Department of Mathematics
Academic Year 2005/6
Old Regulations
Mathematics underlies the pursuit of every scientific endeavour. The Department of Mathematics within
the Faculty of Science satisfies this need by contributing to joint Honours degrees with other science
disciplines such as Physics, Statistics and Computer Science. Besides it also gives courses to other
Faculties, like Engineering, FEMA and Education.
The following is a list of the Full Time Mathematics Academic Staff in the Department of Mathematics:
Dr. J. Sultana
Dr. I. Sciriha
Dr. J. Muscat
Professor J. Lauri
Professor S. Fiorini
Dr. D. Buhagiar
Professor A. Buhagiar, Head of Department
Mr. J. L. Borg
The number of students taking Mathematics as a principal subject for their B.Sc.(Hons.) has now
stabilised at about 30 per year. Student numbers in Mathematics courses for other Faculties vary
greatly from faculty to faculty, with about 420 first year students in FEMA, through Engineering classes
of about 200, to smaller numbers of about 20 in the Science Faculty itself. In order to teach effectively
such large classes, a number of part-timers are employed further to the full-time staff to give small
tutorial classes.
The Department of Mathematics also offers postgraduate (Masters’) courses in Mathematics.
Departmental research centres mainly round Graph Theory and Combinatorics, and Mathematical
Physics.
Definition of Labels for the Study units
Study units in Mathematics carry the prefix MAT, followed by a four digit number:
The first digit indicates the year of the degree course in which the study unit is given:
1
2
3
4
5
if the study unit is given in the 1st year of the B.Sc.(Hons.) course;
if the study unit is given in the 2nd year of the B.Sc.(Hons.) course;
if the study unit is given in the 3rd year of the B.Sc.(Hons.) course;
if the study unit is given in the 4th year of the B.Sc.(Hons.) course;
if the study unit is given in the M.Sc. course.
The first digit of a study unit is 0 if it is only an introductory course or a Foundation course.
The second digit indicates the subject area as follows:
0
1
2
3
4
Mathematical Methods
Algebra
Analysis
Probability
Combinatorics
5
6
7
8
9
Geometry and Vector Analysis
Physical Applied Mathematics
Differential Equations
Engineering Mathematics
Others
The third and fourth digits represent a serial number, numbering courses within the area defined by the
second digit. The third digit is a ‘9’ when two or more study units are amalgamated together.
Except where otherwise stated, study units are assessed by written test or examination according to the
following norms:
1
Number of
Credits
Exam Duration
(Hours)
Number of
Lecturers
Number of
Questions Set
Number of
Questions Chosen
2
1½
-
3
2
4
2½
1
2
5
6
4
4
6
3
1
2
3
7
8
9
5
5
5
Electives
3
-
8
5
Masters
3
-
6
3
The following is a list of study units given by the Department of Mathematics in the second, third and
fourth years of the Mathematics course.
2
PROGRAM OF STUDIES IN MATHEMATICS
BSC(HONS) DEGREE
FOR COURSES WHICH STARTED ON OR BEFORE OCTOBER 2003
ACADEMIC YEAR 2005/6
Year IV
Students choosing to do the elective in Mathematics must take all the twenty elective credits listed
below. Besides, all Mathematics students in the fourth year must take a choice of twelve credits from
the following list of ‘advised’ credits.
