This is an introductory Course on scientific programming using Fortran and C/C++, intended primarily for students in physical and mathematical sciences. The objective is to equip the students with basic programming skills, including the use of existing libraries, useful in the study of physical and mathematical sciences.

1. Fundamental concepts of programming
2. Brief overview of scientific programming languages (Fortran, C/C++)
3. Basic data types, functions, classes, templates, STL (container classes, algorithms), memory management
4. Compilation process, use of existing C/C++/Fortran libraries
5. Algorithmic problem solving and design process, program development, coding and debugging, fundamental programming constructs, data structures, recursions, simple file processing, algorithmic complexity
6. Case studies in physical and mathematical sciences

Prerequisite: A-level Mathematics or equivalent

This Course introduces fundamental ideas and techniques used in many different areas of mathematics.

1. Elementary logic, mathematical statements, quantified statements
2. Sets, operations on sets, Cartesian products, properties of sets
3. Natural numbers, integers, rational numbers, real numbers, complex numbers
4. Relations, equivalence relations, equivalence classes
5. Functions, injective and surjective functions, inverse functions, composition of functions
6. Mathematical proofs, mathematical induction.

Prerequisite: A-level Mathematics or equivalent

This is the first Course on calculus in a sequence of four Courses. The objective is to introduce basic notions of calculus and analytic geometry, including differentiation.

1. Real numbers, functions, their inverses and graphs
2. Transcendental functions: trigonometric and inverse trigonometric, logarithm and exponential, hyperbolic
3. Limits of functions, continuity at a point, continuity on an interval
4. Differentiability, derivatives of functions, chain rule, implicit differentiation, derivatives of higher order
5. Local maxima and local minima, Rolle's Theorem and Mean Value Theorem, points of inflection, first-derivative and second-derivative tests, Netwon's Method
6. Antidifferentiation

Prerequisite: A-level Mathematics or equivalent
Not available to: students who have taken/are taking MAS 181

This is the second Course in the calculus sequence. The objective is to study integration and related topics.

1. Indefinite and definite integrals, Mean Value Theorem for integrals, Fundamental Theorems of Calculus, area of plane regions
2. Parametric equations, polar coordinates
3. Volumes of solids, length of arcs, other applications of the definite integral
4. Techniques of integration
5. Elementary differential equations

Prerequisite: MAS 112
Not available to: students who have taken/are taking MAS 181

This is the first of two Courses on linear algebra. The main objective is to introduce basic notions in linear algebra that are often used in other areas of mathematics and applications.

1. Systems of linear equations, Gaussian elimination
2. Matrices, inverses, determinants
3. Vectors, dot product, cross product
4. Vector spaces, subspaces, linear independence, basis, dimension, row and column spaces, rank

Prerequisite: A-level Mathematics or equivalent
Not available to: students who have taken/are taking MAS 183 or EE2007

This is the first of two Courses on calculus designed to equip students in the sciences with basic working knowledge of calculus. Applications and computer-based learning will be included.

1. Functions and graphs, real numbers
2. Differentiation of functions of one variable, derivative as rate of change, chain rule, implicit functions, inverse functions
3. Local maxima and minima
4. Indefinite and definite integrals, applications of integration
5. Methods of integration
6. Fundamental theorem of calculus

Prerequisite: A-level Mathematics or equivalent
Not available to: students who have taken/are taking MAS 112 or MAS 113

This second Course on calculus is designed to equip students in the sciences with knowledge of further topics in this useful tool for modern science and engineering.

1. Differential equations — first-order and second-order linear differential equations
2. Techniques of solving differential equations, applications
3. Series and power series
4. Taylor's series
5. Fourier series

Prerequisite: MAS 181 or equivalent
Not available to: students who have taken/are taking MAS 113 or MAS 211

The purpose of this Course is to introduce techniques in linear algebra and multivariable calculus which are useful in applications. Applications and computer-based learning will be included.

