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Degree Programme in Financial Mathematics and Economics
Course Descriptions
First
Year
MA180 Mathematics
MA181 Analysis:
Functions
of one real variable: Construction of R. countable and uncountable sets.
Sequences and limits, limits of sums and products, the geometric and Dirichlet
series, continuous and discontinuous functions, the intermediate value
theorem, inverse functions. Mean-value theorem, Taylor?s theorem. Sequences
and Series, tests, power series, products of series, exponential and logarithmic
functions. Riemann Integration.
MA183 Algebra:
Elementary
number theory: primes, factorisation, the division algorithm, greatest
common divisors, arithmetic modulo m, solution of congruences, applications.
Transformations of the plane: linear transformations, isometrics, symmetries.
Introduction to groups, rings, fields: examples from earlier topics, cyclic
groups, Lagrange?s theorem, polynomials. Basic matrix algebra: multiplication,
adjoint, determinant, inverse, eigenvalues and eigenvectors.
EC101/2 Economics
Applied
Economics:
The aim in applied economics
is to discuss how the tools of economic theory can be used to analyse various
issues and problems in everyday economic life. The material covered includes
basic economic concepts, and topics related to microeconomic and macroeconomic
issues.
Microeconomics: This course is an introduction to the principles
of microeconomics. It studies the decisions of individual households and
firms. It also analyses how individual markets operate.
Macroeconomics:
This
is an introductory course in macroeconomics. The interrelationship of the
various actors in the aggregate economy is studied. Also, the derivation
of certain national aggregates is studied using simple models of the macroeconomy.
CS103 Computer Science
Introduction
to programming: Programming in a high level
language (such as C), algorithms, variables, expressions, syntax, implementation
of programs on machines, loops, procedures, function, modular programming,
recursion, introduction to systems software, compilers, batch and on-line
processing modes.
MA110 Statistics &
Probability
Explanation
of statistics through practical examples of its applications.
Data summarisation
and presentation: Numerical measures of location and spread for both
ungrouped and grouped data; graphical methods including histograms, stem-and-leaf
and box plots.
Probability:
The
role of probability theory in modelling random phenomena and in statistical
decision making; sample spaces and events; some basic probability formulae;
conditional probability and independence; Bayes formula; counting techniques;
discrete and continuous random variables; hypergeometric and binomial distributions;
normal distributions; the distribution of the sample mean when sampling
from a normal distribution; the Central Limit Theorem with applications
including normal approximations to binomial distributions.
Statistical Inference:
Concepts
of point and interval estimation; concepts in hypothesis testing including
Type I and Type II errors and power; confidence intervals and hypothesis
tests about a single population mean, a single population proportion, the
difference between two population means, a single population variance and
the ratio of two population variances; the analysis of enumerative data,
including chi-squared goodness-of-fit and contingency table tests; correlation
and linear regression analysis, including least squares estimation of the
parameters of the simple linear regression model, inferences about these
parameters, and prediction.
MA111 Mathematics of Finance I
Simple and
compound interest, annuities certain and variable, perpetuities, amortisation
schedules, sinking funds.
MP191 Mathematical
Methods I
Linear differential
and difference equations. Applications to the modelling of population,
economic, mechanical and financial systems. Introduction to the phase plane
method.
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Second
Year
Semester I
MA286 Functions
of Several Real Variables
Revision of
the main definitions and properties of sequences and series of real numbers.
Lim inf and lim sup, power series, differentiation term by term, Abel?s
limit theorem, Taylor series. Continuity and differentiability of a function
f
: Rm®
Rn,
partial derivatives, directional derivatives, the Chain rule, the mean-value
theorems, maxima and minima, the inverse function theorem.
MA284 Discrete Mathematics
Enumeration:
product rule, sum rule and sieve principle, selections and distributions,
pigeonhole principle. Graphs, the fundamentals (including various notions
of ?path? and ?tree?), plus a study of some of the following topics: colouring
problems, bipartite graphs, Hamiltonian graphs, planar graphs and tournaments.