Type of unit
Code
Elective Credits:
MAT4208
MAT4000
made up of:
MAT4403
Advised Credits:
MAT4207
MAT4505
MAT4603
MAT4703
Title of unit
Credit
Value
Semester
Functional Analysis I
10
1st & 2nd
Graph Theory and
Combinatorics
10
1st & 2nd
Complex Analysis
General Relativity
Classical Mechanics
The Finite Element Method
4
4
4
4
1st
1st
1st
1st
& 2nd
& 2nd
& 2nd
& 2nd
Lecturers
Dr. D. Buhagiar/
Dr. J. Muscat
Prof. S. Fiorini/
Prof. J. Lauri
Dr. J. Sultana
Dr. J. Sultana
Prof. A. Buhagiar
Prof. A. Buhagiar
Advised credits from other departments:
For students not taking Statistics:
SOR2210
or
SOR3110
Families of random variables and
random vectors
4
1st & 2nd
Stochastic Processes I
5
1st & 2nd
COURSE DESCRIPTIONS
Year IV
MAT4207 - Complex Analysis
Lecturer:
Follows from:
Leads to:
Credit value:
Lectures:
Tutorials:
Semesters :
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Dr. J. Sultana
MAT3294, MAT3206
MAT5209
4
18
10
1, 2
Analytic functions;
The Cauchy-Riemann equations;
Exponential and logarithmic functions;
Contour integration;
Cauchy’s theorem;
Cauchy’s integral formulae;
Taylor’s series;
Laurent series;
Liouville’s theorem;
The fundamental theorem of algebra;
Residues;
Evaluation of integrals and summation of series by residues;
Rouche’s theorem;
Conformal transformations.
Textbooks
• Priestley H., Introduction to Complex Analysis, Oxford University Press, Oxford, 1994.
• Osborne A.D., Complex Variables and their Applications, Addison-Wesley, New York, 1999.
3
MAT4505 – General Relativity
Lecturer:
Follows from:
Leads to:
Credit value:
Lectures:
Tutorials:
Semesters:
•
•
•
•
•
Dr. J. Sultana
MAT2292, MAT3504
MAT5604
4
18
10
1, 2
Special Relativity and Flat Spacetime;
Manifolds;
Curvature;
Gravitation;
The Schwarzschild solution and Black Holes.
Recommended Text
• Carroll S.M., Spacetime and Geometry: An Introduction to General Relativity, 1st Edition, Addison
Wesley, New York, 2003.
Suggested Reading
• French A. P., Special Relativity, W. W. Norton & Company, New York, 1968.
• Schutz B. F., A first course in General Relativity, Cambridge University Press, Cambridge, 1985.
• Hughston L. P. and Tod K. P. , An introduction to General Relativity, London Mathematical Society
Student Series, Cambridge University Press, Cambridge, 1991.
• Camilleri C.J., Tensor Analysis, Malta University Press, Malta, 1999.
MAT4603 - Classical Mechanics
Lecturer:
Follows from:
Leads to:
Credit value:
Lectures:
Tutorials:
Semesters:
Prof. A. Buhagiar
MAT1002, MAT1501, MAT3602
MAT5604, MAT5704
4
18
10
1, 2
•
Rotating frames:
• Angular velocity,
• The rotating axes theorem,
• Velocity and acceleration in a rotating frame,
• The rotation of the earth;
•
Systems of many particles:
• The two body problem,
• Rigid bodies;
•
Rigid bodies:
• Angular velocity and angular momentum,
• The equation of motion,
• The inertia tensor,
• Principal axes and principal moments of inertia,
• General rotation of a rigid body fixed at a point,
• Applications;
•
Lagrangian Mechanics:
• Generalised coordinates,
• Virtual displacements,
• Generalised forces,
• Lagrange’s Equations,
• Ignorable coordinates;
•
Application of Lagrangian Mechanics:
• Rigid bodies, the Euler angles,
• Precession of tops and rolling bodies,
• Small Oscillations,
4
• Impulsive Motion.
Method of Assessment:
Examination 85%, Coursework 15%.
Main Texts
• Lunn M., A first Course in Mechanics, Oxford University Press, Oxford, 1991.
• Camilleri C.J., Classical Mechanics, Malta 2004; available from the author.
• Charlton F., Textbook of Dynamics, van Nostrand Company Ltd., London, 1969.
Supplementary Reading
• Marion J.B. and Thornton S.T., Classical Dynamics of Particles and Systems, 4th Edition, Harcourt
College Publishers, New York, 1995.
• Goldstein H., Classical Mechanics, 2nd Edition, Addison-Wesley, New York, 1980.