1. Systems of linear equations
2. Matrices, determinants
3. Vectors in 2- and 3-dimensional Euclidean spaces
4. Vector spaces, linear independence, basis, dimension
5. Linear transformations
6. Eigenvectors and eigenvalues
7. Calculus of functions of several variables, aprtial derivatives, constrained and unconstrained optimization, applications

Prerequisite: MAS 181 or equivalent
Not available to: students who have taken/are taking MAS 114, MAS 212 or MAS 213

This is the third Course in the calculus sequence. The main focus is to explore further topics in one-variable calculus such as sequences and series, as well as rudiments of multi-variable calculus.

1. Indeterminate forms, improper integrals
2. Taylor's formula
3. Sequences, monotonic and bounded sequences
4. Infinite series, tests for convergence and divergence, alternating series, absolute and conditional convergence
5. Power series, differentiation and integration of power series, Taylor series, binomial series, Fourier series
6. Vector-valued functions and parametric equations, calculus of vector-valued functions, solid analytic geometry

Prerequisite: MAS 113
Not available to: students who have taken/are taking MAS 182

This is the last of four Courses in the calculus sequence. In this Course, multi-variable calculus is introduced.

1. Functions of more than one variable, limits, continuity, partial derivatives, differentiability and total differential, chain rule
2. Directional derivatives, gradients, Lagrange multipliers
3. Double integrals, area of a surface, triple integrals
4. Line integrals, Green's Theorem, surface integrals, Gauss' divergence theorem, Stokes' Theorem

Prerequisite: MAS 113
Not available to: students who have taken/are taking MAS 183

This is the second of two Courses on linear algebra. The main focus is on further topics such as eigenvalues and canonical forms.

* Linear transformations, kernels, and images
* Inner products, inner product spaces, orthonormal sets, Gram-Schmidt process
* Eigenvectors and eigenvalues, diagonalization, applications
* Symmetric and Hermitian matrices
* Quadratic forms, bilinear forms
* Jordan Normal Form and other canonical forms

This Course introduces basic notions in discrete mathematics and number theory commonly used in mathematics and computer science.

1. Counting, permutations and combinations, binomial theorem
2. Inclusion-exclusion principle
3. Boolean algebra
4. Recursion
5. Graphs, paths and circuits, isomorphisms, trees, spanning trees
6. Division algorithm, greatest common divisor, Euclidean algorithm, fundamental theorem of arithmetic, modulo arithmetic
7. Diophantine equations ax+by=c

Prerequisite: MAS 111

This Course focuses on probability theory, with the view of probability distributions as models for phenomena with statistical regularity.

1. Discrete distributions (binomial, hypergeometric and Poisson)
2. Continuous distributions (normal, exponential) and densities
3. Random variables, expectation, independence, conditional probability
4. Introduction to the law of large numbers and the central limit theorem

Prerequisite: {MAS 112 and MAS 113} or {MAS 181 and MAS 182}

Some tools in complex methods and special functions that are commonly used in the sciences are introduced in this Course.

1. Complex numbers, Argand diagrams, modulus and arguments, De Moivre's theorem
2. Functions of a complex variable, elementary examples, Cauchy-Riemann equations
3. Contour integrals, Cauchy's theorem and Cauchy's integral formula
4. Taylor series, Laurent series, zeros, poles and essential singularities, residues
5. Fourier transform, inversion, convolution, Parseval's theorem, delta function, applications
6. Elementary partial differential equations, methods of separation
7. Brief introduction to special functions, e.g., gamma function, beta function, Bessel's function, Legendre's function

Prerequisite: MAS 182 and MAS 183
Not available to: students who have taken/are taking MAS 312

This is the first Course in real analysis – the rigorous investigation of calculus. The emphasis is on rigour and proofs.

1. Basic properties of real numbers, supremum and infimum, completeness axiom
2. Limits and convergence of sequences, subsequences, Bolzano-Weierstrass theorem, Cauchy sequences
3. Limits of functions, continuity, intermediate value theorem, extreme-value theorem
4. Differentiability, derivatives, chain rule, Rolle's theorem, mean value theorem, inverse functions, Taylor's theorem, Lagrange's form of the remainder

Prerequisite: MAS 112, MAS 113, and MAS 211

This Course is an introduction to the theory of complex variables that is useful in many branches of pure and applied mathematics.