Algorithms and applications are emphasised throughout.
EC215 Microeconomics:
Review of
microeconomic aspects of the first year course in introductory economics
and in greater detail the following: Demand analysis: Individual consumer
behaviour, market demand, cross demand, elasticity. Utility approach: indifference
curve analysis. Production functions, cost of production, isoquants, applications
of supply and demand analysis. Market structure: purely competitive market,
market equilibrium, the theory of the firm, monopoly pricing and output
decisions under monopoly and under perfect competition, imperfect markets,
monopolistic competition. Income distribution: factor markets and determination
of factor prices. General equilibrium. Welfare economics.
MA227 Probability
Probability
spaces; random variables and vectors, their distributions and moments;
functions of random variables; sampling distributions; limit theorems.
CS204 Algorithms
Theory
of computation: Turing machines, complexity, computability, decidability.
Design
and analysis of algorithms: set operations, tables, stacks, queues,
trees, searching and sorting, file organisation.
MP291 Mathematical
Methods II
Ordinary differential
equations including the Laplace transform. Phase plane analysis. Linear
stability theory. Fourier series. Introduction to partial differential
equations. Optimisation with constraints including the Lagrange multiplier
method.
Semester II
MA287 Functions
of One Complex Variable
Functions
of a complex variable: differentiability, the Cauchy-Riemann equations,
harmonic conjugates, line integrals, log z and ez,
Cauchy?s integral theorem, Cauchy?s formula. Cauchy?s inequalities, the
Laurent series of a function, poles, residues, contour integration, Rouche?s
theorem.
MA283 Linear
Algebra
Vector spaces,
bases, dimension, linear maps, matrix representation of linear maps, matrix
algebra, kernels and images, least squares fitting, inner product spaces,
the Gram-Schmidt process, Fourier series, dual spaces, the rank of a matrix,
determinants, eigenvalues and eigenvectors, the characteristic polynomial,
quadratic forms, diagonalisation of a symmetric or Hermitian liner map,
triangularisation of a linear map, the Hamilton-Cayley theorem, linear
programming.
EC217 Macroeconomics
Basic concepts
of national income. Output and expenditure. Aggregate supply and demand.
Income and output. Equilibrium and disequilibrium. Saving-investment relationship.
Consumption function. Factors influencing consumption demand. The multiplier.
The determinants of investment. Liquidity preference and theory of interest.
The principle of acceleration. The Government sector and National income
and output. Foreign trade and the national income. Terms of trade. Balance
of payments. Exchange rates. Incomes, output, employment, prices. The classical
theory. Keynesian theories. Money. Supply and demand. General price level.
Index numbers. The inflationary process. Economic growth, investment and
employment. Cyclical fluctuations. Monetary and fiscal policies.
EC218 Mathematical
Economics
Applications
of mathematical methods in constructing and analysing economic models,
with an emphasis on methods of constrained optimisation. Topics may include
comparative static analysis, economic dynamics and game-theoretic methods
in economics.
MA228 Statistical Inference
Concepts and
criteria in point and interval estimation and in hypothesis testing; applications
to one- and two-sample problems involving quantitative variables, enumerative
data analysis, and regression.
CS205 Languages and
Operating Systems
Operating
Systems: Introduction to VMS, UNIX, MSDOS. Database management systems:
architecture of DBMS, data sublanguages, commercially available DBMSs.
Study of programming
languages.
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Third
Year
Semester I
MA341 Metric
Spaces
Metric
spaces: continuity, convergence, compactness, sequential compactness,
the Lebesque number of a covering completeness, Cantor's theorem, Baire's
theorem.
MA343 Groups I
Group axioms,
cyclic groups, permutation groups. Normal subgroups, homomorphisms, isomorphism
theorems. Direct products, finite Abelian groups. Automorphisms, groups
of automorphisms.
EC302 Advanced Microeconomics
This course
builds on the foundation of the second year course in microeconomics. A
series of topics will be analysed of a more technical nature than previously
encountered. Students will be introduced to some of the latest development
in microeconomic theory.
