MATHEMATICAL TRIPOS Part IB Friday 10 June 2005 1.30 to 4.30 PAPER 4 Before you begin read these instructions carefully. Each question in Section II carries twice the number of marks of each question in Section I. Candidates may attempt at most four questions from Section I and at most six questions from Section II. Complete answers are preferred to fragments. 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 up your answers in separate bundles labelled A, B, . . . , H according to the examiner letter affixed to each question, including in the same bundle questions from Sections I and II with the same examiner letter. Attach a completed gold cover sheet to each bundle; write the examiner letter in the box marked ‘Examiner Letter’ on the cover sheet. You must also complete a green master cover sheet listing all the questions you have attempted. Every cover sheet must bear your examination number and desk number. STATIONERY REQUIRMENTS SPECIAL REQUIREMENTS Gold cover sheet None Green master cover sheet You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.2 SECTION I 1B Linear Algebra Define what it means for an n × n complex matrix to be unitary or Hermitian. Show that every eigenvalue of a Hermitian matrix is real. Show that every eigenvalue of a unitary matrix has absolute value 1. Show that two eigenvectors of a Hermitian matrix that correspond to different eigenvalues are orthogonal, using the standard inner product on Cn. 2C Groups, Rings and Modules State Eisenstein’s irreducibility criterion. Let n be an integer > 1. Prove that 1 + x + . . . + xn−1 is irreducible in Z[x] if and only if n is a prime number. 3B Analysis II Let V be the vector space of continuous real-valued functions on [0, 1]. Show that the function ||f|| = Z 1 0 |f(x)| dx defines a norm on V . For n = 1, 2, . . ., let fn(x) = e−nx. Is fn a convergent sequence in the space V with this norm? Justify your answer. Paper 43 4A Complex Analysis Let : [0, 1] ! C be a closed path, where all paths are assumed to be piecewise continuously differentiable, and let a be a complex number not in the image of . Write down an expression for the winding number n(, a) in terms of a contour integral. From this characterization of the winding number, prove the following properties: (a) If 1 and 2 are closed paths not passing through zero, and if : [0, 1] ! C is defined by (t) = 1(t)2(t) for all t, then n(, 0) = n(1, 0) + n(2, 0). (b) If : [0, 1] ! C is a closed path whose image is contained in {Re(z) > 0}, then n(, 0) = 0. (c) If 1 and 2 are closed paths and a is a complex number, not in the image of either path, such that |1(t) − 2(t)| < |1(t) − a| for all t, then n(1, a) = n(2, a). [You may wish here to consider the path defined by (t) = 1 − (1(t) − 2(t))/(1(t) − a).] 5H Methods Show how the general solution of the wave equation for y(x, t), 1 c2 @2 @t2 y(x, t) − @2 @x2 y(x, t) = 0 , can be expressed as y(x, t) = f(ct − x) + g(ct + x) . Show that the boundary conditions y(0, t) = y(L, t) = 0 relate the functions f and g and require them to be periodic with period 2L. Show that, with these boundary conditions, 12 Z L 0 1 c2 @y @t 2 + @y @x2dx = Z L −Lg0(ct + x)2 dx , and that this is a constant independent of t. Paper 4 [TURN OVER4 6G Quantum Mechanics Define the commutator [A,B] of two operators, A and B. In three dimensions angular momentum is defined by a vector operator L with components Lx = y pz − z py Ly = z px − x pz Lz = x py − y px . Show that [Lx ,Ly] = i ~Lz and use this, together with permutations, to show that [L2 ,Lw] = 0, where w denotes any of the directions x, y, z. At a given time the wave function of a particle is given by = (x + y + z) exp −px2 + y2 + z2. Show that this is an eigenstate of L2 with eigenvalue equal to 2~2. 