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PROGRAM OF STUDIES IN MATHEMATICS FOR THE B.Sc. (Hons.) degreeACADEMIC YEAR 2007-2008

25th May 2007.

Year I starting 2007-2008

Mathematics students in the first year of the B.Sc.(Hons.) course must take all the compulsory credits and a selection of at most 8 optional credits.

Type of unit Code Title of unit Credit Value

Semester Lecturers

Compulsory Credits:

MAT1090 Mathematical Methods 3 1st Prof. I. Sciriha

Dr. A. Vella

MAT1000 Introductory Mathematics 5 1st Dr. J. L. Borg

MAT1511 Analytical Geometry 4 1st Dr. E. Chetcuti

MAT1101 Groups and Vector Spaces 6 2nd Prof. I. Sciriha

MAT1211 Analysis I 4 2nd Dr. D. Buhagiar

MAT1411 Discrete Methods 4 2nd Prof. J. Lauri

Optional Credits:

A total of 8 credits may be chosen from units designated as such in any program of study within the University.

The following study unit is highly recommended, especially for students who do not have Advanced Applied Mathematics:

MAT1611 Introductory Mechanics 4 1st Mr. J. Borg / Mr. V. Cilia Vincenti

The following study unit/s are highly recommended, especially for students who have not done probability or statistics.

& SOR1110 Probability 4 1st & 2nd Dr. L. Sant

OR& SOR1211 Probability and 2 1st Various& SOR1221 Sampling, Estimation and

Regression2 2nd Various

& These credits cannot be taken by students taking Statistics and Operations Research as a Principal subject area for the B.Sc. (Hons.). The credits SOR1211 and SOR1221 are slightly easier than SOR1110, and cannot be taken with it.

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Other optional credits can be chosen from:

†PHY1020 or/and†PHY1030

Basic Concepts in Physics I

Basic Concepts in Physics II

2

2

1st

2nd

#CIS1003 or/and£SOR1311 or/and~CHE2060

Programming for Scientists

Linear Programming

Principles and Perspectives of Science

2

2

2

1st

2nd

2nd

† Not to be taken by students taking Physics as a principal subject area.

# Not to be taken by students taking Computer Information Systems or Computer Science and AI as a principal subject area.

£ Not to be taken by students taking Statistics as a principal subject area.

~ Not to be taken by students taking Chemistry as a principal subject area.

Optional credits offered to other departments

The Mathematics department offers the following as optional credits to other departments and/or faculties:

# MAT1001 Mathematical Methods I: (i) Matrices 2 1st Prof. I. Sciriha /

Mr. J. Borg

# MAT1002 Mathematical Methods I: (ii) Ordinary

Differential Equations

2 1st Dr. A. Vella /

Ms N. Camillleri # MAT1091 Mathematical Methods I: Matrices and

Differential Equations

4 1st Various

% MAT1611 Introductory Mechanics 4 1st Mr. J. Borg / Mr. V. Cilia Vincenti

& MAT1612 Applied Mathematics for Engineers 4 1st Mr. E. Cardona

$ MAT1900 Elementary Calculus 4 1st Ms G. Galea

# Students can only register for one study unit from MAT1001, MAT1002 and MAT1091. These credits cannot be taken by students in B.E.& A., B.Eng., and B.Sc. with Mathematics as one of the options.

% Given twice in the first semester, once to B.Sc.(Hons.) and B.E.& A. students, and once to B.Eng. and B.Ed.(Hons.) students.

& For B.Eng. and B.E.& A. students only.

$ Cannot be taken by students in B.Com., B.E.& A., B.Eng. or B.Ed. (Mathematics and/or Physics) students.

Year II starting 2007-2008

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Mathematics students in the second year of the B.Sc.(Hons.) course must take all the compulsory credits and a choice of four elective credits from the following list:

Type of unit Code Title of unit Credit

Value

Semester Lecturers

Compulsory Credits:

MAT2112 Linear Algebra I 4 1st Prof. I. Sciriha

MAT2212 Analysis II 4 1st Dr. A. Vella

MAT2412 Introduction to Graph Theory 2 1s t Prof. J. Lauri

MAT2512 Vector Analysis I 4 1st Dr. J. Sultana

MAT2113 Rings 4 2nd Dr. A. Vella

MAT2213 Analysis III 4 2nd Dr. J. L. Borg

MAT2513 Vector Analysis II 4 2nd Dr. J. Sultana

Elective Credits: Students must choose 4 credits from the following list.

£ MAT2912 Computational Mathematics 2 1st & 2nd Various

£ Elective credit offered only to students in the B.Sc. (Hons.) course. This credit is highly recommended to Mathematics students taking Statistics or Physics in the B.Sc.(Hons.) course.

Other elective credits can be chosen from:

* MAT1611 Introductory Mechanics 4 1st Mr. J. Borg / Mr. V. Cilia Vincenti

~ MAT2402 Networks 2 1st Prof. J. Lauri ^ PHY2150 Numerical Analysis 2 2nd Prof. A. Buhagiar

& CHE2060 Principles and perspectives of

Science

2 2nd Prof. F. Ventura

SCI1200 Science of the Earth 4 1st & 2nd Dr. P. Galea /

Mr. J. M. Bonnici % SOR1211 Probability 2 1st Various

% SOR1221 Sampling, Estimation and

Regression

2 2nd Various

%† SOR1231 Hypothesis testing and modeling

using SPSS

2 2nd Various

% SOR1311 Linear Programming 2 2nd Mrs. N. Attard

% SOR 1110 Probability 4 1st & 2nd Dr. L. Sant

* Optional credit offered also to other departments. This credit is given twice in the first semester, once to B.Sc.(Hons.) and B.E.& A. students, and once to B.Eng. and B.Ed.(Hons.) students.

~ This credit is also offered to B.Sc. (I.T.) students. ^ Not to be taken by students taking Physics as a principal subject area. & Not to be taken by students taking Chemistry as a principal subject area.

% Not to be taken by students taking Statistics and Operations Research as a principal subject area. † This study unit needs as prerequisites the credits SOR1211 and SOR1221.

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Year III 2007- 2008 only

Mathematics students in the third year of the B.Sc. (Hons.) course must take the following credits, all of which are compulsory:


Value

Semester Lecturers

Compulsory credits:

MAT3105 Linear Algebra I 4 1st Prof . I. Sciriha

MAT3214 Complex Analysis 4 1st Dr. J. Sultana

MAT3905 Linear Algebra II 2 1st Prof. I. Sciriha

MAT3792 Metric Spaces II &

Uniform Convergence

4 1st Dr. D. Buhagiar

& Dr. J. L. Borg

MAT3206 Lebesgue Integration 4 2nd Dr. D. Buhagiar

MAT3504 Vector Analysis IIA 4 2nd Dr. J. Sultana

MAT3602 Mechanics 4 2nd Prof. A. Buhagiar

MAT3701 Differential Equations 4 2nd Prof. A. Buhagiar

Year III starting in 2008-2009

Mathematics students in the third year of the B.Sc. (Hons.) course must take the following credits, all of which are compulsory:


Value

Semester Lecturers

Compulsory credits:

MAT3114 Linear Algebra II 2 1st Prof. I. Sciriha

MAT3115 Groups 4 1st Prof. J. Lauri

MAT3214 Complex Analysis 4 1st Dr. J. Sultana

MAT3215 Metric Spaces 4 1st Dr. D. Buhagiar

MAT3413 Probabilistic Combinatorics 2 1st Dr. A. Vella

MAT3216 Analysis IV 2 2nd Dr. J. L. Borg

MAT3217 Lebesgue Integration 4 2nd Dr. D. Buhagiar

MAT3612 Mechanics 4 2nd Prof. A. Buhagiar

MAT3711 Differential Equations 4 2nd Dr. J. Muscat

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Year IV starting 2007-2008

One credit in the fourth year of the B.Sc. course is equivalent to five hours of lectures and/or tutorials.

Students taking 34 credits in Mathematics must take: exactly 18 credits from those labelled Project, the compulsory credit MAT3207 Complex Analysis, one elective of 5 credits, and one elective of 6 credits.

Students taking 26 credits in Mathematics must take: the compulsory credits MAT3990 Mathematics Seminar and MAT3207 Complex Analysis, together with a choice of three electives of 5 credits each.


Value

Semester Lecturers

Project MAT3218 Functional Analysis 12 1st & 2nd Dr. D. Buhagiar /

Dr. J. Hamhalter

Project MAT3414 Discrete Mathematics 12 1st & 2nd Prof. J. Lauri /

Prof. I. Sciriha

Project MAT3990 Mathematics Seminar 6 1st & 2nd Various

Compulsory * MAT3207 Complex Analysis 5 1st & 2nd Dr. J. Sultana

Elective ! MAT3613 Classical Mechanics 5 1st & 2nd Prof. A. Buhagiar

Elective ! MAT3116 Group Representations 5 1st & 2nd Prof. J. Lauri

Elective ! MAT3219 General Topology 5 1st & 2nd Dr. D. Buhagiar

Elective ! ^ MAT3712 Partial Differential Equations &

Calculus of Variations

5 2nd Dr. J. Muscat

Elective ! MAT3513 Tensors and Relativity 5 1st & 2nd Dr. J. Sultana

Elective ! MAT3713 The Finite Element Method 5 1st & 2nd Prof. A. Buhagiar

Elective ! MAT3693 Classical Mechanics 6 1st & 2nd Prof. A. Buhagiar

Elective ! MAT3196 Group Representations 6 1st & 2nd Prof. J. Lauri

Elective ! MAT3299 General Topology 6 1st & 2nd Dr. D. Buhagiar

Elective ! ^ MAT3782 Partial Differential Equations &

Calculus of Variations

6 2nd Dr. J. Muscat

Elective ! MAT3593 Tensors and Relativity 6 1st & 2nd Dr. J. Sultana

Elective ! MAT3793 The Finite Element Method 6 1st & 2nd Prof. A. Buhagiar

* Offered for the last time in the fourth year; subsequently offered only in the third year.

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^ Not offered in 2007-2008, as the fourth years had already covered this in the third year.

! MAT3613 cannot be taken with MAT3693, nor MAT3116 with MAT3196, nor MAT3219 with MAT3299, nor MAT3712 with MAT3782, nor MAT3513 with MAT3593, nor MAT3713 with MAT3793.STUDY UNITS TO OTHER FACULTIES

The Department of Mathematics also gives credits in Mathematics to students in other faculties. Optional credits in Mathematics can be taken by students in other departments. Besides, B.Ed.(Hons.) students majoring in Mathematics can choose to follow most study units in the first two years of the B.Sc.(Hons.) course in Mathematics. The following study units are given to students in the Faculty of Engineering, FEMA and also to students reading for a B.Sc. degree in Information Technology. A detailed description of these courses is given in the following pages.

Code Title of unit Credit Value Semester Lecturers

Faculties of Engineering and Architecture

1st Year Optional creditsMAT1611 Introductory Mechanics 4 1st Mr.V.Ciliaor VincentiMAT1612 Applied Mathematics for Engineers 4 1st Mr..E. Cardona

1st Year Core creditsMAT1801 Mathematics for Engineers I 4 1st Dr. J. L. Borg /

VariousMAT1802 Mathematics for Engineers II 4 2nd Dr. A. Vella /

Various2nd YearMAT2803 Laplace and Fourier Transforms 2 1st Dr. J. L. BorgMAT2804 The Eigenvalue problem and multiple 2 2nd Dr. J. L. Borg integrals

Faculty of Engineering3rd YearMAT3805 Optimisation, Complex & Numerical Analysis I 4 1st Dr. J. Sultana /

Prof. A. BuhagiarMAT3806 Optimisation, Complex & Numerical Analysis II 4 2nd Dr. J. Sultana /

Prof. A. Buhagiar

Faculty of Economics, Management and Accountancy (FEMA)1st Year†MAT1991 Mathematics I and II 8 1st & 2nd Prof. A. Buhagiar/

VariousThis study unit comprises two credits as follows:%†MAT1901 Mathematics I 4 1st Dr. A. Vella /and Various† MAT1902 Mathematics II 4 2nd Mr. J. Sciriha /

Various

Board of Studies for Information Technology 1st Year*^ MAT1091 Mathematics Methods I: Matrices 4 1st Prof . I. Sciriha /

and Differential Equations VariousMAT1803 Mathematical Transform Techniques 2 2nd Mr. J. Borg

2nd Year*MAT1401 Discrete Methods 4 2nd Prof. J. LauriMAT2402 Networks 2 1st Prof. J. Lauri

* Students in I.T. attend these Mathematics courses together with students reading Mathematics in the B.Sc. (Hons.) degree. The description of these courses is already given under the Mathematics option in the B.Sc. (Hons.) degree.

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^ Amalgamated version of the credits MAT1001 and MAT1002; these are still individually available.% For students in B.Sc. (Tourism Studies)† Students cannot take MAT1991 with MAT1901 and/or MAT1902.

Brief information on the Department of Mathematics.

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 Education, Engineering, Architecture, FEMA, and I.T.

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. M. Micallef• Prof. J. Lauri• Dr. D. Buhagiar• Prof. A. Buhagiar, Head of Department• Dr. 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. Besides, about 40 students from the Faculty of Education annually follow the same Mathematics courses followed by the Mathematics students in the Faculty of Science. Student numbers in Mathematics courses for other Faculties vary greatly from faculty to faculty, with about 400 first year students in FEMA, to Engineering classes of about 100 to 200. The optional credits offered by the Mathematics Department are also very popular with students from other courses, with audiences of about 100. Part time staff are employed by the Department of Mathematics to give tutorials in Mathematics to small classes.

The Department of Mathematics also offers postgraduate (Masters’) courses in Mathematics. Departmental research centres mainly round Graph Theory and Combinatorics, and Mathematical Physics.

Except where otherwise stated, study units are assessed by written test according to the following norms:

Year of Number of Exam Duration Number of Number of Number ofCourse Credits (Hours) Lecturers Questions Questions

Set Chosen ------------------------------------------------------------------------------------------------------------------------------------------------------------

I-III 2 11/2 - 3 2

I-III 4 2 1 4 32 4 3

I-III 6 3 1 7 52 8 53 9 5

IV 5 3 . 5 4

IV 6 3 - 6 5 IV Project 12 3 - 8 5

Masters 18 3 - 6 3

------------------------------------------------------------------------------------------------------------------------------------------------------------

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The study units given by the Department of Mathematics are described below.COURSE DESCRIPTIONS

B.Sc(Hons.)

Year I

MAT1090 - Mathematical Methods Lecturer: Prof. I. Sciriha, Dr. A. VellaFollows from: A-LevelLeads to: MAT1101, MAT3701Credit value: 3Lectures: 15Tutorials: 6Semester: 1

• Matrices;• Determinants;• Solution of linear equations;• Eigenvalues and diagonalisation.

• Ordinary differential equations of the first order;• Ordinary differential equations of the second order;• Partial differentiation and exact differential equations.

Method of Assessment: Examination 100%

Textbooks

• Gow M., A Course in Pure Mathematics, Arnold, 1960.• Andvilli S. and Hecker D., Elementary Linear Algebra, Harcourt Academic Press, 1999.• Kreyszig E., Advanced Engineering Mathematics, John Wiley & Sons, 8th Edition, 1998.• Derrick W. and Grossman S., Elementary Differential Equations, Addison-Wesley, 4th Edition, 1997.

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MAT1000 - Introductory Mathematics

Lecturers: Dr. J. L. BorgFollows from: A-LevelLeads to: MAT1101, MAT1211Credit value: 5Lectures: 25Tutorials: 10Semester: 1

• Natural numbers and the principle of induction.

• Integers: • Divisibility, • Prime numbers, • The greatest common divisor and the Euclidean algorithm, • The fundamental theorem of arithmetic;

• Rational and real numbers: • The completeness axiom, • The Archimedean property of the real numbers, • Inequalities.

• Sets: • Inclusion, union, intersection, • De Morgan’s laws;

• Ordered pairs and the Cartesian product of sets;

• Relations: • Basic properties, • Equivalence relations and partitions of a set, • Functions: • The function as a mapping, • Injectivity and surjectivity, • Composition of functions, • Inverse functions;

• Cardinality and Countability.

Method of assessment: Examination 100%.

Suggested Readings

• D’Angelo J.P. and West D.B., Mathematical Thinking: Problem-Solving and Proofs, Prentice Hall, 2nd Edition, 2000.• Schumacher C., Chapter Zero: Fundamental Notions of Abstract Mathematics, Addison-Wesley, 1996.• Devlin K.J., Sets, Functions and Logic, Chapman and Hall, 1981.• Epp S., Discrete Mathematics with Applications, Brooks Cole, 1995.

MAT1101 - Groups and Vector Spaces

9

Lecturer: Prof. I. ScirihaFollows from: MAT1000Leads to: MAT2112Credit value: 6Lectures: 30Tutorials: 12Semester: 2

• Introduction to groups: symmetry, axiomatic approach;• Lagrange's theorem;• An introduction to number theory;• Permutations;• Normal subgroups;• Quotient groups;• First isomorphism theorem;• Introduction to vector spaces;• Linear transformations and matrices;• Dimension theorem, nullity, rank;• Change of basis.


