Mathematics for Informatics

Garant předmětu:

Štěpán Starosta

Lecturer:

Jan Spěvák, Štěpán Starosta

Tutor:

Michal Kupsa, Pavel Paták, Jan Spěvák, Štěpán Starosta, Jakub Šístek

Supervisor:

Department of Applied Mathematics

Synopsis:

The course comprises topics from general algebra with focus on finite structures used in computer science. It includes topics from multi-variate analysis, smooth optimization and multi-variate integration. The third large topic is computer arithmetics and number representation in a computer along with error manipulation. The last topic includes selected numerical algorithm and their stability analysis. The topics are completed with demonstration of applications in computer science. The course focuses on clear presentation and argumentation.

Requirements:

linear algebra, elements of discrete mathematics, elements of calculus

Syllabus of lectures:

1. Basic notions of abstract algebra: grupoid, monoid, group, homomorphism.

2. Cyclic and finite groups and their properties.

3. Rings and fields.

4. (2) Finite fields. Applications in cryptography.

5. Multivariate calculus: partial derivative, gradient, and Hessian.

6. Unconstrained extremas of multivariate functions.

7. (2) Constrained extremas of multivariate functions.

8. Mutlivariate integral.

9. Representing numbers in computers, floating point arithmetic and relevant errors.

10. (2) Solving systems of linear equations, finding eigenvalues and stability of numerical algorithms.

Syllabus of tutorials:

1. Fucntions, derivative, polynomials

2. Grupoid, semigroup, monoid, group

3. Cyclic group, generators

4. Homomorphism, discrete logarithm, fields and rings

5. Finite fields

6. Discrete exponenciation, CRT, discrete logarithm

7. Machine numbers.

8. Multivariable functions, partial derivatives

9. Multivariable optimization

10. Constrained multivariable optimization

11. Constrained multivariable optimization with inequality constraints

12. Multivariable integration

Study Objective:

The course covers selected topics from general algebra and number theory with

emphasis on modular arithmetics and finite structures, computer arithmetics and representation of numbers,

multivariable calculus and continuous optimization. It provides some examples of informatics applications of

mathematics.

Study materials:

1. Dummit, D. S. - Foote, R. M. Abstract Algebra. Wiley, 2003. ISBN 978-0471433347.

2. Mareš, J. Algebra. Úvod do obecné algebry. Vydavatelství ČVUT, 1999. ISBN 978-8001019108.

3. Paar, Ch. - Pelzl, J. Understanding Cryptography. Springer, 2010. ISBN 978-3642041006.

4. Cheney, E. W. - Kincaid, D. R. Numerical Mathematics and Computing. Cengage Learning, 2007. ISBN

978-0495114758.

5. Higham, N. J. Accuracy and Stability of Numerical Algorithms. SIAM, 2002. ISBN 978-0898715217.

6. Marsden, J. - Weinstein, A. Calculus III. Springer, 1998. ISBN 978-0387909851.

Note:

Further information:

https://courses.fit.cvut.cz/NI-MPI/

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Master specialization Computer Security, in Czech, 2020 (compulsory course in the program)
• Master specialization Design and Programming of Embedded Systems, in Czech, 2020 (compulsory course in the program)
• Master specialization Computer Systems and Networks, in Czech, 202 (compulsory course in the program)
• Master specialization Management Informatics, in Czech, 2020 (compulsory course in the program)
• Master specialization Software Engineering, in Czech, 2020 (compulsory course in the program)
• Master specialization System Programming, in Czech, version from 2020 (compulsory course in the program)
• Master specialization Web Engineering, in Czech, 2020 (compulsory course in the program)
• Master specialization Knowledge Engineering, in Czech, 2020 (compulsory course in the program)
• Master specialization Computer Science, in Czech, 2020 (compulsory course in the program)
• Mgr. programme, for the phase of study without specialisation, ver. for 2020 and higher (compulsory course in the program)
• Master specialization System Programming, in Czech, version from 2023 (compulsory course in the program)
• Master specialization Computer Science, in Czech, 2023 (compulsory course in the program)