Single subject courses - Mathematics
These courses are only open for exchange students. If you as an exchange student have any questions about admission, prerequisites etc, please contact the International Office at internationaloffice@adm.vxu.se.
If you are a free mover, please contact the Admissions Office regarding possibilities to study at Växjö University: admission@vxu.se.
- MAB726 - Discrete Mathematics
- MAC721 - Analytic Functions
- MAC722 - Theory of Transforms
- MAC723 - Ordinary Differential Equations
- MAC724 - Partial Differential Equations
- MAD710 - Integration Theory
- MAD711 - Distribution Theory
- MAD713 - Topology
- MAD715 - Functional analysis
- MAD720 - Stochastic Analysis
- MSB741 - Multivariate Analysis
- MSC761 - Foundations of Probability
MAB726 - Discrete Mathematics
Contact Person
Lars Gustafsson
Telephone: +46 470 70 86 30
Telefax: +46 470 840 04
E-mail: lars.gustafsson@vxu.se
School of Mathematics and Systems Engineering
January–March, 2007
5 credit points (7.5 ECTS)
This course will introduce the basic concepts and methods of Discrete Mathematics, especially those that are important in Computer Science. The aim of the course is to give an understanding of the applicability of Mathematics in Computer Science.
Contents
Combinatorics, set theory, relations, generating functions, recurrence relations, graphs, trees, optimization and matching.
Recommended reading
Grimaldi R. P., Discrete and Combinatorial Mathematics, 5th Ed., Addison-Wesley, 2004.
Teaching methods
Lectures, exercises.
Prerequisites
A basic course in Algebra including, for example, induction, the binomial theorem, complex numbers and polynomials. Calculus including, for example, elementary functions and power series.
Assessment methods
Written test and oral presentation of group exercises.
MAC721 - Analytic Functions
Contact Person
Lars Gustafsson
Telephone: +46 470 70 86 30
Telefax: +46 470 840 04
E-mail: lars.gustafsson@vxu.se
School of Mathematics and Systems Engineering
September–November, 2006
5 credit points (7.5 ECTS)
This course is an introduction to the Theory of Analytic Functions. It is suitable for mathematics majors, engineering students and majors in the natural sciences. The aim is to provide a basic knowledge of complex analysis.
Contents
Elementary analytic functions, complex power series, Cauchy’s Theorem, zeros of analytic functions, the Maximum-Modulus Theorem, singularities, Laurent Series, Cauchy’s Residue Theorem, Möbius’ Transformations and more general conformal mappings, Riemann’s Mapping Theorem, harmonic functions.
Recommended reading
Saff E. B. & Snider A. D., Fundamentals of Complex Analysis, Prentice-Hall, 1993.
Teaching methods
Lectures, exercises.
Prerequisites
A course in Calculus of one and several variables including uniform convergence. A course in Linear Algebra.
Assessment methods
Written test.
MAC722 - Theory of Transforms
Contact Person
Lars Gustafsson
Telephone: +46 470 70 86 30
Telefax: +46 470 840 04
E-mail: lars.gustafsson@vxu.se
School of Mathematics and Systems Engineering
November, 2006–January, 2007
5 credit points (7.5 ECTS)
This course is an introduction to the Theory of Transforms given at level C. It is suitable for mathematics majors, engineering students and majors in the natural sciences. The aim is to provide a basic knowledge of transforms.
Contents
Theory of Transforms: Laplace Transforms, Fourier Transforms and the Z-Transform.
Recommended reading
Saff E. B. & Snider A. D., Fundamentals of Complex Analysis, Prentice-Hall, 1993.
Teaching methods
Lectures, exercises.
Prerequisites
A course in Calculus of one and several variables including uniform convergence. A course in Linear Algebra. A course in Analytic Functions.
Assessment methods
Written test.
MAC723 - Ordinary Differential Equations
Contact Person
Lars Gustafsson
Telephone: +46 470 70 86 30
Telefax: +46 470 840 04
E-mail: lars.gustafsson@vxu.se
School of Mathematics and Systems Engineering
September–November, 2006
5 credit points (7.5 ECTS)
This course gives an introduction to the theory of ordinary differential equations. It is suitable for mathematics majors, engineering students and majors in the natural sciences. The aim of the course is to provide a knowledge of the theory of ordinary differential equations and to offer important skills in problem solving.