Main Texts

• Herstein I.N., Topics in Algebra, John Wiley & Sons, 3rd Edition, 1996.• Cameron P., Introduction to Algebra, Oxford University Press, Oxford, 1998.• Lipschutz S., Linear Algebra, Schaum’s Outline Series, McGraw-Hill, 2nd Edition, 1991.• Wallace D., Groups, Rings and Fields, Springer, 1998.

Supplementary Readings

• Armstrong M.A., Groups and Symmetry, Springer Verlag, Heidelberg, 1997.• Anton H., Elementary Linear Algebra, John Wiley & Sons, 8th Edition, 2000.

MAT1211 - Analysis I

Lecturer: Dr. D. Buhagiar

10

Follows from: MAT1000Leads to: MAT2212Credit value: 4Lectures: 20Tutorials: 8Semester: 2

• The real number line: • Open and closed sets, • Compact sets;

• Sequences of real numbers: • Convergence, • Basic theorems on limits of sequences, • Lim sup and lim inf, • The Bolzano-Weierstrass theorem, • The Cauchy convergence criterion;

• Sequences in Rn: • The norm of a vector, • Convergence;

• Series in R: • Conditional and absolute convergence, • Tests for convergence.


Suggested reading:

• Spivak M,, Calculus, Publish or Perish, 3rd Edition, 1994 • Abbott S., Understanding Analysis, Springer, 2001• Bartle R. & Sherbert D., Introduction to Real Analysis, Wiley, 3rd Ed., 1999• Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974

MAT1411 - Discrete Methods

Lecturer: Prof. J. LauriFollows from: A-LevelLeads to: MAT2412

11

Credit value: 4Lectures: 20Tutorials: 8Semester: 2

• Permutations,• Combinations,• Partitions of a set,• The inclusion-exclusion principle;

• Recurrence relations,• Generating functions,• Partitions of a positive integer.

Method of Assessment: Examination 100% Textbooks: One of

• Biggs N.L., Discrete Mathematics, Oxford Science Publications, Clarendon Press, 2nd Edition, 2002.or• Dossey J. A., Otto A. D., Spence L.E. and Eynden C.V., Discrete Mathematics, Addison–Wesley, 5th Edition, 2006.

MAT1511 - Analytical Geometry

Lecturer: Dr. E. ChetcutiFollows from: A-LevelLeads to: MAT2512Credit value: 4

12

Lectures: 20Tutorials: 8Semester: 1

• Vector geometry: • Affine spaces, • Position vectors;

• Euclidean geometry: • The dot product, • Angles, • Isometries;

• Coordinates and equations: • Cartesian coordinates, • Curves and equations, • Coordinate form of an isometry, • Change of coordinates;

• Orientation and vector product: • Vector algebra, • Vector equations of lines and planes;

• Conics: • The ellipse, • The parabola, • The hyperbola.


Textbooks

• Roe J., Elementary Geometry, Oxford Science Publications, Clarendon Press, 1997.

Supplemaentary Reading:

• Vaisman I., Analytical Geometry, World Scientific Publishing Company, 1998.• Camilleri C.J., Vector Analysis, Malta University Press, Malta, 1994.

MAT1611 – Introductory Mechanics

Lecturers: Mr. J. Borg; Mr. V. Cilia VincentiFollows from: Advanced Level Pure MathematicsLeads to: MAT3602Credit value: 4Lectures: 20

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Tutorials: 8Semester: repeated twice in Semester 1

• Vectors: • Distance, velocity and acceleration, • Force and Newton’s laws, • Components of a system of forces and their resultant, • Equilibrium and acceleration under concurrent forces;

• Statics: • Systems in equilibrium, • Lami’s theorem, • Friction, • Centroids • Light frameworks, • Shearing force and bending moment in beams;

• Dynamics: • Projectiles, • Circular motion, • Work and energy, • Elasticity, • Simple harmonic motion, • Momentum and impulse.


Textbooks

• Jefferson B. and Beadsworth T., Introducing Mechanics, Oxford University Press, 2000.• Jefferson B. and Beadsworth T., Further Mechanics, Oxford University Press, 2001.• Bostock L. and Chandler S., Applied Mathematics, Volumes 1 and 2, Stanley Thornes, 1975.

Students who have Mathematics as a principal subject area in the B.Sc. course cannot take the following study units.

MAT1612 – Applied Mathematics for Engineers

Lecturers: Mr. E. CardonaFollows from: Advanced Level Pure MathematicsLeads to: MAT1802

14

Credit value: 4Lectures: 20Tutorials: 8Semester: 1

• Vectors: • Addition and subtraction of vectors, • Resolution of a vector, • Rectangular resolution of a vector;

• Geometrical applications: • Vector equation of a straight line, • Vector equation of a plane;

• Products of vectors: • Dot product, • Cross product, • Scalar triple product, • Vector triple product; • Differentiation and integration of vectors: • Derivative of a vector, • Derivative of products, • Integration, • Serret-Frenet formulae of curves: • Curvature, tangent, normal, binormal;

• Coordinate geometry: • Equations of lines and curves, • The circle, the ellipse and the hyperbola;

• Partial differentiation: • Applications; • Integration and applications: • First and second moment of area, • Centroid and centre of mass, • Mass moment of inertia. Method of Assessment: Examination 100%

Suggested Reading:

• Bostock L. and Chandler S., Applied Mathematics, Volumes 1 and 2, Stanley Thornes, 1975.• Jefferson B. and Beadsworth T., Introducing Mechanics, Oxford University Press, 2000.• Jefferson B. and Beadsworth T., Further Mechanics, Oxford University Press, 2001.

MAT1001 - Mathematical Methods I: (i) Matrices Lecturers: VariousFollows from: A-LevelLeads to: MAT1101Credit value: 2Lectures: 10Tutorials: 4Semester: 1

15

• Matrices;• Determinants;• Solution of linear equations;• Eigenvalues and diagonalisation.


Textbooks• Gow M., A Course in Pure Mathematics, Arnold, 1960.• Andvilli S. and Hecker D., Elementary Linear Algebra, Harcourt Academic Press, 1999.

MAT1002 - Mathematical Methods I: (ii) Ordinary Differential Equations Lecturer: VariousFollows from: A-LevelLeads to: MAT3701Credit value: 2Lectures: 10Tutorials: 4Semester: 1 • Ordinary differential equations of the first order;• Ordinary differential equations of the second order with constant coefficients;• Partial differentiation and exact differential equations.

Method of Assessment: Examination 100%Textbooks• Gow M., A Course in Pure Mathematics, Arnold, 1960.• Kreyszig E., Advanced Engineering Mathematics, John Wiley & Sons, 8th Edition, 1998.• Derrick W. and Grossman S., Elementary Differential Equations, Addison-Wesley, 4th Edition, 1997.

MAT1091 - Mathematical Methods I: Matrices and Differential Equations Lecturer: VariousFollows from: A-LevelLeads to: MAT1101, MAT3701Credit value: 4Lectures: 20Tutorials: 8Semester: 1

Method of Assessment: Examination 100%This credit is made up of the two study units MAT1001 and MAT1002 described above.

MAT1900 - Elementary Calculus

Lecturer: Ms. G. GaleaFollows from: O-LevelCredit value: 4Lectures: 20Tutorials: 8Semesters: 1, 2

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• Cartesian coordinates;• Equations of lines and curves;• Coordinate geometry of the circle;• Differentiation;• Partial differentiation;• Integration and applications.


Textbooks

• Bostock L. and Chandler S., The Core Course for A-Level, Stanley Thornes, The Bath Press, Avon, 1990.• Shepperd J.A.H. and Shepperd C.J., Pure Maths for A level, Hodder & Stoughton, 1983.

Year II

MAT2112 - Linear Algebra I

This credit is identical to MAT3105 in the third year. Second and third year students attend together for MAT2112 and/or MAT3105, and they will have the same examination. After 2008, MAT3105 will be discontinued, and only MAT2112 will be given to the second years only.

Lecturer: Prof. I. ScirihaFollows from: MAT1101

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Leads to: MAT3905Credit value: 4Lectures: 20Tutorials: 8Semester: 1

• Vector spaces;• Dimension theorems;• Third isomorphism theorem;• Direct sum of spaces;• Dual spaces;• Inner product spaces;• Gram Schmidt orthogonalisation;• The rank of matrices;• Projection of a vector space onto a subspace;• Similar matrices;• Eigenvalues and eigenvectors;• The characteristic polynomial;• The Cayley-Hamilton theorem;• The minimum polynomial;• Diagonalisation and applications;• The Jordan normal form;• Spectral decomposition.


