4D
Analysis I
Find the radius of convergence of each of the following power series.
X
(i)
n2 z n
n>1
P
Known (Ratio Test): If n an z n is a power series with an 6= 0 for all
n, and there exists l [0, ] such that |an+1 /an | l as n , then
the radius of con
MATHEMATICAL TRIPOS
Part III
Tuesday, 9 June, 2015 1:30 pm to 3:30 pm
PAPER 50
CLASSICAL AND QUANTUM SOLITONS
Attempt no more than TWO questions.
There are THREE questions in total.
The questions carry equal weight.
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MATHEMATICAL TRIPOS
Part III
Wednesday, 3 June, 2015 1:30 pm to 3:30 pm
PAPER 42
CONTEST THEORY
Attempt no more than THREE questions.
There are FIVE questions in total.
The questions carry equal weight.
This is OPEN BOOK examination
Candidates may only br
2E
Numbers and Sets
What is an equivalence relation on a set X? If is an equivalence relation on X,
what is an equivalence class of ? Prove that the equivalence classes of form a partition
of X.
A relation R on a set X is a subset of X X. We write xRy for
MATHEMATICAL TRIPOS
Friday, 29 May, 2015
Part III
9:00 am to 12:00 pm
PAPER 57
ASTROPHYSICAL FLUID DYNAMICS
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
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MATHEMATICAL TRIPOS
Part III
Wednesday, 3 June, 2015 1:30 pm to 4:30 pm
PAPER 63
PLANETARY SYSTEM DYNAMICS
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
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MATHEMATICAL TRIPOS
Part III
Friday, 5 June, 2015 1:30 pm to 4:30 pm
PAPER 55
ADVANCED COSMOLOGY
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There are FOUR questions in total.
The questions carry equal weight.
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MATHEMATICAL TRIPOS
Part III
Tuesday, 2 June, 2015 9:00 am to 12:00 pm
PAPER 53
COSMOLOGY
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
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MATHEMATICAL TRIPOS
Thursday, 28 May, 2015
Part III
9:00 am to 12:00 pm
PAPER 43
QUANTUM FIELD THEORY
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
STATIONERY REQUIREMENTS
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MATHEMATICAL TRIPOS
Thursday, 4 June, 2015
Part III
9:00 am to 12:00 pm
PAPER 54
BLACK HOLES
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
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MATHEMATICAL TRIPOS
Part III
Wednesday, 3 June, 2015 1:30 pm to 4:30 pm
PAPER 46
ADVANCED QUANTUM FIELD THEORY
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
STATIONERY REQUIREMENTS
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MATHEMATICAL TRIPOS
Monday, 8 June, 2015
Part III
9:00 am to 11:00 am
PAPER 56
APPLICATIONS OF DIFFERENTIAL
GEOMETRY TO PHYSICS
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
STATIONERY REQUIREME
MATHEMATICAL TRIPOS
Thursday, 28 May, 2015
Part III
1:30 pm to 4:30 pm
PAPER 58
STRUCTURE AND EVOLUTION OF STARS
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
STATIONERY REQUIREMENTS
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MATHEMATICAL TRIPOS
Friday, 29 May, 2015
Part III
9:00 am to 12:00 pm
PAPER 44
SYMMETRIES, FIELDS AND PARTICLES
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
STATIONERY REQUIREMENTS
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MATHEMATICAL TRIPOS
Monday, 1 June, 2015
Part III
9:00 am to 12:00 pm
PAPER 52
GENERAL RELATIVITY
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There are FOUR questions in total.
The questions carry equal weight.
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MATHEMATICAL TRIPOS
Part III
Tuesday, 2 June, 2015 1:30 pm to 4:30 pm
PAPER 39
STOCHASTIC NETWORKS
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There are FIVE questions in total.
The questions carry equal weight.
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MATHEMATICAL TRIPOS
Thursday, 4 June, 2015
Part III
1:30 pm to 4:30 pm
PAPER 47
THE STANDARD MODEL
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There are FOUR questions in total.
The questions carry equal weight.
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MATHEMATICAL TRIPOS
Part III
Tuesday, 2 June, 2015 9:00 am to 12:00 pm
PAPER 38
MATHEMATICS OF OPERATIONAL RESEARCH
Attempt no more than FOUR questions.
There are SIX questions in total.
The questions carry equal weight.
STATIONERY REQUIREMENTS
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2D
Groups
State and prove Lagranges Theorem.
