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MATHEMATICAL TRIPOS Part IB
Thursday, 4 June, 2009 9:00 am to 12:00 pm
PAPER 3
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 Groups, Rings and Modules Let F be a field. Show that the polynomial ring F [X] is a principal ideal domain. Give, with justification, an example of an ideal in F [X, Y ] which is not principal.
2G Geometry Write down the equations for geodesic curves on a surface. Use these to describe all the geodesics on a circular cylinder, and draw a picture illustrating your answer.
3E Analysis II What is meant by a norm on Rn? For x ∈ Rn^ write
‖x‖ 1 = |x 1 | + |x 2 | + · · · + |xn|,
‖x‖ 2 =
|x 1 |^2 + |x 2 |^2 + · · · + |xn|^2.
Prove that ‖ · ‖ 1 and ‖ · ‖ 2 are norms. [You may assume the Cauchy-Schwarz inequality.]
Find the smallest constant Cn such that ‖x‖ 1 6 Cn‖x‖ 2 for all x ∈ Rn, and also the smallest constant C′ n such that ‖x‖ 2 6 C′ n‖x‖ 1 for all x ∈ Rn.
4F Metric and Topological Spaces Are the following statements true or false? Give brief justifications for your answers. (i) If X is a connected open subset of Rn^ for some n, then X is path-connected. (ii) A cartesian product of two connected spaces is connected. (iii) If X is a Hausdorff space and the only connected subsets of X are singletons {x}, then X is discrete.
5D Complex Methods
Use the residue calculus to evaluate
(i)
C
ze^1 /z^ dz and (ii)
C
zdz 1 − 4 z^2
where C is the circle |z| = 1.
Part IB, Paper 3
8H Statistics In a demographic study, researchers gather data on the gender of children in families with more than two children. For each of the four possible outcomes GG, GB, BG, BB of the first two children in the family, they find 50 families which started with that pair, and record the gender of the third child of the family. This produces the following table of counts: First two children Third child B Third child G GG 16 34 GB 28 22 BG 25 25 BB 31 19
In view of this, is the hypothesis that the gender of the third child is independent of the genders of the first two children rejected at the 5% level?
[Hint: the 95% point of a χ^23 distribution is 7. 8147 , and the 95% point of a χ^24 distribution is 9 .4877.]
9H Markov Chains Let (Xn)n> 0 be a simple random walk on the integers: the random variables ξn ≡ Xn − Xn− 1 are independent, with distribution
P (ξ = 1) = p, P (ξ = −1) = q,
where 0 < p < 1, and q = 1−p. Consider the hitting time τ = inf{n : Xn = 0 or Xn = N }, where N > 1 is a given integer. For fixed s ∈ (0, 1) define ξk = E[sτ^ : Xτ = 0|X 0 = k] for k = 0,... , N. Show that the ξk satisfy a second-order difference equation, and hence find them.
Part IB, Paper 3
SECTION II
10G Linear Algebra For each of the following, provide a proof or counterexample.
(1) If A, B are complex n × n matrices and AB = BA, then A and B have a common eigenvector.
(2) If A, B are complex n × n matrices and AB = BA, then A and B have a common eigenvalue.
(3) If A, B are complex n × n matrices and (AB)n^ = 0 then (BA)n^ = 0.
(4) If T : V → V is an endomorphism of a finite-dimensional vector space V and λ is an eigenvalue of T , then the dimension of {v ∈ V | (T − λI)v = 0} equals the multiplicity of λ as a root of the minimal polynomial of T.
(5) If T : V → V is an endomorphism of a finite-dimensional complex vector space V , λ is an eigenvalue of T , and Wi = {v ∈ V | (T − λI)i(v) = 0}, then Wc = Wc+ where c is the multiplicity of λ as a root of the minimal polynomial of T.
11F Groups, Rings and Modules Let S be a multiplicatively closed subset of a ring R, and let I be an ideal of R which is maximal among ideals disjoint from S. Show that I is prime. If R is an integral domain, explain briefly how one may construct a field F together with an injective ring homomorphism R → F. Deduce that if R is an arbitrary ring, I an ideal of R, and S a multiplicatively closed subset disjoint from I, then there exists a ring homomorphism f : R → F , where F is a field, such that f (x) = 0 for all x ∈ I and f (y) 6 = 0 for all y ∈ S. [You may assume that if T is a multiplicatively closed subset of a ring, and 0 6 ∈ T , then there exists an ideal which is maximal among ideals disjoint from T .]