Textbooks

• Lecture Notes.• Kaye R. and Wilson R., Linear Algebra, Oxford Science Publications, Oxford, 1998.• Herstein I.N., Topics in Algebra, John Wiley & Sons, 3rd Edition, 1996.• Nering E.D., Linear Algebra and Matrix Theory, John Wiley and Sons, 2nd Edition, 1970.• Nicholson R., Linear Algebra with Applications, McGraw-Hill, 2003.

MAT2113 - Rings Lecturer: Dr. A. VellaFollows from: MAT1101Leads to: MAT3905Credit value: 4Lectures: 20Tutorials: 8Semester: 2 • Axioms, examples of and elementary results on rings;• Division rings, integral domains and fields;• Ideals, quotient rings;• Factorisation in rings; • Prime and irreducible elements;

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• Euclidean rings;• Unique factorisation;• Applications to some simple results in number theory;• The ring of polymomials.


Textbooks

• Herstein I.N., Topics in Algebra, John Wiley & Sons, 3rd Edition, 1996.• Cameron P. J., Introduction to Algebra, Oxford University Press, 1998.

MAT2212 Analysis II

Lecturer: Dr. A. VellaFollows from: MAT1211Leads to: MAT2213Credit value: 4Lectures: 20Tutorials: 8Semester: 1

• Limits of functions: • Continuity, • The intermediate value theorem, • Continuous functions on [a, b], • Uniform continuity;

• Differentiability and the derivative: • The mean value theorem, • Classification of critical points, • L’Hospital’s rules, • Taylor’s theorem.


Suggested reading:

• Spivak M., Calculus, Publish or Perish, 3rd Edition, 1994 • Abbott S., Understanding Analysis, Springer, 2001• Bartle R. & Sherbert D., Introduction to Real Analysis, Wiley, 3rd Ed., 1999• Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974

MAT2213 Analysis III

Lecturer: Dr. J. L. BorgFollows from: MAT2212Leads to: MAT3214, MAT3216Credit value: 4Lectures: 20Tutorials: 8Semester: 2

• Riemann integration: • The Riemann integral, • Basic theorems on integrals, • The fundamental theorem of calculus, • Taylor’s theorem in integral form;

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• Uniform convergence: • The Weierstrass approximation theorem, • Applications to power series and Fourier series.


Suggested reading:

• Spivak M., Calculus, Publish or Perish, 3rd Edition, 1994 • Abbott S., Understanding Analysis, Springer, 2001• Bartle R. & Sherbert D., Introduction to Real Analysis, Wiley, 3rd Ed., 1999• Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974

MAT2412 – Introduction to Graph Theory

Lecturer: Prof. J. LauriFollows from: MAT1411Leads to: MAT3414Credit value: 2Lectures: 10Tutorials: 4Semester: 1

• Definitions, examples and elementary results;• Trees;• Connectivity, Max-Flow Min-Cut theorem, Menger's theorem;• Euler tours and Hamilton cycles;• Vector spaces associated with the edge-set of a graph;• Planarity I: Euler's formula, K5 and K3,3, Kuratowski's theorem;• Planarity II: the combinatorial dual, Maclane's characterization of planarity;• Planarity III: the five colour theorem, the four colour theorem, and edge colouring of cubic graphs;• Colouring of graphs on other surfaces.


Textbook:• Bondy J.A. and Murty U. S. R., Graph Theory with Applications, Macmillan, 1978.

MAT2512 - Vector Analysis I

Lecturers: Dr. J. SultanaFollows from: MAT1511Leads to: MAT3504Credit value: 4Lectures: 20Tutorials: 8Semester: 1

• Double integrals;• Triple integrals;• Applications of multiple integrals;• Change of variables in multiple integrals;• Jacobian;• Grad, div and curl operators;• Space curves;

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• Serret-Frenet formulae;• Line integral and its applications;• Conservative vector fields, scalar potential.


Suggested Reading:• Finney R.L., Weir M.D. and Giordano F.R., Thomas Calculus, 10th Edition, Addison-Wesley Longman, New York, 2001.• Camilleri C.J., Vector Analysis, Malta University Press, 1994• Roe J., Elementary Geometry, Oxford Science Publications, Clarendon Press, 1997

MAT2513 - Vector Analysis II Lecturer : Dr. J. SultanaFollows from: MAT2512Leads to: MAT3216Credit value: 4Lectures: 20Tutorials: 8Semester: 2 • The Laplacian and other operators;• Vector identities;• Introduction to surfaces;• Quadrics;• Surface integrals;• Applications of surface integrals;• Green’s theorem;• Gauss’ theorem;• Stokes’ theorem;• Orthogonal curvilinear coordinates.


Suggested Reading:• Finney R.L., Weir M.D. and Giordano F.R., Thomas Calculus, 10th Edition, Addison-Wesley Longman, New York, 2001.• Camilleri C.J., Vector Analysis, Malta University Press, 1994• Roe J., Elementary Geometry, Oxford Science Publications, Clarendon Press, 1997

MAT2912 – Computational Mathematics

Lecturers: VariousCredit value: 2Lectures: 10Tutorials: 4Semesters: 1, 2

• Introduction to mathematical and computer algebra software;• Application to various branches of Mathematics.


Suggested reading

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• Don E., Theory and Problems of Mathematica, Shaum’s Outline Series, McGraw-Hill, 2001.• Kreyszig E. and Normington E.J., Mathematica Computer Guide, John Wiley, 2002.• Wolfram S., Mathematica book, Wolfram Media, 5th Edition, 2003. Available online.

Year III

MAT3105 - Linear Algebra I

This credit is identical to MAT2112 in the second year. Second and third year students attend together for MAT2112 and/or MAT3105, and they will have the same examination. After 2008, MAT3105 will be discontinued, and only MAT2112 will be given to the second years only.

Lecturer: Prof. I. ScirihaFollows from: MAT1101Leads to: MAT3905Credit value: 4Lectures: 20Tutorials: 8Semester: 1

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• Vector spaces;• Dimension theorems;• Third isomorphism theorem;• Direct sum of spaces;• Dual spaces;• Inner product spaces;• Gram Schmidt orthogonalisation;• The rank of matrices;• Projection of a vector space onto a subspace;• Similar matrices;• Eigenvalues and eigenvectors;• The characteristic polynomial;• The Cayley-Hamilton theorem;• The minimum polynomial;• Diagonalisation and applications;• The Jordan normal form;• Spectral decomposition.


Textbooks


MAT3206 - Lebesgue Integration Lecturer: Dr. D. BuhagiarFollows from: MAT3792Leads to: MAT3218Credit value: 4Lectures: 20Tutorials: 8Semester: 2

• algebras;• Measure spaces;• Lebesgue measure on R1 ;• Measurable functions;• Lebesgue integral;• Convergence theorems;• The space .


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Suggested Reading: • Fremlin D.H., Measure Theory; Vol. 1: The Irreducible Minimum, Torres Fremlin, 2000.• Cohn D., Measure Theory, Birkhauser, 1980.• Bear H.S., A Primer of Lebesgue Integration, Academic Press, 2nd Edition, 2001.

MAT3214 - Complex Analysis

This credit will be offered for the first time in the third year, instead of the fourth year where it is offered as MAT3207. This credit will be discontinued from the fourth year after the academic year 2007-2008. Third year and fourth year students attend lectures together for MAT3214 and/or MAT3207.

Lecturer: Dr. J. SultanaFollows from: MAT1211Leads to: MAT3218Credit value: 4Lectures: 20Tutorials: 8Semester : 1

• Continuity and analytic functions;• The Cauchy-Riemann equations;• Exponential, trigonometric, hyperbolic and logarithmic functions;• Harmonic functions;• Contour integration;• Fundamental theorem of calculus;• Cauchy’s theorem;• Cauchy’s integral formulae;• Liouville’s theorem;

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• The fundamental theorem of algebra;• Sequences;• Taylor’s series;• Laurent’s series;• Zeros and poles;• Residues;• Residue theorem and its applications.


Suggested Reading

• Osborne A.D., Complex Variables and their Applications, Addison-Wesley, New York, 1999.• Priestley H., Introduction to Complex Analysis, Oxford University Press, Oxford, 1994.

MAT3504 - Vector Analysis IIA

This credit will be offered for the last time in the academic year 2007-2008. In subsequent years, the subject matter of this credit will be eventually covered in MAT2513 and MAT3216.

