David MacKay

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Mathematical prerequisites for ITPRNN course

The level of Mathematics assumed is 1A Maths for Natural Scientists.

All the following topics will be reviewed.
  • Probability and statistics:
    • definition of conditional probability, joint probability, marginal probability;
    • Bayes' theorem;
    • definition of independence;
    • mean and standard deviation of a sum of independent random variables; law of large numbers.
  • Linear algebra:
    • inner products between vectors;
    • Taylor expansions and gradients of functions over vector spaces;
    • eigenvectors and eigenvalues of symmetric matrices.
  • Differentiation using Chain rule.
  • Statistical physics:
    • definition of partition function;
    • obtaining thermodynamic functions from the partition function;
    • entropy of a set of N binary spins as a function of the fraction x of spins that are +1;
    • relationship between canonical ensemble and microcanonical ensemble, and equivalence of the Boltzmann and Gibbs entropies S=k_B log and S= p_i 1/p_i (for large systems).

Site last modified Sun Aug 31 18:51:05 BST 2014