Contents
Linear differential equations with constant and variable coefficients, existence and uniqueness of solutions, linear systems with constant coefficients, stability of autonomous systems, Liapunov functions.
Recommended reading
Boyce E. & DiPrima R. C., Elemental Differential Equations and Boundary Value Problems, 8th Ed., Wiley, 2004.
Teaching methods
Lectures, exercises.
Prerequisites
A course in Calculus of one and several variables including uniform convergence. A course in Linear Algebra.
Assessment methods
Written test.
MAC724 - Partial Differential Equations
Contact Person
Lars Gustafsson
Telephone: +46 470 70 86 30
Telefax: +46 470 840 04
E-mail: lars.gustafsson@vxu.se
School of Mathematics and Systems Engineering
November, 2006–January, 2007
5 credit points (7.5 ECTS)
This course gives an introduction to the theory of partial differential equations. It is suitable for mathematics majors, engineering students and majors in the natural sciences. The aim of the course is to provide a knowledge of the theory of partial differential equations and to offer important skills in problem solving.
Contents
Boundary value problems, Sturm-Liouville theory, Green’s Function, singular points and Frobenius’ method.
Recommended reading
Boyce E. & DiPrima R. C., Elemental Differential Equations and Boundary Value Problems, 8th Ed., Wiley, 2004.
Teaching methods
Lectures, exercises.
Prerequisites
A course in Calculus of one and several variables including uniform convergence. A course in Linear Algebra. A course in Ordinary Differential Equations.
Assessment methods
Written test and oral presentation.
MAD710 - Integration Theory
Contact Person
Andrei Khrennikov
Telephone: +46 470 70 87 90
Telefax: +46 470 840 04
E-mail: andrei.khrennikov@vxu.se
School of Mathematics and Systems Engineering
February–March, 2007
5 credit points (7.5 ECTS)
This course is an introduction to the measure theory and Lebesgue integration. It is suitable for mathematics majors, engineering students and majors in the natural sciences.
Contents
The set theory, measure theory, measurable functions and Lebesgue integration.
Recommended reading
Kolmogorov A. N. & Fomin S. V., Introductory real analysis, Dover Publication, New York, 1970, ISBN 0-486-61226-0.
Teaching methods
Lectures, exercises, a collection of theoretical and numerical problems as home tasks.
Prerequisites
A course in Calculus of one and several variables including uniform convergence. A course in Linear Algebra.
Assessment methods
Oral examination, home assignments.
MAD711 - Distribution Theory
Contact Person
Andrei Khrennikov
Telephone: +46 470 70 87 90
Telefax: +46 470 840 04
E-mail: andrei.khrennikov@vxu.se
School of Mathematics and Systems Engineering
January–June, 2007
5 credit points (7.5 ECTS)
This is an introduction course in the theory of distributions and applications to linear partial differential equations with constant coefficients.
Contents
Banach and topological linear spaces, spaces of test functions and distributions (generalized functions), derivatives of distributions, Fourier transform, fundamental solution, convolution.
Recommended reading
Vladimirov V. S., Equations of Mathematical Physics, Marcel Dekker, Inc., New York, 1971.
Teaching methods
Lectures, exercises, a collection of theoretical and numerical problems as home tasks.
Prerequisites
A course in Calculus of one and several variables including uniform convergence. A course in Linear Algebra.
Assessment methods
Oral examination, home assignments.
MAD713 - Topology
Contact Person
Andrei Khrennikov
Telephone: +46 470 70 87 90
Telefax: +46 470 840 04
E-mail: andrei.khrennikov@vxu.se
School of Mathematics and Systems Engineering
January–June, 2007
5 credit points (7.5 ECTS)
This course is an introduction to topology. It is suitable for mathematics majors, engineering students and majors in the natural sciences. The aim of the course is to provide a knowledge of metric spaces and their applications, general topology, various topological spaces and their applications, compact sets and continuous functions.
Contents
Metric spaces, examples, complete metric spaces, Baire’s theorem, contraction mappings, fixed point theorem, applications to differential and integral equations, general topology, various topological spaces and their applications, compact sets, continuous functions, Arzela’s theorem, Peano’s theorem, normed spaces, Banach spaces and Hilbert space.
Recommended reading
Kolmogorov A. N. & Fomin S. V., Introductory real analysis, Dover Publication, New York, 1970.