7H Electromagnetism For a static current density J(x) show that we may choose the vector potential A(x) so that −r2A = µ0J . For a loop L, centred at the origin, carrying a current I show that A(x) = µ0I 4IL 1 |x − r| dr −µ0I 41 |x|3 IL 12 x × (r × dr) as |x| ! 1. [You may assume −r2 1 4|x| = 3(x) , and for fixed vectors a, b IL a · dr = 0, IL(a · r b · dr + b · r a · dr) = 0 .# Paper 45 8F Numerical Analysis Define Gaussian quadrature. Evaluate the coefficients of the Gaussian quadrature of the integral Z 1 −1(1 − x2)f(x)dx which uses two function evaluations. 9D Markov Chains Prove that the simple symmetric random walk in three dimensions is transient. [You may wish to recall Stirling’s formula: n! (2) 12 nn+12 e−n.] Paper 4 [TURN OVER6 SECTION II 10B Linear Algebra (i) Let V be a finite-dimensional real vector space with an inner product. Let e1, . . . , en be a basis for V . Prove by an explicit construction that there is an orthonormal basis f1, . . . , fn for V such that the span of e1, . . . , ei is equal to the span of f1, . . . , fi for every 1 6 i 6 n. (ii) For any real number a, consider the quadratic form qa(x, y, z) = xy + yz + zx + ax2 on R3. For which values of a is qa nondegenerate? When qa is nondegenerate, compute its signature in terms of a. 11C Groups, Rings and Modules Let R be the ring of Gaussian integers Z[i], where i2 = −1, which you may assume to be a unique factorization domain. Prove that every prime element of R divides precisely one positive prime number in Z. List, without proof, the prime elements of R, up to associates. Let p be a prime number in Z. Prove that R/pR has cardinality p2. Prove that R/2R is not a field. If p 3 mod 4, show that R/pR is a field. If p 1 mod 4, decide whether R/pR is a field or not, justifying your answer. Paper 47 12A Geometry Given a parametrized smooth embedded surface : V ! U R3, where V is an open subset of R2 with coordinates (u, v), and a point P 2 U, define what is meant by the tangent space at P, the unit normal N at P, and the first fundamental form Edu2 + 2Fdu dv + Gdv2. [You need not show that your definitions are independent of the parametrization.] The second fundamental form is defined to be Ldu2 + 2Mdu dv + Ndv2, where L = uu · N, M = uv · N and N = vv · N. Prove that the partial derivatives of N (considered as a vector-valued function of u, v) are of the form Nu = au + bv, Nv = cu + dv, where −L M M N = a b c dE F F G. Explain briefly the significance of the determinant ad − bc. 13B Analysis II Let F : [−a, a] × [x0 − r, x0 + r] ! R be a continuous function. Let C be the maximum value of |F(t, x)|. Suppose there is a constant K such that |F(t, x) − F(t, y)| 6 K|x − y| for all t 2 [−a, a] and x, y 2 [x0 −r, x0 +r]. Let b < min(a, r/C, 1/K). Show that there is a unique C1 function x : [−b, b] ! [x0 − r, x0 + r] such that x(0) = x0 and dx dt = F(t, x(t)). [Hint: First show that the differential equation with its initial condition is equivalent to the integral equation x(t) = x0 + Z t 0 F(s, x(s)) ds. Paper 4 [TURN OVER8 14A Metric and Topological Spaces Let (M, d) be a metric space, and F a non-empty closed subset of M. For x 2 M, set d(x, F) = inf z2F d(x, z). Prove that d(x, F) is a continuous function of x, and that it is strictly positive for x 62 F. A topological space is called normal if for any pair of disjoint closed subsets F1, F2, there exist disjoint open subsets U1 F1, U2 F2. By considering the function d(x, F1) − d(x, F2), or otherwise, deduce that any metric space is normal. Suppose now that X is a normal topological space, and that F1, F2 are disjoint closed subsets in X. Prove that there exist open subsets W1 F1,W2 F2, whose closures are disjoint. In the case when X = R2 with the standard metric topology, F1 = {(x,−1/x) : x < 0} and F2 = {(x, 1/x) : x > 0}, find explicit open subsets W1,W2 with the above property. 