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PROGRAM OF STUDIES B.Sc. (Hons.) Degree

MATHEMATICS

2010 – 2011

22nd February 2010 Year I Mathematics students in the first year of the B.Sc.(Hons.) course must take all the compulsory credits and a choice of four elective credits.

Elective Credits (for students NOT taking Statistics and Operations Research as a Principal subject): Students must choose either [SOR1110] or [SOR1211 PLUS one of SOR1221 or SOR1311].

SOR1110 Probability 4 1st & 2nd Prof. L. Sant /

Mr. D. Suda

SOR1211 Probability

2

1st

Various

SOR1221

Sampling, Estimation and

Regression

2

2nd

Various

SOR1311 Linear Programming

2 2nd

Various

NOTE: SOR1110 is more demanding than SOR1211 + SOR1221/SOR1311

Type of unit Code Title of unit Credit Value

Semester Lecturers

Compulsory Credits:

MAT1090 Mathematical Methods 3 1st Prof. I. Sciriha /

Ms. C. Zerafa

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

MAT1511 Analytical Geometry 4 1st Dr. J. Muscat

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

MAT1211 Analysis I 4 2nd Prof. D. Buhagiar

MAT1411 Discrete Methods 4 2nd Dr. P. Borg

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Elective Credits (for students taking Statistics and Operations Research as a Principal subject): Students must choose either [MAT1611] or [PHY1020 and PHY1030].

+ MAT1611 Introductory Mechanics 4 1st Mr. V. Mercieca

PHY1020 PHY1030

Basic Concepts in Physics I Basic Concepts in Physics II

2 2

1st

2nd

Various

Various

+ This study unit is highly recommended, especially for students who do not have Advanced Applied Mathematics

Optional credits offered to other departments The Mathematics department offers the following as optional credits to other departments and/or faculties: # MAT1001 Mathematical Methods: Matrices 2 1st Ms. C. Zerafa

# MAT1091 Mathematical Methods: Matrices and

Differential Equations

4 1st Prof. I. Sciriha /

Ms. C. Zerafa # MAT1092 Mathematical Methods for ICT 3 1st Ms. C. Zerafa

% MAT1611 Introductory Mechanics 4 1st Mr. V. Mercieca

$ MAT1951 Elementary Calculus I 2 1st Ms. E. Manicaro

$ MAT1952 Elementary Calculus II 2 2nd Ms. E. Manicaro

# Students can only register for one study unit from MAT1001, MAT1091 and MAT1092. These credits cannot be taken

by students in B.E.& A., B.Eng., B.Sc in Computer and Communications Engineering, and B.Sc. with Mathematics as one of the options.

% This optional credit is intended mostly for B. Sc. (Hons.) and B. Ed. (Hons.) students.

$ Cannot be taken by students in B. Com., B. E. & A., B. Eng. (Hons.) or B. Ed. (Mathematics and/or Physics) students. Repeated twice, once to B. Pharm. students, and once to the other groups. MAT1952 cannot be taken without MAT1951.

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Year II Mathematics students in the second year of the B.Sc.(Hons.) course must take all the compulsory credits and a choice of two elective credits.

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. E. Chetcuti

MAT2413 Introduction to Graph Theory

with Applications

4 1st Prof. J. Lauri

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

MAT2113 Rings 4 2nd Dr. P. Borg

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

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

Elective Credits: Students must choose 2 credits from the following list.

£ MAT2912 Computational Mathematics 2 1st & 2nd Various

* MAT2913 Numerical Analysis 2 1st Prof. A. Buhagiar

£ Elective credit offered only to students in the B.Sc. (Hons.) course. This credit is highly recommended to Mathematics students taking Statistics and Operations Research or Physics in the B.Sc.(Hons.) course. * Optional credit offered also to other departments. MAT2913 cannot be taken together with MAT2814.

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

Type of unit

Code Title of unit Credit

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. E. Chetcuti

MAT3215 Metric Spaces 4 1st Prof. D. Buhagiar

MAT3413 Probabilistic Combinatorics 2 1st Dr. P. Borg

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

MAT3217 Lebesgue Integration 4 2nd Dr. E. Chetcuti

MAT3612

Mechanics 4 2nd Prof. A. Buhagiar

MAT3711 Differential Equations 4 2nd Dr. J. Muscat

Year IV

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 19 credits from those labeled Project, and three electives of 5 credits each.

Students taking 26 credits in Mathematics must take: One elective of 6 credits and four electives of 5 credits each.

Type of unit

Code Title of unit Credit

Value

Semester Lecturers

Project MAT3218 Functional Analysis 12

1st & 2nd Prof. D. Buhagiar

/ Dr. J. Muscat

Project MAT3414 Discrete Mathematics 12 1st & 2nd Prof. I. Sciriha

/ Prof. J. Lauri

Project MAT3997 Mathematics Seminar 7 1st & 2nd Various

continued …

5

… continued

Type of unit

Code Title of unit Credit

Value

Semester Lecturers

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

Elective MAT3219 Topological Spaces 5 1st & 2nd Prof. D. Buhagiar

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

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

Elective MAT3712 Partial Differential Equations &

Calculus of Variations

5 1st & 2nd Dr. J. Muscat

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

Elective MAT3270 Functional Analysis: Normed Spaces 6 1st & 2nd Prof. D. Buhagiar

Elective MAT3271 Functional Analysis: Hilbert Spaces 6 1st & 2nd Dr. J. Muscat

Elective MAT3470 Graph Theory 6 1st & 2nd Prof. I. Sciriha

Elective MAT3471 Combinatorics 6 1st & 2nd Prof. J. Lauri

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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 three years of the B.Sc.(Hons.) course in Mathematics. A selection of 36 ECTS is also offered to Students taking Mathematics as a subsidiary subject in other programs of study. The following study units are given to students in the Faculties of Engineering, ICT, FEMA and also to students reading Mathematics as a Subsidiary subject in other programs of study. A detailed description of these courses is given in the following pages.