Teaching methods
Lectures, exercises, a collection of theoretical and numerical problems as home tasks.
Prerequisites
A course in Calculus of one and several variables including uniform convergence. A course in Linear Algebra.
Assessment methods
Oral examination, home assignments.
MAD715 - Functional analysis
Contact Person
Andrei Khrennikov
Telephone: +46 470 70 87 90
Telefax: +46 470 840 04
E-mail: andrei.khrennikov@vxu.se
School of Mathematics and Systems Engineering
January–June, 2007
5 credit points (7.5 ECTS)
This is an introduction course in functional analysis: normed spaces.
Contents
Banach spaces, dual spaces, theorems of Han-Banach, Banach-Steinhaus, fixed point theorem, iteration algorithms in complete metric spaces, applications to solution of differential and integral equations.
Recommended reading
Kolmogorov A. N. & Fomin S. V., Introductory Real Analysis, Dover Publication, New York, 1970.
Teaching methods
Lectures, exercises, a collection of theoretical and numerical problems as home tasks.
Prerequisites
A course in Calculus of one and several variables including uniform convergence. A course in Linear Algebra.
Assessment methods
Oral examination, home assignments.
MAD720 - Stochastic Analysis
Contact Person
Roger Petersson and Astrid Hilbert
Telephone: +46 470 70 86 55/70 81 12
Telefax: +46 470 840 04
E-mail: roger.petersson@vxu.se / astrid.hilbert@vxu.se
School of Mathematics and Systems Engineering
September 2006–June 2007
5 credit points (7.5 ECTS)
The aim of this course is to provide a basic knowledge of stochastic analysis, in particular stochastic integration and stochastic differential equations, with the emphasis on option pricing.
Contents
Brownian motion, stochastic integrals, Ito’s formula, local time, stochastic differential equations, Girsanov’s formula, numerical methods, Fokker Planck equations, Feynman-Kac formula and option pricing.
Recommended reading
Öksendal B., Stochastic Differential Equations with Applications. An Introduction with Applications, 5th Ed., Springer Verlag, 1998.
Teaching methods
Lectures, exercises, a collection of theoretical and numerical problems as home tasks.
Prerequisites
A basic course in probability and mathematical maturity.
Assessment methods
Oral examination, home assignments.
MSB741 - Multivariate Analysis
Contact Person
Roger Petersson and Astrid Hilbert
Telephone: +46 470 70 86 55 / 70 81 12
Telefax: +46 470 840 04
E-mail: roger.petersson@vxu.se / astrid.hilbert@vxu.se
School of Mathematics and Systems Engineering
September 2006–June 2007
5 credit points (7.5 ECTS)
The aim of this course is to provide knowledge of how to analyse multi-dimensional data. The course should be suitable for anyone working with real statistical data.
Contents
Computation of means and covariances of linear combinations of random variables with matrix operations, multi-dimensional Gaussian distribution, inference of mean vectors, multivariate regression, principal component analysis and discriminant analysis.
Recommended reading
Johnson R. A. & Wichern D. W. A., Applied Multivariate Statistical Analysis, 4th Ed., Prentice Hall, 1998.
Teaching methods
Lectures, exercises.
Prerequisites
A basic course in probability, statistics and linear algebra.
Assessment methods
Oral examination, home assignments.
MSC761 - Foundations of Probability
Contact Person
Roger Petersson and Astrid Hilbert
Telephone: +46 470 70 86 55 / 70 81 12
Telefax: +46 470 840 04
E-mail: roger.petersson@vxu.se / astrid.hilbert@vxu.se
School of Mathematics and Systems Engineering
September 2006–June 2007
5 credit points (7.5 ECTS)
The aim of this course is to present the probability theory in a mathematically sound way. Central concepts such as random variables and means are treated rigorously.
Contents
Measurability, random variables, means and integration, product spaces, basic moment inequalities (Hölder, Minkowski and Markov), different types of convergences of random variables, Borel-Cantelli’s lemma, characteristic functions, the law of large numbers and the central limit theorem.
Recommended reading
Jacod J. & Protter P., Probability Essentials, Springer, 2003.
Teaching methods
Lectures, exercises.
Prerequisites
A course in calculus of one and several variables. A basic course in probability and statistics.
Assessment methods
Oral examination, home assignments.