# Mathematical Sciences Courses

## Ordinary Courses

###### CY1601/RE1001Mathematics I

4 AU

Introductory mathematics course for CN Yang scholars and Renaissance Engineering students. Topics include: limits and continuity; differentiability and differentiation rules; critical points, the mean value theorem, and l'Hospital's rule; inverse functions; trigonometric, logarithm and exponential functions; the Riemann integral; the Fundamental Theorems of Calculus; techniques of integration; infinite sequences and infinite series, power series and convergence criteria, and Taylor series; and ordinary differential equations.

Only offered to CN Yang scholars and Renaissance Engineering students.
Mutually exclusive with MH1100, MH1101, MH1800, and MH1801.

###### CY1602/RE1021Mathematics II

4 AU

Second mathematics course for CN Yang scholars and Renaissance Engineering students. Topics include: systems of linear equations, and the Gaussian elimination algorithm; matrices, and their inverses and determinants; vector spaces, subspaces, linear independence, basis, dimension, row and column spaces, rank, linear transformations, eigenvectors, eigenvalues, and diagonalization; inner products, inner product spaces, orthonormal sets, the Gram-Schmidt process, and Fourier series; calculus of several variables; double and triple integrals; vector calculus, line integrals, Green's Theorem, surface integrals, Gauss's divergence theorem, and Stokes' Theorem.

Prerequisite: CY1601 or RE1001.
Mutually exclusive with MH1200 or MH2100.

###### MH1100Calculus I

4 AU

Introductory course on differential and integral calculus. Topics include: real numbers, functions, their inverses and graphs; trigonometric and inverse trigonometric functions, logarithms and exponentials, and hyperbolic functions; limits of functions, continuity at a point, and continuity on an interval; differentiability, derivatives of functions, the chain rule, implicit differentiation, derivatives of higher order; local maxima and local minima, Rolle's Theorem and the Mean Value Theorem, points of inflection, first-derivative and second-derivative tests, and L'Hospital's Rule; antidifferentiation, indefinite integrals, substitution rule, and integration by parts.

Prerequisite: A level Mathematics or equivalent.
Mutually exclusive with MH1800.

###### MH1101Calculus II

4 AU

Further topics in calculus. Topics include: definite integrals; the Fundamental Theorems of Calculus; area of plane regions, volumes of solids, length of arcs; the Mean Value Theorem for integrals; techniques of integration, numerical integration, and improper integrals; monotonic and bounded sequences, Newton's method, infinite series, tests for convergence and divergence, alternating series, and absolute/conditional convergence criteria; differentiation and integration of power series, Taylor series, binomial series, and Fourier series.

Prerequisite: MH1100
Mutually exclusive with MH1801.

###### MH1200Linear Algebra I

4 AU

Introductory course on linear algebra. Topics include: systems of linear equations; Gaussian elimination; matrices, inverses, and determinants; vectors, dot products, and cross products; vector spaces, subspaces, linear independence, basis, dimension, row and column spaces, and rank.

Prerequisite: A or H2 level Mathematics or equivalent.
Mutually exclusive with MH2800.

###### MH1201Linear Algebra II

4 AU

Further topics in linear algebra. Topics include: linear transformations, kernels and images; inner products, inner product spaces, orthonormal sets, and the Gram-Schmidt process; eigenvectors and eigenvalues; matrix diagonalization and its applications; symmetric and Hermitian matrices; quandratic forms and bilinear forms; Jordan normal form and other canonical forms.

Prerequisite: MH1200.
Mutually exclusive with MH2800.

###### MH1300Foundations of Mathematics

4 AU

Introductory course on core mathematical concepts, including logic and the theory of sets. Topics include: elementary logic, mathematical statements, and quantified statements; sets, operations on sets, Cartesian products, and properties of sets; natural numbers, integers, rational numbers, real numbers, and complex numbers; relations, equivalence relations, and equivalence classes; functions, injective and surjective functions, inverse functions, and composition of functions; division algorithm, greatest common divisor, Euclidean algorithm, fundamental theorem of arithmetic, modulo arithmetic.

Prerequisite: A or H2 level Mathematics or equivalent.

###### MH1301Discrete Mathematics

3 AU

Introduction to discrete mathematics, including: basics of counting; the inclusion-exclusion principle; the pigeonhole principle; permutations and combinations; the binomial theorem; recurrence relations and linear recurrence relations; graph concepts such as Shortest-, Euler-, Hamilton-Paths and Cycles, coloring, planarity, weighted graphs, and directed graphs.

Prerequisite: A or H2 level Mathematics or equivalent.

###### MH1401Algorithms and Computing I

2 AU | No longer offered

Core course introducing fundamentals of programming using the Python programing language. By emphasizing applications to problem-solving, it develops the ability to think algorithmically, which is essential for any professional working in an increasingly computer-driven world. This course is required for future computing courses and for courses using Python as a supporting tool. No prior programming experience is required. Topics include: Python basics, lists, the NumPy module, strings, input/output, selection statements, loop statements, functions, errors and debugging, recursion, algorithm complexity, sorting algorithms, and plotting.

Prerequisite: A or H2 level Mathematics or equivalent.

