Mathematics for Informatics

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 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 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 978-0471433347.

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

3. Cheney, E. W. - Kincaid, D. R. Numerical Mathematics and Computing. Cengage Learning, 2007. ISBN 978-0495114758.

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

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

6. Ross, T. J. Fuzzy Logic with Engineering Applications (3rd Edition). Wiley, 2010. ISBN 978-0470743768.

Note:

Further information:

https://courses.fit.cvut.cz/NIE-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 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)