3rd Year Courses

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.

Content: 

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

Content: A selection from the following topics:

• Exact and asymptotic enumeration of sums;
• Prime numbers and factoring;
• Basic techniques of enumeration, inclusion-exclusion, identities;
• Enumeration under symmetries;
• Continued fractions, arithmetical functions, sums of squares;
• Partitions of integers, q-series.

Content:

• Basic definitions (topological spaces, subspaces, closed sets);
• Basis for a topology;
• Closure, limit points and convergence;
• Continuous functions and homeomorphisms;
• Hausdorff condition and other separation axioms;
• Connectedness and path connectedness;
• Compactness.

Content:

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

Content:

• Classical Cryposystems;
• Ceasar and Affine ciphers;
• Block and Stream ciphers;
• One-time pads;
• Public Key Cryptosystems;
• The RSA cryptosystem;
• Digital signatures;
• Discrete Logs and the ElGamal Cryptosystem;
• Primality Testing and Factoring;
• Pseudorandom numbers;
• Error Detecting Codes.

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.

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.

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.