Download Exam Instructions & Samples: Linear Algebra, Analysis, Complex Analysis, Numerical Analysi and more Exams Mathematics in PDF only on Docsity!
MATHEMATICAL TRIPOS Part IB
Friday 6 June 2008 1.30 to 4.
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 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
1E Linear Algebra
Describe (without proof) what it means to put an n×n matrix of complex numbers into Jordan normal form. Explain (without proof) the sense in which the Jordan normal form is unique.
Put the following matrix in Jordan normal form:
2G Groups, Rings and Modules Let n ≥ 2 be an integer. Show that the polynomial (Xn^ − 1)/(X − 1) is irreducible over Z if and only if n is prime.
[You may use Eisenstein’s criterion without proof.]
3F Analysis II
Let X be the vector space of all continuous real-valued functions on the unit interval [0, 1]. Show that the functions
‖f ‖ 1 =
0
|f (t)| dt and ‖f ‖∞ = sup{|f (t)| : 0 6 t 6 1 }
both define norms on X.
Consider the sequence (fn) defined by fn(t) = ntn(1 − t). Does (fn) converge in the norm ‖ − ‖ 1? Does it converge in the norm ‖ − ‖∞? Justify your answers.
4E Complex Analysis
Suppose that f and g are two functions which are analytic on the whole complex plane C. Suppose that there is a sequence of distinct points z 1 , z 2 ,... with |zi| 6 1 such that f (zi) = g(zi). Show that f (z) = g(z) for all z ∈ C. [You may assume any results on Taylor expansions you need, provided they are clearly stated.]
What happens if the assumption that |zi| 6 1 is dropped?
Paper 4
8D Numerical Analysis
Show that the Chebyshev polynomials, Tn(x) = cos(n cos−^1 x) , n = 0, 1 , 2 ,... obey the orthogonality relation
∫ (^1)
− 1
Tn(x)Tm(x) √ 1 − x^2
dx =
π 2
δn,m(1 + δn, 0 ).
State briefly how an optimal choice of the parameters ak, xk, k = 1, 2... n is made in the Gaussian quadrature formula
∫ (^1)
− 1
f (x) √ 1 − x^2
dx ∼
∑^ n
k=
akf (xk).
Find these parameters for the case n = 3.
9H Markov Chains
A Markov chain on the state–space I = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } has transition matrix
P =
Classify the chain into its communicating classes, deciding for each what the period is, and whether the class is recurrent.
For each i, j ∈ I say whether the limit limn→∞ p( ijn )exists, and evaluate the limit when it does exist.
Paper 4
SECTION II
10E Linear Algebra
What is meant by a Hermitian matrix? Show that if A is Hermitian then all its eigenvalues are real and that there is an orthonormal basis for Cn^ consisting of eigenvectors of A.
A Hermitian matrix is said to be positive definite if 〈Ax, x〉 > 0 for all x 6 = 0. We write A > 0 in this case. Show that A is positive definite if, and only if, all of its eigenvalues are positive. Show that if A > 0 then A has a unique positive definite square root
A.
Let A, B be two positive definite Hermitian matrices with A − B > 0. Writing C =
A and X =
A −
B, show that CX + XC > 0. By considering eigenvalues of X, or otherwise, show that X > 0.
11G Groups, Rings and Modules Let R be a ring and M an R-module. What does it mean to say that M is a free R-module? Show that M is free if there exists a submodule N ⊆ M such that both N and M/N are free.
Let M and M ′^ be R-modules, and N ⊆ M , N ′^ ⊆ M ′^ submodules. Suppose that N ∼= N ′^ and M/N ∼= M ′/N ′. Determine (by proof or counterexample) which of the following statements holds:
(1) If N is free then M ∼= M ′. (2) If M/N is free then M ∼= M ′.
Paper 4 [TURN OVER
14F Metric and Topological Spaces
Explain what is meant by a base for a topology. Illustrate your definition by describing bases for the topology induced by a metric on a set, and for the product topology on the cartesian product of two topological spaces.
A topological space (X, T ) is said to be separable if there is a countable subset C ⊆ X which is dense, i.e. such that C ∩ U 6 = ∅ for every nonempty U ∈ T. Show that a product of two separable spaces is separable. Show also that a metric space is separable if and only if its topology has a countable base, and deduce that every subspace of a separable metric space is separable.
