MASTER OF SCIENCE (MATHEMATICS)

Study Duration
Minimum 2 semesters (1 year)
Maximum 4 semesters (2 years)

Intake
Intake – 2 times a year (March & October)
*subjected to UKM academic calendar

STPD6024 Research Methodology
This course provides guidance in planning, implementing and succeed in scientific research. Students are introduced to the philosophy of science and ethics necessary to be adopted by researchers. Students are given exposure to determine and manage risks in scientific research. Apart from that, issues and rules related to research such as intellectual property, copyright, plagiarism etc. will be discussed.  Subsequently, students are guided to plan their research and prepare a research proposal. For this, students are trained with techniques in information search both manually and on-line, identifying issues and research objectives, planning research and experimental design within their period of study. Students are required to prepare their research proposal according to the format and free from plagiarism. Students are given the opportunity to present and defend their proposal in a seminar. Students will be evaluated based on written and oral presentation of the research proposal, and final examination

STQM6024 Biomechanics
The course begins with the general equations of three-dimensional motion. The variables in these equations are explained and a summary on how these variables can be determined experimentally or theoretically is presented. The course will discuss selected methods for quantifying biomechanical data experimentally, which includes discussion on force measurements, accelerometry, measurement of motion with optical methods, electromyography, and strain measurement. The goal is to explain the principles involved in the experimental techniques, and to compare the different techniques. Next the course will discuss the place of mathematical modeling in biomechanics. The concept of force system analysis is discussed. The course will look into various mathematical models that were determinate. The indeterminate systems are also discussed as well, including solving it using optimization theory. Lastly the course discusses simulation as a tool of biomechanical research.

STQM6034 Decision and Game Analysis
This course aims at showing students that decision problems with limited number of alternatives can be solved by using decision analysis techniques. Instruments that are used to solve these problems depend on the type of problems. Analytic Hierarchy Process, ELECTRE and TOPSIS methods are used to solve problems with certainty.  Decision tree is the best instrument to obtain solution for problems which involve probability. For decision problems with uncertainty, criteria that reflect decision maker’s attitude towards risks are used, while game theory is used to obtain the best decision for one competitors with contradicting goals, under each competitor’s worst condition.

STQM6044 Cryptology
This course serves as an introduction to some of the important topics of cryptology, which is the scientific study of hiding and breaking secret data. Students will be given an overview to the descriptions, theoretical bases, and rigorous protocols of security. This covers the theoretical foundation and mechanisms of symmetric encryption, public key encryption, digital signature, hash functions, message authentication codes, and some advanced topics in cryptology. For each of the cryptology topics discussed, its most common implementation will also be examined and presented for a better understanding. In addition, selected cryptanalysis will be introduced during the course.

STQM6064 Mathematical Modeling and Methods
Mathematical modeling is a process of building mathematical formulation for a physical phenomenon to gain better insights about it. The course intends to train the students in building, analyzing and solving mathematical models for certain complex problems (especially deterministic models of physics). Fundamental concepts of mathematical modeling will be explained. Dimensional procedures, approximation and dimensional analysis will be discussed first. Models introduced are linear and nonlinear. Analytical solution methods discussed include some of the followings: perturbation expansion technique, asymptotic method, transformations, special functions, Fourier series, calculus of variations and integral methods. The usage of computer algebra systems like Maple/Mathematica will be emphasized.

STQM6074 Numerical Analysis
This course covers numerical methods for solving ordinary/partial differential equations (ODEs/PDEs). The problems considered include initial value problems and boundary value problems for ODEs. Numerical methods discussed include one-step and multi-step methods with fixed or variable stepsize for stiff and non-stiff as well as chaotic equations/system of equations. Further, the topics covered include stability and error analysis. Introduction to numerical methods for PDEs such as finite difference/element methods. Analysis of hyperbolic and elliptic equations. Convergence, consistency, order and stability of methods. Applications to certain problems in engineering/science.

STQM6084 Linear Programming
The aim of this course is to introduce to the students the application of mathematical modeling methods for managerial decision makings. Several deterministic models will be discussed with a focus on how to model problems and derive solutions using LINGO computer software. Assignments in the form of case studies require students to use LINGO to obtain the solutions and write short reports. Among the topics discussed include linear programming, integer programming and goal programming problems, while emphasizing on the use of these methods in solving real world problems such as problems in network analysis, transportation and assignment problems, travelling salesman problems and efficiency analysis.

STQM6114 Topology
Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of the great unifying ideas of mathematics. This course concerns with properties that are preserved under continuous deformations of objects that emerges through the development of concepts from geometry and set theory. The most basic and traditional division of topology namely point set topology will be considered.

STQM6124 Algebra
This course begins  by reviewing back one main algebraic structures that is groups and rings together with some concepts related to both. This includes subgroup/subring, ideal, quotient group/ring and mapping of group/ring. Then the extension to polynomial ring and field is introduced. Various extensions of field to domain such as integral domain, Euclidean domain and unique factorisation domain. The embeddedment of domain into field leads to the construction of Galois group. This course ends with reviewing several selected articles on algebraic structures.

STQM6134 Functional Analysis
This course generalizes the study of linear algebra, in particular on finite and infinite dimensional vector spaces. This study is supported by various limit-related structures such as metric, inner product, norm and topology. Then it is added together with linear operators acting upon these spaces. The combination of algebraic and limit-related from new spaces namely Banach and Hilbert space. Thus, this course is basically the study of the properties of these spaces.

STQM6154 Network Science
This course introduces mathematical theories in network science. Network science is a multidiscipline field which investigate problems that can be understood through network approach. Among the aims of network science are to find cross-network equations and increase understanding of systems which are represented by networks through data analysis. The use of network science can be found in mathematics, social networks, biological systems and transportations.

STQM6214 Fuzzy Mathematics
This course introduces fuzzy set as a generalization of classical set. Basic operations on fuzzy sets: fuzzy complement, fuzzy union and fuzzy intersection. s-norm and t-norm. averaging operators. Fuzzy relations. Projection and cylindric extension. Composition of fuzzy relations. Linguistic variables. Fuzzy IF-THEN rule. Interpretation of fuzzy IF-THEN rule. Fuzzy logic and approximate reasoning. Fuzzy rule base and fuzzy inference engine. Fuzzy systems and fuzzy theory.

STQM6224 Complex Analysis
This course gives a view of basic analytic functions such as power series representations, Cauchy-Goursat’s theorem with various versions, maximum modulus theorem with various versions, conformal mappings, Riemann mapping theorem and Phragmen-Lindelof’s theorem. This course also introduces a theorem in analytic function space and shows the application of Runge’s theorem and Mittag-Leffler’s theorem. Harmonic functions including solutions to Dirichlet’s problem, singularities, Picard’s theorem and special functions such as the gamma functions, zeta functions and important theorems for  entire functions are also introduced.

STQM6234 Ergodic Theory
Ergodic theory is a quantitative study of the long term behavior of a system. The collection of all states of a system constitute a space X and the evolution of the system is represented by a transformation whereby if x represents the state of the system at one particular time, then T(x) represents the state of the system after one unit of time. A study will be conducted on X which is a measure space (and T measure-preserving) and topological space (with T continuous). The main objective is to understand  as Tn increases.

STQM6254 Combinatorial Group Theory
The course aims to display geometrical techniques and ideas to study free groups and group presentations, basic to the combinatorial group theory. Variety of equivalent classes will be discussed and related groups will be constructed using various geometrical techniques. Geometrical techniques discussed include graph, group of graph, complexes, picture and diagram. Then several selected and latest articles will be discussed.

STQM6274 Measure Theory and Integration
Firstly the idea of  σ-algebra is introduced. Next a measure is defined as a real valued function with domain σ-algebra. Next outer measures is discussed (including metric outer measures) and with it is defined measurable sets, abstract measure spaces, measurable functions and convergence measurable functions. Using these concepts, properties of integrable functions, convergence theorems,  spaces and important inequalities are discussed. Examples of definite integrals like Riemann integral, Lebesgue integrals and Lebesgue-Stieltjes integrals will be briefly discussed. Finally the concept of measures is extended to sign measures, complex valued measures and integration on product spaces.

STQM6294 History and Philosophy of Mathematics
This course will deepen the history and philosophy of mathematics in various aspects. Topics to be discussed include mathematical reality from metaphysical, epistemological, logical and axiological aspect will be examined together with the relationship between mathematics and belief systems. History and philosophy of the Islamization of knowledge and the indigenization of knowledge will also be discussed.