MAT4703 - The Finite Element Method
Lecturer:
Follows from:
Leads to:
Credit value:
Lectures:
Tutorials:
Semesters:
•
•
Prof. A. Buhagiar
MAT1001, MAT1002, MAT3602
MAT5704
4
18
10
1, 2
Methods for partial differential equations:
• Variational methods: the Rayleigh-Ritz method,
• Galerkin’s weighted residual method;
Discretisation of the domain:
• Linear elements in one and two dimensions;
•
Finite element solution of:
• Poisson’s equation,
• Helmholtz’s equation,
• the biharmonic equation;
•
Applications:
• Axial extension and vibration of bars,
• Bending of beams,
• Torsion,
• Heat Transfer.
Method of Assessment:
Examination 85%, Coursework 15%.
Main Text
• Lewis P. E. and Ward J. P., The Finite Element Method, Principles and Applications, Addison-Wesley,
New York, 1991.
Supplementary Reading
• Dawe D.J., Matrix and Finite Element Displacement Analysis of Structures, Clarendon Press, Oxford,
1984.
• Segerlind L.J. Applied Finite Element Analysis, John Wiley, New York; 1st Edition 1976, 2nd Edition
1984.
• Ottosen N. S. and Petersson H., Introduction to the Finite Element Method, Prentice Hall, New York,
1992.
• Fagan M.J., Finite Element Analysis: Theory and Practice, Longman, Singapore, 1992.
5
Elective Topics
MAT4208 - Functional Analysis I
Lecturers:
Follows from:
Leads to:
Credit value:
Lectures:
Tutorials:
Semesters:
Dr. D. Buhagiar and Dr. J. Muscat
MAT1201, MAT3294
MAT5209
10
36
20
1, 2
•
•
•
•
•
•
•
•
•
Metric Spaces and their Completion;
Normed Vector Spaces;
Examples of Normed Spaces: the spaces l p, Lp and C[a, b];
Dual spaces;
Hahn-Banach Theorem;
Uniform Boundedness Theorem;
Open Mapping Theorem;
Closed Graph Theorem;
Bounded Linear Operators and their Spectra.
•
•
•
•
•
•
Hilbert spaces;
Orthogonal projections;
Fourier theory;
Bessel's inequality;
Self adjoint operators;
Spectral theory.
Main text
• Kreysig E., Introductory Functional Analysis, Wiley, 1989.
Supplementary Reading
• Rudin W., Functional Analysis, Tata McGraw-Hill, 1973.
• Kolmogorov A.N. and Fomin S.V., Elements of the Theory of Functions and Functional Analysis,
Dover, 1957.
MAT4403 - Graph Theory and Combinatorics
Lecturers:
Follows from:
Leads to:
Credit value:
Lectures:
Tutorials:
Semesters:
Prof. S. Fiorini and Prof. J. Lauri
MAT1101, MAT1401, MAT2104
MAT5404
10
36
20
1,2
•
•
•
•
•
•
•
•
Definitions and elementary results;
Trees;
Connectivity;
Euler Tours and Hamilton Cycles;
Vector Spaces associated with graphs;
Cycle-cutset duality;
Graph colourings;
Planar graphs.
•
•
•
•
•
Burnside’s Counting Lemma;
Generating functions;
The cycle index of a permutation and the use of Polya’s Theorem;
Ramsey’s Theorem for Graphs;
An Introduction to error-correcting codes.
Main Texts
• Wilson R.J., Introduction to Graph Theory, Longman, 4th Edition, 1996.
• West D.B., Introduction to Graph Theory, Prentice Hall, 2nd Edition, 2001.
• Biggs N.L., Discrete Mathematics, Oxford Science Publications, Clarendon Press, 1989.
Supplementary Reading
• Cameron P.J., Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press,
Cambridge, 1994.
• Bryant V., Aspects of Combinatorics: A Wide Ranging Introduction, Cambridge University Press,
Cambridge, 1993.
6