1st February 2007.
Department of Mathematics
M.Sc. Courses
Academic Year 2007-2008
Mathematics undertakes the pursuit of every scientific endeavour. The Department of Mathematics
satisfies this need by contributing to joint Honours degrees with other Science departments within the
Faculty of Science. Besides it also offers a postgraduate program in Mathematics with the purpose of
consolidating Mathematical knowledge obtained in the Bachelor’s course, and of specialising and doing
research in select branches of Mathematics.
The following is a list of the full time Mathematics academic staff in the Department of Mathematics:
Dr. A. Vella
Dr. J. Sultana
Prof. I. Sciriha
Dr. J. Muscat
Prof. J. Lauri
Dr. D. Buhagiar
Prof. A. Buhagiar, Head of Department
Dr. J. L. Borg
On average there are four students reading for the Master’s degree in Mathematics in a given year.
In the Department of Mathematics, research centres round two main areas: Graph Theory and
Combinatorics, and Mathematical Physics. To further the advancement of these specialities there are
on-going M.Sc. and Ph.D. programmes:
(i)
Graph Theory & Combinatorics
The main research areas include:

Reconstruction, pseudosimilarity and related problems on graph symmetries;

The chromatic index and, in particular, unique colourability;

Graph spectra: singular graphs, polynomial reconstruction conjecture; line graphs of trees; possible
equations in stoichiometry; applications to the structure of fullerenes; main eigenvalues, walks of
graphs and self complementary graphs;

Euler trails and their connexion with other areas of graph theory and mathematics.
M.Sc. students have recently worked on topics such as adversary and ally reconstruction numbers, selfcomplementary graphs, measures of non-planarity, the polynomial reconstruction for disconnected
graphs, and graphs with end-vertices.
(ii) Mathematical Physics and Applications
The main research areas include:

Random Schrödinger operators with the presence of a random potential in a magnetic field; the
theory of self-adjoint operators and probability theory as applied to operators; application to the
quantum Hall effect;

Quantum logics in inner product spaces; completeness criteria; completeness type properties: Cech
completeness and ultra completeness;

Semi uniform convergence structures and other types of convergence and filter structures;

The general theory of relativity;

The mathematics of fusion plasmas;

Numerical solution of partial differential equations using the finite element and similar methods;
M.Sc. students have also worked on other topics such as the Schwarzschild metric in general relativity,
minimal surfaces, stable embeddability of graphs on surfaces, free vibrations in rectangular plates, finite
element solution of non-linear problems in elasticity, and self-similar teletraffic models.
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Assessment
The Masters’ degree in the Department of Mathematics is assessed by
(i)
a three hour examination on one or more study units; in this examination, six questions are
usually set from which the candidate chooses three questions,
(18 ECTS credits),
(ii)
a dissertation on a subject related to the topic in (i),
(38 ECTS credits),
(iii)
a seminar discussing the content of the dissertation in (ii),
(4 ECTS credits),
for a total of 60 ECTS credits.
Study units in the Masters’ Course
The following is a list of study units for the Masters’ Course in Mathematics:
Code
Credit Value
MAT5211
9
MAT5212
9
MAT5411
MAT5412
MAT5611
MAT5711
9
9
18
18
Study-unit
Semester
Lecturers
General Topology I
1st & 2nd
Dr. D. Buhagiar
General Topology II
1st & 2nd
Dr. D. Buhagiar
Graph Theory I
1st
Graph Theory II
1st
General Relativity
1st
The Finite Element Method
1st
A detailed description of these study units is given below.
MAT5211 General Topology I
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&
2nd
Prof. J. Lauri
&
2nd
Prof. I. Sciriha
&
2nd
Dr. J. Sultana
&
2nd
Prof. A. Buhagiar
Tutor:
Follows from:
Scheduled:
ECTS credits:

Dr. D. Buhagiar
MAT3218, MAT3219
1st semester
9
General Topology I:

Cardinals and ordinals;

Subspaces, product spaces and quotient spaces;

Separation axioms and countability axioms;

Connectedness, compactness and paracompactness;

Further topics.
Recommended Texts



Munkres J.R., Topology, Prentice Hall, 2nd Edition, 2000.
Nagata J., Modern General Topology, North-Holland, 2nd revised edition, 1985.
Engelking R., General Topology, revised and completed edition, Helderman Verlag, 1989.
MAT5212 General Topology II
Tutor:
Follows from:
Scheduled:
ECTS credits:

Dr. D. Buhagiar
MAT5211.
2nd semester
9
General Topology II:

Compactifications;

Compactness type properties;

Paracompact spaces and their characterisation;

Metrisation theorems;

Further topics.
Recommended Texts


Nagata J., Modern General Topology, North-Holland, 2nd revised edition, 1985.
Engelking R., General Topology, revised and completed edition, Helderman Verlag, 1989.
MAT5411 Graph Theory I
Tutor:
Follows from:
Scheduled:
ECTS credits:
Prof. J. Lauri
MAT3414
1st and 2nd semesters
9

Graphs and groups:
 Isomorphims, the automorphism group, vertex-and edge-transitivity;
 Cayley colour graphs, Cayley graphs, and applications;
 The automorphism group and the spectrum;
 Graphical regular representations and pseudosimilarity.