Lagranges Theorem: If G is a finite group and H a subgroup then |H|
divides |G|.
Proof: Define a relation on G by g1 g2 iff g11 g2 H. This is
reflexive (since H contains the identity), symmetric (since g21 g1 =
MATHEMATICAL TRIPOS
Monday, 8 June, 2015
Part III
1:30 pm to 4:30 pm
PAPER 48
SUPERSYMMETRY AND EXTRA DIMENSIONS
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
STATIONERY REQUIREMENTS
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MATHEMATICAL TRIPOS
Friday, 29 May, 2015
Part III
1:30 pm to 3:30 pm
PAPER 61
THE ORIGIN AND EVOLUTION OF GALAXIES
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There are THREE questions in total.
The questions carry equal weight.
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MATHEMATICAL TRIPOS
Part III
Wednesday, 3 June, 2015 9:00 am to 12:00 pm
PAPER 49
STRING THEORY
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
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MATHEMATICAL TRIPOS
Friday, 29 May, 2015
Part III
1:30 pm to 3:30 pm
PAPER 45
STATISTICAL FIELD THEORY
Attempt no more than TWO questions.
There are THREE questions in total.
The questions carry equal weight.
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MATHEMATICAL TRIPOS
Part III
Friday, 5 June, 2015 1:30 pm to 3:30 pm
PAPER 41
OPTIMAL INVESTMENT
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
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2A
Differential Equations
dy
(a) For a differential equation of the form dx
= f (y), explain how f 0 (y) can be used
to determine the stability of any equilibrium solutions and justify your answer.
If f 0 (y0 ) > 0 then the equilibrium point y0 is unstabl
MATHEMATICAL TRIPOS
Thursday, 4 June, 2015
Part III
9:00 am to 12:00 pm
PAPER 62
GALACTIC ASTRONOMY AND DYNAMICS
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
STATIONERY REQUIREMENTS
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MATHEMATICAL TRIPOS
Monday, 8 June, 2015
Part III
1:30 pm to 3:30 pm
PAPER 76
DIRECT AND INVERSE SCATTERING OF WAVES
Attempt no more than TWO questions.
There are THREE questions in total.
The questions carry equal weight.
STATIONERY REQUIREMENTS
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MATHEMATICAL TRIPOS
Part III
Tuesday, 9 June, 2015 9:00 am to 12:00 pm
PAPER 80
COMPLEX AND BIOLOGICAL FLUIDS
There are THREE questions in total.
Attempt all THREE.
The questions carry equal weight.
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MATHEMATICAL TRIPOS
Monday, 8 June, 2015
Part III
9:00 am to 11:00 am
PAPER 70
BOUNDARY VALUE PROBLEMS FOR LINEAR PDES
Attempt no more than TWO questions.
There are THREE questions in total.
The questions carry equal weight.
STATIONERY REQUIREMENTS
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MATHEMATICAL TRIPOS
Part III
Wednesday, 3 June, 2015 9:00 am to 11:00 am
PAPER 77
SOUND GENERATION AND PROPAGATION
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There are THREE questions in total.
The questions carry equal weight.
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3F
Probability
Let X be a random variable taking non-negative integer values and let Y be a
random variable taking real values.
(a) Define the probability-generating function GX (s). Calculate it explicitly for a
Poisson random variable with mean > 0.
P
MATHEMATICAL TRIPOS
Part III
Friday, 5 June, 2015 9:00 am to 11:00 am
PAPER 67
ADVANCED QUANTUM INFORMATION THEORY
Attempt no more than TWO questions.
There are FOUR questions in total.
The questions carry equal weight.
STATIONERY REQUIREMENTS
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MATHEMATICAL TRIPOS
Thursday, 28 May, 2015
Part III
9:00 am to 11:00 am
PAPER 60
MAGNETOHYDRODYNAMICS
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
Extra credit will be given for complete answer
MATHEMATICAL TRIPOS
Part III
Tuesday, 9 June, 2015 1:30 pm to 4:30 pm
PAPER 81
QUANTUM CONDENSED MATTER FIELD THEORY
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
STATIONERY REQUIREMENTS
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MATHEMATICAL TRIPOS
Part III
Friday, 5 June, 2015 9:00 am to 11:00 am
PAPER 64
DYNAMICS OF ASTROPHYSICAL DISCS
Attempt no more than TWO questions.
There are THREE questions in total.
The questions carry equal weight.
STATIONERY REQUIREMENTS
Cover sheet
Tr