This module introduces the theory of metric spaces. Following on from an undergraduate course on analysis of the real numbers, this module considers convergence and continuity in a more general framework. This central result addressed is the Contraction Mapping Principal, which will be developed and proved in a rigorous, abstract, manner, and then used in applied settings, such as in the theory of differential equations.
Aims
To gain an in depth understanding of limiting processes in a general framework, and to be able to specialise to concrete settings. To understand central theoretical results, such as the Contraction Mapping Principal, and to use them to solve applied problems.
Learning Outcomes
On completion of the module students should:
1. Be able to work with abstract metric spaces and concrete examples of them.
2. Be able to perform abstract arguments involving metric spaces.
3. Understand notions such as openness, closedness, continuity, completeness, equivalence of metrics, compactness, contractions.
4. Be able to prove the Contraction Mapping theorem and use it in applied situations, such as solving differential equations.
5. Have a thorough understanding of limiting processes in a general framework.
Syllabus
1. Sequences, iteration. Review of epsilon delta analysis.
2. Metric spaces. Definition and examples; sequences.
3. Closed, complete, and compact sets.
4. Fixed points, contractions. The Contraction Mapping Principal.
5. Applications of the Contraction Mapping Principal to solving differential equations.
6. Continuity. Open and Closed sets.
The module requires basic mathematical knowledge of algebra as permutations and combinations. Partial fractions, including quadratic factors. Solution of linear and quadratic equations. Manipulations and solution of simple inequalities. Trigonometric functions and their inverses. Summation of series with _notation. Limits of sequences and functions. Moreover, differential and integral calculus and matrix algebra.
The R package is used to analyse data for illustration. The module introduces descriptive statistics and the concepts of probability, distribution theory, estimation, hypothesis tests and confidence intervals.
On completion of the course students should be able to:
* Apply, use and understand descriptive statistics using R;
* Understand the concept of probability;
* Understand the concept of distribution theory;
* Understand the concept of estimation and properties of estimators;
* Understand hypothesis testing, exact and asymptotic tests, properties of tests and confidence intervals.
* Understand and apply correlation and regression analysis, one-way analysis of variance.
Syllabus:
Descriptive statistics
Measurement scales, univariate characteristics of a sample, visualisation of univariate, bivariate and multivariate data.
Probability
Sampling with and without replacement. Elementary problems involving urn models.
Joint probability, marginal and conditional probability, independence. Law of total probability. Bayes' Theorem.
Distribution theory
Random variables. Discrete and continuous random variables. The probability mass function and probability density function. Cumulative distribution function.
Expectation as a linear operator. Expectation of functions of a random variable. Mean and variance. Approximate mean and variance of a function of a random variable. Variance-stabilising transformations.
Standard distributions and their use in modelling, including Bernoulli, binomial, Poisson, geometric, hypergeometric, discrete uniform, Normal, exponential, gamma, continuous uniform.
Joint, marginal and conditional distributions. Independence. Covariance and correlation.
Probability generating function. Moment generating function. Applications of generating functions. Distribution of sums of random variables, and of sample mean.
Central limit theorem. Joint distribution of mean and variance from a Normal random sample. The t, hisquare and F distributions, and their use as sampling distributions.
Estimation
Unbiasedness, mean square error, consistency, relative efficiency, sufficiency, minimum variance. Fisher information for a function of a parameter, Cramér-Rao lower bound, efficiency. Maximum likelihood estimation: finding estimators analytically and numerically, invariance.
Hypothesis testing
Simple and composite hypotheses, types of error, power, operating characteristic curves, p-value. Use of asymptotic results to construct tests. Central limit theorem, asymptotic distributions of maximum likelihood estimator and generalised likelihood ratio test statistic.
Confidence intervals and sets
Random intervals and sets. Use of pivotal quantities. Relationship between tests and confidence intervals. Use of asymptotic results.
Linear models
Correlation and regression analysis, one-way analysis of variance.
The R package is used to analyse data for illustration. The module introduces descriptive statistics and the concepts of probability, distribution theory, estimation, hypothesis tests and confidence intervals.