Now let X = R with the topology T having as a base the set of all half-open intervals [a, b) = {x ∈ R : a 6 x < b}
with a < b. Show that X is separable, but that the subspace Y = {(x, −x) : x ∈ R} of X × X is not separable.
[You may assume standard results on countability.]
15C Complex Methods
Let H be the domain C − {x + iy : x ≤ 0 , y = 0} (i.e., C cut along the negative x-axis). Show, by a suitable choice of branch, that the mapping
z 7 → w = −i log z
maps H onto the strip S = {z = x + iy, −π < x < π}.
How would a different choice of branch change the result?
Let G be the domain {z ∈ C : |z| < 1 , |z + i| >
2 }. Find an analytic transformation that maps G to S, where S is the strip defined above.
Paper 4 [TURN OVER
16A Methods
Assume F (x) satisfies (^) ∫ ∞
−∞
|F (x)|dx < ∞ ,
and that the series
g(τ ) =
∑^ ∞
n=−∞
F (2nπ + τ )
converges uniformly in [0 6 τ 6 2 π].
If F˜ is the Fourier transform of F , prove that
g(τ ) =
2 π
∑^ ∞
n=−∞
F^ ˜ (n)einτ^.
[Hint: prove that g is periodic and express its Fourier expansion coefficients in terms of F^ ˜ ].
In the case that F (x) = e−|x|, evaluate the sum
∑^ ∞
n=−∞
1 + n^2
Paper 4
19H Statistics
(i) Consider the linear model
Yi = α + βxi + εi ,
where observations Yi, i = 1,... , n, depend on known explanatory variables xi, i = 1,... , n, and independent N (0, σ^2 ) random variables εi, i = 1,... , n.
Derive the maximum-likelihood estimators of α , β and σ^2.
Stating clearly any results you require about the distribution of the maximum-likelihood estimators of α , β and σ^2 , explain how to construct a test of the hypothesis that α = 0 against an unrestricted alternative.
(ii) A simple ballistic theory predicts that the range of a gun fired at angle of elevation θ should be given by the formula
Y =
V 2
g
sin 2θ ,
where V is the muzzle velocity, and g is the gravitational acceleration. Shells are fired at 9 different elevations, and the ranges observed are as follows:
θ (degrees) 5 15 25 35 45 55 65 75 85 sin 2θ 0. 1736 0. 5 0. 7660 0. 9397 1 0. 9397 0. 7660 0. 5 0. 1736 Y (m) 4322 11898 17485 20664 21296 19491 15572 10027 3458
The model Yi = α + β sin 2θi + εi (∗)
is proposed. Using the theory of part (i) above, find expressions for the maximum- likelihood estimators of α and β.
The t-test of the null hypothesis that α = 0 against an unrestricted alternative does not reject the null hypothesis. Would you be willing to accept the model (∗)? Briefly explain your answer.
[You may need the following summary statistics of the data. If xi = sin 2θi, then x ¯ ≡ n−^1
xi = 0. 63986 , Y¯ = 13802, Sxx ≡
(xi − x¯)^2 = 0. 81517 , Sxy =
Yi(xi − x¯) =
- ]
Paper 4
20H Optimization
(i) Suppose that f : Rn^ → R, and g : Rn^ → Rm^ are continuously differentiable. Suppose that the problem
max f (x) subject to g(x) = b
is solved by a unique ¯x = ¯x(b) for each b ∈ Rm, and that there exists a unique λ(b) ∈ Rm such that ϕ(b) ≡ f (¯x(b)) = sup x
f (x) + λ(b)T^ (b − g(x))
Assuming that ¯x and λ are continuously differentiable, prove that
∂ϕ ∂bi
(b) = λi(b). (∗)
(ii) The output of a firm is a function of the capital K deployed, and the amount L of labour employed, given by
f (K, L) = KαLβ^ ,
where α, β ∈ (0, 1). The firm’s manager has to optimize the output subject to the budget constraint K + wL = b ,
where w > 0 is the wage rate and b > 0 is the available budget. By casting the problem in Lagrangian form, find the optimal solution and verify the relation (∗).
END OF PAPER
Paper 4