###### MH1402Algorithms and Computing II

2 AU | No longer offered

Further topics in algorithms and computing. Topics include: the concept of an algorithm; the basic structures of a C/C++ program; debugging and good programming style; vectors and arrays; algorithms for searching and sorting vectors and arrays; basic concepts of algorithm efficiency; functions, classes, and libraries in C/C++; recursion and the divide-and-conquer paradigm.

Prerequisite: MH1401.

###### MH1403Algorithms and Computing

3 AU

Systematic introduction to data structures and algorithms for constructing efficient computer programs. Topics include: data abstraction in the program development process; design of efficient algorithms; simple algorithmic paradigms such as greedy algorithms, divide-and-conquer algorithms and dynamic programming; and elementary analyses of algorithmic complexity.

Prerequisite: PS0001 or BS1009 or CV1014 or MS1008 or MA1008 or {CB0494, CH2107} or {CB0494, BG2211}

###### MH1800Calculus for the Sciences I

3 AU | No longer offered

First of two courses on calculus for students in the sciences. Applications and computer-based learning are included. Topics include: functions and graphs; real numbers; differentiation of functions of one variable, derivative as rate of change, chain rule, implicit functions, and inverse functions; local maxima and minima; indefinite and definite integrals, and applications of integration; methods of integration; and the Fundamental Theorem of Calculus.

Prerequisite: A or H2 level Mathematics or equivalent.
Mutually exclusive with MH1100 and MH1101.

###### MH1801Calculus for the Sciences II

3 AU | No longer offered

Second of two courses on calculus for students in the sciences. Applications and computer-based learning are included. Topics include: differential equations; first-order and second-order linear differential equations; techniques of solving differential equations, and applications; series and power series; Taylor series; Fourier series.

Prerequisite: MH1800 or equivalent.
Mutually exclusive with MH1101.

###### MH1802Calculus for the Sciences

4 AU

Introductory course in calculus, for students majoring in the physical sciences. Topics include: types of Numbers; functions and graphs; algebraic, trigonometric, logarithmic and exponential functions and identities; complex numbers; limits and continuity; derivatives and techniques of differentiation; applications of differentiation; indefinite integrals and definite integrals; the Fundamental Theorem of Calculus; techniques of integration; applications of integration in science; differential equations; and power series.

Prerequisite: A or H2 level Mathematics or equivalent.
Mutually exclusive with MH1800 and MH1801.

###### MH1803Calculus for Physics

4 AU

Additional topics in calculus, for students majoring in physics. Topics include: vectors and multivariable calculus; vector analysis; ordinary differential equations; an partial differential equations.

Prerequisite: A or H2 level Mathematics or equivalent.
Mutually exclusive with MH1800 and MH1801.

###### MH1804Mathematics for Chemistry

2 AU

Detailed Course Information

Additional topics in calculus, for students majoring in chemistry. Topics include: Cartesian and spherical coordinates; complex numbers; vectors; linear algebra and matrices; summation, series, and expansions of functions; Fourier series and Fourier transforms.

Prerequisite: MH1802.

###### MH1805Calculus

4 AU

Further topics in calculus, including: sets and functions; limits and continuity; differentiation and optimization; the Riemann integral; the Fundamental Theorem of Calculus; applications of integration; methods of integration; series, power series, and Taylor series; and ordinary differential equations

Mutually exclusive with CY1601, MH1100, MH1101, MH1802, and RE1011.

###### MH2100Calculus III

4 AU

Intermediate course in calculus. Topics include: parametric equations; polar coordinates; vector-valued functions, calculus of vector-valued functions, and analytic geometry; functions of more than one variable, limits, continuity, partial derivatives, differentiability, total differentials, the chain rule, and the implicit function theorem; directional derivatives, gradients, and Lagrange multipliers; double and triple integrals; line integrals, Green's theorem, surface integrals, the Gauss divergence theorem, and Stokes' theorem.

Prerequisite: MH1101 or MH1802 or MH1805.
Mutually exclusive with MH2800.

###### MH2200Groups and Symmetry

3 AU

Introductory course on group theory, with emphasis on symmetry groups of geometric structures. Topics include: symmetries of 2D and 3D objects (e.g. quadrangles, tetrahedrons); group axioms; cyclic and dihedral groups; permutation groups; representation of rotations and reflections by matrices; wallpaper groups; and puzzles such as the 15-puzzle and Rubik's cube.

Prerequisite: MH1200 or CY1602.

###### MH2220Algebra I

3 AU

This MAS course aims to introduce group theory that is essential for more advanced algebra courses and applications. The axiomatic concepts serve as a language to study concrete examples in broader sense and helps in developing logical thinking.

Prerequisite: (MH1200 or CY1602) and MH1300.

###### MH2401Algorithms and Computing III

2 AU | No longer offered

Application of computing skills and previously-learnt mathematical topics (Linear Algebra, Calculus, Discrete Mathematics, etc.) for solving real-world problems. This course emphasizes group project work, and assessments are based substantially on a term project.

Prerequisites: MH1100, MH1101, MH1200, MH1401, and MH1402.

###### MH2500Probability & Introduction to Statistics

4 AU

Introductory course on probability and statistics. Topics include: discrete distributions (binomial, hypergeometric and Poisson); continuous distributions (normal, exponential) and densities; random variables, expectation, independence, conditional probability; the law of large numbers and the central limit theorem; sampling distributions; and elementary statistical inference (confidence intervals and hypothesis tests).