Lecturer : Dr. J. SultanaFollows from: MAT2503Leads to: MAT3713Credit value: 4Lectures: 20Tutorials: 8Semester: 2

• Scalar and vector fields: gradient, divergence, curl;• Vector operators and identities;• Line, surface and volume integrals;• Integral theorems of Gauss, Green and Stokes;• General orthogonal curvilinear coordinates.

• The inverse function theorem;• The implicit function theorem.


Suggested Reading

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• Finney R.L., Weir M.D. and Giordano F.R., Thomas’ Calculus, 10th Edition, Addison-Wesley Longman, New York, 2001.• Camilleri C.J., Vector Analysis, Malta University Press, Malta, 1994.• Camilleri C.J., Tensor Analysis, Malta University Press, Malta, 1999.• Rudin W., Principles of Mathematical Analysis, McGraw-Hill, 3rd Edition, 1976.• Apostol T., Calculus Vol. II: Calculus of Several Variables with applications to vector analysis, Blaisdell, 1961.• Thurston H., Intermediate Mathematical Analysis, Clarendon Press, Oxford, 1988.• Edwards C.H., Advanced Calculus of Several Variables, Academic Press, 1973.• Webb J.R.L., Functions of Several Variables, Ellis Horwood, 1991.

MAT3602 - Mechanics

Lecturer: Prof. A. BuhagiarFollows from: MAT1511, MAT1611Leads to: MAT3603Credit value: 4Lectures: 20Tutorials: 8Semester: 2

• Newton’s laws of motion:

• Cartesian coordinates: • Motion in a straight line and a plane, • Projectiles, • Simple harmonic motion;

• Intrinsic coordinates: • The catenary, • The cycloid, • The isochronous pendulum;

• Plane polar coordinates: • Central forces, • Binary systems, • Disturbed orbits and pedal coordinates;

• Principle of conservation of energy: • Conservative forces, • Equivalent conditions, • Examples: • Orbital motion,

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• Motion on a surface under gravity;

• Many-particle systems: • Motion of centroid, • Angular momentum and moment of force;

• Rotational motion: • Rotation of a rigid body about a fixed axis, • Bodies rolling in a straight line;

• Impulsive motion.


Main Text

• Lunn M., A first course in Mechanics, Oxford University Press, Oxford, 1991.

Supplementary Reading:

• Charlton F., Textbook of Dynamics, van Nostrand Company Ltd., London, 1969.• Camilleri C.J., Classical Mechanics, Malta 2004.

MAT3701 - Differential Equations

Lecturer: Prof. A. BuhagiarFollows from: MAT1211, MAT1090Leads to: MAT3712Credit value: 4Lectures: 20Tutorials: 8Semester: 2

• Picard's theorem;• Linear differential equations;• Green’s function;• Simple dynamical systems;• First order quasi-linear equations.


Suggested Reading

• Burkill J.C., The Theory of Ordinary Differential Equations, Oliver & Boyd, 1962.• Arnold V.I., Ordinary Differential Equations, MIT Press, Cambridge, Massachusetts, 1978.• Hirsch M. and Smale S., Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, 1997.• Ayres F., Differential Equations, Schaum’s Outline Series, McGraw-Hill, 1972.• Tenebaum M. and Pollard H., Ordinary Differential Equations, Dover Publications, 1985.

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MAT3792 – Metric Spaces II and Uniform Convergence

Lecturers: Dr. D. Buhagiar, Dr. J. L. BorgFollows from: MAT1211, MAT2292Leads to: MAT3206, MAT3218Credit value: 4Lectures: 20Tutorials: 8 Semester: 1

• Metric spaces: • Compactness, • Connectedness, • Completeness.

• Sequences and series of functions;• Uniform convergence: • Continuity, • Differentiability, • Integration, • Applications.


Textbook

• Kolmogorov A. N. and Fomin S.V., Introductory Real Analysis, Dover, 1975.

Suggested Reading

• Sutherland W., Introduction to Metric and Topological spaces, Clarendon Press, Oxford, 1975.• Bryant V., Metric Spaces, Cambridge University Press, 1995.• Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974.

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• Spivak M., Calculus, Publish or Perish, 3rd Edition, 1994.• Rudin W., Principles of Mathematical Analysis, McGraw-Hill, 3rd Edition, 1976.

MAT3905 - Linear Algebra II

Lecturer: Prof. I. ScirihaFollows from: MAT3105Leads to: MAT3218, MAT3414Credit value: 2Lectures: 10Tutorials: 4Semester: 1

• Algebra of linear transformations;• Dual spaces and annihilators;• The transpose;• The minimum polynomial;• T-invariant spaces;• Spectral decomposition;• Criteria for diagonalisation;• Jordan normal forms.


Textbooks


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Credits to be offered in the third year in and after 2008-2009.

MAT3115 - Groups

This credit will not be offered in 2007-2008.

Lecturer: Prof. J. LauriFollows from: MAT1101Leads to: MAT3116Credit value: 4Lectures: 20Tutorials: 8Semester: 1

• Groups acting on finite sets: the orbit-stabiliser theorem;• Conjugacy;• Strong form of Cayley's theorem;• Sylow’s theorems;• Application to the classification of groups of low order;• Automorphisms;• Permutation groups;• Burnside’s counting theorem.


Main Text

• Ledermann W. and Weir A.J., Introduction to Group Theory, Addison-Wesley Longman, 2nd Edition, 1996.

Supplementary Reading • Armstrong M.A., Groups and Symmetry, Springer Verlag, Heidelberg, 1997.• Biggs N.L., Discrete Mathematics, Oxford Science Publications, Clarendon Press, 1989.• Herstein I.N., Topics in Algebra, Wiley, 3rd Edition, 1996.

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MAT3215 – Metric Spaces


Lecturer: Dr. D. BuhagiarFollows from: MAT1211Leads to: MAT3216, MAT3217Credit value: 4Lectures: 20Tutorials: 8Semester: 1

• Axioms and examples of metric spaces; • Subspaces; • Open and closed sets; • Continuity; • Equivalent metrics; • Compactness and sequential compactness: their equivalence; • Connectedness; • Completeness.


Textbook

• Kolmogorov A. N. and Fomin S.V., Introductory Real Analysis, Dover, 1975.

Suggested Reading

• Sutherland W., Introduction to Metric and Topological spaces, Clarendon Press, Oxford, 1975.• Bryant V, Metric Spaces, Cambridge University Press, 1995.• Rudin W., Principles of Mathematical Analysis, McGraw-Hill, 3rd Edition, 1976.• Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974.

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MAT3216 – Analysis IV


Lecturer: Dr. J. L. BorgFollows from: MAT2213Leads to: MAT3218, MAT3513Credit value: 4Lectures: 20Tutorials: 8Semester: 2

• Functions of several variables: • Directional derivatives and partial derivatives;

• Vector-valued functions: • Differentiability and the total derivative; • The chain rule.

• The inverse and implicit function theorems.


Suggested reading:

• Webb J.R.L., Functions of Several Variables, Ellis Horwood, 1991• Thurston H., Intermediate Mathematical Analysis, Clarendon Press, Oxford, 1988• Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974• Rudin W., Principles of Mathematical Analysis, McGraw-Hill, 3rd Edition, 1976

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MAT3413 – Probabilistic Combinatorics


Lecturer: Dr. A. VellaFollows from: MAT2412Leads to: MAT3414Credit value: 2Lectures: 10Tutorials: 4Semester: 1

• The Monty Hall problem and the birthday paradox,• Discrete sample spaces and discrete random variables,• Bernoulli, binomial and geometric random variables,• Random graphs,• Simple lower bound for the Ramsey number ;

• Expectation and linearity of expectation,• Coupon collector’s problem,• Szele’s theorem on Hamiltonian paths,• Existence of large cuts in graphs,• Existence of large sum-free subsets within sets of positive integers;

• Randomized algorithms,• Existence of large independent sets in graphs,• Existence of graphs with large girth,• Improved lower bound for ;

• Variance and standard deviation,• Chebyshev’s inequality,• Threshold functions for balanced subgraphs,• Discussion of phase transition in random graphs;

• Conditional probability and independence,• Erdos-Ko-Rado theorem,• Lovasz local lemma,• Applications: edge-disjoint paths in a graph, - SAT, lower bound for the van der Waerden function;

• Turan’s theorem.

Method of Assessment: 80% Examination, 20% Oral

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Textbooks

• Mitzenmacher M. and Upfal E., Probability and Computing. Randomized Algorithms and Probabilistic Analysis, Cambridge University Press, 2005.• Alon N. and Spencer J. H., The Probabilistic Method, John Wiley and Sons, 1992.

