Courses of Master Program

Grade One

 

MMA602  ALGEBRA(1)

@Grade one  required  (3-0)

@FieldsGalgebraic extensions, splitting fields, normal extensions, finite fields. Galois TheoryGautomorphisms of field extensions, Galois extensions, fundamental theorem of Galois theory, solvability of polynomials by radicals, Galois groups of polynomials with degree up through 4, insolvability of the quintic.

@GroupsGpermutation groups, alternating groups, simple group, finite groups, the Sylow theorems.

 

MMA627  ALGEBRA (2)

@Grade one  elective  (0-3)

@The structure of modulesGfree modules, semisimple modules, projective modules, injective modules.

@The structure of ringsGlocalization, Hilbert basis theorem, Hilbert Nullstellensatz, matrix rings, simple rings, prime rings, semiprime rings, Wedderburn-Artin theorem.

 

MMA601  ANALYSIS (1)

@Grade one  required  (3-0)

@Lebesgue measureBouter measureBmeasurable functionsBLesgue integralBLebesgues dominated convergence theoremBmonotone convergence theorem and uniform convergence theorem.

@PrerequisiteGAdvanced Calculus.

 

MMA626  ANALYSIS (2)

@Grade one  elective  (0-3)

@Fubinis theoremBLebesgues differentiation theoremBL^p spacesBJordan decompositionB Hahn decomposition and Radon-Nikodym theorem.

@PrerequisiteGAnalysis(1)

 

MMA621  PROBABILITY THEORY

@Grade one  elective  (3-3)

@In this course, measure theory is used to discuss probability theory. Main contents are basic measure theory, random variables, distribution function, characteristic functions, integration, conditioning, law of large number, law of iterated logarithm, central limit theorem and other frequently used convergence theorems.

 

MMA624/MMA625  TOPICS IN MATHEMATICAL BIOLOGY(1)(2)

@Grade one  elective  (3-0)(0-3)

@Introduction of various biological models and their mathematical structures. Study of selected papers.

@PrerequisiteGAdvanced CalculusBOrdinary Differential EquationsBPartial Differential Equations.

 

MMA622  MATRIX THEORY

@Grade one  elective  (3-3)

@This course covers classical and modern results in matrix theory, and its important applications in different areas. Topics contain eigenvalues, decomposition, canonical form, norms and positive-definiteness.

 

CODING THEORY

@Grade one  elective  (3-3)

@Coset decoding, perfect codes, Hamming codes, Golay codes, a double-error-correcting BCH code, cyclic codes, weight enumerators, and existence of extremal self-dual codes.

 

MMA630  PARTIAL DIFFERENCE EQUATIONS

@Grade one  elective  (3-3)

@First order partial differential equations, Cauchy-Kowalevski theorem and characteristics, Classification of second order linear differential operators, LaplaceBheat and wave equations.

@PrerequisiteGAdvanced CalculusBOrdinary Differential Equations.

 

MMA629  GAME THEORY

@Grade one  elective  (3-3)

@Definition of A Game, Two-Person Zero-Sum Games, Infinite Games, Infinite Games, Multistage Games.

@Two-Person General Sum Game, Two-Person Cooperative Games, n-Person Games, Indices of Power.

 

MMA637  RELIABILITY THEORY

@Grade one  elective  (3-3)

@Reading the most recent results on Coherent System, Importance of Coherent System.

@Reading the most recent results on NBU, NBUE Distribution Functions, Multi-State Coherent Systems.

 

MMA632  ORDINARY DIFFERENTIAL EQUATIONS

@Grad one  elective  (3-3)

@Existence and uniqueness theorems, dependence of solutions on parameters, Linear systems, autonomous systems, stability of equilibria, Lyapunov function, Poincaré-Bendixon theorem, perturbation theory, boundary value problems.

 

MMA730/MMA635  EQUILIBRIUM ANALYSIS(1)B(2)

@Grade one  elective  (3-0) (0-3)

@Fixed point theorems with applications to economics and game theory. The main content includes :Sperner's Lemma,Continuity of correspondences, Fan-Browder theorem, Walrasian equilibrium, and so on.

 

MMA633  GRAPH THEORY

@Grade one  elective (3-3)

@Topics of Graph theory contain the basic definitions of graph and introductions to the properties of connectivity, Eulerian, Hamiltonian graph, planar graph, graph coloring, graph decompositions and Ramsey theory.

 

MMA639/ MMA640  HISTORY of MATHEMATICS (1)(2)

@Grade one  elective  (2-0)(0-2)

@The development of meditation originated from the three mathematics events: discovery of irrational numbers, definition of limit, paradox in set theory

 

Grad Two

 

MMA723/MMA724  MATRIX ANALYSIS (1)(2)

@Grad two  elective  (2-0)(0-2)

@Interesting topics include combinatorial matrix theory, sign pattern matrices and numerical ranges. Related research papers will be studied.

 

MMA721  ADVANCED GAME THEORY

@Grade two  elective  (3-3)

@Reading the most recent papers on Cooperative Games, Multi-Choice Cooperative Games.

@Reading the most recent papers on Multi-Choice Shapley Value, Continuously-Many-Choice Cooperative Games.

 

MMA725  INTRODUCTION TO STOCHASTIC PROCESSES AND APPLICATIONS

@Grade two  elective  (3-3)

@In this course, basic theories of stochastics processes, including Markov chain, Markov processes and martingales, and discussed. Stochastic integral, stochastic differential equations and their applications are also discussed.

 

MMA739  TOPICS IN CODING THEORY

@Grade two  elective  (3-3)

@Codes and designs, hand decoding of Golay codes, duadic codes, self-dual codes, Reed-Muller codes, and covering radius.

 

MMA731  DIFFERENTIAL TOPOLOGY(1)

@Grade two  elective  (3-0)

@We discuss differential structures on manifolds.  The tools include fibre bundle and characteristic classes, for instance, tangent bundle and Euler class.  We also discuss maps between manifolds, their homotopy approximation, and degrees.

 

MMA732  DIFFERENTIAL TOPOLOGY (2)

@Grade two  elective  (0-3)

@We discuss variational problems and Morse theory.  These lead to the computation of homotopy and homolgy groups.  The application of the theory is the complete classification of compact surfaces.

 

MMA727  FUNCTIONAL ANALYSIS

@Grade two  elective  (3-3)

@Normed Space, Banach Space, Inner Product Space, Hilbert Space, and related Theorems such as Open Mapping Theorem, Closed Graph Theorem.

Banach Fixed Point Theorem, Spectral, Resolvent of Linear Operator, Compact Operator.

 

MMA631  TOPICS IN MATHEMATICAL ECONOMICS]Original: Mathematical Economics^

@Grade two  elective  (3-3)

@There are two kinds of contents in this program. One is microeconomic: supply, demand and equilibrium, factories and markers, consumer behavior, general equilibrium K. The other we read some papers about static equilibrium.

 

MMA740  COMBINATORIAL DESIGN

@Grade two  elective (3-3)

@Topics include Block designs, Orthogonal Latin squares, Symmetric designs, Steiner systems and Tournament designs.