Prerequisite: (MH1100 & MH1101) or (MH1800 & MH1801) or (MH1101 & MH110S) or (MH1100 & MH111S) or MH1802 or CY1601 or MH1805.

###### MH2800Linear Algebra and Multivariable Calculus

4 AU | No longer offered

Techniques in linear algebra and multivariable calculus, and their applications. This course includes computer-based learning. Topics include: systems of linear equations; matrices and determinants; vectors in 2- and 3-dimensional euclidean spaces; vector spaces, linear independence, basis, and dimension; linear transformations; eigenvectors and eigenvalues; calculus of functions of several variables; partial derivatives; and constrained and unconstrained optimization.

Prerequisite: MH1800 or equivalent.
Mutually exclusive with MH1200, MH1201, and MH2100.

###### MH2801Complex Methods for the Sciences

3 AU

Detailed Course Information

Introduction to complex numbers and their applications in physics and the other sciences. Topics include: complex numbers, the argand diagram, modulus and argument; complex representations of waves and oscillations; functions of a complex variable, analyticity, and the Cauchy-Riemann equations; contour integration, Cauchy's integral formula, and the residue theorem; Fourier series and Fourier transformations, and their applications; and Green's functions methods.

Prerequisites: (MH1801 and MH2800) or (MH1101 and MH1200) or (MH1802 and MH1803 and MH1200) or (MH1802 and MH1803 and MH2802) or (CY1601 and CY1602).
Mutually exclusive with MH3101.

###### MH2802Linear Algebra for Scientists

3 AU

Introduction to linear algebra and its applications in physics and the other sciences. Topics include: vector algebra and analytical geometry; linear spaces; linear transformations and matrices; eigenvalues and eigenvectors; and applications of linear algebra to problems in physics and computing.

Prerequisite: A or H2 level Mathematics or equivalent.

###### MH3100Real Analysis I

4 AU

Introduction to real analysis. Topics include: properties of real numbers, supremum and infimum, completeness axiom, open and closed sets, compact sets, countable sets; limits and convergence of sequences, subsequences, Bolzano-Weierstrass theorem, Cauchy sequences, infinite series, double summations, products of infinite series; limits of functions, continuity, uniform continuity, intermediate value theorem, extreme-value theorem; differentiability, derivatives, intermediate value property, cauchy mean value theorem, Taylor's theorem, Lagrange's form of the remainder; sequence and series of functions, uniform convergence and differentiation; power series, radius of convergence, and local uniform convergence of power series.

Prerequisites: (MH1100 & MH1101) or CY1601 or MH1802.

###### MH3101Complex Analysis

4 AU

Introduction to complex analysis. Topics include: analytic functions of one complex variable, the Cauchy-Riemann equations; contour integrals, Cauchy's theorem and Cauchy's integral formula, maximum modulus theorem, Liouville's theorem, fundamental theorem of algrebra, Morera's theorem; Taylor series, Laurent series, and singularities of analytic functions; the residue theorem and the calculus of residues; Fourier transforms, inversion formula, convolution, and Parseval's formula.

Prerequisite: MH2100.
Mutually exclusive with MH2801.

###### MH3110Ordinary Differential Equations

4 AU

The course builds on Calculus and Linear Algebra. It aims to equip students with useful solution methods for solving various types of ordinary differential equations (ODEs), introduce the fundamental theory of ODEs, and develop skills for modeling real phenomena by ODEs.

Prerequisite: MH2100 or CY1602

###### MH3200Abstract Algebra I

3 AU

Introduction to modern algebra, including basic algebraic structures such as groups, rings and fields. Topics include: groups, subgroups, cyclic groups, groups of permutations, cosets, Lagrange's Theorem, homomorphism, and factor groups; rings and fields, ideals, integral domains, quotient fields, rings of polynomials, and factorization of polynomials over a field.

Prerequisites: MH1201, MH1300, and MH2200.

###### MH3220Algebra II

4 AU

This MAS course aims to introduce ring theory that is essential for more advanced algebra courses and applications. The axiomatic concepts serve as a language to study concrete examples in broader sense and helps in developing logical thinking.

Prerequisites: MH1201 and MH2220.

###### MH3210Number Theory

4 AU

Introduction to basic number theory, including modern applications. Topics include: modular arithmetic; the Chinese remainder theorem, Fermat's little theorem, and Wilson's theorem; number-theoretic functions such as the τ, σ, Euler's φ-function; the Möbius inversion formula; applications to cryptography; primitive roots and indices; Legendre's symbols; the quadratic reciprocity law; continued fractions and Pell's equations; primality tests, factorization of integers, and the RSA cryptosystem.

Prerequisite: MH1300.

###### MH3300Graph Theory

4 AU

Topics in graph theory, including: connectivity and matchings, Hall's theorem, Menger's theorem, network flows; paths and cycles, complete subgraphs and Turán's theorem, and the Erdös-Stone theorem; graph colouring and the four-colour theorem; Ramsey theory; probabilistic methods in graph theory; and the use of software to solve graph-theoretic problems.

Prerequisites: MH1201 and MH1301 or MH2802.

