H Galois Theory Let M/K be a finite Galois extension of fields. Explain what is meant by the Galois correspondence between subfields of M containing K and subgroups of Gal(M/K). Show that if K L M then Gal(M/L) is a normal subgroup of Gal(M/K) if and only if L/K is normal. What is Gal(L/K) in this case? Let M be the splitting field of X4 −3 over Q. Prove that Gal(M/Q) is isomorphic to the dihedral group of order 8. Hence determine all subfields of M, expressing each in the form Q(x) for suitable x 2 M. 4H Differentiable Manifolds Define what it means for a manifold to be oriented, and define a volume form on an oriented manifold. Prove carefully that, for a closed connected oriented manifold of dimension n, Hn(M) = R. [You may assume the existence of volume forms on an oriented manifold.] If M and N are closed, connected, oriented manifolds of the same dimension, define the degree of a map f : M ! N. If f has degree d > 1 and y 2 N, can f−1(y) be (i) infinite? (ii) a single point? (iii) empty? Briefly justify your answers. Paper 43 5G Algebraic Topology Write down the definition of a covering space and a covering map. State and prove the path lifting property for covering spaces and state, without proof, the homotopy lifting property. Suppose that a group G is a group of homeomorphisms of a space X. Prove that, under conditions to be stated, the quotient map X ! X/G is a covering map and that 1(X/G) is isomorphic to G. Give two examples in which this last result can be used to determine the fundamental group of a space. 6H Number Fields Let K be a finite extension of Q, and O the ring of integers of K. Write an essay outlining the proof that every non-zero ideal of O can be written as a product of non-zero prime ideals, and that this factorisation is unique up to the order of the factors. 7G Hilbert Spaces Suppose that T is a bounded linear operator on an infinite-dimensional Hilbert space H, and that hT(x), xi is real and non-negative for each x 2 H. (a) Show that T is Hermitian. (b) Let w(T) = sup{hT(x), xi : kxk = 1}. Show that kT(x)k2 6 w(T)hT(x), xi for each x 2 H. (c) Show that kTk is an approximate eigenvalue for T. Suppose in addition that T is compact and injective. (d) Show that kTk is an eigenvalue for T, with finite-dimensional eigenspace. Explain how this result can be used to diagonalise T. Paper 4 [TURN OVER4 8H Riemann Surfaces Let be a lattice in C, = Z!1 + Z!2, where !1, !2 6= 0 and !1/!2 62 R. By constructing an appropriate family of charts, show that the torus C/is a Riemann surface and that the natural projection : z 2 C ! z + 2 C/is a holomorphic map. [You may assume without proof any known topological properties of C/.] Let 0 = Z!01 + Z!02 be another lattice in C, with !01 , !02 6= 0 and !01 /!02 62 R. By considering paths from 0 to an arbitrary z 2 C, show that if f : C/! C/0 is a conformal equivalence then f(z + ) = (az + b) + 0 for some a, b,2 C, with a 6= 0. [Any form of the Monodromy Theorem and basic results on the lifts of paths may be used without proof, provided that these are correctly stated. You may assume without proof that every injective holomorphic function F : C ! C is of the form F(z) = az + b, for some a, b 2 C.] Give an explicit example of a non-constant holomorphic map C/! C/that is not a conformal equivalence. 9H Algebraic Curves Let F(X, Y,Z) be an irreducible homogeneous polynomial of degree n, and write C = {p 2 P2 | F(p) = 0} for the curve it defines in P2. Suppose C is smooth. Show that the degree of its canonical class is n(n − 3). Hence, or otherwise, show that a smooth curve of genus 2 does not embed in P2. 10F Logic, Computation and Set Theory Write an essay on recursive functions. Your essay should include a sketch of why every computable function is recursive, and an explanation of the existence of a universal recursive function, as well as brief discussions of the Halting Problem and of the relationship between recursive sets and recursively enumerable sets. [You may assume that every recursive function is computable. You do not need to give proofs that particular functions to do with prime-power decompositions are recursive.] Paper 45 11I Probability and Measure Let (,F, P) be a probability space and let X,X1,X2, . . . be random variables. Write an essay in which you discuss the statement: if Xn ! X almost everywhere, then E(Xn) ! E(X). You should include accounts of monotone, dominated, and bounded convergence, and of Fatou’s lemma. [You may assume without proof the following fact. Let (,F, µ) be a measure space, and let f : ! R be non-negative with finite integral µ(f). If (fn : n > 1) are non-negative measurable functions with fn(!) " f(!) for all ! 