Year Code Title of unit Credit Value

Semester Lecturers

Faculty of Economics, Management and Accountancy (F EMA)

†MAT1991 Mathematics I and II 8 1st & 2nd Dr. P. Borg /

Various

This study unit comprises two credits as follows: %†MAT1901 Mathematics I 4 1st Dr. P. Borg /

Various

And

Year 1

† MAT1902 Mathematics II 4 2nd Dr. P. Borg /

Various

Faculty of ICT / Board of Studies for Information T echnology

^@MAT1001 Mathematics Methods: Matrices 2 1st Ms. C. Zerafa

Or

Year I

@MAT1092 Mathematical Methods for ICT 3 1st Ms. C. Zerafa

MAT1411 Discrete Methods 4 2nd Dr. P. Borg

MAT2413 Introduction to Graph Theory

with Applications

4 1st Prof. J. Lauri

Year II

MAT2814 Numerical Analysis with Matlab 4 2nd Prof. A. Buhagiar

Please turn over for other study units to other faculties.

† Students cannot take MAT1991 with MAT1901 and/or MAT1902. % For students in B.Sc. (Tourism Studies) ^ This credit is part of MAT1092, which is also available to students in ICT. @ Students cannot take MAT1001 with MAT1092 or vice-versa.

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Year Code Title of Unit Credit Value

Semester Lecturers

Faculties of Engineering and Architecture

Core Credits: MAT1801 Mathematics for Engineers I 4 1st Dr. J. L. Borg /

Various

MAT1802 Mathematics for Engineers II 4 2nd Dr. E. Chetcuti / Various

Core Credits: MAT2803 Laplace and Fourier Transforms 2 1st Dr. J. B. Gauci

MAT2804 The Eigenvalue problem and multiple integrals

2 2nd Dr. J. B. Gauci

Year II

MAT2814 Numerical Analysis with Matlab 4 2nd Prof. A. Buhagiar

Faculty of Engineering only

Core Credits: Year III MAT3815 Mathematics for Engineers III 4 1st Dr. J. Sultana

Mathematics as a Subsidiary Subject A total of 36 ECTS credits are selected from the B.Sc. credits in Mathematics to form a subsidiary option in Mathematics within other programs of study. The following are the study units offered:

Year Code Title of unit Credit Value

Semester Lecturers

Credits for Subsidiary Mathematics:

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

MAT1090 Mathematical Methods 3 1st Prof. I. Sciriha /

Ms. C. Zerafa

MAT1411 Discrete Methods 4 2nd Dr. P. Borg

Year I

MAT2913 Numerical Analysis 2 1st Prof. A. Buhagiar

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

MAT1211 Analysis I 4 2nd Prof. D. Buhagiar

Year II

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

MAT2212 Analysis II 4 1st Dr. E. Chetcuti

Year III

MAT2413 Introduction to Graph Theory with

Applications

4 1st Prof. J. Lauri

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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 ICT. The following is a list of the Full Time Mathematics Academic Staff in the Department of Mathematics: • Dr. J. L. Borg • Dr. P. Borg • Prof. A. Buhagiar • Prof. D. Buhagiar, Head of Department • Dr. E. Chetcuti • Prof. J. Lauri • Dr. J. Muscat • Prof. I. Sciriha • Dr. J. Sultana The number of students taking Mathematics as a principal subject for their B.Sc.(Hons.) has now stabilised at about 50 per year. Besides, about 20 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 300 first year students in FEMA, to Engineering classes of about 250. 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, Functional Analysis, General Topology and Mathematical Physics. Except where otherwise stated, study units in the B.Sc. and M.Sc. in Mathematics are assessed by written test according to the following norms: Year of Course Number of

Credits Exam Duration

(Hours) Number of Lectures

Number of questions set

Number of questions Chosen

I – IV 2 1½ - 3 2

I – III 3 / 4 / 5 2 - 4 3

I – III 6 3 - 5 4

IV 5 2 - 4 3

IV 6 3 - 5 4

IV Project 12

( = 6 + 6 )

3

3

-

-

5

5

4

4

Masters Total = 24

(= 12 + 12)

3

3

-

-

4

4

3

3

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

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COURSE DESCRIPTIONS B.Sc. (Hons.) Year I

�MAT1090 - Mathematical Methods Lecturers: Prof. I. Sciriha / Ms. C. Zerafa Follows from: A-Level Leads to: MAT1101, MAT3711 Credit value: 3 Lectures: 18 Tutorials: 3 Semester: 1 • Matrices; • Determinant, rank, trace and inverse of a matrix, • Solution of linear equations, • Eigenvalues and diagonalisation, • Applications; • Ordinary differential equations: • First order differential equations, • Second order differential equations; • Partial differentiation. Method of Assessment: Examination 100%, Duration 1½ hours Textbooks • Andvilli S. and Hecker D., Elementary Linear Algebra, Harcourt Academic Press, 1999. • Nagle K.B., Saff E.B., and Snider A.D., Fundamentals of Differential Equations and Boundary Value Problems, Addison-Wesley, 4th Edition, 2003. • Kreyszig E., Advanced Engineering Mathematics, John Wiley & Sons, 8th Edition, 1998.

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�MAT1000 - Introductory Mathematics Lecturer: Dr. J. L. Borg Follows from: A-Level Leads to: MAT1101, MAT1211 Credit value: 5 Lectures: 30 Tutorials: 5 Semester: 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%, Duration 2 hours. 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, 2nd Edition, 2000. • Devlin K.J., Sets, Functions and Logic, Chapman and Hall, 3rd Edition, 2003. • Epp S., Discrete Mathematics with Applications, Brooks Cole, 3rd Edition, 2003. • Daepp U. and Gorkin P., Reading, Writing and proving: A Closer Look at Mathematics, Springer, 2003.

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�MAT1101 - Groups and Vector Spaces Lecturer: Prof. I. Sciriha Follows from: MAT1000 Leads to: MAT2112 Credit value: 6 Lectures: 36 Tutorials: 6 Semester: 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; • Steinitz replacement process; • Linear transformations and matrices; • Dimension theorem, nullity, rank; • Change of basis; • Transition matrices.

Method of Assessment: Examination 100%, Duration 3 hours 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.

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�MAT1211 - Analysis I Lecturer: Prof. D. Buhagiar Follows from: MAT1000 Leads to: MAT2212 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 2 • Countability: • Countable and uncountable sets, • Unions and products of countable sets, • The power set; • The real number line: • Open and closed sets, • Compact sets, • Perfect 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; • Series in � : • Conditional and absolute convergence, • Tests for convergence. Method of Assessment: Examination 100%, Duration 2 hours Suggested reading: • Spivak M,, Calculus, Publish or Perish, 3rd Edition, 1994 • Abbott S., Understanding Analysis, Springer, 2001 • Bartle R. and Sherbert D., Introduction to Real Analysis, Wiley, 3rd Ed., 1999 • Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974

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�MAT1411 - Discrete Methods Lecturer: Dr. P. Borg Follows from: A-Level Leads to: MAT2412 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 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%, Duration 2 hours 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.

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�MAT1511 - Analytical Geometry Lecturer: Dr. J. Muscat Follows from: A-Level Leads to: MAT2512 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 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. Method of Assessment: Examination 100%, Duration 2 hours Textbooks • Roe J., Elementary Geometry, Oxford Science Publications, Clarendon Press, 1997. Supplementary Reading: • Vaisman I., Analytical Geometry, World Scientific Publishing Company, 1998. • Camilleri C.J., Vector Analysis, Malta University Press, Malta, 1994.

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�MAT1611 – Introductory Mechanics Lecturer: Mr. V. Mercieca Follows from: Advanced Level Pure Mathematics Leads to: MAT3612 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 1 • Basic terms and concepts used in Mechanics: • Motion, equilibrium, force, mass, momentum, • Introduction to Newton’s laws of motion, • Use of simple models in problem solving; • Vector applications to geometry; • Coplanar forces acting at a point: • Resultant of 2 or more forces: the parallelogram rule, the triangle rule, equivalent systems, • Equilibrium of 3 forces: the triangle of forces, Lami’s theorem, • Equilibrium of 3 or more forces: the polygon of forces, resolution of forces in perpendicular directions, • Elastic strings and Hooke’ law, • Friction: limiting and non-limiting equilibrium of rough objects in contact, coefficient and angle of friction; • Coplanar forces acting on a rigid body: • Moment of a force, couples, • Resultant of systems of forces and couples, • Conditions for a general system to reduce to equilibrium, or a force and/or a couple; • Vector application to statics: • Vector, parametric and Cartesian equation of a straight line, circle, parabola and helix, • Equation of the line of action of a resultant force; • Kinematics: • Displacement, velocity and acceleration, • Relative motion of two particles, problems on interception and closest distance of approach, • Constant acceleration: equations of motion, velocity-time and velocity-displacement diagrams, • Projectiles; • Fundamental dynamics: • Vector expressions for force, momentum and Newton’s laws, • Work done by a force, kinetic and potential energy, • The work-energy principle and the principle of conservation of energy. Method of Assessment: Examination 100%, Duration 2 hours Textbooks • 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.

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

�MAT1001 - Mathematical Methods: Matrices Lecturers: Ms. C. Zerafa Follows from: A-Level Leads to: MAT1101 Credit value: 2 Lectures: 12 Tutorials: 2 Semester: 1 • Matrices; • Determinants; • Solution of linear equations; • Eigenvalues and diagonalisation; • Applications. Method of Assessment: Examination 100%, Duration 1½ hours Textbooks • Andvilli S. and Hecker D., Elementary Linear Algebra, Harcourt Academic Press, 1999.

�MAT1091 - Mathematical Methods: Matrices and Differ ential Equations Lecturer: Prof. I. Sciriha / Ms. C. Zerafa Follows from: A-Level Leads to: MAT1101, MAT3701 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 1 • Matrices: • Determinant, rank, trace and inverse of a matrix, • Solution of linear equations, • Eigenvalues and diagonalisation, • Applications. • Ordinary differential equations: • 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%, Duration 2 hours Textbooks • Andvilli S. and Hecker D., Elementary Linear Algebra, Harcourt Academic Press, 1999. • Nagle K.B., Saff E.B., and Snider A.D., Fundamentals of Differential Equations and Boundary Value Problems, Addison-Wesley, 4th Edition, 2003. • Kreyszig E., Advanced Engineering Mathematics, John Wiley & Sons, 8th Edition, 1998.