###### MH3310Mathematical Foundations of Game Theory

4 AU | No longer offered

Introduction to game theory. Topics include: games of normal form and extensive form, and their applications in economics, relations between game theory and decision making; games of complete information: static games with finite or infinite strategy spaces, Nash equilibrium of pure and mixed strategy, dynamic games, backward induction solutions, information sets, subgame-perfect equilibrium, finitely and infinitely-repeated games; games of incomplete information: Bayesian equilibrium, first price sealed auction, second price sealed auction, and other auctions, dynamic Bayesian games, perfect Bayesian equilibrium, signaling games; cooperative games: bargaining theory, cores of n-person cooperative games, the Shapley value and its applications in voting, cost sharing, etc.

Prerequisite: MH2500.
Mutually exclusive with HE302 / HE3002.

###### MH3400Algorithms for the Real World

4 AU

Applications of algorithms. Topics include: mathematical concepts for analysis of algorithms; fundamental algorithm design techniques, with applications to various problems: network algorithms, matrix algorithms, optimization algorithms, and algorithms for data analysis and machine learning; and applications to problems in combinatorial optimization, networks, operations research, data analysis and machine learning.

Prerequisites: (MH1201, MH1301, MH1402, and MH2500) or (MH1201, MH1301, MH1403 and MH2500).

###### MH3500Statistics

4 AU

Further topics in statistics, including: random samples, sample mean and sample variance, distributions derived from the normal distribution, the central lmit theorem; parameter estimation and quality criteria for parameter estimators; the construction of good estimators; method of moments and maximum likelihood method; asymptotic properties of estimators, Cramer-Rao bound and efficient estimators; confidence intervals for estimators; hypothesis testing and Fisher-type tests; Neyman-Pearson tests and Neyman-Pearson Lemma.

Prerequisite: MH2500.

###### MH3510Regression Analysis

4 AU

Introduction to regression analysis, one of the most widely-used statistical techniques. Topics include: simple and multiple linear regression, nonlinear regression, analysis of residuals and model selection; one-way and two-way factorial experiments, random and fixed effects models.

Prerequistes: MH2500 and MH3500.

###### MH3511Data Analysis with Computer

3 AU

Data collection and analysis processes; graphical and numerical methods for describing data; summarizing bivariate data; probability and population distributions; estimation and hypothesis testing using a single sample; comparing two population or treatments; analysis of categorical data and goodness-of-fit tests.

Prerequisite: MH2500 or BS1008

###### MH3512Stochastic Processes

4 AU

Introduction to the theory of stochastic processes, including: gambling problems; random walks; discrete-time Markov chains; first step analysis (hitting probabilities and mean hitting times); classification of states; branching processes; and continuous-time Markov chains.

Prerequisite: MH2500.

###### MH3520Mathematics of Deep Learning

4 AU

This course investigates deep learning from the perspectives of several mathematical theories: numerical optimisation, statistical learning, function approximation, and coding theory. The aim is to shed some light on why and under what circumstances deep learning can be expected to work well - or not.

Prerequistes: MH2100 and MH3500 and MH3600 and PS0001.

###### MH3600Topology and Manifolds

4 AU

Detailed Course Information

Previously listed as "Knots & Surfaces: Introduction to Topology"

This is a first introduction to topology and calculus on manifolds. The tools introduced in this course are the natural framework for the generalization of the ideas that you learnt in Calculus I, II, and III to infinite-dimensional and non-Euclidean spaces. These methods open the door to other fields in mathematics like algebraic topology, functional analysis, differential/Riemannian/symplectic/Poisson geometry, or Lie theory, to name a few. They also have strong ties with important applications in the physical sciences and engineering like dynamical systems, mechanics, symmetry analysis, or control theory.

The aim of this course is to enable you to formulate and solve mathematical problems using the ideas and the formalism coming from topology and global analysis.

Prerequisites: MH1803 or MH2100

###### MH3700Numerical Analysis I

3 AU

Introduction to the theory and applications of numerical approximation techniques. Topics include: commonly used numerical algorithms; computational errors; numerical methods for solving systems of linear equations; iterative methods for systems of linear equations; polynomial interpolation; numerical integration; and numerical solutions of nonlinear equations.

Prerequisites: (MH1200 & MH1201) or (MH1800 & MH2800) or CY1602 or MH2802.

###### MH3701Basic Optimization

4 AU

Introduction to the mathematics of optimization. Topics include: geometric simplex method; algebraic simplex method in tabular form; revised simplex method; the network simplex method; linear programming duality; sensitivity and post-optimality analysis;and Lagrange duality and the Karush-Kuhn-Tucker conditions.

Prerequisite: MH1201 or MH2800 or MH2802.

###### MH4100Real Analysis II

4 AU

Detailed Course Information

Basic topology on the real line and extended real line; measurable sets and measurable functions; Lebesgue integration; differentiation, bounded variation, absolute continuity, and convex functions; classical Banach spaces.

Prerequisites: (MH2100 and MH3100) or (CY1602 and MH3100) or (MH1803 and MH3100).