Year IV

MAT3116 – Group Representations

Lecturer: Prof. J. LauriFollows from: MAT1101, MAT3115Leads to: MAT5411 Credit value: 5Lectures: 18Tutorials: 7Semester: 1, 2

• Quick review of groups, vector spaces and linear transformations;• Group representation over the complex field: matrix and module point of view;• Maschke’s theorem and Schur’s lemma; reducibility and complete reducibility;• Characters, inner products of characters and character tables;• Some applications.


Main Text

• James G. and Liebeck M., Representations and Characters of Groups, Cambridge University Press, Cambridge, 2001.

Supplementary Reading

• Serre J. P., Linear representations of Finite Groups, 4th Edition, Springer, 1977. • Nering E. D., Linear Algebra and Matrix Theory, 2nd Edition, Wiley, 1970.

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MAT3207 - Complex Analysis

This credit will be given in the fourth year for the last time in 2007-2008. Subsequently it will be offered in the third year as MAT3214.

Lecturer: Dr. J. SultanaFollows from: MAT1211Leads to: MAT3218Credit value: 5Lectures: 18Tutorials: 7Semesters : 1, 2

• Continuity and analytic functions;• The Cauchy-Riemann equations;• Exponential, trigonometric, hyperbolic and logarithmic functions;• Harmonic functions;• Contour integration;• Fundamental theorem of calculus;• Cauchy’s theorem;• Cauchy’s integral formulae;• Liouville’s theorem;• The fundamental theorem of algebra;• Sequences;• Taylor’s series;• Laurent’s series;• Zeros and poles;• Residues;• Residue theorem and its applications.


Textbooks

• Osborne A.D., Complex Variables and their Applications, Addison-Wesley, New York, 1999.• Priestley H., Introduction to Complex Analysis, Oxford University Press, Oxford, 1994.

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MAT3219 – General Topology

Lecturers: Dr. D. BuhagiarFollows from: MAT3792Leads to: MAT5211Credit value: 5Lectures: 18Tutorials: 7Semesters: 1, 2

• Cardinals and ordinals;• Subspaces, product spaces and quotient spaces;• Separation axioms and countability axioms;• Connectedness, compactness and paracompactness.


Main Text

• 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.

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MAT3513 – Tensors and Relativity

Lecturer: Dr. J. SultanaFollows from: MAT3504Leads to: MAT5611Credit value: 5Lectures: 18Tutorials: 7Semesters: 1, 2

• Differentiable manifolds;• Maps of manifolds;• Tangent and cotangent spaces;• Bases;• Tensors and tensor algebra;• The metric tensor;• Tensor transformation law;• Tensor fields;• Christoffel symbols;• Covariant derivative;• Parallel propagation and geodesics;• The Riemann tensor and its symmetries;• The Ricci tensor, curvature scalar and Weyl tensor;• The Bianchi identities;• Principle of equivalence;• Gravitation as space-time curvature;• Energy-momentum tensor;• Perfect fluids;• Einstein’s field equations;• Schwarzschild black hole solution.


Suggested Reading

• Carroll S.M., Spacetime and Geometry: An Introduction to General Relativity, 1st Edition, Addison Wesley, New York, 2003.

• Schutz B. F., A first course in General Relativity, Cambridge University Press, Cambridge, 1985.• Camilleri C.J., Tensor Analysis, Malta University Press, Malta, 1999.

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MAT3613 - Classical Mechanics

Lecturer: Prof. A. BuhagiarFollows from: MAT1501, MAT3602Leads to: MAT5611, MAT5711Credit value: 5Lectures: 18Tutorials: 7 Semesters: 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, • Impulsive motion.


Main Texts

• Lunn M., A first Course in Mechanics, Oxford University Press, Oxford, 1991.


• Camilleri C.J., Classical Mechanics, Malta, 2004.• Charlton F., Textbook of Dynamics, van Nostrand Company Ltd., London, 1969.• 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.

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MAT3713 - The Finite Element Method

Lecturer: Prof. A. BuhagiarFollows from: MAT3602, MAT3701Leads to: MAT5711Credit value: 5Lectures: 18Tutorials: 7Semesters: 1, 2

• Approximate solution of boundary value problems: • Variational methods, • Weighted residual methods;

• Variational methods: • The Rayleigh-Ritz method, • Minimisation of the energy functional, • Application to bars and beams, • Discretisation of region, • The piecewise Rayleigh-Ritz method, • One dimensional elements, • The element shape functions, • The linear bar element, • The cubic beam element, • Element stiffness and consistent loading, • Assembly and solution of stiffness equations, • Estimate of accuracy of approximate solution;

• Weighted residual methods: • The Galerkin method, • The one-dimensional Poisson equation: • Dirichlet, derivative or mixed boundary conditions,

• Solution with linear or quadratic elements; • The two-dimensional Poisson equation: • Solution with linear triangular elements;

• Applications of above: • Axial extension and vibration of bars, • Bending of beams, • Torsion, • Heat Transfer, • Groundwater flow.

Method of Assessment: Examination 100%.

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.

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MAT3712 – Partial Differential Equations and Calculus of Variations


Lecturers: Dr. J. MuscatFollows from: MAT3701Leads to: MAT3613, MAT5711Credit value: 5Lectures: 18Tutorials: 7Semesters: 1, 2

• Partial differential equations: • Separable solutions; • Elliptic equations: • Gravitation,

• Laplace’s equation, • Poisson’s equation,

• Harmonic functions; • Hyperbolic equations:

• Waves, • D’Alembert solution;

• Parabolic equations: • Heat, • Fundamental solution.

• Calculus of variations: • Motivating examples, • Continuous and piecewise differentiable solution, • The Euler-Lagrange equation, • Problems with fixed and non-fixed endpoints, • Constraints, • Necessary conditions for a minimum and corner conditions.


Suggested Reading

• Sneddon I., Elements of Partial Differential Equations, McGraw-Hill, 1957.• Haberman R., Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value

Problems, Prentice Hall, New Jersey, 3rd Edition, 1998.• Farbon S., Partial Differential Equations for scientists and Engineers, Dover Publications, 1993.

• Clegg J.C., Calculus of Variations, Oliver and Boyd, 1968.• Pars L.A., An Introduction to the Calculus of Variations, Heinemann, 1962.• Pinch E.R., Optimal Control and the Calculus of Variations, Oxford University Press, 1993.

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MAT3196 – Group Representations

Lecturer: Prof. J. LauriFollows from: MAT1101, MAT3115Leads to: MAT5411 Credit value: 6Lectures: 22Tutorials: 8Semester: 1, 2

• Quick review of groups, vector spaces and linear transformations;• Group representation over the complex field: matrix and module point of view;• Maschke’s theorem and Schur’s lemma; reducibility and complete reducibility;• Characters, inner products of characters and character tables;• Some applications;• Further topics.


Main Text

• James G. and Liebeck M., Representations and Characters of Groups, Cambridge University Press, Cambridge, 2001.


• Serre J. P., Linear representations of Finite Groups, 4th Edition, Springer, 1977. • Nering E. D., Linear Algebra and Matrix Theory, 2nd Edition, Wiley, 1970.

MAT3299 – General Topology

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Lecturers: Dr. D. BuhagiarFollows from: MAT3792Leads to: MAT5211Credit value: 6Lectures: 22Tutorials: 8Semesters: 1, 2

• Cardinals and ordinals;• Subspaces, product spaces and quotient spaces;• Separation axioms and countability axioms;• Connectedness, compactness and paracompactness;• Further topics.


Main Text

• 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.

MAT3593 – Tensors and Relativity

Lecturer: Dr. J. SultanaFollows from: MAT3504Leads to: MAT5611Credit value: 6

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Lectures: 22Tutorials: 8Semesters: 1, 2

• Differentiable manifolds;• Maps of manifolds;• Tangent and cotangent spaces;• Bases;• Tensors and tensor algebra;• The metric tensor;• Tensor transformation law;• Tensor fields;• Christoffel symbols;• Covariant derivative;• Parallel propagation and geodesics;• The Riemann tensor and its symmetries;• The Ricci tensor, curvature scalar and Weyl tensor;• The Bianchi identities;• Principle of equivalence;• Gravitation as space-time curvature;• Energy-momentum tensor;• Perfect fluids;• Einstein’s field equations;• Schwarzschild black hole solution;• Further topics.


Suggested Reading

• Carroll S.M., Spacetime and Geometry: An Introduction to General Relativity, 1st Edition, Addison Wesley, New York, 2003.

• Schutz B. F., A first course in General Relativity, Cambridge University Press, Cambridge, 1985.• Camilleri C.J., Tensor Analysis, Malta University Press, Malta, 1999.