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�MAT1092 - Mathematical Methods for ICT Lecturers: Ms. C. Zerafa Follows from: A-Level Leads to: MAT1101, MAT3711 Credit value: 3 Lectures: 18 Tutorials: 3 Semester: 1 • Matrices; • Determinant, rank, trace and inverse of a matrix, • Solution of linear equations, • Eigenvalues and diagonalisation, • Applications. • Ordinary differential equations: • First order differential equations, • Second order differential equations. Method of Assessment: Examination 100%, Duration 1½ hours Textbooks • Andvilli S. and Hecker D., Elementary Linear Algebra, Harcourt Academic Press, 1999. • Nagle K.B., Saff E.B., and Snider A.D., Fundamentals of Differential Equations and Boundary Value Problems, Addison-Wesley, 4th Edition, 2003. • Kreyszig E., Advanced Engineering Mathematics, John Wiley & Sons, 8th Edition, 1998.

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�MAT1951 - Elementary Calculus I Lecturer: Ms. E. Manicaro Follows from: O-Level Credit value: 2 Lectures: 12 Tutorials: 2 Semester: 1 • Cartesian coordinates; • Equations of lines and curves; • Coordinate geometry of the circle; • Differentiation. Method of Assessment: Examination 100%, Duration 1½ hours 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.

�MAT1952 - Elementary Calculus II Lecturer: Ms. E. Manicaro Follows from: MAT1951 Credit value: 2 Lectures: 12 Tutorials: 2 Semester: 2 • Further differentiation; • Partial differentiation; • Integration and applications. Method of Assessment: Examination 100%, Duration 1½ hours 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.

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Year II

� MAT2112 - Linear Algebra I Lecturer: Prof. I. Sciriha Follows from: MAT1101 Leads to: MAT3905 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 1 • Vector spaces; • Dimension theorems; • Third isomorphism theorem; • Direct sum of spaces; • Dual spaces; • Inner product spaces; • Orthogonal projection and orthogonal complement; • Gram Schmidt orthogonalisation; • The rank of matrices; • Projection of a vector space onto a subspace; • Similar matrices and transition matrices; • Eigenvalues and eigenvectors; • The characteristic polynomial; • The Cayley-Hamilton theorem; • The minimum polynomial; • Diagonalisation and applications; • The Jordan normal form. Method of Assessment: Examination 100%, Duration 2 hours 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. • Moore H.G. and Yaqub A., A First Course in Linear Algebra with Applications, Academic Press, 3rd Edition,1998. • Leon S., Linear Algebra with Applications, Prentice Hall, 6th Edition, 2002.

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�MAT2113 - Rings Lecturer: Dr. P. Borg Follows from: MAT1101 Leads to: MAT3114 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 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; • Euclidean rings; • Unique factorisation; • Applications to some simple results in number theory; • The ring of polymomials. Method of Assessment: Examination 100%, Duration 2 hours 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. E. Chetcuti Follows from: MAT1211 Leads to: MAT2213 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 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. Method of Assessment: Examination 100%, Duration 2 hours Suggested reading: • Abbott S., Understanding Analysis, Springer, 2001. • Spivak M., Calculus, Publish or Perish, 3rd Edition, 1994. • Bartle R. and Sherbert D., Introduction to Real Analysis, Wiley, 3rd Ed., 1999. • Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974.

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�MAT2213 Analysis III Lecturer: Dr. J. L. Borg Follows from: MAT2212 Leads to: MAT3214, MAT3216 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 2 • Riemann integration: • The Riemann integral, • Basic theorems on integrals, • The fundamental theorem of calculus, • Taylor’s theorem in integral form; • Uniform convergence: • The Weierstrass M-test, • Applications to power series and Fourier series. Method of Assessment: Examination 100%, Duration 2 hours Suggested reading: • Spivak M., Calculus, Publish or Perish, 3rd Edition, 1994 • Abbott S., Understanding Analysis, Springer, 2001 • Bartle R. and Sherbert D., Introduction to Real Analysis, Wiley, 3rd Ed., 1999 • Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974

�MAT2512 - Vector Analysis I Lecturer: Dr. J. Sultana Follows from: MAT1511 Leads to: MAT2513 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 1 • Double integrals; • Triple integrals; • Applications of multiple integrals; • Change of variables in multiple integrals; • Jacobian; • Grad, div and curl operators; • Space curves; • Serret-Frenet formulae; • Line integral and its applications; • Green’s theorem; • Conservative vector fields, scalar potential. Method of Assessment: Examination 100%, Duration 2 hours

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

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�MAT2413 – Introduction to Graph Theory with Applica tions Lecturer: Prof. J. Lauri Follows from: MAT1411 Leads to: MAT3413 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 1 THEORY: • Introductory concept: vertices, edges, paths, cycles, connectedness and special classes of graphs, such

as complete graphs and bipartite graphs. The maximum and minimum number of edges in a connected graph on a given number of vertices.

• Simple examples of the relation between the spectrum of a graph and its structure. • Trees. Equivalent definitions of trees. • Connectivity and Menger's Theorem. • Bipartite graphs seen as the easiet instance of graph colouring, namely the 2-colourable graphs.