###### MH4110Partial Differential Equations

4 AU

Advanced course on partial differential equations. Topics include: first-order equations, quasi-linear equations, general first-order equation for a function of two variables, Cauchy problem; wave equation, wave equation in two independent variables, Cauchy problem for hyperbolic equations in two independent variables; the heat equation, the weak maximum principle for parabolic equations, Cauchy problem for heat equation, regularity of solutions to heat equation; the Laplace equation, Green's formulas, harmonic functions, maximum principle for Laplace equation, Dirichlet problem, Green's function and Poisson's formula.

Prerequisite: (MH3100 and MH3110) or (MH1803 and MH3100). (MH4100 is useful but not required.)

###### MH4200Abstract Algebra II

4 AU

Unique factorization domains, Euclidean domains, principal ideal domains; modules, submodules, homomorphisms, quotient modules, modules over principal ideal domains; field extensions, automorphisms of fields, spilitting fields, normal and separable extensions; Galois extensions, Galois groups, Galois correspondence, and finite fields.

Prerequisites: MH1201 and MH3200.

###### MH4300Combinatorics

4 AU

Recursions and generating functions; partitions and tableaux; designs, Latin squares, combinatorial designs and projective geometries; extremal combinatorics, asymptotic analysis.

Prerequisite: (MH1101 and MH1201 and MH1301) or (MH1802 and MH1201 and MH1301).

###### MH4301Set Theory and Logic

4 AU

Partially-ordered sets, well-orderings and order-types, induction and recursion on ordinals, ordinal arithmetic, cardinals, cardinal arithmetic; the axiom of choice and its equivalences; axiom of determinacy; propositional calculus, truth tables, validity and contradictions; predicate calculus with equality, completeness and compactness theorems; the Löwenheim-Skolem theorem.

Prerequisites: MH1300 and MH1301.

###### MH4302Theory of Computing

4 AU

Models of computation and finitary representations. Topics include: formal languages and Chomsky's grammars; finite automata, regular expressions, regular grammars, and their equivalence; properties of regular languages: pumping lemma for regular languages and its applications; pushdown automata, context free languages and context free grammars; properties of context free languages: pumping lemma for context free languages; Turing machines: definition and construction for simple problems; the Church-Turing thesis and computability; uncountable numbers and the diagonalization argument; computably enumerable sets and Post’s problem; nondeterministic Turing machines and the classes P and NP; polynomial-time reductions and Cook’s Theorem; sSatisfiability and other NP-complete problems;and Co-NP space.

Prerequisites: MH1300 and MH1301 and (MH1402 or MH1403 or CZ001).

###### MH4310Coding Theory

4 AU

The definition of a linear code, its dimension and its length; generator matrix, parity check matrix and dual code; Hamming distance/weight; Hamming codes; perfect codes; Golay codes; maximum distance separable (MDS) codes; Reed-Mueller codes; BCH codes; Reed-Solomon codes; and bounds on code parameters.

Prerequisite: MH2200 and MH1301.

###### MH4311Cryptography

4 AU

Classical ciphers, cryptanalysis, linear complexity; the Data Encryption Standard (DES); the RSA cryptosystem, primality testing and factorization of integers; discrete logarithms; signatures; the Digital Signature Standard.

Prerequisites: MH1301 and MH2200.

###### MH4312Topics in Mathematics of Information and Communication

4 AU

Introduction to specialized advanced topics related to information theory, coding theory and cryptography. The choice of the topic depends on the instructor.

Prerequisite: division approval.

###### MH4320Computational Economics

4 AU

Topics included: strategic-form games and domination, Nash Equilibria and mixed strategies, evolutionary game theory, maxmin strategies, zero-sum games, extensive-form games, Zermelo's theorem, subgame-perfect equilibrium, games of incomplete information, single-item auctions, single-parameter environment, Myerson's lemma, VCG mechanisms and combinatorial auctions and revenue equivalence.

Prerequisite: (MH1200 and MH2500) or (MH2500 and MH2802).

###### MH4500Time Series Analysis

4 AU

Introduction to time series models and their applications in economics, engineering and finance. Topics include: trend fitting, autoregressive and moving average models, spectral analysis; seasonality, forecasting and estimation; and the use of computer package to analyze real data sets.

Prerequisites: MH2500, MH3500, and MH3510.

###### MH4501Multivariate Analysis

4 AU

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.

Prerequisites: MH2500, MH3500, and MH3510.

###### MH4510Statistical Learning and Data Mining

4 AU

Introduction to data analytics; optimal decision rules; K-nearest neighbors methods; linear models for regression; generalized linear models for classification; cross-validation and bootstrap methods; subset selection, ridge regression and lasso; artificial neural networks; classification and regression trees; ensemble methods; support vector machines; association analysis.

Prerequisites: MH2500, MH3500, MH3510, and MH3511.

###### MH4511Sampling and Survey

4 AU

Ratio and regression estimators under simple random sampling, separate and combined estimators for stratified random sampling; systematic sampling and its relationship with stratified and cluster sampling; further aspects of stratified sampling, cluster sampling with clusters of unequal sizes; subsampling; multi-stage sampling; complex sample designs.

Prerequisites: MH2500 and MH3500.

###### MH4512Clinical Trials

4 AU

Introduction to the design and analysis of clinical trials, with emphasis on the statistical aspects. Topics include: phases of clinical trials; objectives and endpoints, the study cohort, controls, randomization and blinding, sample size determination, treatment allocation; monitoring trial progress: compliance 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; good practice versus misconduct.

