Advanced Mathematical Methods

1. Advanced Mathematical Methods COMP3006 Introduction to the course

2. Introduction • 2 sections • Maths-Dr. Karen Page & Statistics –Dr. Simon Prince • Maths until reading week

3. Course contact details • All communication concerning this course will be done via the email list. • Please join by sending an email with Subject: join • to 3006-request@cs.ucl.ac.uk • Information also on the websites: http://www.cs.ucl.ac.uk/staff/K.Page/maths.html http://www.cs.ucl.ac.uk/staff/S.Prince/3006.htm

4. Lectures and examples classes • Check the website for timetable changes • Until reading week: lectures Thurs 9-10, MPEB 1.04 Fri 9-10, MPEB 1.13 Fri 12-1, MPEB 1.13 examples class Thurs 10-11, MPEB 1.04 (with Dr. Ged Ridgway); starting 12th October

5. Coursework • 85% exam • 15% coursework • Maths coursework= average of homework grades

6. Homework • I’ll set several exercises per lecture • To help pass exam you should try to do all of these before the exam • 2 per lecture = 6 per week are mandatory for coursework • You will get credit for serious attempts • Bring solutions for the week to the next examples class, attach coursework coversheet (http://www.cs.ucl.ac.uk/teaching/cwsheet.htm) • I will attend examples classes to mark your work (for undergraduates only)

7. Notes • Handouts will be partial copies of overheads • They will contain spaces which you’ll need to fill

8. Useful books • Axler “Linear algebra done right” 2nd edition (Springer) • Boas “Mathematical methods in the physical sciences” 2nd edition (Wiley) • **Bourne and Kendall “Vector analysis and Cartesian tensors” 3rd edition (Chapman and Hall) • ***Kreyszig “Advanced Engineering Mathematics” 8th edition (Wiley) • Pinkus and Zafrany "Fourier Series and Integral Transforms" 1st edition (Cambridge University Press) • **Any books in the Schaum series on relevant topics

9. Motivation- Section 1 mathematics Syllabus consists of two areas: • Linear algebra & calculus These build on courses B45 & B46 and are designed to give a general education in mathematics which will be useful for further courses in fourth year : intelligent systems machine vision and virtual environments many other useful applications: financial world, game theory in economics, bioinformatics, mathematical and computational biology

11. Option pricing: Black-scholes’ stochastic differential equation Bioinformatics: Sequence comparison and microarray expression matrices

12. Topics • Week 1: Basic topics in linear algebra, Gaussian elimination, complex numbers, eigenvalues and eigenvectors (easy stuff) • Week 2: Differential vector calculus, including method of steepest descents • Week 3: Integral vector calculus- Green’s theorem, Divergence theorem, Stokes’ theorem • Week 4: Fourier series (complex), Fourier transforms, Laplace transforms

13. Topics • Week 5: Further linear algebra- Gram-Schmidt, special complex matrices, orthogonal diagonalisation, spectral decomposition, singular values decomposition • Note: The 2nd lecture will be on complex numbers. If you haven’t done this before, try to do lots of exercises (you’ll need to be familiar with this for later lectures)

14. Down to business…