Characterisation in terms of not having odd cycles and edge-set partitioned into cutsets. • Eulerian graphs. Equivalent characterisations. Also, characterisations in terms of no odd cutsets and

edge-set partitioned into cycles. • Hamiltonian graphs. Ore's Theorem. • Chromatic number of a graph. Brook's Theorem for graphs which are not regular. • Planarity. Euler's polyhedral formula. Non-planarity of K_5 and K_{3,3}. Kuratowski's Theorem (without

proof). • The Five Colour Theorem for planar graphs. APPLICATIONS: • A brief look at ranking players in a tournament or web-pages. • Kruskal's Algorithm for finding a minimum weight spanning tree. Brief introduction to the greedy

algorithm on matroids. • A brief look at trees as data-structures. • Dijkstra's Algorithm. • Activity networks. • The difficulties of scheduling. • Introduction to the Travelling Salesman Problem. Connection with scheduling and a brief discussion of

computationally hard problems. The twice-around-the-spanning-tree heuristic. • Flow networks. The Minimum-Cut-Maximum-Flow Theorem. Application: Proof of Menger's Theorem. Method of Assessment: Examination 100%, Duration 2 hours Suggested Reading:

• Course notes will be handed out for most of the topics. • Bondy J.A. and Murty U.S.R., Graph Theory with Applications, Macmillan, 1978 (available with

permission online at Professor Bondy's website). • Balakrishnan, V.K., Schaum's Outline of Graph Theory.

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�MAT2513 - Vector Analysis II Lecturer : Dr. J. Sultana Follows from: MAT2512 Leads to: MAT3216 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 2 • The Laplacian and other operators; • Vector identities; • Introduction to surfaces; • Quadrics; • Surface integrals; • Applications of surface integrals; • Gauss’ theorem; • Stokes’ theorem; • Orthogonal curvilinear coordinates.

Method of Assessment: Examination 100%, Duration 2 hours

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: Various Credit value: 2 Lectures: 14 Semesters: 1, 2 • Introduction to mathematical and computer algebra software; • Application to various branches of Mathematics. Method of Assessment: Essay / Seminar 100% Suggested reading • 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.

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�MAT2913 – Numerical Analysis Lecturer: Prof. A. Buhagiar Follows from: MAT1090 Credit value: 2 Lectures: 12 Tutorials: 2 Semester: 1 • Locating roots of equations: • The Newton-Raphson method in one and two dimensions: rate of convergence, • The variable secant method, • The fixed point theorem for equations like ( )x f x= ,

• The method of steepest descent; • Solution of linear equations: • Gaussian elimination,

• Cholesky’s LU and TLL methods,

• Iterative methods: Jacobi, Gauss-Seidel and the SOR methods; • Interpolation: • Lagrangian interpolation, • 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 38 rules,

• Errors for the local and global versions of these rules; • 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; • Introduction to the Matlab language: • Implementation of some numerical procedures in Matlab. Method of Assessment: Examination 100%, Duration 1½ hours Main Texts • Gerald C.F. and Wheatley P.O., Applied Numerical Analysis, 6th Edition, Addison-Wesley, 1997. • Hahn B.D. and Valentine D.T., Essential Matlab for Engineers and Scientists, 3rd Edition, Elsevier, 2007. Supplementary Reading • Sauer T.D., Numerical Analysis, Pearson, 2006. • Fausett L.V., Applied Numerical Analysis using Matlab, 2nd Edition, Pearson, 2008. • Grasselli M. and Pelinovsky D., Numerical Mathematics, Jones and Bartlett Publishers, 2008. • Kiusalaas J., Numerical Methods in Engineering with Matlab, Cambridge University Press, 2005. • Press W.H., Flannery B.P., et al., Numerical Recipes in Fortran, Cambridge University Press, 1989. • Burden R.L. and Faires J.D., Numerical Analysis, 7th Edition, Brooks Cole, London, 2001.

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Year III

�MAT3114 - Linear Algebra II Lecturer: Prof. I. Sciriha Follows from: MAT2112 Leads to: MAT3218, MAT3414 Credit value: 2 Lectures: 12 Tutorials: 2 Semester: 1 • Algebra of linear transformations; • Dual spaces and annihilators; • The transpose; • The minimum polynomial; • Primary decomposition theorem; • T-invariant spaces; • Schur’s theorem; • Criteria for diagonalisation; • Jordan normal forms; • Hermitian matruces, isometries and normal matrices; • Real symmetric matrices: • Spectral decomposition; • Positive semidefinite matrices. Method of Assessment: Examination 100%, Duration 2 hours 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. • Spence L.E., Insel A.J. and Friedberg S.H., Linear Algebra - a Matrix Approach, Pearson, 2nd Edition, 2007.

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� MAT3115 - Groups Lecturer: Prof. J. Lauri Follows from: MAT1101 Leads to: MAT3116 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 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. Method of Assessment: Examination 100%, Duration 2 hours 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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����MAT3214 - Complex Analysis Lecturer: Dr. E. Chetcuti Follows from: MAT1211 Leads to: MAT3218 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 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; • The fundamental theorem of algebra; • Sequences; • Taylor’s series; • Laurent’s series; • Zeros and poles; • Residues; • Residue theorem and its applications. Method of Assessment: Examination 100%, Duration 2 hours 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.

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�MAT3215 – Metric Spaces Lecturer: Prof. D. Buhagiar Follows from: MAT1211 Leads to: MAT3216, MAT3217 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 1 • Axioms and examples of metric spaces; • Subspaces; • Open and closed sets; • Continuity and uniform continuity; • Equivalent metrics; • Compactness and sequential compactness: their equivalence; • Connectedness; • Completeness; • Separability. Method of Assessment: Examination 100%, Duration 2 hours 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. • Rudin W., Principles of Mathematical Analysis, McGraw-Hill, 3rd Edition, 1976. • Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974.