2 , then µ(fn) ! µ(f) as n ! 1.] 12I Applied Probability Consider an M/G/1 queue with = ES < 1. Here is the arrival rate and ES is the mean service time. Prove that in equilibrium, the customer’s waiting time W has the moment-generating function MW(t) = EetW given by MW(t) = (1 − )t t + (1 −MS(t)) where MS(t) = EetS is the moment-generating function of service time S. [You may assume that in equilibrium, the M/G/1 queue size X at the time immediately after the customer’s departure has the probability generating function EzX = (1 − )(1 − z)MS((z − 1)) MS((z − 1)) − z , 0 6 z < 1 .] Deduce that when the service times are exponential of rate µ then MW(t) = 1 − + (1 − ) µ − − t , −1 < t < µ − . Further, deduce that W takes value 0 with probability 1 − and that P(W > x|W > 0) = e−(µ−)x, x > 0 . Sketch the graph of P(W > x) as a function of x. Now consider the M/G/1 queue in the heavy traffic approximation, when the service-time distribution is kept fixed and the arrival rate ! 1/ES, so that ! 1. Assuming that the second moment ES2 < 1, check that the limiting distribution of the re-scaled waiting time ˜W= (1 − ES)W is exponential, with rate 2ES/ES2. Paper 4 [TURN OVER6 13J Information Theory Define a cyclic code of length N. Show how codewords can be identified with polynomials in such a way that cyclic codes correspond to ideals in the polynomial ring with a suitably chosen multiplication rule. Prove that any cyclic code X has a unique generator, i.e. a polynomial c(X) of minimum degree, such that the code consists of the multiples of this polynomial. Prove that the rank of the code equals N − deg c(X), and show that c(X) divides XN + 1. Let X be a cyclic code. Set X? = {y = y1 . . . yN : N Xi=1 xiyi = 0 for all x = x1 . . . xN 2 X} (the dual code). Prove that X? is cyclic and establish how the generators of X and X? are related to each other. Show that the repetition and parity codes are cyclic, and determine their generators. 14I Optimization and Control Consider the deterministic dynamical system ˙ xt = Axt + But where A and B are constant matrices, xt 2 Rn, and ut is the control variable, ut 2 Rm. What does it mean to say that the system is controllable? Let yt = e−tAxt − x0. Show that if Vt is the set of possible values for yt as the control {us : 0 x t} is allowed to vary, then Vt is a vector space. Show that each of the following three conditions is equivalent to controllability of the system. (i) The set {v 2 Rn : v>yt = 0 for all yt 2 Vt} = {0}. (ii) The matrix H(t) = Rt 0 e−sABB>e−sA>ds is (strictly) positive definite. (iii) The matrix Mn = [B AB A2B · · · An−1B] has rank n. Consider the scalar system n Xj=0 ajd dtn−jt = ut , where a0 = 1. Show that this system is controllable. Paper 47 15J Principles of Statistics Suppose that 2 Rd is the parameter of a non-degenerate exponential family. Derive the asymptotic distribution of the maximum-likelihood estimator ˆn of based on a sample of size n. [You may assume that the density is infinitely differentiable with respect to the parameter, and that differentiation with respect to the parameter commutes with integration.] 16J Stochastic Financial Models What is Brownian motion (Bt)t>0? Briefly explain how Brownian motion can be considered as a limit of simple random walks. State the Reflection Principle for Brownian motion, and use it to derive the distribution of the first passage time a inf{t : Bt = a} to some level a > 0. Suppose that Xt = Bt + ct, where c > 0 is constant. Stating clearly any results to which you appeal, derive the distribution of the first-passage time (c) a inf{t : Xt = a} to a > 0. Now let a sup{t : Xt = a}, where a > 0. Find the density of a. 17B Nonlinear Dynamical Systems (a) Consider the map G1(x) = f(x+a), defined on 0 6 x < 1, where f(x) = x [mod 1], 0 6 f < 1, and the constant a satisfies 0 6 a < 1. Give, with reasons, the values of a (if any) for