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MATH III

Course Coordinator: Dr Wen-Chi Kuo

Possible destinations / required for: further postgraduate study, mathematics teaching.

Points: 72 (at third-year level) | Prerequisite: MATH II

First semester courses:
  • Core: MATH3048A Real Analysis III
  • Core: MATH3006A Group Theory III
  • Choice: MATH3001A Number Theory III or MATH3010A Topology III
Second semester courses:
  • Core: MATH3004A Complex Analysis III
  • Choice: MATH3003A Coding and Cryptography III or MATH 3047A Advanced Real Analysis III
  • Choice: MATH3049A Positive Linear Systems III or MATH3009A Rings and Fields III

Course descriptions:

MATH3048A Real Analysis III

Content: This course includes the following topics:   Riemann sums;   refinements; Riemann integrals; metric spaces; completeness; open and closed sets; power series;  existence and uniqueness of solutions to ordinary differential equations; improper integrals and fixed point theorems.

MATH3006A Group Theory III

Content: 1. Groups and subgroups (revision), cyclic groups, quotient groups, direct product of groups; 2. Homomorphism and isomorphism theorems; 3. Group action, orbits, stabilizers, conjugacy; 4. Cauchy's theorem and the structure of p-groups; 5. Sylow theorems.

MATH3001A Number Theory III

Content: A selection from the following topics: 1. Exact and asymptotic enumeration of sums; 2. Prime numbers and factoring; 3. Basic techniques of enumeration, inclusion-exclusion, identities; 4. Enumeration under symmetries; 5. Continued fractions, arithmetical functions, sums of squares; 6. Partitions of integers, q-series.

MATH3010A Topology III

Content: 1. Basic definitions (topological spaces, subspaces, closed sets); 2. Basis for a topology; 3. Closure, limit points and convergence; 4. Continuous functions and homeomorphisms; 5. Hausdorff condition and other separation axioms; 6. Connectedness and path connectedness; 7. Compactness.

MATH3004A Complex Analysis III

Content: 1. Complex Differentiability, the Cauchy-Riemann Equations and Analytic Functions; 2. Functions Defined by Power Series; 3. Path Integrals in the Complex Domain; 4. The Index of a Closed Curve; 5. Cauchy's Integral Theorem, Cauchy's Integral Formula and Taylor Series; 6. Singularities and Laurent Series; 7. The Residue Theorem and Rouch'e's Theorem; 8. Evaluation of Integrals of Real Valued Functions via Complex Methods; 9. Open Mapping Theorem, Maximum Modulus Theorem, Schwarz's Lemma.

MATH3003A Coding and Cryptography III

Content: 1. Classical Cryposystems; 2. Ceasar and Affine ciphers; 3. Block and Stream ciphers; 4. One-time pads; 5. Public Key Cryptosystems; 6. The RSA cryptosystem; 7. Digital signatures; 8. Discrete Logs and the ElGamal Cryptosystem; 9. Primality Testing and Factoring; 10. Pseudorandom numbers; 11. Error Detecting Codes.

MATH 3047A Advanced Real Analysis III

Content: This course is a continuation of Real Analysis III (MATH3048A), further developing students' understanding of analytical properties of real functions and analysis of metric spaces. Topics include limit superior and limit inferior; applications to convergence of series; differentiability of functions of several variables; the   Implicit and   Inverse Function Theorems; completeness and compactness in metric spaces; uniform convergence in metric spaces; Fourier Series and the Weierstrass Approximation Theorem; and an introduction to  Lebesgue integration.

MATH3009A Rings and Fields III

Content: Topics include rings, subrings, ideals, factor rings, homomorphisms, integral domains, Euclidean domains, principal ideal domains, unique factorisation domains, Eiseinstein's criterion, Gauss' Lemma and field extensions.

MATH3049A Positive Linear Systems III

Content: Topics covered include: Equilibrium in linear economic models; Hawkings-Simon condition; outputs and prices; profit rate; matrices and linear mappings; convergence in $\R^n$; irreducible matrices; product planning in activity analysis; convex sets; and Koopman's postulates.

EXAMS 

Final exams for Math III courses are written in June and November. Each exam will be one and a half hours long and will count 50 percent of the final mark per topic. Failure to write the exam will earn a zero mark for the student unless a valid medical or other certificate excusing the student is produced. In these cases, a deferred exam, either written or oral, will be set.

PLEASE NOTE THAT DEFERRED EXAMS ARE NOT GIVEN AUTOMATICALLY, BUT AT THE DISCRETION OF THE MATHEMATICS DEPARTMENT AND / OR THE DEAN.

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