Prerequisites: MH3500 and MH3510.

###### MH4513Survival Analysis

4 AU

Introduction to survival 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.

Prerequisites: MH2500, MH3500, and MH3510.

###### MH4514Financial Mathematics

4 AU

Discrete-Time martingales; assets, portfolios, and arbitrage; discrete-time models; pricing in discrete time; hedging in discrete time; Brownian motion; stochastic calculus; the Black-Scholes equation; Martingale approach to pricing and hedging; estimation of volatility; basic numerical methods.

Prerequisites: MH2500 and MH3512.

###### MH4517Data Applications in Natural Sciences

4 AU

Detailed Course Information

Topological data analysis models, including simplicial complex, nerve theorem, homology, cohomology, filtration, persistent homology, Morse theory, Hodge-Laplacian, Reeb graph. Geometric data analysis models, including multidimensional scaling, isomap, diffusion map, spectral graph, manifold learning, differential forms. Geometry and topology based learning, including data representation, feature engineering, molecular/chemical descriptors, graph neural network.

Prerequisites: MH1402 or MH1403 or CZ2001.

###### MH4518Simulation Techniques in Finance

4 AU

Detailed Course Information

Generating random numbers and random variables, generating Brownian motion and other diffusion processes, variance reduction techniques, introduction to futures, options, and other derivatives, pricing exotic options with simulations, estimating sensitivities of derivatives with simulations, and applications in risk management

Prerequisites: MH2500 and MH3511 .

###### MH4600Algebraic Topology

4 AU

Detailed Course Information

Advanced course in algebraic topology. Topics covered include: point-set topology; homotopy; CW- or simplicial complexes; fundamental group; and homology groups.

Prerequisites: MH3200 and MH3600.

###### MH4601Differential Geometry

4 AU

Advanced course in differential geometry. Topics include: curves; surfaces and curvature; manifolds; differential forms.

Prerequisites: MH3100 and MH3600.

###### MH4700Numerical Analysis II

4 AU

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; matrix method and von Neumann method for stability analysis, Lax's equivalence theorem for convergence, method of characteristics; application to heat equation, wave equation and Poisson's equation.

Prerequisites: MH3700 and MH3110. (MH4110 is useful but not required.)

###### MH4701Mathematical Programming

4 AU

One-dimensional optimization: sectioning methods, Newton’s method; unconstrained optimization: optimality conditions, steepest descent method, Newton descent method; set-constrained optimization: optimality conditions, conditional gradient method; constrained optimization: Lagrange multiplier theory, Karush-Kuhn-Tucker theory, augmented Lagrangian method, and barrier method.

Prerequisites: (MH2100 and MH3701) or (MH1803 and MH3701).

###### MH4702Probabilistic Methods in OR

4 AU

Introduction to probabilistic methods used in operations research and statistics. Topics include: basic models of queueing, performance analysis, simulation of queueing systems; stochastic programming, modeling and algorithms for stochastic optimization, Markov decision process, and stochastic approximation.

Prerequisites: (MH2500 and MH3701 and MH3512).

###### MH4710Topics in Scientific Computing

4 AU

Specialized advanced topics in scientific computation and continuous applied mathematics. The choice of the topic depends on the instructor.

Prerequisite: division approval.

###### MH4711Mathematical Modeling in Imaging, Vision and Graphics

4 AU | No longer offered

Calculus of variations; convexity and differential geometry, level set method; phase field method; image denoising and deblurring; image segmentation; image inpainting and registration; curve reconstruction and smoothing; and surface reconstruction and smoothing.

Prerequisites: (MH2100 and MH3100) or (MH1803 and MH3100).

###### MH4720Logistics and Supply Chain Management

4 AU | No longer offered

Overview of supply chains: components of a supply chain, material and information flow, supplier-retailer-customer interaction, e-business; inventory and materials management: economic order quantity model, Lot sizing models, models with uncertain demands, MRP/JIT; facility location and transportation - single-source capacitated facility location, vehicle routing problems with equal, unequal demands and time-window constraints.

Prerequisite: MH2500 and MH3701.

###### MH4730Mathematics in Biology and Medicine

4 AU | No longer offered

Mathematical models and methods often used in bioinformatics, computational biology, and medicine. Topics include: model-based data clustering; maximum likelihood method; hidden Markov models; regression analysis.

Prerequisite: MH2500.

###### PS0001Introduction to Computational Thinking

3 AU

Detailed Course Information

This course aims to take students from having no prior experience of thinking in a computational manner to a point where they can derive simple algorithms and code the programs to solve some basic problems in mathematics and science in general. It will include topics to appreciate the internal operations of a processor and raise awareness of the socio-ethical issues arising from the pervasiveness of computing technology.

Mutually exclusive with CE1003 and CZ1003

###### PS0002Introduction to Data Science and Artificial Intelligence

3 AU

Detailed Course Information

This course aims to provide students with an understanding of basic techniques for data analysis, machine learning and dimension reduction for big data and expose them to hands-on computational tools that are fundamental for data science. Besides supervised and unsupervised learning, another fundamental technology of Artificial Intelligence - reinforcement learning will also be introduced, including Markov decision process and Q-learning. It will also show students how they could apply various methods to data examples and case studies from both research and industrial sources in the Singapore context.

