Group theory tutorial, TT week 1 (=HT week 9)

I got quite confused with one of the problems in tutorial today. I had a glimmer of the right answer, but then misled myself into saying something quite silly. Anyways, here is a short note on this.

Probability, week 7

I’ve asked David Steinsaltz about problem 5 (b) in sheet 3, and he tells me that he stated a theorem (13.7) last week that allows you to compute long term averages using stationary states even in the non-ergodic case. Recall that the problem is to compute The probability can be written as , where and [...]

Symmetries

I have written some notes on symmetries following the Group Theory tutorials this week.

Linear algebra II, week 4

Here are some brief remarks on the questions. For the question on the determinant of it is probably best to write at once so that by the multiplicativity of the determinant. Then and are immediate, by induction on and , for example, using a column expansion. [A word on induction: I'm sure you can see [...]

Yet more remarks on Probability A, sheet 1

I am adding yet more remarks on sheet 1 of Probability A, this time on question 6.

Probability, more remarks

I am adding now some comments on problem 5 of the probability sheet 1.

Probability tutorials, week 3

Here are some remarks on the probability tutorials last week. I didn’t have time to comment on any but the most elementary problems, but I thought it might be good to review the basics anyways. I’ll try to write more this weekend.

Tutorials, Hilary term, week 2

Here are some remarks on problem 7 (i) of Linear Algebra II, sheet 1. Let me know if you have a simpler proof. ———————————— Here are some further remarks. Recall that the key statement is this: For if then It is perhaps better to write it in the form This is giving a lower bound [...]

Collections, Hilary Term, 2012

I’ve marked the scripts for the algebra portion of the Merton collections for Mods Pure Maths, Part A AC1 and Part A AC2. I am including here brief remarks on the questions. Later, if I have the energy, I will also upload comments on the Banach Spaces paper for Part B students.

POSTECH notes, algebra.

Here are some notes on Automorphisms of algebraic number fields to accompany the lecture on 5 December, 2011.