Description:

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.

Contents:

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.

Seminar contents:

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.

Recommended literature:

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.

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar