APPENDIX A
Introduction
To prepare students adequately for graduate studies in Industrial Engineering in GT, a good grounding in Mathematics is required. In particular, students should have a strong foundation in Linear Algebra and Calculus as well as Probability Theory along with a good exposure to apply these concepts in Stochastic Processes and Modeling as well as Optimization problems.
Prescribed Mathematics Modules
MA1101R Linear Algebra or MA1506 Mathematics II
MA1507 Advanced Calculus or MA1505 Mathematics I
MA2216 Probability (or precluded modules in MA2216)
MA3238 Stochastic Processes I (or precluded modules of MA3238)
MA3252 Network Optimization (or precluded modules in MA3252)
Module Code / MA1101RModule Title / Linear Algebra I
Description / This module is a first course in linear algebra. Fundamental concepts of linear algebra will be introduced and investigated in the context of the Euclidean spaces R^n. Proofs of results will be presented in the concrete setting. Students are expected to acquire computational facilities and geometric intuition with regard to vectors and matrices. Some applications will be presented. Major topics: Systems of linear equations, matrices, determinants, Euclidean spaces, linear combinations and linear span, subspaces, linear independence, bases and dimension, rank of a matrix, inner products, eigenvalues and eigenvectors, diagonalization, linear transformations between Euclidean spaces, applications.
Module Credit / 4
Workload / 3-1-1-0-6
Prerequisites / 'A' level Mathematics or [GM1101 and GM1102] or MA1301
Preclusions / GM1302, GM1306, GM1308, EG1401, EG1402, MQ1101, MQ1103, MA1101, MA1306, MA1311, MA1508, MPE students, CVE students, CHE students (for breadth requirement), EVE students (for breadth requirement), FASS students from 2003 cohort onwards who major in Mathematics (for breadth requirement), ISE students admitted in 2002.
Module Code / MA1505
Module Title / Mathematics I
Description / This module provides a basic foundation for calculus and its related subjects required by engineering students. The objective is to train students to be able to handle calculus techniques, arising in their engineering courses. In addition to the standard calculus material, the course covers simple mathematical modeling techniques in connection with ordinary differential equations, basic Fourier series methods and an introduction to Laplace transform. Major topics: Sets, complex numbers, Calculus, sequences and series, Laplace transform, Differential equations.
Module Credit / 4
Workload / 3-1-1-0-6
Prerequisites / 'A' level Mathematics or [GM1101 and GM1102] or MA1301
Preclusions / MA1102, MA1102R, GM1306, GM1307, EE1401, EE1461, MA1306, MA1505C, MA1312, MA1507, MA2311, MA2501, MQ1102, MQ1103, MPE students (for breadth requirement), FASS students from 2003 cohort onwards who major in Mathematics (for breadth requirement).
Module Code / MA1506
Module Title / Mathematics II
Description / This module, together with MA1505, introduces students of engineering and physical sciences to those areas of mathematics which are important in connection with practical problems. The primary objective of this module is to give students in these disciplines a firm grounding in the fundamental mathematical principles needed for their further study of engineering and physical sciences. The module emphasizes problem solving and mathematical techniques and covers three important areas: (a) multi-variate calculus and vector analysis, (b) linear algebra; (c) partial differential equations. Major topics: Vectors, equations of lines and planes, functions of several variables, geometric interpretation, directional derivatives, extrema of functions, vector calculus, partial differential equations, Laplace transform solution to partial diffential equations, Linear algebra, Matrix algebra, matrix inversion, eigenvalues and eigenvectors.
Module Credit / 4
Workload / 3-1-1-0-6
Prerequisites / 'MA1102R or MA1505 or MA1505C or GM1307
Preclusions / MA1311, MA1507, MA2501, MA2221, MA2311, MA2312, PC1134, PC2201, EE1461, MQ2102, MQ2202, MPE students (for breadth requirement), FASS students from 2003 cohort onwards who major in Mathematics (for breadth requirement).
Module Code / MA1507
Module Title / Advanced Calculus
Description / The objective of this module is to provide a foundation for calculus of one and several variables. The module is targeted at students in the Engineering Science Programme. Topics: brief review of one variable calculus, sequences and series, tests of convergence and divergence, power series in one variable, interval of convergence, Maclaurin and Taylor series, Taylor's theorem with remainder, lines and planes, functions of several variables, continuity of functions of several variables, partial derivatives, chain rule, directional derivatives, normal lines and tangent planes to surfaces, extrema of functions, vector-valued functions, curves, tangents and arc length, gradient, divergence and curl, line, surface and volume integrals, Green's theorem, divergence theorem, Stokes' theorem.