Prerequisites: PS0001 or CZ1003

## Math Courses For Non-Math Majors

###### MH1810Mathematics 1

3 AU

Detailed Course Information

In this course, the basic concepts of limits, differentiation and integration are introduced. Applications of differential and integral calculus are included. In addition, the course also covers topics on complex numbers, vectors and matrices.

Mutually exclusive with MH2813, CE1011, CZ1011, MH1100, MH1101, MH1800 and MH1801

###### MH1811Mathematics 2

3 AU

Detailed Course Information

This course extends the basic concepts of differentiation and integration learned in Mathematics 1 to the operations on functions of multiple variables. Advanced applications of differential and integral calculus are included. In addition, the course covers topics on sequences, series and ordinary differential equations.

Prerequisite: MH1810
Mutually exclusive with MS2900, FE1007, MH1100MH1101 and MH1801

###### MH1812Discrete Mathematics

3 AU

Detailed Course Information

This course serves as an introduction to various topics in discrete mathematics. Topics included: number theory, logic, combinatorics and graph theory.

Mutually exclusive with CE1001, CZ1001 and MH1301

###### MH1820Introduction to Probability and Statistical Methods

3 AU

Detailed Course Information

This course provides a good foundation in probability and statistical inference. Basic ideas and methodologies in probability and statistics which are useful for economics students are introduced. It also aims to prepare students for higher level applied and theoretical econometric courses.

Prerequisite: HE1004
Mutually exclusive with MH2500, HE1005, MH1800, MH2814, MT2001, AB1202

###### MH2810Mathematics A

4 AU

Detailed Course Information

This course aims to provide a mathematical foundation to those who start the B. Eng programme directly from the 2nd year, and to ensure students have necessary mathematical capability for their study in all other courses in the subsequent semesters. Topics covered include vectors, functions and limits, differentiation, integration, sequences and power series, Taylor series, ordinary differential equations, partial differentiation and multiple integrals.

Mutually exclusive with EE2090 and EE2092

###### MH2811Mathematics II

3 AU

Detailed Course Information

This course prepares students for the solution and interpretation of practical problems encountered in engineering disciplines with emphasis given to strengthening problem-solving abilities. Topics covered include Fourier series, Fourier integrals, partial differentiation, chain rule for partial derivatives, double integrals, ordinary differential equations, partial differential equations, wave and heat equations and vector calculus.

Prerequisite: MH1810

###### MH2814Probability & Statistics

3 AU

Detailed Course Information

This course provides the basics of probability and statistical concepts in terms that are more easily understood by engineering students and presents probability and statistical concepts through problems that are meaningful to engineering science. This course should motivate the recognition of the significant roles of the relevance mathematical concepts in engineering.

Prerequisite: MH1810 or MT1001
Mutually exclusive with MT2001, CV2001, CV2018, HE1005, MH2500

## Projects and Internships

###### MH4900Final Year Project

8 AU

Semester-long research course on an advanced topic, under the supervision of a faculty member, leading to a research thesis. Must be taken over two consecutive semesters.

Prerequisite: division approval.
Mutually exclusive with MH4903.

###### MH4901Professional Attachment

8 AU | Pass/Fail | For students who matriculated in AY15/16 or earlier

12-week job placement for acquiring practical working experience and exposure to the workplace.

Prerequisite: division approval.

###### MH4903Professional Internship

11 AU | Pass/Fail | For students who matriculated in AY16/17 to AY18/19

22-week job placement for acquiring practical working experience and exposure to the workplace.

Prerequisite: division approval.
Mutually exclusive with MH4900 or MH4907.

###### PS4001Overseas Entrepreneurship Programme

10 AU | For students who matriculated in AY21/22 and after

Detailed Course Information

###### MH4905Overseas Entrepreneurship Programme

10 AU | For students who matriculated in AY19/20 and AY20/21

###### MH4906Overseas Entrepreneurship Programme

11 AU | For students who matriculated in AY16/17 to AY18/19

Detailed Course Information

6-month internship with a startup company, located in a major global innovation hub (remote internship can be arranged if there are travel restrictions).

The Overseas Entrepreneurship Programme (OEP) allows entrepreneurially inclined students to intern at a startup company, so as to experience the process and challenges that entrepreneurs face in building and growing their companies. The startup companies are located in major global innovation hubs such as Silicon Valley, New York, Shanghai and Berlin. Students can also opt to work for a startup company that is based in Singapore, but with ambitions to grow their business globally.

###### MH4907Professional Attachment

6 AU | Pass/Fail | For students who matriculated in AY16/17 till AY20/21.

12-week job placement for acquiring practical working experience and exposure to the workplace.

Prerequisite: division approval.
Mutually exclusive with MH4903.

###### MH4910Undergraduate Research Experience in Mathematical Sciences I

4 AU

Research on a specific mathematical topic, under the supervision of a faculty member.

Prerequisite: division approval.

###### MH4911Undergraduate Research Experience in Mathematical Sciences II

4 AU

Further research on a specific mathematical topic, under the supervision of a faculty member.

Prerequisite: division approval and MH4910.