MAT3693 - Classical Mechanics

Lecturer: Prof. A. BuhagiarFollows from: MAT1501, MAT3602Leads to: MAT5611, MAT5711Credit value: 6Lectures: 22Tutorials: 8

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Semesters: 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, • Impulsive motion;

• Further topics.


Main Texts

• Lunn M., A first Course in Mechanics, Oxford University Press, Oxford, 1991.


• Camilleri C.J., Classical Mechanics, Malta, 2004.• Charlton F., Textbook of Dynamics, van Nostrand Company Ltd., London, 1969.• 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.

MAT3793 - The Finite Element Method

Lecturer: Prof. A. BuhagiarFollows from: MAT3602, MAT3701Leads to: MAT5711Credit value: 6Lectures: 22Tutorials: 8 Semesters: 1, 2

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• Approximate solution of boundary value problems: • Variational methods, • Weighted residual methods;

• Variational methods: • The Rayleigh-Ritz method, • Minimisation of the energy functional, • Application to bars and beams, • Discretisation of region, • The piecewise Rayleigh-Ritz method, • One dimensional elements, • The element shape functions, • The linear bar element, • The cubic beam element, • Element stiffness and consistent loading, • Assembly and solution of stiffness equations, • Estimate of accuracy of approximate solution;

• Weighted residual methods: • The Galerkin method, • The one-dimensional Poisson equation: • Dirichlet or derivative boundary conditions,

• Solution with linear or quadratic elements; • The two-dimensional Poisson equation: • Solution with linear triangular elements;

• Applications of above: • Axial extension and vibration of bars, • Bending of beams, • Torsion, • Heat Transfer, • Groundwater flow;

• Further topics.

Method of Assessment: Examination 100%.

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.

MAT3782 – Partial Differential Equations and Calculus of Variations


Lecturers: Dr. J. MuscatFollows from: MAT3701Leads to: MAT3613, MAT5711Credit value: 6

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Lectures: 22Tutorials: 8Semesters: 1, 2

• Partial differential equations: • Separable solutions; • Elliptic equations: • Gravitation,

• Laplace’s equation, • Poisson’s equation,

• Harmonic functions; • Hyperbolic equations:

• Waves, • D’Alembert solution;

• Parabolic equations: • Heat, • Fundamental solution.

• Calculus of variations: • Motivating examples, • Continuous and piecewise differentiable solution, • The Euler-Lagrange equation, • Problems with fixed and non-fixed endpoints, • Constraints, • Necessary conditions for a minimum and corner conditions.

• Further topics.


Suggested Reading

• Sneddon I., Elements of Partial Differential Equations, McGraw-Hill, 1957.• Haberman R., Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value

Problems, Prentice Hall, New Jersey, 3rd Edition, 1998.• Farbon S., Partial Differential Equations for scientists and Engineers, Dover Publications, 1993.

• Clegg J.C., Calculus of Variations, Oliver and Boyd, 1968.• Pars L.A., An Introduction to the Calculus of Variations, Heinemann, 1962.• Pinch E.R., Optimal Control and the Calculus of Variations, Oxford University Press, 1993.

THE PROJECT MODULES

For the project modules, students choose either MAT3218 or MAT3414 (12 credits), together with MAT3990, the Mathematics Seminar (6 credits), for a total of 18 credits. These credits are described next.

MAT3218 - Functional Analysis

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Lecturers: Dr. D. Buhagiar, Dr. J. HamhalterFollows from: MAT3206Leads to: MAT5211 Credit value: 12Lectures: 45Tutorials: 15Semesters: 1, 2

• Metric spaces and their completion;• Normed vector spaces, Banach spaces;• Examples of normed spaces: the spaces ;• Bounded linear operators, dual spaces;• Hilbert spaces;• Orthogonal projections.

• Hahn-Banach theorem;• Uniform boundedness theorem;• Open mapping theorem, closed graph theorem;• Spectral theory;• Compact and self adjoint operators.


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.

MAT3414 - Discrete Mathematics

Lecturers: Prof. J. Lauri, Prof. I. Sciriha.Follows from: MAT2412Leads to: MAT5411 Credit value: 12 Lectures: 45Tutorials: 15Semesters: 1, 2

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• Definitions and elementary results on graphs;• Trees;• Connectivity;• Euler tours and Hamilton cycles;• The cycle index of a permutation group and the use of Polya’s theorem;• Ramsey’s theorem for graphs;

• Vector spaces associated with graphs;• Cycle-cutset duality;• Graph colourings;• Planar graphs;• Introduction to error-correcting codes;• Introduction to combinatorial designs.


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.


• Cameron P.J., Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, 1994.• Bryant V., Aspects of Combinatorics: A Wide Ranging Introduction, Cambridge University Press, Cambridge, 1993.

MAT3990 – Mathematics Seminar

Lecturers: VariousCredit value: 6Semesters: 1, 2

• In this study unit, a long essay and a seminar are presented by the student on a suitable topic in Mathematics.

Method of Assessment: Essay/Seminar 100%

FACULTIES OF ENGINEERING & ARCHITECTURE

Year I

MAT1801 – Mathematics for Engineers I

Lecturer: Dr. J. L. Borg, Dr. A. Vella, Mr. J. B. GauciFollows from: A-LevelLeads to: MAT1802Credit value: 4Lectures: 20Tutorials: 8Semester: 1

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• Matrices and determinants;• Systems of linear equations;• Matrices: eigenvalues and eigenvectors.

• First order differential equations;• Second order linear differential equations with constant coefficients;• Partial differentiation.


Textbooks

• Zill D.G. and Cullen M.R., Advanced Engineering Mathematics, Jones and Bartlett Publishers, 3rd Edition, 2006.

• Spiegel M.R., Advanced Calculus, Schaum’s Outline Series, McGraw-Hill, 1981.• Finney R.L., Weir M.D., Giordano F.R., Thomas’ Calculus, Addison-Wesley Longman, New York, 10th

Edition, 2001.

MAT1802 - Mathematics For Engineers II

Lecturer: Dr. J. L. Borg, Dr. A. Vella, Mr. J. B. GauciFollows from: A-level, MAT1801Leads to: MAT2803, MAT2804Credit value: 4Lectures: 20Tutorials: 8Semester: 2

• Vector algebra and introduction to vector spaces;• Transformation of rectangular Cartesian coordinates on a plane and in space; linear transformations;• Linear objects: lines (in 2D and 3D), planes (in 3D).

• Double integrals;• Fourier series;• Sequences and series


Textbooks

• Zill D.G. and Cullen M.R., Advanced Engineering Mathematics, Jones and Bartlett Publishers, 3rd Edition, 2006.

• Spiegel M.R., Advanced Calculus, Schaum’s Outline Series, McGraw-Hill, 1981.• Finney R.L., Weir M.D., Giordano F.R., Thomas’ Calculus, Addison-Wesley Longman, New York, 10th

Edition, 2001.Year II

MAT2803 - Laplace and Fourier Transforms

Lecturer: Dr. J. L. BorgFollows from: MAT1802Credit value: 2Lectures: 10Tutorials: 4Semester: 1

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• Laplace transforms: • The delta function and the unit step function, • Transforms of elementary functions, • Properties including linearity, scaling, shift, modulation, convolution and correlation, • Transforms of integrals and derivatives, • Differential equations; solution of initial and boundary value problems, • Impulse response and the transfer function;

• Fourier transforms: • Fourier’s identity, • Transform pairs, duality and symmetry, • Properties including linearity, scaling, shift, modulation, convolution and correlation, • Rayleigh’s theorem and the power theorem, • Transforms of derivatives, • Solution of boundary value problems.


Main Text• Zill D.G. and Cullen M.R., Advanced Engineering Mathematics, Jones and Bartlett Publishers, 3rd Edition, 2006.

Supplementary Reading• Spiegel M.R., Laplace Transforms, Schaum’s Outline Series, McGraw-Hill, New York, 1994.• Senior T.B., Mathematical Methods in Electrical Engineering, Cambridge University Press, Cambridge, 1986.• Bracewell R.N., The Fourier Transform and its Applications, 3rd Edition, McGraw-Hill, New York, 2000.

MAT2804 - The Eigenvalue Problem and Multiple Integrals

Lecturer: Dr. J. L. BorgFollows from: MAT1801, MAT1802Credit value: 2Lectures: 10Tutorials: 4Semester: 2

• Eigenvalues and eigenvectors with application to simultaneous linear differential equations;• Transformations of double and triple integrals.


