Mathematics for Informatics

Garant předmětu:

Lecturer:

Tutor:

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. 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. Representing numbers in computers, floating point arithmetics.

7. Solving systems of linear equations, finding eigenvalues and stability of numerical algorithms.

8. Multivariable calculus: partial derivative and gradient.

9. Continuous optimization methods. Selected optimization problems in informatics.

10. Constrained multivariable optimization.

11. Integration of multivariable functions.

12. Discrete Fourier transform.

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 aritmetics and finite structures, computer aritmetics and representation of numbers,

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

mathematics.

Study materials:

1. Schroeder, R. M.: ''Number Theory in Science and Communication'', Springer, Berlin, 2006.

2. Graham, R., Knuth, D., Patashnik, O.: ''Concrete Mathematics: A Foundation for Computer Science'', Addison-Wesley, Reading, Mass., 1989.

Note:

Further information:

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

No time-table has been prepared for this course

The course is a part of the following study plans: