| STUDENTS
> Advanced Mathematical Methods
|
Advanced Mathematical Methods
Note:
Whilst every effort is made to keep the syllabus and assessment records correct
for this course, the precise details must be checked with the lecturer(s).
| Code: | 3006 |
| Year: | 3 |
| Prerequisites: | MATH6301, COMP2003. Students should be familiar with partial differentiation in order to do this course. In addition that should be familiar with determinants in 2 and 3d and with matrix multiplication. They should recall LU decomposition from MATH6301. |
| Term: | 1 |
| Taught By: | Kevin Bryson (50%)
Simon Prince (50%)
|
| Aims: |
This mathematics module is primarily intended to give students the background to allow them to
take the more mathematical options in the fourth year. However, it is of general benefit to anyone who
expects to handle data in the real world.
|
| Learning Outcomes: |
Students will be proficient in linear algebra, calculus and statistical topics to prepare them for 4th year CS courses |
Content:
| Mathematics: Linear algebra topics | 1. Recap: solving Ax=b; Gaussian elimination
2. Complex numbers
3. Eigenvalues and eigenvectors; Characteristic polynomial
4. Vector spaces, Spanning sets, linear independence, basis and dimension(n-D),
5. Inner products, orthogonality, Metric- Euclidean (n_D), l_2 and other examples (including infinite dimensional spaces), Schwarz and triangle inequalities
6. Gram-Schmidt orthogonalisation, special complex matrices (eg. unitary and hermitian),
7. Orthogonal diagonalisation, Spectral decomposition, Singular values decomposition
|
| Mathematics: Calculus topics | 1. Differential vector calculus: Gradient of a scalar function, Divergence, operator del (3-d and n-d Euclidean)
2. Method of steepest descents solution of Ax=b
3. Line integrals, double integrals, Jacobian of adouble integral
4. complex Fourier series
5. Fourier transforms
6. Laplace Transforms |
| Statistics | 1. Traditional scientific method based on falsifiability, and the role of statistical methods in putting this into practice. A range of skills in statistical analysis of data, including the use of appropriate software systems.
2. Design one-way and multi-way simple factorial experiments in order to test hypotheses. Be able to identify appropriate statistical procedures and tests. Be able to use statistical data to explore relationship between several variables. Understand the basis of statistical methods in probability theory.
3.The axioms of probability theory, and interpretations of probability as frequency based or subjective assessment. Standard probability distributions, such as binomial, normal, and the distributions that are most common in statistical testing. Estimation theory for the parameters of a probability distribution. Standard statistical tests for population means and variances. Linear models and their application to analysis of variance and regression. Questionnaire design and experimental design principles. Data reduction techniques such as principle components. Generalized linear models.
|
Method of Instruction:
Lecture presentations, programming lab classes, exercise
questions, mathematics problem classes.
Assessment:
The course has the following assessment components:
- Written Examination ( 2.5 hours, 85%)
- Coursework Section (2 pieces, 15%)
To pass this course, students must:
- Obtain an overall pass mark of 40% for all sections combined
The examination rubric is:
Answer THREE questions, including at least one (out of 3) from Section One, and one (out of 3) from Section Two.All questions carry equal marks.Resources:
1. Axler "Linear algebra done right" 2nd edition (Springer)
2. Boas "Mathematical methods in the physical sciences" 2nd edition (Wiley)
3. Bourne and Kendall "Vector analysis and Cartesian tensors" 3rd edition (Chapman and Hall)
4. Kreyszig "Advanced Engineering Mathematics" 8th edition (Wiley)
5. Pinkus and Zafrany "Fourier Series and Integral Transforms" 1st edition (Cambridge University Press)
6. Protter and Morrey "A first course in real analysis" 2nd edition (Springer) [short excerpts only]
7. A.Papoulis "Probability and Statistics" (Prentice Hall) 1990.
8. R J Freund and W J Wilson "Regression Analysis" (Academic Press) 1998
9. C M Grinstead and J L Snell "Introduction to Probability" (see Simon Prince's web link)
Any books in the Schaum series on relevant topics
Web resources (Simon Prince)
Web resources (Karen Page)
|
+44 (0)20 7679 7214 - Copyright © 1999-2007
UCL