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�MAT3413 – Probabilistic Combinatorics Lecturer: Dr. P. Borg Follows from: MAT2412 Leads to: MAT3414 Credit value: 2 Lectures: 12 Tutorials: 2 Semester: 1 • Finite Probability Spaces • The Birthday Paradox • A simple lower bound for the Ramsey number R(k,k) • A lower bound for the van der Waerden number W(k), • 2-colourable uniform families • (r,s)-systems • Random graphs. • Conditional probability and independence • An application to tournaments • Random variables, expectation and linearity of expectation • Szele’s theorem on the number of Hamiltonian paths in a tournament • Existence of large cuts in graphs • Turán’s theorem: a lower bound for the independence number of a graph • Moments and variances • Markov’s inequality and Chebyshev’s inequality • An example of threshold behaviour in random graphs • Alterations • Improved lower bound for R(k,k) • Existence of graphs with large girth • The Lovasz local lemma • Applications: edge-disjoint paths in a graph, k-satisfiability Method of Assessment: Examination 100%, Duration 1½ hours 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.

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�MAT3216 – Analysis IV Lecturer: Dr. J. L. Borg Follows from: MAT2213 Leads to: MAT3218, MAT3513 Credit value: 2 Lectures: 12 Tutorials: 2 Semester: 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. Method of Assessment: Examination 100%, Duration 1½ hours Suggested reading: • Marsden J.E. and Tromba A.J., vector Calculus, W.H.Freeman, 5th Edition, 2003. • 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.

�MAT3217 - Lebesgue Integration Lecturer: Dr. E. Chetcuti Follows from: MAT3215 Leads to: MAT3218 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 2 • σ - algebras; • Measure spaces; • Lebesgue measure on � ; • Measurable functions; • Lebesgue integral; • Convergence theorems;

• The space , 1pL p≤ ≤ ∞ .

Method of Assessment: Examination 100%, Duration 2 hours 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.

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�MAT3612 - Mechanics Lecturer: Prof. A. Buhagiar Follows from: MAT1511, MAT1611 Leads to: MAT3613 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 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, • 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. Method of Assessment: Examination 100%, Duration 2 hours Main Text • Lunn M., A first course in Mechanics, Oxford University Press, Oxford, 1991. Supplementary Reading: • Chorlton F., Textbook of Dynamics, van Nostrand Company Ltd., London, 1969. • Camilleri C.J., Classical Mechanics, Malta 2004.

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�MAT3711 - Differential Equations Lecturer: Dr. J. Muscat Follows from: MAT1211, MAT1090 Leads to: MAT3712 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 2 • Picard's theorem; • Linear differential equations; • Green’s function; • Simple dynamical systems; • Power series solutions. Method of Assessment: Examination 100%, Duration 2 hours Suggested Reading • Derrick W. and Grossman S., Elementary Differential Equations, Addison Wesley, 4th Edition, 1997. • 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. • Bronson R., Differential Equations, Schaum’s Outline Series, McGraw-Hill, 2009. • Tenebaum M. and Pollard H., Ordinary Differential Equations, Dover Publications, 1985.

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Year IV

�MAT3116 – Group Representations Lecturer: Prof. J. Lauri Follows from: MAT1101, MAT3115 Leads to: MAT5411 Credit value: 5 Lectures: 21 Tutorials: 4 Semester: 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. Method of Assessment: Examination 100%, Duration 2 hours 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.

�MAT3219 – Topological Spaces Lecturer: Prof. D. Buhagiar Follows from: MAT3215 Leads to: MAT5211 Credit value: 5 Lectures: 21 Tutorials: 4 Semesters: 1, 2 • Cardinals and ordinals; • Topological spaces and examples; • Subspaces, product spaces and quotient spaces; • Separation axioms and countability axioms; • Connectedness and compactness. Method of Assessment: Examination 100%, Duration 2 hours Main Text • Munkres J.R., Topology, Prentice Hall, 2nd Edition, 2000. Supplementary Reading • 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. Sultana Follows from: MAT3216 Leads to: MAT5613 Credit value: 5 Lectures: 21 Tutorials: 4 Semesters: 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. Method of Assessment: Examination 100%, Duration 2 hours 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. Buhagiar Follows from: MAT3612 Leads to: MAT5613, MAT5711 Credit value: 5 Lectures: 21 Tutorials: 4 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. Method of Assessment: Examination 100%, Duration 2 hours Main Text • Lunn M., A first Course in Mechanics, Oxford University Press, Oxford, 1991. Supplementary Reading • Chorlton F., Textbook of Dynamics, van Nostrand Company Ltd., London, 1969. • Camilleri C.J., Classical Mechanics, Malta, 2004. • 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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�MAT3712 – Partial Differential Equations and Calcul us of Variations Lecturer: Dr. J. Muscat Follows from: MAT3711 Leads to: MAT5711, MAT5614 Credit value: 5 Lectures: 21 Tutorials: 4 Semester: 1, 2 • Partial differential equations: • First-order quasi-linear equations; • Separable solutions; • Elliptic equations: • Gravitation, • Laplace’s equation, • Poisson’s equation, • Harmonic functions. • 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. Method of Assessment: Examination 100%, Duration 2 hours 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. • DuChateau P. and Zachmann D., Partial differential Equations, Schaum Outline Series, McGraw-Hill, 1986.

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����MAT3713 - The Finite Element Method Lecturer: Prof. A. Buhagiar Follows from: MAT3711 Leads to: MAT5711 Credit value: 5 Lectures: 21 Tutorials: 4 Semesters: 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%, Duration 2 hours. 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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�MAT3270 - Functional Analysis: Normed Spaces Lecturer: Prof. D. Buhagiar Follows from: MAT3217 Leads to: MAT5211, MAT5311 Credit value: 6 Lectures: 26 Tutorials: 4 Semesters: 1, 2 • Metric spaces and their completion; • Normed vector spaces, Banach spaces;

• Examples of normed spaces: the spaces ,p pl L and [ , ];C a b

• Bounded linear operators, dual spaces; • Hahn-Banach theorem; • Open mapping and closed graph theorems. Method of Assessment: Examination 100%, Duration 3 hours 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.

