Title	Theory of Information
Code	ÚMV/TIN1/03	Teacher	Bukovský Lev
ECTS credits	4	Hrs/week	2/-
Assessment	Examination	Semester	1
T/L method	Lecture
Objective	To introduce students to mathematical attempts at solving selected problems of computer science.
Content	Measurement of information. Entropy and its properties. Shanon’s theorems. Coding and basic types of codes. Using of algebraic structures in construction of codes. Kolmogorov complexity. Basic properties and relation to the notion of entropy. Complexity and randomness.
Recommended reading	J. H. van Lint, Introduction to Coding Theory, Springer 1992 M. Li and P. Vitanyi, Kolmogorov Complexity and its Applications, Handbook of Theoretical Computer Science, Elsevier, 1990, p. 188-252

Title	Mathematical Foundations of Cryptography
Code	ÚINF/MZK/06	Teacher	Geffert Viliam, Lacko Vladimír
ECTS credits	6	Hrs/week	3/2
Assessment	Examination	Semester	1
T/L method	Lecture, Practical
Content	Classical cryptography. Steganography. Conventional symmetric cryptography. Feistel networks. Encryption modes: ECB, CBC, CFB, OFB. Security definitions. Models of an adversary. Cryptoanalysis. Asymmetric cryptosystems. One-way functions with trapdoors. RSA cryptosystem. El-Gamal cryptosystem. Generating large prime numbers. Elliptic curves cryptography. Keyed Hash functions. Collision resistant functions. Birthday paradox. SHA-1. Message authentication codes. Password security. Digital signatures. Blind signatures. Key management. X509. Certificates. Certification authorities. Electronic payments. Electronic cash.
Exclusive courses	ÚINF/UKR1/03

Title	Theory of Codes
Code	ÚMV/TKO1/04	Teacher	Horňák Mirko
ECTS credits	6	Hrs/week	4/-
Assessment	Examination	Semester	1, 3
T/L method	Lecture
Objective	To acquaint students with the basic principles and theoretical bases of text coding and possibilities for their application.
Content	Monoids. Basic notions of theory of codes. Examples of codes. Important classes of codes. Maximal codes. Submonoids generated by codes. Stable submonoids. Group codes. Free hull of a set of words. Test for recognising codes. Measure of a code. Bernoulli distribution. Dyck code. Complete sets in monoids. Thin codes. Composition of codes. Indecomposable codes.
Recommended reading	J. Berstel and D. Perrin, Theory of Codes, Academic Press 1985

Title	Matroid Theory
Code	ÚMV/TMT/04	Teacher	Horňák Mirko
ECTS credits	5	Hrs/week	3/-
Assessment	Examination	Semester	1, 3
T/L method	Lecture
Objective	To acquaint students with basic notions of matroid theory and possibilities of using matroids in various disciplines of discrete mathematics.
Content	Independent sets and bases. Properties of rank function. Closure operator. Circuits. Duality in matroids. Hyperplanes. Submatroids. Restriction, contraction, minor of a matroid. Transversals. Radó-Hall's Theorem and its generalisations. Greedy algorithm versus matroids.
Recommended reading	D. J. A. Welsh, Matroid Theory, Academic Press, 1976

Title	Computational Complexity
Code	ÚINF/VYZ1/04	Teacher	Geffert Viliam
ECTS credits	4	Hrs/week	2/-
Assessment	Examination	Semester	3
T/L method	Lecture
Objective	To give students background in the computational complexity and theory of NP-completeness.
Content	Deterministic and nondeterministic algorithms with polynomial time; NP-completeness. Deterministic simulation of a nondeterministic Turing machine. Satisfiability of Boolean formulae. Other NP-complete problems: satisfiability of a formula in a conjunctive normal form, 3-satisfiability, 3-colorability of a graph, 3-colorability of a planar graph, knapsack problem, balancing, etc. Space bounded computations, classes LOG-space and P-space. Deterministic simulation: Savitch’s theorem. Closure under complement. Classification of computational complexity of problems.
Alternate courses	ÚINF/VYZ1/03 orÚINF/VYZ1/00
Recommended reading	A.V.Aho and J.D.Ullman. The design and analysis of computer algorithms. Addison-Wesley, 1974. P.van Emde Boas. Machine models and simulations. In J.van Leeuwen (ed.): Handbook of theoretical computer science. North-Holland, 1990. Ch.K.Yap. Introduction to the theory of complexity classes. To be published by Oxford Univ. Press. (Electronic version available via anonymous ftp://cs.nyu.edu/pub/local/yap/complexity-bk).