15F Complex Methods Determine the Fourier expansion of the function f(x) = sin x, where −6 x 6 , in the two cases where is an integer and is a real non-integer. Using the Parseval identity in the case = 12 , find an explicit expression for the sum 1Xn=1 n2 (4n2 − 1)2 . Paper 49 16H Methods Define an isotropic tensor and show that ij , ijk are isotropic tensors. For ˆx a unit vector and dS(ˆx) the area element on the unit sphere show that Z dS(ˆx) ˆxi1 . . . ˆxin is an isotropic tensor for any n. Hence show that Z dS(ˆx) ˆxiˆxj = aij , Z dS(ˆx) ˆxiˆxj ˆxk = 0 , Z dS(ˆx) ˆxiˆxj ˆxk ˆxl = bijkl + ikjl + iljk, for some a, b which should be determined. Explain why ZV d3x x1 + p−1 x2nf(|x|) = 0 , n = 2, 3, 4 , where V is the region inside the unit sphere. [The general isotropic tensor of rank 4 has the form a ijkl + b ikjl + c iljk.] Paper 4 [TURN OVER10 17G Special Relativity Obtain the Lorentz transformations that relate the coordinates of an event measuure in one inertial frame (t, x, y, z) to those in another inertial frame moving with velocity v along the x axis. Take care to state the assumptions that lead to your result. A star is at rest in a three-dimensional coordinate frame S0 that is moving at constant velocity v along the x axis of a second coordinate frame S. The star emits light of frequency 0, which may considered to be a beam of photons. A light ray from the star to the origin in S0 is a straight line that makes an angle 0 with the x0 axis. Write down the components of the four-momentum of a photon in this light ray. The star is seen by an observer at rest at the origin of S at time t = t0 = 0, when the origins of the coordinate frames S and S0 coincide. The light that reaches the observer moves along a line through the origin that makes an angle to the x axis and has frequency . Make use of the Lorentz transformations between the four-momenta of a photon in these two frames to determine the relation = 0 1 − v2 c2 −1/2 1 + vc cos . where is the observed wavelength of the photon and 0 is the wavelength in the star’s rest frame. Comment on the form of this result in the special cases with cos = 1, cos = −1 and cos = 0. [You may assume that the energy of a photon of frequency is hand its threemomeentu is a three-vector of magnitude h/c.] 18E Fluid Dynamics A fluid of density 1 occupies the region z > 0 and a second fluid of density 2 occupies the region z < 0. State the equations and boundary conditions that are satisfied by the corresponding velocity potentials 1 and 2 and pressures p1 and p2 when the system is perturbed so that the interface is at z = (x, t) and the motion is irrotational. Seek a set of linearised equations and boundary conditions when the disturbances are proportional to ei(kx−!t), and derive the dispersion relation !2 = 2 − 1 2 + 1 gk, where g is the gravitational acceleration. Paper 411 19D Statistics Let Y1, . . . , Yn be observations satisfying Yi = xi + i, 1 6 i 6 n, where 1, . . . , n are independent random variables each with the N(0, 2) distribution. Here x1, . . . , xn are known but and 2 are unknown. (i) Determine the maximum-likelihood estimates (b, b2) of (, 2). (ii) Find the distribution of b. (iii) By showing that Yi − bxi and bare independent, or otherwise, determine the joint distribution of band b2. (iv) Explain carefully how you would test the hypothesis H0 : = 0 against H1 : 6= 0. 20D Optimization Describe the Ford–Fulkerson algorithm for finding a maximal flow from a source to a sink in a directed network with capacity constraints on the arcs. Explain why the algorithm terminates at an optimal flow when the initial flow and the capacity constraints are rational. Illustrate the algorithm by applying it to the problem of finding a maximal flow from S to T in the network below. H G F E K J T B A S D C 5 8 76 10 8 9 6 17 11 10 12 4 5 4 10 5 5 7 8 11 END OF PAPER Paper 4