Module Credit / 4
Workload / 3-1-0-0-6
Prerequisites / A' level or H2 Mathematics or equivalent
Preclusions / MA1102R, MA1104, MA1104S, MA1505, MA1505C, MA1506, MA2221, MA2311
Module Code / MA2216
Module Title / Probability
Description / The objective of this course is to give an elementary introduction to probability theory for science (including computing science, social sciences and management sciences) and engineering students with a knowledge of elementary calculus. It will cover not only the mathematics of probability theory but will work through many diversified examples to illustrate the wide scope of applicability of probability. Topics covered are: combinatorial analysis, axioms of probability, conditional probability and independence, random variables, distributions and joint distributions, expectations, central limit theorem.
Module Credit / 4
Workload / 3-1-0-0-6
Prerequisites / MA1102 or MA1102R or MA1312 or MA1507 or MA1505 or MA1505C or GM1307 or GM1304 or EG1402 or EE1401 or EE1461
Preclusions / GM2303, MQ2205, ST2131, ST2334, ST2201, ST2203, SA2101, CE2407, FASS students from 2003 cohort onwards who major in Mathematics (for breadth requirement)
Cross-listing / ST2131
Module Code / MA2215
Module Title / Linear Programming
Description / The objective of this course is to work on optimization problems which can be formulated as linear programming problems. We formulate LP problems and solve them by the simplex method (algorithm). The simplex algorithm is the fundamental method of the other algorithms to be introduced in subsequent topics such as transportation problems. We shall also look at the geometrical aspect and develop the mathematical theory of the subject. This course should help the student in developing confidence in solving many similar problems in daily life that require much computing. Major topics: Introduction to LP: solving 2-variable LP via graphical methods. Minimize max( ). Geometry of LP: polyhedron, concept of corners: equivalence of extreme point, vertex and basic feasible solution existence of optimal solution at extreme point. Development of simplex method: standard form LP: basic solution, reduced costs and optimality condition. Iterative steps in a simplex method, carry out the algorithms via simplex tableau and revised simplex method, 2-phase method and Big-M method. Duality: dual LP, duality theory, dual simplex method. Sensitivity Analysis. Transportation problems, transportation algorithm.
Module Credit / 4
Workload / 3-1-0-0-6
Prerequisites / MA1101 or MA1101R or MA1306 or MA1311 or MA1508 or MA1506 or GM1306
Preclusions / GM2302, MQ2204, CS3252, IC2231, ISE students, FASS students from 2003 cohort onwards who major in Mathematics (for breadth requirement), all students in 2007 and later cohorts.
Module Code / MA3238
Module Title / Stochastic Processes I
Description / This module introduces the concept of modeling dependence and focuses on discrete-time Markov chains. Major topics: discrete-time Markov chains, examples of discrete-time Markov chains, classification of states, irreducibility, periodicity, first passage times, recurrence and transience, convergence theorems and stationary distributions.
Module Credit / 4
Workload / 3-1-0-0-6
Prerequisites / {MA1101 or MA1101R or MA1311 or MA1508 or GM1302} and {MA2216 or ST2131}
Preclusions / ST3236, FASS students from 2003 cohort onwards who major in Mathematics (for breadth requirement), ISE students.
Cross-listing / ST3236
Module Code / MA3252
Module Title / Network Optimization
Description / The objective of this course is to work on optimization problems which can be formulated as linear and network optimization problems. We formulate linear programming (LP) problems and solve them by the simplex method (algorithm). We also look at the geometrical aspect and develop the mathematical theory of the simplex method. We further study problems which may be formulated using graphs and networks. These optimization problems can be solved by using linear or integer programming approaches. However, due to its graphical structure, it is easier to handle these problems by using network algorithmic approaches. Applications of LP and network optimization will be demonstrated. This course should help the student in developing confidence in solving many similar problems in daily life that require much computing. Major topics: Introduction to LP: solving 2-variable LP via graphical methods. Geometry of LP: polyhedron, extreme points, existence of optimal solution at extreme point. Development of simplex method: basic solution, reduced costs and optimality condition, iterative steps in a simplex method, 2-phase method and Big-M method. Duality: dual LP, duality theory, dual simplex method. Sensitivity Analysis. Network optimization problems: minimal spanning tree problems, shortest path problems, maximal flow problems, minimum cost flow problems, salesman problems and postman problems.
Module Credit / 4
Workload / 3-1-0-0-6
Prerequisites / MA1101 or MA1101R or MA1306 or MA1311 or MA1508 or MA1506 or GM1306
Preclusions / GM2302, MQ2204, CS3252, IC2231, DSC3214, GM3308, MA3235, BH3214, ISE students, FASS students from 2003 cohort onwards who major in Mathematics (for breadth requirement).