Title	Classical and Quantum Computations
Code	ÚINF/KKV1/06	Teacher	Semanišin Gabriel
ECTS credits	6	Hrs/week	3/1
Assessment	Examination	Semester	3
T/L method	Lecture, Practical
Objective	To provide students information on quantum computers and quantum computations. To allow students to compare classical and quantum models and methods.
Content	The basics of classical theory of computation: Turing machines, Boolean circuits, parallel algorithms, probabilistic computation, NP-complete problems, and the idea of complexity of an algorithm.  Introduction of general quantum formalism (pure states, density matrices, and superoperators), universal gate sets and approximation theorems. Grover's algorithm, Shor's factoring algorithm, and the Abelian hidden subgroup problem. Parallel quantum computation, a quantum analogue of NP-completeness, and quantum error-correcting codes.
Recommended reading	1. A. Yu. Kitaev, Classical and Quantum Computation, American Mathematical Society, Graduate Studies in Mathematics 47 (2002), ISBN 0-8218-3229-8 2. Gruska, J: Quantum Computing. McGraw-Hill Londýn 1999

Title	Combinatorial Algorithms
Code	ÚMV/KOA1/04	Teacher	Jendroľ Stanislav, Lacko Vladimír
ECTS credits	5	Hrs/week	3/-
Assessment	Examination	Semester	2
T/L method	Lecture
Content	Introduction to graphs. Introduction to algorithms and complexity. Sorting algorithms. Search algorithms. Greedy algorithms. NP-completeness. Trees and rooted trees. Generating all spanning trees of a graph. Minimum spanning tree problem. Distance in graphs. Shortest path problem and its analogues. Location centres. Networks. Eulerian gaphs and Chinese Postman's Problem. Hamiltonian graphs. Travelling Salesman Problem. Matchings. Transportation and assignment problems.
Exclusive courses	ÚMV/KOO/04
Recommended reading	1. N. Christophides, Graph Theory: An Algorithmic Approach,   Academic Press, New York 1975 2. G. Chartrand, O. R. Oellermann, Applied and Algorithmic Graph Theory, McGraw-Hill, Inc., New York 1993

Title	Lattice Theory
Code	ÚMV/TZ1/04	Teacher	Lihová Judita
ECTS credits	5	Hrs/week	2/1
Assessment	Examination	Semester	2
T/L method	Lecture, Practical
Content	Distributive and modular lattices, Boolean algebras. Ideals, representation of distributive lattices and Boolean algebras. Metric lattices. Congruence relations of lattices.
Recommended reading	G.Grätzer: General Lattice Theory, Birkhäuser, 1998

Title	Algorithmically Unsolvable Problems
Code	ÚMV/ANP/03	Teacher	Bukovský Lev
ECTS credits	4	Hrs/week	2/-
Assessment	Examination	Semester	4
T/L method	Lecture
Objective	To introduce students to the most important results about the non-existence of an algorithm for solving a given problem.
Content	Axiomatic theories of natural numbers. Definability of recursive functions. Tarski’s theorem on undefinability of truth in formalised arithmethic. Godel incompleteness theorem. Algorithmic unsolvability of particular mathematical problems. Non-existence of an algorithm for finding a solution to Diophantine equations. Reduction of problems and degrees of unsolvability.
Recommended reading	J. Barwise ed., Handbook of Mathematical Logic, North Holland 1977S. C. Kleene, Introduction to the Metamathematics, Van Nostrand 1952, ruský preklad Moskva 1957. E. Mendelson, Introduction to Mathematical Logic, Van Nostrand 1963. M. Davis, Hilbert's Tenth Problem is Unsolvable, Amer. Math. Monthly,1973, 233--269.