Textbook• Zill D.G. and Cullen M.R., Advanced Engineering Mathematics, Jones and Bartlett Publishers, 3rd Edition, 2006.Year III

MAT3805 – Optimisation, Complex & Numerical Analysis I

Lecturer: Dr. J. Sultana, Prof. A. BuhagiarFollows from: MAT1801, MAT1802Credit value: 4Lectures: 20Tutorials: 8Semester: 1

• Optimisation: • Local and global extrema of functions of several variables,

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• Lagrange’s method for constrained problems, • Lagrange’s method with two constraints;

• Functions of a complex variable: • Notation and definitions, • Limits, continuity and analytic functions, • The Cauchy-Riemann equations, • Exponential, trigonometric, hyperbolic, logarithmic and power functions, • Harmonic functions.

• Numerical Analysis:

• Locating roots of equations: • The Newton Raphson in one and two dimensions: rate of convergence, • The variable secant method, • The fixed point theorem, x = f (x ), • The method of steepest descent, • Bairstow’s method for the quadratic factors of a polynomial with real coefficients, • The quotient difference method for the roots of a polynomial;

• Solution of linear equations: • Gaussian elimination, • Cholesky’s LU and LLT methods, • Iterative methods: Jacobi, Gauss-Seidel and the SOR methods;

• Eigenvalues and eigenvectors: • Bounds for eigenvalues using Gershgorin’s theorem, • The power method for the largest and smallest eigenvalues, and the corresponding eigenvectors, • Rayleigh’s quotient for Hermitean matrices, • Jacobi’s method of rotations for real symmetric matrices, • Applications to quadratic forms and normal modes.


Textbooks• Zill D.G. and Cullen M.R., Advanced Engineering Mathematics, Jones and Bartlett Publishers, 3rd Edition, 2006.• Gerald C.F. and Wheatley P.O., Applied Numerical Analysis, 6th Edition, Addison-Wesley, New York, 1997.

Supplementary Reading• Press W.H., Flannery B.P., Teukolsky S.A. and Vetterling W.T., Numerical Recipes in Fortran, Cambridge University Press, Cambridge, 1989. • Rajasekaran S., Numerical Methods in Science and Engineering, 2nd Edition, Wheeler Publishing, New Delhi, 1999.• Burden R. L. and Faires J. D., Numerical Analysis, 7th Edition, Brooks Cole, London, 2001.• Cheney E. W. and Kincaid D. R., Numerical Mathematics and Computing, 4th Edition, Brooks Cole, 1999.

MAT3806 - Optimisation, Complex & Numerical Analysis II

Lecturer: Dr. J. Sultana, Prof. A. BuhagiarFollows from: MAT1801, MAT1802Credit value: 4Lectures: 20Tutorials: 8Semester: 2

• Functions of a complex variable:

• Contour integration, • Fundamental theorem of calculus, • Cauchy’s theorem,

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• Cauchy’s integral formulae, • Sequences and series, • Taylor’s series, • Laurent’s series, • Zeros and poles, • Residues.

• Numerical analysis:

• Interpolation: • Lagrangian interpolation, • The divided difference table, • The difference table and the Newton Gregory forward polynomial, • Inverse interpolation, • Spline interpolation;

• Numerical differentiation: • Central difference formulae for the first and second derivatives, • Forward difference formulae for the first derivative, • Improvement by extrapolation;

• Numerical integration: • Trapezoidal rule and Simpson’s 1/3 and 3/8 rules, • Errors for the local and global versions of these rules, • Gaussian quadrature;

• Ordinary differential equations: • Euler’s method, • The modified Euler method, • The Runge-Kutta method;

• The finite difference method for partial differential equations: • Laplace’s equation, • Poisson’s equation, • The transient heat equation.


Textbooks• Zill D.G. and Cullen M.R., Advanced Engineering Mathematics, Jones and Bartlett, 3rd Edition, 2006.• Gerald C.F. and Wheatley P.O., Applied Numerical Analysis, 6th Edition, Addison-Wesley, 1997.

Supplementary Reading• Press W.H., Flannery B.P., et al., Numerical Recipes in Fortran, Cambridge University Press, 1989. • Rajasekaran S., Numerical Methods in Science and Engineering, 2nd Edition, Wheeler Publishing, 1999.• Burden R. L. and Faires J. D., Numerical Analysis, 7th Edition, Brooks Cole, London, 2001.• Cheney E. W. and Kincaid D. R., Numerical Mathematics and Computing, 4th Edition, Brooks Cole, 1999.

FACULTY OF ECONOMICS, MANAGEMENT & ACCOUNTANCY

Year I

MAT1991 - Mathematics I and II

Lecturers: Prof. A. Buhagiar, Dr. A. Vella, Mr. J. ScirihaFollows from: Intermediate MathematicsLeads to: MAT1902Credit value: 8Lectures: 40Tutorials: 16

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Semester: 1, 2

This credit comprises the two study units MAT1901 and MAT1902 which are described below. Students cannot take MAT1991 with MAT1901 and / or MAT1902.

MAT1901 - Mathematics I

Lecturers: Prof. A. Buhagiar, Dr. A. Vella, Mr. J. ScirihaFollows from: Intermediate MathematicsLeads to: MAT1902Credit value: 4Lectures: 20Tutorials: 8Semester: 1

• Rectangular Cartesian coordinates;

• Linear equalities and inequalities: • The straight line, • Systems of linear equalities and inequalities and their representation on the plane, • Introduction to linear programming in two variables;

• Functions: • The linear, quadratic, exponential and logarithmic functions and their graphs, • Functions related to business: cost, revenue, profit, demand and supply functions;

• Matrices: • Basic properties, • Elementary row operations, • Determinants, • The matrix inverse, • Systems of linear equations, • Easy applications;

• Differential calculus: • Differentiation of simple algebraic, exponential and logarithmic functions, • Maxima and minima, • Applications to business problems.


Recommended Text

• Barnett R.A. and Ziegler M.R., College Mathematics, Prentice Hall, New York, 1996.

MAT1902 - Mathematics II

Lecturers: Dr. A. Vella, Prof. A. Buhagiar, Mr. J. ScirihaFollows from: MAT1901Credit value: 4Lectures: 20Tutorials: 8Semester: 2

• Mathematics of finance: • Simple and compound Interest, • Present and future value, • Effective rate of Interest, • Annuities and sinking funds;

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• Integration: • Integral calculus, • Antiderivatives, • Integration by substitution, • Integration by parts, • The definite integral, • Calculation of areas, • The trapezoidal rule, • Applications like consumers’ surplus and the Lorenz curve, • Solution of differential equations using separation of variables;

• Matrices: • The input-output model, • The brand-switching model;

• Functions of two variables: • Partial derivatives, • Maxima, minima and saddle points, • Optimisation with constraints, Lagrange multipliers, • Applications to problems in business and economics.


Recommended Text

• Barnett R.A. and Ziegler M.R., College Mathematics, Prentice Hall, New York, 1996.

BOARD OF STUDIES FOR IINFORMATION TECHNOLOGY

Year I

MAT1803 - Mathematical Transform Techniques

Lecturer: Mr. J. BorgFollows from: MAT1001, MAT1002Credit value: 2Lectures: 10Tutorials: 4Semester: 2

• Laplace transforms;• Fourier series;• Fourier transforms.

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Textbooks

• Spiegel M.R., Laplace Transforms, Schaum’s Outline Series, McGraw-Hill, New York, 1994.• Senior T.B., Mathematical Methods in Electrical Engineering, Cambridge University Press, Cambridge, 1986.

MAT2402 - Networks

Lecturer: Prof. J. LauriFollows from: A-LevelLeads to: MAT3490Credit value: 2Lectures: 10Tutorials: 4Semester: 1

• Basic graph theoretical concepts;• Eigenvector methods for ranking vertices in a graph;• Shortest paths and minimum spanning tree;• Flow in networks, max-flow-min-cut theorem;• Matching in bipartite graphs;• Critical path and PERT techniques;• Scheduling and sequencing problems;• Travelling salesman problem.

Method of Assessment: Examination 100% Main Text: One of • Dolan A., Aldous J., Networks and Algorithms, An Introductory Approach, John Wiley & Sons, 1994.or• Dossey J. A., Otto A. D., Spence L.E. and Eynden C.V., Discrete Mathematics, Addison–Wesley, 5th Edition, 2006.

Supplementary reading• Biggs N.L., Discrete Mathematics, Oxford Science Publications, Clarendon Press, Oxford, 1989.• Gibbons A., Algorithmic Graph Theory, Cambridge University Press, 1985.• Wilson R.J., Introduction to Graph Theory, Longman, 4th Edition, 1996.

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introduction - university of maltahome.um.edu.mt/maths/math0708.doc · web view• haberman r.,...

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