EC311 Economics of
Financial Markets
The objective
of the course is to appraise students of the concepts of financial markets
(bond, equity and foreign exchange) and financial instruments as well as
interest rate determination for various financial instruments. In addition,
economic theories of short term and long term interest rate determination
will be covered.
MA313 Applied Statistics
Sampling techniques;
various designs and analyses for survey data; applied regression analyses;
simple linear regression, multiple linear regression, diagnostics and remedial
measures for violations of assumptions; experimental design and analysis
of variance; principles of experimental designs, completely randomised,
randomised block and Latin square designs and their analyses; use of computer
software.
MA311 Annuities and
Life Insurance
Elements of
probability and applications to life contingencies; force of mortality;
construction of mortality tables; types of life annuities and life insurance;
commutation functions; net premiums.
MP391 Mathematical
Modelling
A selection
of diverse areas are modelled mathematically and computationally, including
population models, network maximal flow/minimal cost models, diffusion
processes, linear and non-linear programming, queuing theory via Markov
processes, random number generators and Monte Carlo methods, Steiner tree
problems, continuous and discrete applications to financial and economic
models.
MG Industrial Management
------ (Semester II also)
Management
and organisation theory. Marketing, personnel and financial functions,
financial aspects of production, statement of account, working capital,
cost accounting, budgets, budgetary control, standard costs, variance analysis,
performance analysis, financial decision making, management and modelling,
industrial relations. The cultural context of industrial relations, 19th
century developments, the 1906 Trade Disputes Acts, picketing, trade union
organisation, the ICTU, employer organisations, the Labour Court and its
functioning, the Employer Labour Conference, Rights Commissioners, substantive
and procedural agreements, unofficial action, inter-union disputes, recognition,
the law and industrial relations, diagnosis and proposals of the Commission
of Inquiry on Industrial Relations.
Semester II
MA342 Topology
Topology:
Continuity and homeomorphism, subspaces, bases and sub-bases, product spaces,
compactness, connected and path-connected spaces, paths in a topological
space, the fundamental group of the circle.
MA344 Groups II
Group actions,
automorphism groups of graphs, applications to enumeration. Sylow's Theorems,
groups of small order, simple groups. Frattini subgroup. Semi-groups, machines.
EC304 Advanced Macroeconomics
This builds
on previous macroeconomics courses and provides a rigorous survey of selected
key themes in modern macroeconomics, which may include growth and technological
change, business cycles, interactions between the real and monetary sectors
of the economy and institutional aspects of monetary and fiscal policy.
EC321 Money and Banking
The objective
of the course is to discuss the significance of financial intermediaries
in modern financial structures and the issues arising from bank regulation
and deregulation. In addition, theories of money supply, money demand and
the impact of monetary policy on economic activity and inflation will be
discussed.
MA310 Actuarial Mathematics
I
Net premium
reserves; multiple life functions; multiple decrement models; valuation
theory for pension plans.
EC323 Advanced Econometrics
Building on
previous statistcs courses in this programme, this course introduces students
to the concepts and problems of multicollinearity, heteroscedasticity,
autocorrelation and model specification in the context of applied economic
modelling. Other topics may include simultaneous equation models and dummy
variables.
(See also MG?Industrial
Management outline above)
(In addition, one optional
course from list below in Semester II)
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Fourth
Year
Semester I
MA416 Rings
Commutative
rings: ideals, prime ideals, maximal ideals, integral domains, unique factorisation
domains, principal ideal domains, Euclidean domains, polynomial rings,
rings of fractions with respect to a multiplicative set.
MA490 Measure Theory
The Lebesgue
integral: the deficiencies of the Riemann integral, Lebesgue measure, measurable
functions, the Lebesgue integral. Convergence theorems, functions of bounded
variatiton and absolutely continuous functions, Vitali?s Covering Theorem,
integration and differentiation. General measure and integration theory:
outer measures, measures, measurable functions, modes of convergence.
