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## Mathematical Methods for Computer Science

Lecturer: Dr R.J. Gibbens

No. of lectures: 12

Prerequisite course: Probability

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 that glimpse ahead at applications developed 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 related algorithms and applications. [2 lectures]

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

• 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

• understand the fundamental properties of inner product spaces and orthonormal systems

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

• understand discrete transform techniques and their applications

• 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: Prolog Up: Michaelmas Term 2008: Part Previous: Logic and Proof   Contents