Mathematics for Informatics

Lecturer:

Štěpán Starosta, Martin Holeňa (gar.), Karel Klouda

Tutor:

Štěpán Starosta, Tomáš Kalvoda, Karel Klouda, Michal Kupsa

Supervisor:

Department of Applied Mathematics

Synopsis:

Students will master advanced topics from various fields of mathematics and learn mathematical methods that are useful in applications in modern informatics.

Requirements:

Syllabus of lectures:

1. [2] Universal algebra: groups, finite groups, Cayley tables, group types, permutation, alternating, cyclic, and symmetry groups, normal subgroups.

2. Finite fields, prime order of field, rings and their properties, integral domain, ideal. Lattices.

3. Introduction to category theory, classes of objects, classes of morphisms and its properties, examples of categories: grupoid, category of all lattices, category of all commutative groups, category of all integral domains, category of all relations. Homomorphisms.

4. Selected problems of graph theory, types of Hamiltonian problems. Algebraic solutions of combinatorial problems, Polya enumeration theorem.

5. Algebra and algorithms (Algorithms for calculations of polynom roots - Newton' method, Lehmer-Schur's method, etc.).

6. Convex sets, convex hull, pure convex set, theorem on partition of convex sets, Minkowski theorem on projection.

7. Selected problems of number theory, quadratic congruence, Gauss algorithms. Special primes - factorial, palindromic, cyclic, Gauss', Eisenstein's primes. Examples of applications.

8. Properties of Fermat primes, Little Fermat Theorem, primality tests, Pépin test, number theory and geometry, constructability of polygons.

9. Selected numerical methods, Lagrange and Hermite interpolation, numerical integration, numerical solution of ordinary differential equations, calculating of eigenvalues of matrices, methods of solving of linear equations systems.

10. Fast algorithms: multiplication, numerical searching of square roots, Fourier transformation, Fermat transformation.

11. Axiomatic systems and their properties, recursive functions, proofs in the axiomatic system, examples of axiomatic systems, Peano's arithmetics, von Neumann's model of numbers.

12. Special logics, multi-valued logics, modal logics, fuzzy logics.

Syllabus of tutorials:

Study Objective:

Mathematics as a language for description of the world is a key discipline for an informatics engineer. The aim of this module is introduce students to the relevant parts of modern mathematics that form the theoretical background of many informatics disciplines.

Study materials:

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

2. Křížek, M., Luca, F., Somer, L.: ''17 Lectures on Fermat Numbers: From Number Theory to Geometry'', Springer, New York, 2001.

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

4. Lovász, L.: ''Combinatorial Problems and Exercises'', 2nd Ed., Akademiai Kiadó Budapest and North- Holland, Amsterdam, 1993.

Note:

Time-table for winter semester 2011/2012:

Time-table is not available yet

Time-table for summer semester 2011/2012:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon	roomTK:BS Klouda K. 12:45–14:15 ODD WEEK (lecture parallel1) Dejvice NTK Ballingův sálroomT9:346 Kalvoda T. 14:30–16:00 (lecture parallel1 parallel nr.106) Dejvice NBFIT učebnaroomT9:346 Klouda K. 16:15–17:45 (lecture parallel1 parallel nr.107) Dejvice NBFIT učebna
Tue	roomT9:347 Kalvoda T. 09:15–10:45 (lecture parallel1 parallel nr.101) Dejvice NBFIT učebnaroomT9:347 Kalvoda T. 11:00–12:30 (lecture parallel1 parallel nr.102) Dejvice NBFIT učebnaroomT9:105 Klouda K. 12:45–14:15 (lecture parallel1) Dejvice Posluchárna
Fri	roomT9:344 Starosta Š. 11:00–12:30 (lecture parallel1 parallel nr.109) Dejvice NBFIT ucebnaroomT9:347 Klouda K. 12:45–14:15 (lecture parallel1 parallel nr.108) Dejvice NBFIT učebnaroomT9:347 Klouda K. 14:30–16:00 (lecture parallel1 parallel nr.110) Dejvice NBFIT učebna
Thu
Fri	roomT9:344 Kupsa M. 09:15–10:45 (lecture parallel1 parallel nr.104) Dejvice NBFIT ucebnaroomT9:344 Kupsa M. 11:00–12:30 (lecture parallel1 parallel nr.105) Dejvice NBFIT ucebna