Title	Databases Systems
Code	ÚMV/DBS/04	Teacher	Soták Roman, Horňák Mirko
ECTS credits	6	Hrs/week	3/1
Assessment	Examination	Semester	4
T/L method	Lecture, Practical
Objective	To teach students to use and understand the principles of database modelling and the theory of databases through work with a selected concrete database system.
Content	Formal foundations of database systems: logic of database systems, relation algebra. Functional dependencies, decompositions. Normal forms. SQL: aggregate functions, nested queries, modification of data, integrity constraints, modification of the table structure, transitive closure, next aspects of SQL.
Recommended reading	J. Ullman: Principles of database and knowledge – base systems, Comp. Sci. Press., 1988

Compulsory elective courses

Title	Taxes and Information Systems
Code	ÚMV/DIS/04	Teacher	Cechlárová Katarína, Semanišin Gabriel, Soták Roman
ECTS credits	5	Hrs/week	3/2
Assessment	Examination	Semester	1
T/L method	Lecture, Practical
Content	Information system, subsystem, information system development life cycle. Visual modelling, overview of modelling techniques. Structured methodologies. Algorithms in taxes.
Exclusive courses	ÚMV/DIS/00
Alternate courses	ÚMV/DIS/00
Recommended reading	Booch G., Jacobson I., Rumbaugh J.: The Unified Modelling Language user Guide, Addison-Wesley Pub. Co. 1998, ISBN 0-20157168-4

Title	Geometric Transformations
Code	ÚMV/GZ/04	Teacher	Ivančo Jaroslav
ECTS credits	5	Hrs/week	2/1
Assessment	Examination	Semester	1, 3
T/L method	Lecture, Practical
Objective	To provide deeper knowledge of projective spaces and transformation groups.
Content	Projective spaces, Projective transformations, collineations. Fixed elements of a collineation. A clasification of collineations.
Recommended reading	S. V. Duzhin, B. D. Chebotarevsky: Transformation Groups for Beginers, AMS  2004

Title	Control Theory
Code	ÚMV/TS1/04	Teacher	Cechlárová Katarína
ECTS credits	6	Hrs/week	3/1
Assessment	Examination	Semester	1
T/L method	Lecture, Practical
Objective	To teach students the basic concepts of controllable systems.
Content	Controllable systems. Pontrjagin maximum principle. Linear systems, bang-bang controls,  singular controls.. Discrete systems, dynamic programming, Bellmann’s optimality principle. Practical applications of theoretical results.
Alternate courses	ÚMV/TS1/99 or ÚMV/TS1/00
Recommended reading	K. Macki, A. Strauss: Introduction to Optimal Control Theory, Springer, 1980 G. Feichtinger, R.F. Hartl: Optimale Kontrolle okonomischer Prozesse, Berlin, 1986

Title	Graph Theory
Code	ÚMV/TGT/04	Teacher	Jendroľ Stanislav, Madaras Tomáš
ECTS credits	4	Hrs/week	2/-
Assessment	Examination	Semester	1, 3
T/L method	Lecture
Objective	To provide students deeper knowledge of graph theory.
Content	Connectivity of graphs. Hamiltonian graphs. Colouring of graphs. Planar graphs. Oriented graphs. Automorphism of graphs. Snarks. Minors of graphs.
Recommended reading	J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications. Springer-Verlag London 2001 J. A. Bondy and U.S.R. Murty, Graph Theory with Applications. North Holland, Amsterdam 1976. R. Diestel, Graph Theory, Springer-Verlag. New York 2000, 2nd edition.