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�MAT3271 - Functional Analysis: Hilbert Spaces Lecturer: Dr. J. Muscat Follows from: MAT3217 Leads to: MAT5311, MAT5614 Credit value: 6 Lectures: 26 Tutorials: 4 Semesters: 1, 2 • Hilbert spaces, Riesz representation theorem; • Orthonormal bases; • Least squares approximation; • Adjoints; • Spectral theory; • Self adjoint and normal operators. Method of Assessment: Examination 100%, Duration 3 hours Main text • Kreysig E., Introductory Functional Analysis, Wiley, 1989. Supplementary Reading • Rynne B. and Youngson M., Linear Functional Analysis, Springer, 2nd Edition, 2008. • 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.

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�MAT3470 – Graph Theory Lecturer: Prof. I. Sciriha Follows from: MAT3413 Leads to: MAT5411 Credit value: 6 Lectures: 26 Tutorials: 4 Semesters: 1, 2 • Definitions and elementary results on graphs; • Graphical degree sequences, Ryser switch; • Trees; • Cayley’s theorem on the number of spanning trees, • The matrix-tree theorem; • Connectivity; • Euler tours and Hamilton cycles; • Vector spaces associated with graphs; • Cycle-cutset duality; • Vertex and edge colourings: • Chromatic polynomials; • Planar graphs; • Matchings; • Independence number. Method of Assessment: Examination 100%, Duration 3 hours Main Texts • Gross J.L. and Yellen J., Handbook of Graph Theory, CRC Press, 2nd Edition, 2004. • 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. • Agnarsson G. and Greenlaw R., Graph Theory: Modelling, Applications and Algorithms, Pearson, 2006. Supplementary Reading • Diestel R., Graph Theory, Springer-Verlag, 3rd Edition, 2006. • 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.

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�MAT3471 – Combinatorics Lecturer: Prof. J. Lauri Follows from: MAT3413 Leads to: MAT5411 Credit value: 6 Lectures: 26 Tutorials: 4 Semesters: 1, 2 • The cycle index of a permutation group and the use of Polya’s theorem; • Ramsey’s theorem for graphs; • Introduction to error-correcting codes; • Introduction to combinatorial designs. Method of Assessment: Examination 100%, Duration 3 hours Main Texts • Gross J.L. and Yellen J., Handbook of Graph Theory, CRC Press, 2nd Edition, 2004. • 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. • Agnarsson G. and Greenlaw R., Graph Theory: Modelling, Applications and Algorithms, Pearson, 2006. Supplementary Reading • Diestel R., Graph Theory, Springer-Verlag, 3rd Edition, 2006. • 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.

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THE PROJECT MODULES For the project modules, students choose either MAT3218 or MAT3414 (12 credits), together with MAT3997, the Mathematics Seminar (7 credits), for a total of 19 credits. These credits are described next.

�MAT3218 - Functional Analysis Lecturers: Prof. D. Buhagiar / Dr. J. Muscat Follows from: MAT3217 Leads to: MAT5311, MAT5614 Credit value: 12 Lectures: 52 Tutorials: 8 Semesters: 1, 2 • Metric spaces and their completion; • Normed vector spaces, Banach spaces;

• Examples of normed spaces: the spaces ,p pl L and [ , ];C a b

• Bounded linear operators, dual spaces; • Hahn-Banach theorem; • Open mapping and closed graph theorems. • Hilbert spaces, Riesz representation theorem; • Orthonormal bases; • Least squares approximation; • Adjoints; • Spectral theory; • Self adjoint and normal operators. Method of Assessment: Examination 100%; 2 papers of 3 hours each 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. • Rynne B. and Youngson M., Linear Functional Analysis, Springer, 2nd Edition, 2008.

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�MAT3414 - Discrete Mathematics Lecturers: Prof. I. Sciriha / Prof. J. Lauri Follows from: MAT3413 Leads to: MAT5411 Credit value: 12 Lectures: 52 Tutorials: 8 Semesters: 1, 2 • Definitions and elementary results on graphs; • Graphical degree sequences, Ryser switch; • Trees; • Cayley’s theorem on the number of spanning trees, • The matrix-tree theorem; • Connectivity; • Euler tours and Hamilton cycles; • Vector spaces associated with graphs; • Cycle-cutset duality; • Vertex and edge colourings: • Chromatic polynomials; • Planar graphs; • Matchings; • Independence number. • The cycle index of a permutation group and the use of Polya’s theorem; • Ramsey’s theorem for graphs; • Introduction to error-correcting codes; • Introduction to combinatorial designs. Method of Assessment: Examination 100%; 2 papers of 3 hours each Main Texts • Gross J.L. and Yellen J., Handbook of Graph Theory, CRC Press, 2nd Edition, 2004. • 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. • Agnarsson G. and Greenlaw R., Graph Theory: Modelling, Applications and Algorithms, Pearson, 2006. Supplementary Reading • Diestel R., Graph Theory, Springer-Verlag, 3rd Edition, 2006. • 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.

