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Course Modules for Fourth Year B.A. Mathematics

MODULE CODE: MH4718

MODULE TITLE: Numerical Methods and Computing

PREREQUISITE MODULE(S): MH4712, MH4723, MH4714

OBJECTIVES:

To explore the capabilities of computers in mathematical problem solving and to enable students to develop expertise in such applications.

LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

  • be competent in the programming language C++ and competent in the use of the software package Maple;
  • know how common functions such as cos, sin, exp and log are approximated by computers and be able to write their own programs to approximate such functions;
  • be able to write programs which use numerical methods to estimate solutions to linear, non-linear and differential equations;
  • to estimate areas bounded by functions using Simpson's Rule and Romberg Integration;
  • to manipulate and analyse matrices;
  • to use the method of Divided Differences for Newton's form of the Interpolating Polynomial;
  • to use Chebychev polynomials in Least Squares Regression.

MODULE CONTENT:

Revision of number representation; number bases, machine representable numbers, floating point arithmetic, error analysis.

Introduction to Maple.

Non-linear algebraic equations; iteration, Accelerated Iterative Formula, Secant method.

Numerical integration; Simpson's Rule and Romberg Formula.

Polynomial interpolation; Lagrange and Newton interpolation, errors in interpolation, numerical differentiation, spline functions.

Systems of linear equations; Gaussian elimination, band systems, Gauss Seidel Iteration for diagonally dominant matrices.

Ordinary differential equations; Taylor Series and Runge-Kutta methods, stability.

PRIME TEXT:

ABERTH, O., Precise Numerical Methods using C++, Volume 1, (Academic Press), 1998.

OTHER RELEVANT TEXTS:

CHENEY, W., KINCAID, D., Numerical Mathematics and Computing, (Brooks Cole), 1980.
GARNIER, R., TAYLOR, J., Discrete Mathematics for New Technology, (Adam Hilger), 1992.
MORRIS, J.L., Computational Methods in Elementary Numerical Methods, (Wiley), 1983.
QUINNEY, D., An Introduction to the Numerical Solution of Differential Equations, (Wiley), 1987.

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MODULE CODE: MH4749

MODULE TITLE: Geometry

PREREQUISITE MODULE(S): MH4713

OBJECTIVES: 

Geometry is a core part of mathematical education, because it provides a paradigm of rigorous mathematical reasoning. This module equips students with sound knowledge of euclidean geometry and introduces them to non-euclidean geometry.

LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

  • express, justify and establish geometric results;
  • determine geometric quantities using theorems and techniques of euclidean geometry;
  • demonstrate understanding of geometry beyond classical euclidean geometry;
  • carry out and describe geometric constructions.

MODULE CONTENT:

angle, distance, length, area;
coordinates;
lines, triangles and circles;
geometric constructions;
symmetry, congruence and similarity;
vectors and dot product;
non-euclidean geometry.

PRIME TEXT:

BYER, O., LAZEBNIK, F., SMELTZER, D., Methods for Euclidean Geometry, (MAA) 2010.

OTHER RELEVANT TEXTS:

GARDINER, A.D., BRADLEY, C.J., Plane Euclidean Geometry: Theory and Problems, (UKMT) 2005.
LANG, S., MURROW, G., Geometry, A High School Course, (Springer) 1983
BARKER, W., HOWE, R., Continuous Symmetry, From Euclid to Klein, (AMS) 2007.
COXETER, H.S.M., Introduction to Geometry, (John Wiley & Sons) 1961.

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MODULE CODE: MH4724

MODULE TITLE: Introduction to Statistics

PREREQUISITE MODULE(S): MH4712

OBJECTIVES:

To provide an introduction to the theory of probability and to statistical techniques in a manner which will foster understanding of concepts and development of expertise in their applications.

MODULE CONTENT:

Description of sample data; graphical representation, measures of location and dispersion, mean and standard deviation. Probability theory; applications.
Random variables; discrete and continuous, expectation and variance. Probability distributions; binomial, poisson and normal distributions. Sampling theory; random sampling, sampling distributions.
Estimation; point and interval estimates, Student's t distribution. Hypothesis testing; error types.
Correlation and regression; least squares, errors. Testing methods; chi square test, F test, non parametric tests.

PRIME TEXT:

HOEL, P., Elementary Statistics, (Wiley), 1976.

OTHER RELEVANT TEXTS:

FRANK, H., Introduction to Probability and Statistics, (Wiley), 1974.
HOGG, R., CRAIG, A., Introduction to Mathematical Statistics, (Collier Macmillan), 1978.
MENDENHALL, SCHEAFFER, WACKERLY, Mathematical Statistics with Applications, (Duxbury), 1986.

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MODULE CODE: MH4728

MODULE TITLE: Abstract Algebra

PREREQUISITE MODULE(S): MH4711

OBJECTIVES: 

To present the algebraic structures of groups, rings and fields in order to foster an understanding of their central importance in modern mathematics and of their relevance to engineering and science.

MODULE CONTENT:

Groups; axioms and examples, subgroups, mappings and symmetries, applications of symmetry groups.
Subgroups; cosets, Lagrange's theorem. Groups of small order; isomorphism. Binary codes; application of group codes, error correction.
Conjugacy; normal subgroups, factor groups, homomorphism, isomorphism.  Permutation groups; Cayley's theorem.
Rings; axioms and examples, polynomial rings. Subrings; ideals, quotient rings, ring homomorphisms, isomorphisms.
Integral domains; integers. Congruences; Fermat's theorem, Euler's theorem, application of Euler's theorem to public key codes.
Fields; axioms and examples, polynomials over a field.

PRIME TEXT:

DURBIN, J.R., Modern Algebra, (Wiley), 1979.

OTHER RELEVANT TEXTS:

FRALEIGH, J.B., A First Course in Abstract Algebra, (Addison Wesley), 1976.
KIM, K.H., ROUSH, F.W., Applied Abstract Algebra, (Ellis Horwood), 1983.
LEDERMANN, W., Introduction to Group Theory, (Oliver and Boyd), 1973.
WHITELAW, T.A., Introduction to Abstract Algebra, (Blackie), 1988.

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