Title	Introductory Course in Quantum Computers
Code	ÚFV/KVP/02	Teacher	Mockovčiak Samuel
ECTS credits	3	Hrs/week	2/-
Assessment	Examination	Semester	1, 3
T/L method	Lecture
Content	Reasons to study quantum computers (QC): microtechnology, dissipation of energy, "classical PC" is time-demanding. Quantum mechanics for QC. Hilbert space of quantum states. Operators of observables. EPR paradox. Reversible gates. Qubits as quantum states, their evolution. Quantum memory registers. Logic circuits. Quantum algorithms. Superposition of states and parallelism of computations. Entaglement of quantum states. Teleportation. Quantum information..
Recommended reading	J. Gruska: Quantum Computing, McGraw Hill, Maidenhead, 1999 C.PWilliams, S.H. Clearwater: Explorations in Quantum Computing, Springer Verlag, New York, 1998 G.Birkhoff, T.C.Bartee: Aplikovaná Algebra, Alfa, Bratislava, 1981

Title	Foundations of Knowledge Systems
Code	ÚINF/ZNA1/06	Teacher	Vojtáš Peter
ECTS credits	4	Hrs/week	2/-
Assessment	Examination	Semester	1, 3
T/L method	Lecture
Content	Relations of formal models: DBMS, SQL, and logic programming. Summary of different formal models of computational processes, connections among them and translations. Gentzenov systems, semantics and verification of programs. Formal specifications, temporal logics: formulas, models, tableaux.
Recommended reading	M. Ben-Ari. Mathematical logic for Computer Science 2ed. Springer Verlag London 2001 J. Ullman. Principles of database and knowledge based systems. Comp. Sci. Press 1988 J. W. Lloyd. Foundations of logic programming. Springer Berlin 1987

Title	Applied Linear Algebra
Code	ÚMV/ALA1/04	Teacher	Studenovská Danica
ECTS credits	5	Hrs/week	2/1
Assessment	Examination	Semester	2
T/L method	Lecture, Practical
Objective	To teach students basic knowledge about linear algebra; to make students able to apply the theory in concrete excercises.
Content	Matrices over Euclidean rings, canonical forms. Polynomial matrices. Similar matrices. Jordan normal form. Functions of matrices, sequences, series. Inversion of singular matrices, pseudoinverse matrices and their application.
Alternate courses	ÚMV/ALA1/99
Recommended reading	H.E.Rose: Linear algebra, A pure mathematical approach, Birkhäuser   Verlag, 2002 D.Serre: Matrices, Theory and applications, Springer Verlag, 2002. http://www.cs.ut.ee/~toomas_l/linalg/

Title	Fractal Geometry
Code	ÚMV/FRG1/03	Teacher	Bukovský Lev
ECTS credits	4	Hrs/week	2/-
Assessment	Examination	Semester	2
T/L method	Lecture
Objective	To introduce students to the mathematical approach to analysing the concept of a fractal with the possibility of using the results for the construction of fractals.
Content	Concept of a fractal. Basic topology of metric spaces. Self-similarity of a fractal. Fractal as fixpoint of a mapping. Construction of a fractal by iteration. Topological dimension and basic properties. Hausdorff measure and Hausdorff dimension. Topological and Hausdorff dimension of particular fractals. Some methods of construction of a fractal (topological dynamics).
Recommended reading	G. A. Edgar: Measure, Topology and Fractal Geometry, Springer 1990 K. Falconer, Fractal Geometry, John Willey 1992 H. O. Peitgen, H. Jurgens and D. Saupe, Fractals for Classroom, I, II, Springer Verlag, Berlin 1991