MA418 Differential
Equations with Financial Derivatives
Introduction
to Continuous Stochastic Processes. General probability spaces and information
structures. The Wiener process. Stochastic processes as solutions of Stochastic
Differential Equations. Ito process, Ito?s lemmas an anlogue of the chain
rule. Application to the Block-Scholes Model. Derivation of the Block-Scholes
Partial Differential Equation.
MP491 Non Linear Systems
Non-linear
stability and phase plane methods. Chaotic dynamical systems, bifurcation
theory, period doubling and strange attractors. Models with chaotic behaviour
in mechanics, economics and financial systems are studied mathematically
and computationally.
EC410/1 Seminar
in Economics of Financial Markets I and II
The aim of
this course, over both Semesters, is to provide an opportunity for students
to integrate the diverse material in other courses in the context of developments
in financial markets and institutions and related policy debates. This
may in particular involve further explorations in the literature of the
economics of financial markets, and contributions from a number of sources,
including financial market participants. These seminars may also provide
a platform from which the project/minor theses which students undertake
in this programme will be advanced.
MA(?) Actuarial Mathematics
II
Insurance
models and economics of insurance; nonforfeiture benefits; dividends; risk
models, independent increment processes; Markov processes.
CS421/3 Neural Networks
I and II
Training and
optimization. Perceptrons, linear and sigmoid neurons. Single and multi
layer networks. Various architectures. Model Identification Non-linear
time series, parameter estimation and predictions.
(In addition, one optional
course from list below, as well as project/minor thesis
work).
Semester II
MA491 Fields
and Codes
Finite field
extensions, splitting fields of polynomials, normal extension, separable
extensions, Galois extensions, the fundamental theorem of Galois Theory.
Solubility of equations by radicals, ruler and compass constructions. Coding
theory, polynomial codes, Hamming codes.
MA494 Stochastic
Processes
Discrete Models
of Financial Markets. Information structures, trading strategies. Completeness
of markets. Adapted processes, conditional expectations, martingales. Discrete
versions of the stochastic integral, Ito?s lemma, Girsanov?s theorem. Application
to option pricing models.
MA Actuarial
Mathematics III
Premium calculations;
retentions and reserves; stability; dividend policy; utility; applications
of risk theory.
EC320 International
Monetary Economics
This course
aims to introduce students to international economic transactions, as summarised
in the balance of payments. Having taken the course, students should be
in a position to follow contemporary discussion of exchange rates, the
current account, and the overall balance of payments, and to appreciate
the impact of policy changes on such variables.
(In addition, project/minor
thesis work).
See also entries above
for CS421/3 Neural Networks I and II and for EC410/1 Seminar in Economics
of Financial Markets I and II.
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Third
and Fourth Year Optional Courses
One optional
course (two hours per week, 4 credits) is to be taken in semester two of
Third Year and in semester one of Fourth Year. Options must be chosen in
consultation with the relevant Departments. Not all options may be available
in any year and additions may be made to this list.
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Linear Programming/Operations
Research/Game Theory I
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Linear Programming/Operations
Research/Game Theory II
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Applied Multivariate
Analysis
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AY206
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Financial Accounting
I
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AY208
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Business Finance
I or AY872 Financial Management I
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AY307
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Business Taxation
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AY314
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Business Finance
II or AY875 Financial Management II
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CS304
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Mathematical and
logical aspects of computing
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CS403
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Cryptography
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CS407
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Computer Algebra
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EC301
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Irish Economy
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EC310
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International
Trade
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EC353
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European Economy
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MA317
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Advanced Applied
Statistics I
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MA318
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Advanced Applied
Statistics II
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MA410
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Artificial Intelligence
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MA421
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Mathematics of
risk
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MA422
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Theory of Interest
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MA457
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Data Analysis
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ME502
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Expert System
Application
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MG315
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Organisational
Change
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MK204
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Marketing Principles
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MS303
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Management Decision
Systems I
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MS306
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Management Decision
Systems II
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