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MATHEMATICAL TRIPOS Part IB
Tuesday, 1 June, 2010 9:00 am to 12:00 pm
PAPER 1
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.
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 REQUIREMENTS 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.
SECTION I
1F Linear Algebra Suppose that V is the complex vector space of polynomials of degree at most n − 1
in the variable z. Find the Jordan normal form for each of the linear transformations d dz
and z d dz acting on V.
2A Complex Analysis or Complex Methods (a) Write down the definition of the complex derivative of the function f (z) of a single complex variable.
(b) Derive the Cauchy-Riemann equations for the real and imaginary parts u(x, y) and v(x, y) of f (z), where z = x + iy and
f (z) = u(x, y) + iv(x, y).
(c) State necessary and sufficient conditions on u(x, y) and v(x, y) for the function f (z) to be complex differentiable.
3F Geometry (i) Define the notion of curvature for surfaces embedded in R^3. (ii) Prove that the unit sphere in R^3 has curvature +1 at all points.
4D Variational Principles (a) Define what it means for a function f : Rn^ → R to be convex and strictly convex. (b) State a necessary and sufficient first-order condition for strict convexity of f ∈ C^1 (Rn), and give, with proof, an example of a function which is strictly convex but with second derivative which is not everywhere strictly positive.
Part IB, Paper 1
SECTION II
9F Linear Algebra Let V denote the vector space of n × n real matrices.
(1) Show that if ψ(A, B) = tr(ABT^ ), then ψ is a positive-definite symmetric bilinear form on V.
(2) Show that if q(A) = tr(A^2 ), then q is a quadratic form on V. Find its rank and signature.
[Hint: Consider symmetric and skew-symmetric matrices.]
10H Groups Rings and Modules Prove that the kernel of a group homomorphism f : G → H is a normal subgroup of the group G.
Show that the dihedral group D 8 of order 8 has a non-normal subgroup of order
- Conclude that, for a group G, a normal subgroup of a normal subgroup of G is not necessarily a normal subgroup of G.
11G Analysis II State and prove the contraction mapping theorem. Demonstrate its use by showing that the differential equation f ′(x) = f (x^2 ), with boundary condition f (0) = 1 , has a unique solution on [0, 1), with one-sided derivative f ′(0) = 1 at zero.
12H Metric and Topological Spaces Let f : X → Y and g : Y → X be continuous maps of topological spaces with f ◦ g = idY.
(1) Suppose that (i) Y is path-connected, and (ii) for every y ∈ Y , its inverse image f −^1 (y) is path-connected. Prove that X is path-connected.
(2) Prove the same statement when “path-connected” is everywhere replaced by “connected”.
Part IB, Paper 1
13A Complex Analysis or Complex Methods Calculate the following real integrals by using contour integration. Justify your steps carefully.
(a) I 1 =
0
x sin x x^2 + a^2
dx, a > 0 ,
(b) I 2 =
0
x^1 /^2 log x 1 + x^2
dx.
14A Methods (a) A function f (t) is periodic with period 2π and has continuous derivatives up to and including the kth derivative. Show by integrating by parts that the Fourier coefficients of f (t)
an =
π
∫ (^2) π
0
f (t) cos nt dt,
bn =
π
∫ (^2) π
0
f (t) sin nt dt,
decay at least as fast as 1/nk^ as n → ∞. (b) Calculate the Fourier series of f (t) = | sin t| on [0, 2 π]. (c) Comment on the decay rate of your Fourier series.
Part IB, Paper 1 [TURN OVER
17B Fluid Dynamics Starting with the Euler equations for an inviscid incompressible fluid, derive Bernoulli’s theorem for unsteady irrotational flow.
Inviscid fluid of density ρ is contained within a U-shaped tube with the arms vertical, of height h and with the same (unit) cross-section. The ends of the tube are closed. In the equilibrium state the pressures in the two arms are p 1 and p 2 and the heights of the fluid columns are ℓ 1 , ℓ 2.
The fluid in arm 1 is displaced upwards by a distance ξ (and in the other arm downward by the same amount). In the subsequent evolution the pressure above each column may be taken as inversely proportional to the length of tube above the fluid surface. Using Bernoulli’s theorem, show that ξ(t) obeys the equation
ρ(ℓ 1 + ℓ 2 )ξ¨ + p 1 ξ h − ℓ 1 − ξ
p 2 ξ h − ℓ 2 + ξ
Now consider the special case ℓ 1 = ℓ 2 = ℓ 0 , p 1 = p 2 = p 0. Construct a first integral of this equation and hence give an expression for the total kinetic energy ρℓ 0 ξ˙^2 of the flow in terms of ξ and the maximum displacement ξmax.
Part IB, Paper 1 [TURN OVER
18C Numerical Analysis Let 〈f, g〉 =
−∞
e−x
2 f (x) g(x) dx ,
be an inner product. The Hermite polynomials Hn(x), n = 0, 1 , 2 ,... are polynomials in x of degree n with leading term 2nxn^ which are orthogonal with respect to the inner product, with
〈Hm, Hn〉 =
γm > 0 if m = n , 0 otherwise,
and H 0 (x) = 1. Find a three-term recurrence relation which is satisfied by Hn(x) and γn for n = 1, 2 , 3. [You may assume without proof that
〈 1 , 1 〉 =
π , 〈x, x〉 = (^12)
π , 〈x^2 , x^2 〉 = (^34)
π , 〈x^3 , x^3 〉 = (^158)
π .]
Next let x 0 , x 1 ,... , xk be the k + 1 distinct zeros of Hk+1(x) and for i, j = 0, 1 ,... , k define the Lagrangian polynomials
Li(x) =
j 6 =i
x − xj xi − xj
associated with these points. Prove that 〈Li, Lj 〉 = 0 if i 6 = j.
19E Statistics Consider the the linear regression model
Yi = β xi + ǫi,
where the numbers x 1 ,... , xn are known, the independent random variables ǫ 1 ,... , ǫn have the N (0, σ 2 ) distribution, and the parameters β and σ 2 are unknown. Find the maximum likelihood estimator for β.
State and prove the Gauss–Markov theorem in the context of this model.
Write down the distribution of an arbitrary linear estimator for β. Hence show that there exists a linear, unbiased estimator β̂ for β such that
Eβ, σ 2 [(β̂ − β)^4 ] 6 Eβ, σ 2 [( β˜ − β)^4 ]
for all linear, unbiased estimators β˜.
[Hint: If Z ∼ N (a, b 2 ) then E [(Z − a)^4 ] = 3 b^4 .]
Part IB, Paper 1