Title	Encoding and Transfer of Information
Code	ÚINF/KPI1/01	Teacher	Geffert Viliam, Jirásek Jozef
ECTS credits	4	Hrs/week	2/1
Assessment	Examination	Semester	1
T/L method	Lecture, Practical
Objective	To provide students knowledge of basic principles of information theory, coding and data compression.
Content	Introduction to information theory: entropy, Markov models. Huffman coding, adaptive Huffman coding, applications. Arithmetic coding, dictionary techniques, applications. Lossless image compression. Scalar and vector quantisations. Differential encoding, delta modulation, subband coding, wavelets. Transform coding, DFT, DCT, application to JPEG. Analysis/synthesis schemes; fractal compression. Video compression.
Alternate courses	ÚINF/KPI1/00
Recommended reading	D. Hankersson, G. Harris, P. Johnson: Introduction to Information Theory and Data Compression, CRC Pr.,1998 K. Sayood: Introduction to Data Compression, Morgan Kaufmann, 1996 J. Adámek: Coding and Inormation Theory, ČVUT, 1994 (Czech)

Title	Theory of Groups
Code	ÚMV/TGR1/04	Teacher	Lihová Judita
ECTS credits	5	Hrs/week	2/1
Assessment	Examination	Semester	2
T/L method	Lecture, Practical
Content	Cyclic groups; quotient groups. Finitely generated Abelian groups. Groups of permutations and their applications.
Recommended reading	M.Hall: The Theory of Groups, New York, 1959 L.Fuchs: Abelian groups, Akadémiai Kiadó,Budapest, 1966

Title	Game Theory
Code	ÚMV/TH1/04	Teacher	Cechlárová Katarína, Hajduková Jana
ECTS credits	6	Hrs/week	3/1
Assessment	Examination	Semester	3
T/L method	Lecture, Practical
Objective	To teach the basic methods of game theory and to have students model situations from everyday life as simple games.
Content	Examples of games. Extensive form of a game, value of the game. Von Neumann Morgenstern theory of utility. Matrix games and their solution. Bimatrix games.Theory of negotiations. n-person games: core, Shapley value. Economic applications of game theory.
Alternate courses	ÚMV/TH1/99
Recommended reading	K. Binmore, Fun and games, D.C. Heath, 1992 G. Owen: Game Theory, Academic Press. L.C. Thomas:  Games, Theory and Applications,  Wiley, New York. H.S. Bierman, L.Fernandez, Game Theory with Economic Applications, Addison-Wesley, 1998

Elective courses

Title	Queueing Theory
Code	ÚMV/THO1/04	Teacher	Horňák Mirko
ECTS credits	6	Hrs/week	4/-
Assessment	Examination	Semester	1, 3
T/L method	Lecture
Objective	To acquaint students with functioning of simple queuing systems and with the analysis of corresponding stochastic processes and Markov chains.
Content	Queuing system. Input request stream. Intensity and parameter of input request stream. Stationarity. Memoryless stochastic process. Markov's theorem. Ergodic theorem. Markov chain.
Recommended reading	B.V. Gnedenko and I.N. Kovalenko, Introduction to Queueing Theory, Second Edition, Birkhauser Boston, Cambridge MA 1989

Title	Probability and Statistics
Code	ÚMV/PST1b/04	Teacher	Skřivánková Valéria, Ohriska Ján
ECTS credits	6	Hrs/week	2/2
Assessment	Examination	Semester	1
T/L method	Lecture, Practical
Content	Random vectors, their distributions and characteristics. Correlation and regression. Random sample, sampling distributions and characteristics. Point estimates and their properties. Maximum likelihood method. Interval estimates; confidence interval construction. Testing of statistical hypothesis; critical region and level of significance. Parametric and nonparametric tests.
Recommended reading	Mandenhall W.: Introduction to probability and statistics,PWS Publishers, Boston, 1987 Sincich T.: Statistics by example, Dellen Publishing Company, New Jersey, 1990

Title	Mathematical Economics
Code	ÚMV/MAE1/04	Teacher	Cechlárová Katarína
ECTS credits	5	Hrs/week	3/-
Assessment	Examination	Semester	2
T/L method	Lecture
Objective	To teach the basic concepts and methods of modern mathematical economics.
Content	The notion of exchange economy. Edgeworth box. Preferences and utility functions. Optimality in exchange economies. Existence of core. Walrasian equilibrium. Optimality and decentralisation. Production economies.
Alternate courses	ÚMV/MAE1/99
Recommended reading	C.D. Aliprantis, D.J. Brown, O. Burkinshaw: Existence and optimality of competitive equilibria, Springer 1989 W. Hildenbrand, A.P. Kirman: Equilibrium analysis, North Holland, A. Takayama: Mathematical economics, Cambridge University Press, 1985

