Mathematics for Informatics
Code  Completion  Credits  Range  Language 

NIEMPI  Z,ZK  7  3P+2C  English 
 Garant předmětu:
 Štěpán Starosta
 Lecturer:
 Francesco Dolce
 Tutor:
 Francesco Dolce
 Supervisor:
 Department of Applied Mathematics
 Synopsis:

The course focuses on selected topics from general algebra with emphasis 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 the 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: groupoid, monoid, group, homomorphism.
2. Cyclic and finite groups and their properties.
3. Discrete logarithm problem in various groups and its applications in cryptography.
4. Rings and fields and their properties.
5. Modular arithmetics and equations in finite fields.
6. Multivariable calculus: partial derivative and gradient.
7. Geometrical interpretation of partial derivatives, tangent spaces.
8. Continuous optimization methods. Selected optimization problems in informatics.
9. Constrained multivariable optimization.
10. Integration of multivariable functions.
11. Representation of numbers in computers, floating point arithmetics and related errors.
12. Solving systems of linear equations, finding eigenvalues and stability of numerical algorithms.
13. Error estimation in numerical algorithms. Numerical differentiation.
 Syllabus of tutorials:

1. Functions, 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. Paar, Ch.  Pelzl, J. Understanding Cryptography. Springer, 2010. ISBN 9783642041006.
3. Cheney, E. W.  Kincaid, D. R. Numerical Mathematics and Computing. Cengage Learning, 2007. ISBN 9780495114758.
4. Higham, N. J. Accuracy and Stability of Numerical Algorithms. SIAM, 2002. ISBN 9780898715217.
5. Marsden, J.  Weinstein, A. Calculus III. Springer, 1998. ISBN 9780387909851.
6. Ross, T. J. Fuzzy Logic with Engineering Applications (3rd Edition). Wiley, 2010. ISBN 9780470743768.
 Note:
 Further information:
 https://courses.fit.cvut.cz/NIEMPI
 Timetable for winter semester 2024/2025:
 Timetable is not available yet
 Timetable for summer semester 2024/2025:
 Timetable is not available yet
 The course is a part of the following study plans:

 Master specialization Software Engineering, in English, 2021 (compulsory course in the program)
 Master specialization Computer Security, in English, 2021 (compulsory course in the program)
 Master specialization Computer Systems and Networks, in English, 2021 (compulsory course in the program)
 Master specialization Design and Programming of Embedded Systems, in English, 2021 (compulsory course in the program)
 Master specialization Computer Science, in English, 2021 (compulsory course in the program)
 Master Specialization Digital Business Engineering, 2023 (compulsory course in the program)