����MAT3997 – Mathematics Seminar Lecturers: Various Credit value: 7 Semesters: 1, 2 • In this study unit, a long essay and a seminar are presented by each student on a suitable topic in

Mathematics. Method of Assessment: Essay / Seminar 100%

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FACULTIES OF ENGIINEERING & ARCHITECTURE Year I

����MAT1801 – Mathematics for Engineers I Lecturers: Dr. J. L. Borg, Dr. E. Chetcuti, Dr. J. B. Gauci Follows from: A-Level Leads to: MAT1802 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 1 • First order differential equations; • Second order linear differential equations with constant coefficients; • Partial differentiation • Sequences and series • Fourier series; • Double integrals; Method of Assessment: Examination 100%, Duration 2 hours 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 Lecturers: Dr. J .L. Borg, Dr. E. Chetcuti, Dr. J. B. Gauci Follows from: A-level, MAT1801 Leads to: MAT2803, MAT2804 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 2 • Matrices and determinants; • Systems of linear equations; • Matrices: eigenvalues and eigenvectors. • 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). Method of Assessment: Examination 100%, Duration 2 hours Textbooks • Zill D.G. and Cullen M.R., Advanced Engineering Mathematics, Jones and Bartlett Publishers, 3rd Edition, 2006. • Vaisman I., Analytical Geometry, World Scientific Publishing Company, 1998. • Roe J., Elementary Geometry, Oxford Science Publications, Clarendon Press, 1997.

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Year II

�MAT2803 - Laplace and Fourier Transforms Lecturers: Dr. J. B. Gauci Follows from: MAT1802 Credit value: 2 Lectures: 12 Tutorials: 2 Semester: 1 • 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. Method of Assessment: Examination 100%, Duration 1½ hours 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 Integ rals Lecturers: Dr. J. B. Gauci Follows from: MAT1801, MAT1802 Credit value: 2 Lectures: 12 Tutorials: 2 Semester: 2 • Eigenvalues and eigenvectors with application to simultaneous linear differential equations; • Transformations of double and triple integrals. Method of Assessment: Examination 100%, Duration 1½ hours Textbook • Zill D.G. and Cullen M.R., Advanced Engineering Mathematics, Jones and Bartlett, 3rd Edition, 2006.

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�MAT2814: Numerical Analysis with Matlab Lecturers: Prof. A. Buhagiar Follows from: MAT1801, MAT1802 Leads to: MAT3815 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 2 • The Matlab language: • Description of the main commands, • Introduction to programming in Matlab; • Locating roots of equations: • The Newton Raphson in one and two dimensions: rate of convergence, • The variable secant method, • The fixed point theorem for equations like ( )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 TLL methods,

• Iterative methods: Jacobi, Gauss-Seidel and the SOR methods; • 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 38 rules,

• Errors for the local and global versions of these rules, • Gaussian quadrature; • Ordinary differential equations: • Euler, modified Euler and Runge-Kutta methods; • The finite difference method for partial differential equations: • Equations of Laplace and Poisson, • The transient heat equation. • Implementation of most of the above numerical methods in Matlab. Method of Assessment : Coursework 15%, Examination 85%, Duration 2 hours

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Main Texts • Gerald C.F. and Wheatley P.O., Applied Numerical Analysis, 6th Edition, Addison-Wesley, 1997. • Hahn B.D. and Valentine D.T., Essential Matlab for Engineers and Scientists, 3rd Edition, Elsevier, 2007. Supplementary Reading • Sauer T.D., Numerical Analysis, Pearson, 2006. • Fausett L.V., Applied Numerical Analysis using Matlab, 2nd Edition, Pearson, 2008. • Grasselli M. and Pelinovsky D., Numerical Mathematics, Jones and Bartlett Publishers, 2008. • Kiusalaas J., Numerical Methods in Engineering with Matlab, Cambridge University Press, 2005. • 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.

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Year III

�MAT3815: Mathematics for Engineers III Lecturers: Dr. J. Sultana Follows from: MAT1801, MAT1802, MAT2814 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 1 • Optimisation: • Local and global extrema of functions of several variables, • Lagrange’s method for constrained problems,

• Lagrange’s method with two constraints

• Vector Calculus: • The gradient, streamlines and contours, • Divergence and curl, • Vector Identities, • Double and Triple Integrals,

• Change of variables in Multiple Integrals, • Jacobian, • Line Integrals • Surface Integrals • Integral Theorems of Gauss, Green and Stokes • Applications of Integral Theorems.

Method of Assessment: Examination 100%, Duration 2 hours 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.

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FACULTY OF ECONOMICS, MANAGEMENT & ACCOUNTANCY Year I

�MAT1991 - Mathematics I and II

Lecturers: Dr. P. Borg, Mr. V. Mercieca, Mr. E. Cardona, Prof. A. Buhagiar Follows from: Intermediate Mathematics Leads to: MAT1902 Credit value: 8 Lectures: 48 Tutorials: 8 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: Dr. P. Borg, Mr. V. Mercieca, Mr. E. Cardona, Prof. A. Buhagiar Follows from: Intermediate Mathematics Leads to: MAT1902 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 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.

Method of Assessment: Examination 100% Recommended Text • Barnett R.A., Ziegler M.R. and Byleen K.E., College Mathematics: for Business, Economics, Life Sciences and Social Sciences,10th Edition, Prentice Hall, New York, 2004.

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�MAT1902 - Mathematics II Lecturers: Dr. P. Borg, Mr. V. Mercieca, Mr. E. Cardona, Prof. A. Buhagiar Follows from: MAT1901 Credit value: 4 Lectures: 24 Tutorials: 4 Semester: 2 • Mathematics of finance: • Simple and compound Interest, • Present and future value, • Effective rate of Interest, • Annuities and sinking funds; • 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. Method of Assessment: Examination 100% Recommended Text • Barnett R.A., Ziegler M.R. and Byleen K.E., College Mathematics: for Business, Economics, Life Sciences and Social Sciences,10th Edition, Prentice Hall, New York, 2004.