[The University of Leicester]

Department of Mathematics



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MA1221 Pure Mathematics at Work

MA1221 Pure Mathematics at Work

Credits: 10 Convenor: Dr. N. Snashall Semester: 1 (weeks 7 to 12)
Prerequisites: desirable: MA1101, MA1102
Assessment: Project and coursework: 100% Examination: 0%
Lectures: 18 Problem Classes: 10
Tutorials: none Private Study: 47
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 75

Subject Knowledge

Aims

This module aims to introduce and study various aspects of pure mathematics which arise in the diverse settings of biology, chemistry, communication/IT, to introduce some new ways of using the mathematics learned elsewhere and to enhance the understanding of that mathematics.

Learning Outcomes

Students should know the definitions of and understand the key mathematical concepts and their relationship to the general problems which motivate their consideration in each of the topics of this module, that is, in the Fibonacci sequence and its relation to population growth in mathematical biology, in the paper folding topic to construct regular polygons, in the graph theoretic topics motivated by the Königsberg bridge problem and the utilities problem, and in Public Key Encryption codes.

Students should be able to explain the main proofs given in the lectures, and be able to apply this knowledge to solve problems in number theory using proof by induction, second order recurrence relations, elementary graph theory, and congruence arithmetic.

Methods

Class sessions and workshops together with some handouts.

Assessment

Marked problem sheets, class test, individual project.

Subject Skills

Aims

To provide students with team working skills and develop written communication skills and problem solving skills.

Learning Outcomes

Students will have worked in a group context to investigate a problem, draw conclusions and make conjectures, and have written a short individual project using library and other external resources. Students will be able to use the techniques taught within the module to solve problems, and be able to present arguments and solutions in a coherent and logical form.

Methods

Class sessions, workshops.

Assessment

Marked problem sheets, team work project, class test, individual project.

Explanation of Pre-requisites

Use is made of the following concepts from the modules MA1101 and MA1102: the notion of proof in general; proof by induction; congruence arithmetic.

Course Description

The aim of this module is to introduce and study various aspects of pure mathematics which are used in real life situations, to introduce some new ways of using the mathematics learned elsewhere and to enhance the understanding of that mathematics. The mathematics studied is motivated from from a wide range of problems in biology (Fibonacci series and population growth), communication/IT (graph theory and number theory in relation to the travelling salesperson problem, design of electronic circuitry and public key encryption codes), and chemistry (convex polyhedra and molecular structure). The module is structured to encourage the mathematical qualities of investigation, spotting patterns and making conjectures, and applying previous knowledge to new situations through formal proofs.

Syllabus

The Fibonacci sequence and its relation to population growth in mathematical biology, second order recurrence relations.

Paper folding to construct regular polygons, concept of an approximate solution and convergence.

The Königsberg bridge problem and its formal setting within graph theory, connected graphs, spanning trees, Euler circuits, Euler walks, relation to the travelling salesperson problem.

The utilities problem as a motivation for planar graphs, Euler characteristic of a planar graph and convex polyhedra, complete graphs, complete bipartite graphs.

The concept of Public Key Encryption codes, Euler's totient, Euler's theorem.

Reading list

Recommended:

Background:

N. L. Biggs, Discrete Mathematics, Oxford University Press, 1994.

P. Hilton, D. Holton and J. Pedersen, Mathematical Reflections: In a room with many mirrors, Springer-Verlag, 1997.

I. Stewart, The Problems of Mathematics, Oxford University Press, 1987.

R. J. Wilson and J. J. Watkins, Graphs (An Introductory Approach), Wiley, 1990.

Resources

Problem sheets and workshops, additional handouts, polyhedra kit, lecture rooms.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA1251 Chaos and Fractals Up: ModuleGuide03-04 Previous: MA1201 Mathematical Modelling

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Last updated: 2004-02-21
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