1. Analytic functions of one complex variable, Cauchy-Riemann equations
2. Contour integrals, Cauchy's theorem and Cauchy's integral formula, maximum modulus theorem, Liouville's theorem, fundamental theorem of algrebra, Morera's theorem
3. Taylor series, Laurent series, singularities of analytic functions
4. Residue theorem, calculus of residues
5. Fourier transforms, inversion formula, convolution, Parseval's formula
6. Applications

Prerequisite: MAS 211 and MAS 212
Not available to: students who have taken/are taking MAS 281

This is the first Course on modern algebra, introducing basic algebraic structures such as groups, rings and fields.

1. Groups, subgroups, cyclic groups, groups of permutations, Cosets, Lagrange's Theorem, homomorphism, factor groups
2. Rings and fields, ideals, integral domains, quotient fields, rings of polynomials, factorization of polynomials over a field

Prerequisite: MAS 111, MAS 213, MAS 214

1. Basics on computational errors, basic numerical methods for solutions of systems of linear equations, iterative methods for systems of linear equations, polynomial interpolation, numerical integration, numerical solutions of nonlinear equations, implementation of algorithms using MATLAB

Prerequisite: {MAS 114 and MAS 213} or {MAS 181 and MAS 183}

The purpose of this Course is to introduce modern statistical concepts and procedures derived from a mathematical framework.

1. Statistical inference, decision theory, point and interval estimation, tests of hypotheses, Neyman-Pearson lemma
2. Bayesian analysis, maximum likelihood, large sample theory

Prerequisite: MAS 215

The object of study in this Course is regression analysis – one of the most widely used statistical techniques.

1. Simple and multiple linear regression, nonlinear regression, analysis of residuals and model selection
2. One-way and two-way factorial experiments, random and fixed effects models

Prerequiste: MA215 And MA315

1. Pseudorandom number generation, generating discrete and continuous random variables, data access, transformations, estimation, testing hypotheses, ANOVA, resampling methods and simulations

Prerequisite: MAS 215

The Course is to provide the most useful methods and techniques of solving typical ordinary differential equations, to introduce the fundamental theory of ODEs, and to develop methods to analyze given equations.

1. First order equations, exact equations, integrating factors, separable equations, linear homogenous and non-homogenous equations, variation of parameters, Principle of superposition
2. Second order equations, Wronskian, Abel's formula, variation of parameters, exact equations, adjoint and self-adjoint equations, Lagrange and Green's identities, Sturm's comparison and separation theorems
3. First order linear systems, Wronskian, Abel's formula, variation of parameters, systems with constant coefficients
4. First order nonlinear equations, initial value problem
5. Use of ODE in simple modeling problems

Prerequisite: MAS 212

1. Review of modular arithmetic, Chinese remainder theorem, Fermat's little theorem, Wilson's theorem
2. Number-theoretic functions: τ, σ, Euler's φ-function, Möbius inversion formula, applications to cryptography
3. Primitive roots, indices
4. Legendre's symbols, quadratic reciprocity law
5. Continued fractions, Pell's equations
6. Primality tests and factorization of integers, RSA cryptosystem

Prerequisite: MAS 214

This Course provides an introduction to working with the most accessible discrete structures, i.e., graphs.

1. Review of introductory graph theory from MAS 214
2. Connectivity and matchings, Hall's theorem, Menger's theorem, Network flows
3. Paths and cycles, complete subgraphs and Turán's theorem, Erdös-Stone theorem
4. Graph colouring, four-colour theorem
5. Ramsey theory
6. Probabilistic methods in graph theory
7. Use of software to solve graph-theoretic problems

Prerequisite: MAS 214 and MAS 215

This is the first Course in optimization and operations research. Basic methods and concepts are introduced.

1. Introduction of optimization models: objective and constraints, convex sets and functions, polyhedron and extreme points
2. Introduction to LP: solving 2-variable LP via graphical methods; simplex method; dual LP and sensitivity analysis
3. Karush-Kuhn-Tucker optimality conditions, optimal solution via optimality conditions, Duality theory
4. Network optimization: Shortest path, maximum flow, minimum cost flow, assignment problem, transportation problem, network simplex method

Prerequisite: either MAS 213 or MAS 183

The objective of this Course is to introduce modeling dependence.

