On successful completion of this module, students will be able to
Construct reasoned logical arguments to identify and justify
basic properties of mathematical objects that are specified
as sets, relations on sets, functions between sets,
Identify formal languages generated by simple context-free
grammars, and construct specifications of context-free grammars
and finite state machines that generate and/or determine formal
languages, given specifications of such formal languages.
Recognize and identify properties of undirected graphs
that are networks consisting of vertices together with edges
joining pairs of vertices, and find examples of isomorphisms
between such graphs satisfying given criteria.
Find solutions to certain types of homogeneous and
inhomogeneous linear ordinary differential equations
of degree at least two, using methods based on the
use of power series, and also methods based on the
identification of particular integrals and complementary
functions, where the coefficients of the differential equation
are constants and the forcing function is typically constructed
from polynomial, exponential and trigonometric functions.
Expound and apply basic properties of exponential and
trigonometric functions, where the arguments of those functions
are complex numbers and variables, and thereby obtain results
that are relevant to the basic implementation of the
Discrete Fourier Transform.
Perform calculations within the algebra of vectors
in three-dimensional space, and the algebra of quaternions,
and apply the results of such calculations to the solution
of simple geometrical problems.
Perform calculations in basic number theory, justified
on the basis of theorems explicitly presented and proved
within the module, that have relevance to the implementation
of public key cryptographic systems such as the
Rivest-Shamir-Adelman (RSA) public key cryptosystem.
Specific topics addressed in this module include the following:
The Principle of Mathematical Induction
Sets, Relations and Functions
Introduction to Abstract Algebra
Introduction to Formal Languages and Context-Free Grammars
Introduction to Graph Theory
Ordinary Differential Equations
Trigonometric Identities, Complex Exponentials and Periodic Sequences
Module CS1001 (Mathematics I), or an equivalent module
developing the necessary mathematical skills in areas such
as calculus and linear algebra.
This module will be examined in a 3 hour examination in Trinity
term. Also students should complete a small number of assignments during
the academic year. The final grade at the annual examination session will
be a weighted average over the examination mark (90%) and the continuous
assessment mark (10%). The final grade at the supplemental examination
session will be wholly determined by the supplemental examination paper.