Part IB, Paper 3 [TURN OVER
14E Complex Analysis For each positive real number R write BR = {z ∈ C : |z| 6 R}. If F is holomorphic on some open set containing BR, we define
J(F, R) =
2 π
∫ (^2) π
0
log |F (Reiθ^ )| dθ.
- If F 1 , F 2 are both holomorphic on some open set containing BR, show that J(F 1 F 2 , R) = J(F 1 , R) + J(F 2 , R).
- Suppose that F (0) = 1 and that F does not vanish on some open set containing BR. By showing that there is a holomorphic branch of logarithm of F and then considering z−^1 log F (z), prove that J(F, R) = 0.
- Suppose that |w| < R. Prove that the function ψW,R(z) = R(z − w)/(R^2 − wz) has modulus 1 on |z| = R and hence that it satisfies J(ψW,R, R) = 0.
Suppose now that F : C → C is holomorphic and not identically zero, and let R be such that no zeros of F satisfy |z| = R. Briefly explain why there are only finitely many zeros of F in BR and, assuming these are listed with the correct multiplicity, derive a formula for J(F, R) in terms of the zeros, R, and F (0). Suppose that F has a zero at every lattice point (point with integer coordinates) except for (0, 0). Show that there is a constant c > 0 such that |F (zi)| > ec|zi| 2 for a sequence z 1 , z 2 ,... of complex numbers tending to infinity.
Part IB, Paper 3 [TURN OVER
15A Methods A function g(r) is chosen to make the integral ∫ (^) b
a
f (r, g, g′)dr
stationary, subject to given values of g(a) and g(b). Find the Euler–Lagrange equation for g(r).
In a certain three-dimensional electrostatics problem the potential φ depends only on the radial coordinate r, and the energy functional of φ is
E[φ] = 2π
∫ R 2
R 1
[
dφ dr
2 λ^2
φ^2
]
r^2 dr ,
where λ is a parameter. Show that the Euler–Lagrange equation associated with minimizing the energy E is equivalent to
1 r
d^2 (rφ) dr^2
λ^2
φ = 0. (1)
Find the general solution of this equation, and the solution for the region R 1 6 r 6 R 2 which satisfies φ(R 1 ) = φ 1 and φ(R 2 ) = 0.
Consider an annular region in two dimensions, where the potential is a function of the radial coordinate r only. Write down the equivalent expression for the energy functional E above, in cylindrical polar coordinates, and derive the equivalent of (1).
Part IB, Paper 3
17A Electromagnetism Two long thin concentric perfectly conducting cylindrical shells of radii a and b (a < b) are connected together at one end by a resistor of resistance R, and at the other by a battery that establishes a potential difference V. Thus, a current I = V /R flows in opposite directions along each of the cylinders.
(a) Using Amp`ere’s law, find the magnetic field B in between the cylinders. (b) Using Gauss’s law and the integral relationship between the potential and the electric field, or otherwise, show that the charge per unit length on the inner cylinder is
λ = 2 πǫ 0 V ln(b/a)
and also calculate the radial electric field.
(c) Calculate the Poynting vector and by suitable integration verify that the power delivered by the system is V 2 /R.
Part IB, Paper 3
18D Fluid Dynamics
Starting from Euler’s equations for an inviscid incompressible fluid of density ρ with no body force, undergoing irrotational motion, show that the pressure p is given by
p ρ
∂φ ∂t
(∇φ)^2 = F (t),
for some function F (t), where φ is the velocity potential.
The fluid occupies an infinite domain and contains a spherical gas bubble of radius R(t) in which the pressure is pg. The pressure in the fluid at infinity is p∞.
Show that
R R¨ +^3
R˙^2 = pg^ −^ p∞ ρ
The bubble contains a fixed mass M of gas in which
pg = C
M/R^3
for a constant C. At time t = 0, R = R 0 , R˙ = 0 and pg = p∞/2. Show that
R˙^2 R^3 = p∞ ρ
[
R^30 −
R^60
3 R^3
R^3
]
and deduce that the bubble radius oscillates between R 0 and R 0 / 21 /^3.
Part IB, Paper 3 [TURN OVER
20H Optimization Four factories supply stuff to four shops. The production capacities of the factories are 7, 12, 8 and 9 units per week, and the requirements of the shops are 8 units per week each. If the costs of transporting a unit of stuff from factory i to shop j is the (i, j)th element in the matrix
find a minimal-cost allocation of the outputs of the factories to the shops.
Suppose that the cost of producing one unit of stuff varies across the factories, being 3, 2, 4, 5 respectively. Explain how you would modify the original problem to minimise the total cost of production and of transportation, and find an optimal solution for the modified problem.
END OF PAPER
Part IB, Paper 3