1. 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
2. Random walk, Poisson processes

Prerequisite: MAS 215

This is a research-based Course where the student works on a specific topic under the supervision of a faculty member.

Prerequisite: MAS331, Subject to approval of the Head of Division

This is a continuation of MAS 311 where further topics in real analysis are investigated with rigour.

1. Riemann integral, integrability, fundamental theorem of calculus, improper integrals
2. Convergent series, absolute convergence, tests of convergence
3. Sequence and series of functions, uniform convergence
4. Power series, radius of convergence, local uniform convergence of power series

Prerequisite: MAS 212 and MAS 311

This is a continuation of MAS 421 where further topics in real analysis are investigated with rigour.

1. Normed space Rn, Lipschitz mappings, Bolzano-Weierstrass theorem in Rn, open and closed sets, sequences, Cauchy sequences, completeness, continuity of functions, compactness, Heine-Borel Theorem, Bolzano-Weierstrass Theorem, connectedness
2. Introduction to metric spaces, limits, continuity, balls, neighbourhoods, open and closed sets, completeness, compactness, space of continuous functions, contraction mapping principle, Arzela-Ascoli Theorem, Weierstrass Approximation Theorem

Prerequisite: MAS 311 and MAS 421

The objective of this introductory Course on partial differential equations is to provide their basic properties as well as the techniques to solve some such equations.

1. First-order equations, Quasi-linear equations, general first-order equation for a function of two variables, Cauchy problem
2. Wave equation, wave equation in two independent variables, Cauchy problem for hyperbolic equations in two independent variables
3. Heat equation, the weak maximum principle for parabolic equations, Cauchy problem for heat equation, regularity of solutions to heat equation
4. Laplace equation, Green's formulas, harmonic functions, maximum principle for Laplace equation, Dirichlet problem, Green's function and Poisson's formula

Prerequisite: MAS 311 and MAS 321
Remark: it is useful to take MAS 421 before taking MAS 423

Further topics in groups, rings and fields are discussed in this Course.

1. Sylow's Theorems, Abelian groups
2. Homomorphisms of rings, factor rings, prime and maximal ideals
3. Unique factorization domains, Euclidean domains, principal ideal domains
4. Modules, submodules, homomorphisms, quotient modules, modules over principal ideal domains

Prerequisite: MAS 213 and MAS 313

This Course deals with the theory of fields, culminating in Galois theory, the most famous of whose application is the proof that the general quintic equation with rational coefficients cannot be solved by radicals.

1. Field extensions, algebraic extensions, geometric constructions
2. Finite fields
3. Automorphisms of fields, splitting fields, normal and separable extensions, Galois extensions, Galois groups
4. Galois correspondence
5. Solution of equations by radicals, insolvability of the quintic equation
6. Fundamental Theorem of Algebra

Prerequisite: MAS 313 and MAS 425

This Course introduces the notions of validity and provability in formal logic, the concepts of ordinals and cardinals as well as some formal set theory.

1. Partially-ordered sets, well-orderings and order-types, induction and recursion on ordinals, ordinal arithmetic, cardinals, cardinal arithmetic
2. Axiom of choice and its equivalences, axiom of determination
3. Propositional calculus, truth tables, validity and contradictions
4. Predicate calculus with equality, completeness and compactness theorems, Löwenheim-Skolem theorem

Prerequisite: MAS 111 and MAS 214

The purpose of this Course is to study some topics in combinatorics and their connections with other branches of mathematics and theoretical computer science.

1. Recursions and generating functions
2. Partitions and tableaux
3. Designs, Latin squares, combinatorial designs and projective geometries
4. Extremal combinatorics, asymptotic analysis

Prerequisite: MAS 211, MAS 213, and MAS 214

This Course is to introduce basic notions in the theory of error-correcting codes which is used in data storage and telecommunication.

1. Error detection, correction and decoding, Hamming distance
2. Basic facts on finite fields
3. Linear codes, Hamming weight, generator and parity-check matrices, encoding, decoding
4. Bounds, Hamming codes, Golay codes, perfect codes, MDS codes
5. Construction of codes, Reed-Muller codes
6. Cyclic codes, generator polynomials, BCH codes, Reed-Solomon codes
7. Computer implementation of efficient coding and decoding

Prerequisite: MAS 213 and MAS 214

This Course is an introduction to some basic issues in cryptography, especially the underlying mathematical concepts.

