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## Mathematical Methods for Computer Science

Lecturer: Dr R.J. Gibbens

No. of lectures: 16

This course is a prerequisite for Computer Graphics and Image Processing (Part IB) and the following Part II courses: Artificial Intelligence II, Bioinformatics, Computer Systems Modelling, Computer Vision, Digital Signal Processing, Information Theory and Coding, Quantum Computing.

Aims

The aim of this course is to introduce and develop mathematical methods that are key to many modern applications in Computer Science. The course proceeds on two fronts: (i) Fourier methods and their generalizations that lie at the heart of modern digital signal processing, coding and information theory and (ii) probability modelling techniques that allow stochastic systems and algorithms to be described and better understood. The style of the course is necessarily concise but will attempt to blend a mix of theory with examples and glimpse ahead at applications taken up in Part II courses.

Lectures

• Fourier methods. Inner product spaces and orthonormal systems. Periodic functions and Fourier series. Results and applications. The Fourier transform and its properties. [3 lectures]

• Discrete Fourier methods. The Discrete Fourier transform and applications. The Fast Fourier Transform algorithm. [2 lectures]

• Wavelets. Introduction to wavelets with computer science applications. [1 lecture]

• Review of elementary probability theory. Random variables. Discrete and continuous distributions. Means and variances, independence, conditional probabilities. Bayes's theorem. [1 lecture]

• Probability generating functions. Definitions and properties. Use in calculating moments of random variables and for finding the distribution of sums of independent random variables. [1 lecture]

• Elementary stochastic processes. Random walks. Recurrence and transience. The Gambler's ruin problem. Solution using difference equations. [2 lectures]

• Inequalities and limit theorems. Bounds on tail probabilities, moment generating functions, notions of convergence, weak and strong laws of large numbers, the central limit theorem, statistical applications, Monte Carlo simulation. [3 lectures]

• Markov chains. Discrete-time Markov chains, Chapman-Kolmogorov equations, classifications of states, limiting and stationary behaviour, time-reversible Markov chains. Examples and applications. [3 lectures]

Objectives

At the end of the course students should

• grasp key properties and uses of Fourier analysis, transforms, and wavelets

• understand discrete transform techniques

• understand elementary probability theory

• understand basic probabilistic inequalities and limit results and be able to apply them to commonly arising models

• be familiar with the fundamental properties and uses of discrete-time Markov chains

Reference books

Oppenheim, A.V. & Willsky, A.S. (1997). Signals and systems. Prentice Hall.
* Pinkus, A. & Zafrany, S. (1997). Fourier series and integral transforms. Cambridge University Press.
Mitzenmacher, M. & Upfal, E. (2005). Probability and computing: randomized algorithms and probabilistic analysis. Cambridge University Press.
* Ross, S.M. (2002). Probability models for computer science. Harcourt/Academic Press.    Next: Programming in C and Up: Lent Term 2008: Part Previous: Foundations of Functional Programming   Contents