The course is a part of the following study plans:

• Branch Software Engineering, Version for Students who Enrolled in 2010, Presented in Czech (compulsory course in the program)
• Master Branch Web Engineering - Version for Students who Enrolled in 2010, Presented in Czech (compulsory course in the program)
• Computer Security, Version for Students who Enrolled in 2010, Presented in Czech (compulsory course in the program)
• Design of Digital Systems, Version for Students who Enrolled in 2010, Presented in Czech (compulsory course in the program)
• Computer Systems and Networks, Version for Students who Enrolled in 2010, Presented in Czech (compulsory course in the program)
• Branch System Programming, Version for Students who Enrolled in 2010, Presented in Czech (compulsory course in the program)
• Branch System Programming, Version for Students who Enrolled in 2010, Presented in Czech (compulsory course in the program)
• Knowledge Engineering, Version for Students who Enrolled in 2010, Presented in Czech (compulsory course in the program)
• Branch Web and Software Engineering, Version for Students who Enrolled in 2010, Presented in Czech (compulsory course in the program)
• Master Informatics, Version for Students who Enrolled in 2010, Presented in Czech (compulsory course in the program)
• Master Informatics, Presented in Czech, Version for Students who Enrolled in 2011 (compulsory course in the program)
• Branch Web and Software Engineering, Presented in Czech, Version for Students who Enrolled in 2011 (compulsory course in the program)
• Branch Web and Software Engineering, Presented in Czech, Version for Students who Enrolled in 2011 (compulsory course in the program)
• Web and Software Engineering, Presented in Czech, Version for Students who Enrolled in 2011 (compulsory course in the program)
• Knowledge Engineering, Presented in Czech, Version for Students who Enrolled in 2011 (compulsory course in the program)
• Computer Security, Presented in Czech, Version for Students who Enrolled in 2011 (compulsory course in the program)
• Design of Digital Systems, Presented in Czech, Version for Students who Enrolled in 2011 (compulsory course in the program)
• Computer Systems and Networks, Presented in Czech, Version for Students who Enrolled in 2011 (compulsory course in the program)
• Branch System Programming, Presented in Czech, Version for Students who Enrolled in 2011 (compulsory course in the program)
• Branch Computer Science, Presented in Czech, Version for Students who Enrolled in 2011 (compulsory course in the program)
• Master Informatics, Presented in Czech, Version for Students who Enrolled in 2012 (compulsory course in the program)
• Computer Security, Presented in Czech, Version for Students who Enrolled in 2012 (compulsory course in the program)
• Design of Digital Systems, Presented in Czech, Version for Students who Enrolled in 2012 (compulsory course in the program)
• Branch System Programming, Presented in Czech, Version for Students who Enrolled in 2012 (compulsory course in the program)
• Branch Computer Science, Presented in Czech, Version for Students who Enrolled in 2012 (compulsory course in the program)
• Branch Web and Software Engineering, Presented in Czech, Version for Students who Enrolled in 2012 (compulsory course in the program)
• Branch Web and Software Engineering, Presented in Czech, Version for Students who Enrolled in 2012 (compulsory course in the program)
• Web and Software Engineering, Presented in Czech, Version for Students who Enrolled in 2012 (compulsory course in the program)
• Knowledge Engineering, in Czech, Presented in Czech, Version for Students who Enrolled in 2012 (compulsory course in the program)
• Computer Systems and Networks, Presented in Czech, Version for Students who Enrolled in 2012 (compulsory course in the program)