The reconstruction problem:
 Introduction: the edge and the vertex problems, Kelly's lemma,
regular graphs, disconnected graphs, maximal planar graphs;
 Counting arguments;
 Connections with topics from the previous section.
Recommended Texts
 Lauri J. and Scopellato R., Topics in Graph Automorphisms and Reconstruction, Cambridge
University Press, 2003.
 Biggs N.L., Algebraic Graph Theory, Cambridge University Press, 2nd Edition, 1994.
 Beineke L.W. and Wilson R.J., Selected Topics in Graph Theory, Academic Press, 1988.
MAT5412 Graph Theory II
Tutor:
Prof. I. Sciriha
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Follows from:
Scheduled:
ECTS credits:
MAT5411
1st and 2nd semesters
9

Graph Spectra:
 Relations to graph structure;
 Line graphs, characterizations;
 Strong regularity, Moore graphs;
 Reconstruction;

Decomposition of graphs:
 The chromatic number and index;
 The role of line-graphs.
Recommended Texts
 Biggs N.L., Algebraic Graph Theory, Cambridge University Press, 2nd Edition, 1994.
 Beineke L.W. and Wilson R.J., Selected Topics in Graph Theory, Academic Press, 1988.
MAT5611 General Relativity
Tutors:
Follows from:
Scheduled:
ECTS credits:










Dr J. Sultana
MAT3513, MAT3613
1st and 2nd semesters
18
Manifolds, tensors;
First fundamental form;
Covariant derivatives;
Torsion, curvature tensors;
Geodesics;
General Relativity, Einstein’s equations;
Homogeneous and isotropic universe, cosmological solution;
Schwarzschild solution;
Weak gravitational field, post-Newtonian metric, post-Newtonian equations of motion for test
particles;
Equations in variations for the spherically symmetric metric.
Recommended Texts




do Carmo A., Differential Geometry, Prentice Hall, 1976.
Camilleri C.J., Tensor Analysis, Malta University Press, 1999.
Wald R.M., General Relativity, University of Chicago Press, 1984.
Brumberg V.A., Essential Relativistic Celestial Mechanics, Adam Hilger, 1991.
MAT5711 The Finite Element Method
Tutor:
Prof. A. Buhagiar
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Follows from:
Scheduled:
ECTS credits:
MAT3712, MAT3713
1st and 2nd semesters
18

Finite element concepts:

The Rayleigh-Ritz variational method,

Galerkin’s weighted residual method,

the weak form of the equilibrium equations;

Element shape functions:

Linear and higher order shape functions
for linear, triangular and rectangular elements,

Axisymmetric elements,

Mapped elements:
the serendipity quadrilateral and numerical integration,

Non-conforming elements,

The patch test;

Steady state field problems:
 Torsion,
 Heat transfer,
 Groundwater flow,
 Fluid flow,
 Acoustics;

Problems in elasticity:

Plane stress and structural problems,

Axisymmetric stress analysis,

Plates and shells,

Vibrational analysis;

Further topics such as:

Time dependent problems,

Non-linear problems,

Numerical software.
Suggested Reading







Segerlind L.J., Applied Finite Element Analysis, John Wiley, New York, 2nd Edition 1984.
Dawe D.J., Matrix and Finite Element Displacement Analysis of Structures, Clarendon Press,
Oxford, 1984.
Lewis P.E. and Ward J.P., The Finite Element Method, Principles and Applications, AddisonWesley, New York, 1991.
Ottosen N.S. and Petersson H., Introduction to the Finite Element Method, Prentice Hall, New York,
1992.
Bathe K.J., Finite Element Procedures, Prentice Hall, New York, 1996.
Reddy J.N. and Gartling D.K., The Finite Element Method in Heat Transfer and Fluid Dynamics, 2nd
Edition, CRC Press, New York, 2001.
Zienkiewicz O.C. and Taylor R.L., The Finite Element Method; 5th Edition
Volume 1: The Basis; Volume 2: Solid Mechanics; Volume 3: Fluid Mechanics; Butterworth –
Heinemann, Oxford, 2000.
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