On completion of the course students should be able to:
* Apply, use and understand descriptive statistics using R;
* Understand the concept of probability;
* Understand the concept of distribution theory;
* Understand the concept of estimation and properties of estimators;
* Understand hypothesis testing, exact and asymptotic tests, properties of tests and confidence intervals.
* Understand and apply correlation and regression analysis, one-way analysis of variance.
Syllabus:
Descriptive statistics
Measurement scales, univariate characteristics of a sample, visualisation of univariate, bivariate and multivariate data.
Probability
Sampling with and without replacement. Elementary problems involving urn models.
Joint probability, marginal and conditional probability, independence. Law of total probability. Bayes' Theorem.
Distribution theory
Random variables. Discrete and continuous random variables. The probability mass function and probability density function. Cumulative distribution function.
Expectation as a linear operator. Expectation of functions of a random variable. Mean and variance. Approximate mean and variance of a function of a random variable. Variance-stabilising transformations.
Standard distributions and their use in modelling, including Bernoulli, binomial, Poisson, geometric, hypergeometric, discrete uniform, Normal, exponential, gamma, continuous uniform.
Joint, marginal and conditional distributions. Independence. Covariance and correlation.
Probability generating function. Moment generating function. Applications of generating functions. Distribution of sums of random variables, and of sample mean.
Central limit theorem. Joint distribution of mean and variance from a Normal random sample. The t, hisquare and F distributions, and their use as sampling distributions.
Estimation
Unbiasedness, mean square error, consistency, relative efficiency, sufficiency, minimum variance. Fisher information for a function of a parameter, Cramér-Rao lower bound, efficiency. Maximum likelihood estimation: finding estimators analytically and numerically, invariance.
Hypothesis testing
Simple and composite hypotheses, types of error, power, operating characteristic curves, p-value. Use of asymptotic results to construct tests. Central limit theorem, asymptotic distributions of maximum likelihood estimator and generalised likelihood ratio test statistic.
Confidence intervals and sets
Random intervals and sets. Use of pivotal quantities. Relationship between tests and confidence intervals. Use of asymptotic results.
Linear models
Correlation and regression analysis, one-way analysis of variance.
The module provides an introduction to the principal research tools for the students on postgraduate courses in Mathematical Sciences, including practice in the mathematical word-processing language LaTeX.
Syllabus:
Sources of information Library, WoS, Reading papers, Periodicals, Blackwells, JSTOR, PubMed, Zentralblatt, ResearchIndex, CiteULike, Google, Google Scholar, Wikipedia) Introduction to \LaTeX (Crimson Editor, Brackets, Sections, Spacing, Fonts, Equations, Left and Right brackets, Tables, Matrices, Symbols, Styles) TeXnicCenter Postscript (Commands, importing and viewing .ps files) Essays, dissertations, papers and theses. (References, Presentation, Plagiarism) Unix/Linux and the vi editor
Syllabus:
Sources of information Library, WoS, Reading papers, Periodicals, Blackwells, JSTOR, PubMed, Zentralblatt, ResearchIndex, CiteULike, Google, Google Scholar, Wikipedia) Introduction to \LaTeX (Crimson Editor, Brackets, Sections, Spacing, Fonts, Equations, Left and Right brackets, Tables, Matrices, Symbols, Styles) TeXnicCenter Postscript (Commands, importing and viewing .ps files) Essays, dissertations, papers and theses. (References, Presentation, Plagiarism) Unix/Linux and the vi editor
This is a dissertation module for MSc students.
Students will be provided with a list of dissertation titles or topics proposed by members of staff. It may also be possible to propose a topic of your own, provided a member of staff agrees it is of a suitable standard and is able to supervise it. We hope there will be a mechanism for expressing preferences about which topic to do, and that this will be reflected in the allocation of topics to candidates. However, it must be pointed out that the exact nature of the procedure cannot be guaranteed because of staff numbers and availability, staff interests etc.
Students will be provided with a list of dissertation titles or topics proposed by members of staff. It may also be possible to propose a topic of your own, provided a member of staff agrees it is of a suitable standard and is able to supervise it. We hope there will be a mechanism for expressing preferences about which topic to do, and that this will be reflected in the allocation of topics to candidates. However, it must be pointed out that the exact nature of the procedure cannot be guaranteed because of staff numbers and availability, staff interests etc.