Mathematics for Informatics
Code  Completion  Credits  Range  Language 

NIMPI  Z,ZK  7  3P+2C  Czech 
 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 multivariate analysis, smooth optimization and multivariate 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 9780471433347.
2. Mareš, J. Algebra. Úvod do obecné algebry. Vydavatelství ČVUT, 1999. ISBN 9788001019108.
3. Paar, Ch.  Pelzl, J. Understanding Cryptography. Springer, 2010. ISBN 9783642041006.
4. Cheney, E. W.  Kincaid, D. R. Numerical Mathematics and Computing. Cengage Learning, 2007. ISBN
9780495114758.
5. Higham, N. J. Accuracy and Stability of Numerical Algorithms. SIAM, 2002. ISBN 9780898715217.
6. Marsden, J.  Weinstein, A. Calculus III. Springer, 1998. ISBN 9780387909851.
 Note:
 Further information:
 https://courses.fit.cvut.cz/NIMPI/
 Timetable for winter semester 2023/2024:
 Timetable is not available yet
 Timetable for summer semester 2023/2024:
 Timetable 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)