which the map has (i) a fixed point, (ii) a cycle of least period n, (iii) an aperiodic orbit. Does the map exhibit sensitive dependence on initial conditions? Show (graphically if you wish) that if the map has an n-cycle then it has an infinite number of such cycles. Is this still true if G1 is replaced by f(cx + a), 0 < c < 1? (b) Consider the map G2(x) = f(x + a + b 2sin 2x), where f(x) and a are defined as in Part (a), and b > 0 is a parameter. Find the regions of the (a, b) plane for which the map has (i) no fixed points, (ii) exactly two fixed points. Now consider the possible existence of a 2-cycle of the map G2 when b 1, and suppose the elements of the cycle are X, Y with X < 12 . By expanding X, Y, a in powers of b, so that X = X0 + bX1 + b2X2 + O(b3), and similarly for Y and a, show that a = 12 + b2 8sin 4X0 + O(b3). Use this result to sketch the region of the (a, b) plane in which 2-cycles exist. How many distinct cycles are there for each value of a in this region? Paper 4 [TURN OVER8 18D Partial Differential Equations (a) State a theorem of local existence, uniqueness and C1 dependence on the initial data for a solution for an ordinary differential equation. Assuming existence, prove that the solution depends continuously on the initial data. (b) State a theorem of local existence of a solution for a general quasilinear first– order partial differential equation with data on a smooth non-characteristic hypersurface. Prove this theorem in the linear case assuming the validity of the theorem in part (a); explain in your proof the importance of the non-characteristic condition. 19D Methods of Mathematical Physics Let h(t) = i(t+t2) . Sketch the path of Im(h(t)) = const. through the point t = 0, and the path of Im(h(t)) = const. through the point t = 1 . By integrating along these paths, show that as ! 1 Z 1 0 t−1/2ei(t+t2)dt c1 1/2 + c2e2i, where the constants c1 and c2 are to be computed. 20D Numerical Analysis Write an essay on the method of conjugate gradients. You should define the method, list its main properties and sketch the relevant proof. You should also prove that (in exact arithmetic) the method terminates in a finite number of steps, briefly mention the connection with Krylov subspaces, and describe the approach of preconditioned conjugate gradients. Paper 49 21C Electrodynamics Using Lorentz gauge, Aa,a = 0, Maxwell’s equations for a current distribution Ja can be reduced to Aa(x) = µ0Ja(x). The retarded solution is Aa(x) = µ0 2Z d4y (z0)(zczc)Ja(y), where za = xa − ya. Explain, heuristically, the rˆole of the -function and Heaviside step function in this formula. The current distribution is produced by a point particle of charge q moving on a world line ra(s), where s is the particle’s proper time, so that Ja(y) = q Z ds V a(s)(4)(y − r(s)), where V a = ˙ ra(s) = dra/ds. Show that Aa(x) = µ0q 2Z ds (X0)(XcXc)V a(s), where Xa = xa − ra(s), and further that, setting = XcV c, Aa(x) = µ0q 4V a s=s, where sshould be defined. Verify that s,a = Xa s=s. Evaluating quantities at s = sshow that V a ,b = 1 2 [−V aVb + SaXb] , where Sa = ˙Va + V a(1 − Xc ˙Vc)/. Hence verify that Aa,a(x) = 0 and Fab = µ0q 42 (SaXb − SbXa) . Verify this formula for a stationary point charge at the origin. [Hint: If f(s) has simple zeros at si, i = 1, 2, . . . then (f(s)) =Xi (si) |f0(si)|. ] Paper 4 [TURN OVER10 22E Foundations of Quantum Mechanics The states of the hydrogen atom are denoted by |nlmi with l < n,−l m l and associated energy eigenvalue En, where En = − e2 80a0n2 . A hydrogen atom is placed in a weak electric field with interaction Hamiltonian H1 = −eEz . a) Derive the necessary perturbation theory to show that to O(E2) the change in the energy associated with the state |100i is given by E1 = e2E2 1Xn=2 n−1 Xl=0 l X m=−l ???h100|z|nlmi???2 E1 − En . () The wavefunction of the ground state |100i is n=1(r) = 1 (a30 )1/2 e−r/a0 . By replacing En, 8 n > 1, in the denominator of () by E2 show that |E1| < 323 0E2a30 . b) Find a matrix whose eigenvalues are the perturbed energies to O(E) for the states |200i and |210i. Hence, determine these perturbed energies to O(E) in terms of the matrix elements of z between these states. [Hint: hnlm|z|nlmi = 0 8 n, l,m hnlm|z|nl0m0i = 0 8 n, l, l0, m,m0, m 6= m0 ] Paper 411 23E Statistical Physics Derive the Bose-Einstein expression for the mean number of Bose particles ¯n occupying a particular single-particle quantum state of energy ", when the chemical potential is µ and the temperature is T in energy units. Why is the chemical potential for a gas of photons given by µ = 0? Show that, for black-body radiation in a cavity of volume V at temperature T, the mean number of photons in the angular frequency range (!, ! + d!) is V 2c3 !2d! e~!