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MATHEMATICAL TRIPOS Part IB
Friday 9th June, 2006 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 sheets 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
1H Linear Algebra
Suppose V is a vector space over a field k. A finite set of vectors is said to be a basis for V if it is both linearly independent and spanning. Prove that any two finite bases for V have the same number of elements.
2E Groups, Rings and Modules How many elements does the ring Z[X]/(3, X^2 + X + 1) have?
Is this ring an integral domain?
Briefly justify your answers.
3F Analysis II Let V be the vector space of all sequences (x 1 , x 2 ,.. .) of real numbers such that xi converges to zero. Show that the function
|(x 1 , x 2 ,.. .)| = max i> 1
|xi|
defines a norm on V.
Is the sequence (1, 0 , 0 , 0 ,.. .), (0, 1 , 0 , 0 ,.. .),...
convergent in V? Justify your answer.
4H Complex Analysis
State the principle of isolated zeros for an analytic function on a domain in C.
Suppose f is an analytic function on C \ { 0 }, which is real-valued at the points 1 /n, for n = 1, 2 ,.. ., and does not have an essential singularity at the origin. Prove that f (z) = f (¯z) for all z ∈ C \ { 0 }.
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8D Numerical Analysis
(a) Given the data
xi − 1 0 1 3
f (xi) − 7 − 3 − 3 9
find the interpolating cubic polynomial p ∈ P 3 in the Newton form, and transform it to the power form.
(b) We add to the data one more value f (xi) at xi = 2. Find the power form of the interpolating quartic polynomial q ∈ P 4 to the extended data
xi − 1 0 1 2 3
f (xi) − 7 − 3 − 3 − 7 9
9C Markov Chains A game of chance is played as follows. At each turn the player tosses a coin, which lands heads or tails with equal probability 1/2. The outcome determines a score for that turn, which depends also on the cumulative score so far. Write Sn for the cumulative score after n turns. In particular S 0 = 0. When Sn is odd, a head scores 1 but a tail scores 0. When Sn is a multiple of 4, a head scores 4 and a tail scores 1. When Sn is even but is not a multiple of 4, a head scores 2 and a tail scores 1. By considering a suitable four-state Markov chain, determine the long run proportion of turns for which Sn is a multiple of 4. State clearly any general theorems to which you appeal.
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SECTION II
10E Linear Algebra
Suppose that α is an orthogonal endomorphism of the finite-dimensional real inner product space V. Suppose that V is decomposed as a direct sum of mutually orthogonal α-invariant subspaces. How small can these subspaces be made, and how does α act on them? Justify your answer.
Describe the possible matrices for α with respect to a suitably chosen orthonormal basis of V when dim V = 3.
11E Groups, Rings and Modules (a) Suppose that R is a commutative ring, M an R-module generated by m 1 ,... , mn and φ ∈ EndR(M ). Show that, if A = (aij ) is an n × n matrix with entries in R that represents φ with respect to this generating set, then in the sub-ring R[φ] of EndR(M ) we have det(aij − φδij ) = 0.
[Hint: A is a matrix such that φ(mi) =
aij mj with aij ∈ R. Consider the matrix C = (aij − φδij ) with entries in R[φ] and use the fact that for any n × n matrix N over any commutative ring, there is a matrix N ′^ such that N ′N = (det N )1n.]
(b) Suppose that k is a field, V a finite-dimensional k-vector space and that φ ∈ Endk(V ). Show that if A is the matrix of φ with respect to some basis of V then φ satisfies the characteristic equation det(A − λ1) = 0 of A.
Paper 4 [TURN OVER
14F Metric and Topological Spaces
(a) Show that every compact subset of a Hausdorff topological space is closed.
(b) Let X be a compact metric space. For F a closed subset of X and p any point of X, show that there is a point q in F with
d(p, q) = inf q′∈F
d(p, q′).
Suppose that for every x and y in X there is a point m in X with d(x, m) = (1/2)d(x, y) and d(y, m) = (1/2)d(x, y). Show that X is connected.
15D Complex Methods
Denote by f ∗ g the convolution of two functions, and by f̂ the Fourier transform, i.e.,
f ∗ g =
−∞
f (t)g(x − t) dt, f̂ (λ) =
−∞
f (x)e−iλx^ dx.
(a) Show that, for suitable functions f and g, the Fourier transform F̂ of the
convolution F = f ∗ g is given by F̂ = f̂ · ̂g.
(b) Let
f 1 (x) =
1 |x| 6 1 /2 , 0 otherwise.
and let f 2 = f 1 ∗ f 1 be the convolution of f 1 with itself. Find the Fourier transforms of f 1 and f 2 , and, by applying Parseval’s theorem, determine the value of the integral
∫ (^) ∞
−∞
sin y y
dy.
Paper 4 [TURN OVER
16B Methods
The integral
I =
∫ (^) b
a
F (y(x), y′(x))dx ,
where F is some functional, is defined for the class of functions y(x) for which y(a) = y 0 , with the value y(b) at the upper endpoint unconstrained. Suppose that y(x) extremises the integral among the functions in this class. By considering perturbed paths of the form y(x) + η(x), with 1, show that
d dx
( ∂F
∂y′
∂F
∂y
and that ∂F ∂y′
x=b
Show further that
F − y′^
∂F
∂y′^
= k
for some constant k.
A bead slides along a frictionless wire under gravity. The wire lies in a vertical plane with coordinates (x, y) and connects the point A with coordinates (0, 0) to the point B with coordinates (x 0 , y(x 0 )), where x 0 is given and y(x 0 ) can take any value less than zero. The bead is released from rest at A and slides to B in a time T. For a prescribed x 0 find both the shape of the wire, and the value of y(x 0 ), for which T is as small as possible.
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19C Statistics
Two series of experiments are performed, the first resulting in observations X 1 ,... , Xm, the second resulting in observations Y 1 ,... , Yn. We assume that all observa- tions are independent and normally distributed, with unknown means μX in the first series and μY in the second series. We assume further that the variances of the observations are unknown but are all equal.
Write down the distributions of the sample mean X¯ = m−^1
∑m i=1 Xi^ and sum of squares SXX =
∑m i=1(Xi^ −^ X¯)
Hence obtain a statistic T (X, Y ) to test the hypothesis H 0 : μX = μY against H 1 : μX > μY and derive its distribution under H 0. Explain how you would carry out a test of size α = 1/.
20C Optimization Use a suitable version of the simplex algorithm to solve the following linear programming problem:
maximize 50 x 1 − 30 x 2 + x 3 subject to x 1 + x 2 + x 3 ≤ 30 2 x 1 − x 2 ≤ 35 x 1 + 2 x 2 − x 3 ≥ 40 and x 1 , x 2 , x 3 ≥ 0.
END OF PAPER
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