###### MH4912Professional Internship

10 AU | Pass/Fail | For students who matriculated in AY19/20 and later.

20-week job placement for acquiring practical working experience and exposure to the workplace.

Prerequisite: division approval.
Mutually exclusive with MH4900 and MH4907 for students admitted in AY19/20 and AY20/21.
Mutually exclusive with MH4913 for students admitted in AY21/22 and later.

###### MH4913Professional Attachment

5 AU | Pass/Fail | For students who matriculated in AY21/22 and later.

10-week job placement for acquiring practical working experience and exposure to the workplace.

Prerequisite: division approval.
Mutually exclusive with MH2900, MH3900, MH4901MH4903, MH4906, MH4907, MH4912

## Supervised Study and Special Topics

###### MH4920Supervised Independent Study I

4 AU

Independent reading on a mathematical topic, under the supervision of a faculty member.

Prerequisite: division approval.

###### MH4921Supervised Independent Study II

4 AU

Further independent reading on a mathematical topic, under the supervision of a faculty member.

Prerequisite: division approval and MH4920.

###### MH4930Special Topics in Mathematics

4 AU

Advanced topics in mathematics, not normally covered in the regular courses. The choice of topics is determined by the instructor.

Prerequisite: division approval.

###### MH4931Special Topics in Applied Mathematics

4 AU

Advanced topics in mathematics, not normally covered in the regular courses. The choice of topics is determined by the instructor.

Prerequisite: division approval.

###### MH4932Special Topics in Statistics

4 AU

Advanced topics in statistics, not normally covered in the regular courses. The choice of topics is determined by the instructor.

Prerequisite: division approval.

## General Education Requirement (GER) Courses

###### MH5301Modern Cryptography: Real-World Applications and Impact

3 AU

Detailed Course Information

Previously listed as MH8301

Introduction to the key concepts of cryptography. Topics include: encryption algorithms; message authentication codes; authenticated encryptions; public-key schemes; random number generators; digital signatures; and hash functions.

Prerequisite: AO or H1 level Mathematics or equivalent.

###### MH8300It’s a Discreetly Discrete World: Mathematics in Real-life Applications

3 AU | No longer offered

Introduction to some simple and yet useful mathematics. Important applications of mathematics are discussed, which demonstrate the influence of mathematics on our everyday life. Topics include: coding theory — detecting and correcting errors in data, basic modular arithmetic used in the design of codes, basic issues in theory and applications, 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, and the RSA cryptosystem; graph theory — basic notions and algorithms, the travelling salesman problem, computational complexity, brute force methods, tour construction heurists, and applications; probability and statistics — examples, visualization, counterintuitive results, coincidences, paradoxes; searching for information on the web — applications of probability and linear algebra, especially eigenvalues, underlying search engines such as Google.

Prerequisite: AO or H1 level Mathematics or equivalent.

###### MH8500Tackling the Odds: Inside Statistics

3 AU | No longer offered

Overview of statistics and its applications in other disciplines, with emphasis on statistics methodology and how to evaluate statistical studies that students may encounter in some other courses, their future career, or everyday life. Topics include: measurement; visual displays; data descriptions; probability and risk; correlation and causality; statistical methodologies; statistical modeling.

Prerequisite: AO or H1 level Mathematics or equivalent.

## Special Problem-Solving Modules

###### MH5000Mathematical Problem-Solving

2 AU

Detailed Course Information

Previously listed as MH9000

A course about solving challenging non-standard problems from various areas, including calculus, linear algebra, algebra, differential equations, probability, discrete mathematics, etc., with the aim of developing creative thinking and exposition skills.

Prerequisites: (MH1100, MH1101, MH1200, MH1201 and MH1300) OR (CY1601, CY1602, MH1201, and MH1300) OR division approval.

###### MH5100Advanced Investigations in Calculus I

1 AU

Detailed Course Information

Previously listed as MH9100

A course where students are given challenging problems in calculus. Supplement to MH1100 for students who want to be challenged.

Prerequisites: division approval; must be read alongside MH1100.

###### MH5101Advanced Investigations in Calculus II

1 AU

Detailed Course Information

Previously listed as MH9101

A course where students are given challenging problems in calculus. Supplement to MH1101 for students who want to be challenged.

Prerequisites: division approval; must be read alongside MH1101.

###### MH5200Advanced Investigations in Linear Algebra I

1 AU

Detailed Course Information

Previously listed as MH9200

A course where students are given challenging problems in linear algebra. Supplement to MH1200 for students who want to be challenged.

Prerequisites: division approval; must be read alongside MH1200.

###### MH5201Advanced Investigations in Linear Algebra II

1 AU

Detailed Course Information

Previously listed as MH9201

A course where students are given challenging problems in linear algebra. Supplement to MH1201 for students who want to be challenged.

Prerequisites: division approval; must be read alongside MH1201.

###### MH9102Advanced Investigations in Calculus III

1 AU | No longer offered

A course where students are given challenging problems in calculus. Supplement to MH2100 for students who want to be challenged.

Prerequisites: division approval; must be read alongside MH2100.

###### MH9300Advanced Investigations in Discrete Mathematics

1 AU | No longer offered

A course where students are given challenging problems in discrete mathematics and number theory. Supplement to MH1301 for students who want to be challenged.

Prerequisites: division approval; must be read alongside MH1301.