1. Classical ciphers, cryptanalysis, linear complexity
2. Data Encryption Standard
3. RSA cryptosystem, primality testing and factorization of integers
4. Discrete logarithms
5. Signatures, Digital Signature Standard

Prerequisite: MAS 214 and MAS 313

This Course introduces the notions of metric and topological spaces.

1. Metric spaces, limits, continuity, balls, neighbourhoods, open and closed sets
2. Topology, metric topologies, convergence, Hausdorff spaces, homeomorphisms, topological and non-topological properties, subspace, quotient and product topologies
3. Connectedness, components, path-connectedness
4. Compactness, sequential compactness
5. Contraction mapping theorem

Prerequisite: MAS 311 and MAS 421

Basic ideas in algebraic topology will be introduced in this Course.

1. Simplicial complexes, subdivisions, simplicial approximation theorem, classification of surfaces
2. Fundamental groups, homotopy of continuous functions and homotopy equivalence, change of base point, van Kampen's theorem
3. Euler characteristic, Lefschetz fixed point theorem
4. Covering spaces and covering maps

Homology Prerequisite: MAS 313 and MAS 421
Remarks: it is useful to take MAS 436 before taking MAS 437

This is an introduction to differential geometry, with curves and surfaces in the Euclidean 3-dimensional space as the focus

1. Metrics, Lie brackets, connections, curvature and torsion of curves, the Frenet-Serret equations, Gaussian and mean curvature of surfaces, geodesics, isometries and Gauss's Theorem Egregium, tensors
2. Gauss-Bonnet theorem

Prerequisite: MAS 311 and MAS 421
Remark: it is useful to take MAS 422 or MAS 436 before taking MAS 438

This Course is to provide numerical solutions of differential equations using finite difference methods and to understand the implementation of the numerical computations using computer software packages such as MATLAB.

1. Finite difference formulae, consistency of difference schemes, finite difference methods for ordinary differential equations, classification of second-order partial differential equations, first and second order characteristics
2. Matrix method and von Neumann method for stability analysis, Lax's equivalence theorem for convergence, method of characteristics
3. Application to heat equation, wave equation and Poisson's equation

Prerequisite: MAS 314 and MAS 321
Remark: it is useful to take MAS 423 before taking MAS 441

This Course is to provide the necessary mathematical tools for image and signal processing.

1. Fourier transform, Window Fourier transform, Fourier series, discrete Fourier transform and discrete Window Fourier transform, orthonormal basis and tight frames
2. Splines, approximation by splines
3. Refinable functions, subdivision scheme
4. Multiresolution analysis, orthonormal wavelets, spline tight frame wavelets, discrete wavelet transform, analysis and synthesis algorithms

Prerequisite: MAS 311 and MAS 441
Remark: it is useful to take MAS 421 and MAS 422 before taking MAS 442

This Course introduces some deterministic methods commonly used in operations research.

1. Unconstrained optimization: one-dimensional search, gradient method, Newton-Raphson method
2. Constrained optimization: feasible direction methods, penalty/barrier function methods, modern interior point methods for convex programming
3. Discrete optimization: formulations, cutting plane methods, branch-and-bound methods, Lagrangian relaxation, dynamic programming approach

Prerequisite: MAS 212 and MAS 326

This Course introduces some useful probabilistic methods commonly used in operations research and statistics.

1. Queueing: basic models, performance analysis, simulation of queueing systems
2. Stochastic optimization: Stochastic programming, modeling and algorithms, Markov decision process, stochastic approximation

Prerequisite: MAS 215 and MAS 326

This Course focuses on issues which arise in the integrated design and management of the entire logistics network.

1. Overview of supply chain - components of a supply chain, material and information flow, supplier-retailer-customer interaction, e-business
2. Inventory and materials management - Economic order quantity model, Lot sizing models, models with uncertain demands, MRP/JIT
3. Facility location and transportation - single-source capacitated facility location, vehicle routing problem with equal, unequal demands and time-window constraints

Prerequisite: MAS 215 and MAS 326

This Course introduces time series models used in economics, engineering and finance.