/T − 1 . Hence, show that the total energy E of the radiation in the cavity is E = KV T4 , where K is a constant that need not be evaluated. Use thermodynamic reasoning to find the entropy S and pressure P of the radiation and verify that E − TS + PV = 0 . Why is this last result to be expected for a gas of photons? Paper 4 [TURN OVER12 24E Applications of Quantum Mechanics Describe briefly the variational approach to determining approximate energy eigenvallue for a Hamiltonian H. Consider a Hamiltonian H and two states | 1i, | 2i such that h 1|H| 1i = h 2|H| 2i = E , h 2|H| 1i = h 1|H| 2i = " , h 1| 1i = h 2| 2i = 1 , h 2| 1i = h 1| 2i = s . Show that, by considering a linear combination | 1i + | 2i, the variational method gives E − " 1 − s , E + " 1 + s , as approximate energy eigenvalues. Consider the Hamiltonian for an electron in the presence of two protons at 0 and R, H = p2 2m + e2 401R − 1 |r| − 1 |r − R|, R = |R| . Let 0(r) = e−r/a/(a3) 12 be the ground state hydrogen atom wave function which satisfies p2 2m − e2 40|r| 0(r) = E0 0(r) . It is given that S = Z d3r 0(r) 0(r − R) = 1 + Ra + R2 3a2e−R/a , U = Z d3r 1 |r| 0(r) 0(r − R) = 1a1 + Ra e−R/a , and, for large R, that Z d3r 1 |r − R| 0(r)2 − 1R = Oe−2R/a. Consider the trial wave function 0(r)+ 0(r−R). Show that the variational estimate for the ground state energy for large R is E(R) = E0 + e2 40RS − RU) + Oe−2R/a. Explain why there is an attractive force between the two protons for large R. Paper 413 25C General Relativity Starting from the Ricci identity Va;b;c − Va;c;b = VeReabc, give an expression for the curvature tensor Reabc of the Levi-Civita connection in terms of the Christoffel symbols and their partial derivatives. Using local inertial coordinates, or otherwise, establish that Reabc + Rebca + Recab = 0. () A vector field with components V a satisfies Va;b + Vb;a = 0. () Show, using equation () that Va;b;c = VeRecba, and hence that Va;b;b + RacVc = 0, where Rab is the Ricci tensor. Show that equation () may be written as (@cgab)V c + gcb@aV c + gac@bV c = 0. () If the metric is taken to be the Schwarzschild metric ds2 = −1 − 2Mr dt2 + 1 − 2Mr −1 dr2 + r2(d2 + sin2 d2), show that V a = a0 is a solution of (). Calculate V a;a. Electromagnetism can be described by a vector potential Aa and a Maxwell field tensor Fab satisfying Fab = Ab;a − Aa;b and Fab;b = 0. () The divergence of Aa is arbitrary and we may choose Aa;a = 0. With this choice show that in a general spacetime Aa;b;b − RacAc = 0. Hence show that in the Schwarzschild spacetime a tensor field whose only non-trivial components are Ftr = −Frt = Q/r2, where Q is a constant, satisfies the field equations (). 26A Fluid Dynamics II Write an essay on the Kelvin-Helmholtz instability of a vortex sheet. Your essay should include a detailed linearised analysis, a physical interpretation of the instability, and an informal discussion of nonlinear effects and of the effects of viscosity. Paper 4 [TURN OVER14 27A Waves in Fluid and Solid Media A plane shock is moving with speed U into a perfect gas. Ahead of the shock the gas is at rest with pressure p1 and density 1, while behind the shock the velocity, pressure and density of the gas are u2, p2 and 2 respectively. Derive the Rankine-Hugoniot relations across the shock. Show that 1 2 = 2c21 + (− 1)U2 (+ 1)U2 , where c21 = p1/1 and is the ratio of the specific heats of the gas. Now consider a change of frame such that the shock is stationary and the gas has a component of velocity V parallel to the shock. Deduce that the angle of deflection of the flow which is produced by a stationary shock inclined at an angle = tan−1(U/V ) to an oncoming stream of Mach number M = (U2 + V 2) 12 /c1 is given by tan = 2 cot (M2 sin 2 − 1) 2 +M2(+ cos 2) . [Note that tan(+ ) = tan + tan 1 − tan tan . ] Paper 4