STQM6324 Numerical Methods for Heat Transfer and Fluid Flow
This course will present heat transfer and fluid flow models and their numerical solutions. The course begins with heat transfer and fluid flow model formulations. Steady and unsteady heat conduction up to three dimensions will be discussed. Next, the course discusses Crank-Nicholson method, steady and unsteady convection and diffusion up to three dimensions, and their numerical solution schemes include hybrid and power laws. Flow regimes and numerical solution methods will also be presented.

STQM6414 Dynamical Systems
The course aims to introduce basic concepts in discrete-time and continuous-time dynamical systems. These include discussion on some topics such as locally property, stability comprises structural stability, hyperbolic and homoclinic point, strange attractor, Lyapunov exponent etc. Some other concepts such as bifurcation, chaos and fractal will also be explored.

STQM6524 Linear Modeling of Non-Deterministic Dynamical System
This course is designed to exhibit the capability to model the dynamical system with non-deterministic condition as a stochastic process which fulfills the linear stochastic differential equations. It, furthermore can lead to the stochastic integral. This includes various Newtonian dynamical systems with noise, planning of monitoring system, management and screening of information. From this model, the definition of the concept of stochastic process is exhibited and in addition the analytical and numerical Ito’s Stochastic Calculus is constructed to solve the mentioned model. The relationship between the stochastic differential equations and the diffusion process is discusses to the research boundary.

STQM6534 Fluid Mechanics
The aim of this course is to show how the ideal and viscous fluids can be modelled mathematically, and further, to investigate the behaviour of the fluids analytically and numerically, especially towards the Navier-Stokes equation. This course starts with general introduction to fluid and the principle of fluid static and kinematic. Discussion on ideal fluid includes continuity, Euler and Bernoulli equations. Potential flow and incompressible flow will also be discussed. Most parts of this course discuss viscous fluid, which leads to Navier-Stokes equation, its derivations and exact solutions, as well as steady and unsteady flows. Basic flows, Stokes flow, laminar and turbulent flows, dimensional analysis, similarity method as well as Reynolds number and its importance will also be discussed. In addition, boundary layer theory and fluid instabilities will also be discussed in detail.

STQM6624 Simulation
This course introduces the students to the concepts and usage of simulation systems. It aims to enable the students to quickly perform modelling, simulation and analysis on simple but representative systems, as well encourages the students to further investigates the results experimentally.  Simulation model; static, discrete and dynamics systems are discussed.  Topics include system concept, modelling, simulation and analysis of various systems especially those that are related to Excel, @Risk and Arena. The science of managing simulation projects are discussed.

STQM6988 Research Project
Research project is a compulsory course, which is either a practical training, an  industrial training, a literature review or a research. Every student does this project under the supervision of a supervisor. Each student must choose a suitable topic within his/her programme module and it must be approved by the supervisor. The student must complete a report, which is either a critical review to the selected topic, a new theory or a new model in its own way

• Bachelor’s Degree in relevant field with minimum CGPA 2.50 or equivalent, from any institutions of higher learning  recognized by the UKM Senate; or
• Bachelor’s Degree in relevant field with minimum CGPA 2.00 – 2.49 or equivalent, with minimum of 5 years working experience or research project in relevant field. Proof of the working experience of a foreign candidate should be acknowledge by the embassy of the respective country; or
• Fulfill Accreditation of Prior Experiential Learning (APEL A) for local candidates only:
  • should be more than 30 years of age in the year of application; and
  • possess at least STPM or diploma in relevant field or other equivalent qualification recognized by the Government of Malaysia and approved by the UKM Senate; and
  • possess a certificate of MQA APEL with MQF Level 7
• An international student shall obtain minimum results of TOEFL iBT score 46 / IELTS band 5.5 / PTE score 51 / CEFR band B2 / MUET band 4 (An international student who comes from a country where English is the official language, or who has obtained academic qualifications from any institution of higher learning that uses English as the medium of instruction may be exempted from the requirement for TOEFL / IELTS / MUET); and
• Fulfill other requirements prescribed by the UKM Senate from time to time.

1. Mathematician 
2. Actuary
3. Market Research Analyst
4. Data Scientist
5. Risk Analyst