1. Trend fitting, autoregressive and moving average models, spectral analysis
2. Seasonality, forecasting and estimation
3. Use of computer package to analyze real data sets

Prerequisite: MAS 215, MAS 315, and MAS 316

1. Distribution theory: multivariate normal distribution, Hotelling's T2 and Wishart distributions, inference on the mean and covariance, principal components and canonical correlation, factor analysis, discrimination and classification

Prerequisite: MAS 215, MAS 315, and MAS 316

1. Neural networks, support vector machines, classification trees and boosting

This Course gives an introduction to sampling and the design of sample surveys.

1. Ratio and regression estimators under simple random sampling, separate and combined estimators for stratified random sampling
2. Systematic sampling and its relationship with stratified and cluster sampling
3. Further aspects of stratified sampling, cluster sampling with clusters of unequal sizes
4. Subsampling; multi-stage sampling
5. Complex sample designs

Prerequisite: MAS 215 and MAS 315

1. Phases of clinical trials, objectives and endpoints, the study cohort, controls, randomization and blinding, sample size determination, treatment allocation, monitoring trial progress: complient effects, ethical issues, quality of life assessment, data analysis involving multiple treatment groups and endpoints, stratification and subgroup analysis, intent to treat analysis, analysis of compliance data, surrogate endpoints, multi-centre trials and good practice versus misconduct.

Prerequisite: MAS 215 and MAS 315

This Course focuses on the standard methods of survival data analysis.

1. Examples of survival data analysis, types of censoring, parametric survival distributions (exponential, Weibull, lognormal), nonparametric methods, Kaplan-Meier estimator, tests of hypotheses, graphical methods of survival distribution fitting, goodness of fit tests.

Prerequisite: MAS 215 and MAS 315

Prerequisite: subject to approval of the Head of Division

Some advanced topics in statistics not normally covered in the regular Courses may be offered.

Prerequisite: subject to approval of the Head of Division

Prerequisite: subject to approval of the Head of Division

This is a one-semester research Course on an advanced topic leading to an Honours thesis, under the supervision of a faculty member.

Prerequisite: Subject to approval of the Head of Division
Not available to: students who have taken/are taking MAS 492

Error-detecting and error-correcting codes – detecting and correcting errors in data: basic modular arithmetic used in the design of such codes, basic issues in theory and applications, well-known examples, real-life applications such as NRIC numbers, ISBN, CD, telecommunications, etc. Cryptography – ensuring security of information: basic issues and use in applications such as electronic transactions and communication, Euclidean algorithm, congruences, Chinese Remainder Theorem, the RSA cryptosystem. Travelling Salesman Problem – finding optimal routes: basic concepts in graph theory and linear programming, simplex algorithm, relationship to applications, e.g., wiring a chip, scheduling airline crews. P vs NP – understanding computational complexity: complexity classes, NP-complete problems and links with other applications such as the RSA cryptosystem and the Travelling Salesman Problem. Google – search for information on the Web: basic concepts in graph theory, probability and linear algebra, especially eigenvalues, underlying the Google search engine.

Pre-requisite: AO Level Mathematics or equivalent

The Course provides an overview of statistics and its applications in other disciplines. In particular, this Course provide students with the understanding of statistics methodology and necessary skills of evaluating statistical studies that they may encounter in some other Courses, future career, or even their everyday lives.

Topics includes overview of statistics; measurement; visual displays; data descriptions; probability and risk; correlation and causality; statistical methodologies; statistical modeling.

Pre-requisite: AO Level Mathematics or equivalent

Prerequisite: Subject to approval of the Head of Division, may only be taken concurrently with MAS 113

Prerequisite: Subject to approval of the Head of Division, may only be taken concurrently with MAS 211

The main focus of this Course, conducted in seminar-style, is to develop students’ problem-solving skills and experience through attempting the solution of challenging mathematical problems. The emphasis is on solving non-standard problems from usual courses like calculus, linear algebra, algebra, differential equations, probability, discrete mathematics, learning to think creatively, getting exposition skills, rather than studying theory or specialized techniques for solving specific problems.

Prerequisite: Approval by the Division of Mathematical Sciences. Applicants may be required to sit for a qualifying test.