Title	Polyhedral Theory
Code	ÚMV/POT1/04	Teacher	Jendroľ Stanislav
ECTS credits	4	Hrs/week	2/-
Assessment	Examination	Semester	2
T/L method	Lecture
Objective	To teach students basic knowledge of the theory of convex polyhedra and polyhedral maps.
Content	Combinatorial and geometric properties of three-dimensional convex polyhedra and their analogues: polyhedral maps. Euler’s theorem; Steinitz’s theorem. Light subgraphs. Face and vertex vectors. Groups of symmetries of polyhedra. Applications in optimisation and chemistry.
Recommended reading	B. Grunbaum: Convex polytopes (2nd edition), Springer New York 2003 G.M. Ziegler: Lectures on Polytopes, Springer-Verlag, New York, 1996

Title	Number Theory
Code	ÚMV/TCI/04	Teacher	Harminc Matúš
ECTS credits	4	Hrs/week	2/-
Assessment	Examination	Semester	2
T/L method	Lecture
Objective	To familiarise students with the divisibility and congruences of integers, linear and quadratic congruences and arithmetic functions.
Content	Euclidean algorithm. Fundamental theorem of arithmetic. Primes, composites, canonical form and its applications. Congruences, criteria of divisibility. Arithmetic functions. Euler’s theorem. Fermat’s theorem. Wilson’s theorem. Linear and quadratic congruences.
Recommended reading	M. B. Nathanson: Elementary Methods in Number Theory. Springer, 2000 H. E. Rose: A Course in Number Theory. Clarendon Press, Oxford, 1994

Title	Universal Algebra
Code	ÚMV/UAL/04	Teacher	Studenovská Danica
ECTS credits	5	Hrs/week	3/-
Assessment	Examination	Semester	2
T/L method	Lecture
Objective	To provide basic knowledge of universal algebra and to make students able to apply this knowledge in concrete situations.
Content	Algebraic structures. Homomorphisms and congruences. Direct and subdirect products. Terms. Free algebras. Birkhoff’s theorems about varieties.
Exclusive courses	ÚMV/KAL1/04
Recommended reading	S.Burris, H.P.Sankappanavar: A Course in Universal Algebra. Springer-Verlag, 1981 B. Jónsson: Topics in universal algebra, Springer-Verlag 1972 G. Grätzer: Universal Algebra, 2nd edition, Springer Verlag, 1979

Study programme  Manager Mathematics

(Full-time master)

      Code  Title  ECTS  Credit  Hours/week Assessment   Recommended Year/Semester

Compulsory courses

ÚMV/DPM1a/03	Diploma Work	2	-/-	Recognition	1/1
ÚMV/PST1b/04	Probability and Statistics	6	2/2	Examination	1/1
ÚMV/TH1/04	Game Theory	6	3/1	Examination	1/1
ÚMV/TS1/04	Control Theory	6	3/1	Examination	1/1
KFaDF/DF2p/07	History of Philosophy	4	2/1	Examination	1/1
ÚMV/DPM1b/04	Diploma Work	2	-/-	Recognition	1/2
ÚMV/KOO/04	Combinatorial Optimisation	6	3/1	Examination	1/2
ÚMV/DBS/04	Databases Systems	6	3/1	Examination	1/2
ÚMV/MAE1/04	Mathematical Economics	5	3/-	Examination	1/2
ÚINF/KKV1/06	Classical and quantum Computations	6	3/1	Examination	2/3
ÚMV/THO1/04	Queueing Theory	6	4/-	Examination	2/3
ÚMV/DPM1c/04	Diploma Work	5	-/-	Recognition	2/3
ÚMV/DPM1d/04	Diploma Work	15	-/-	Recognition	2/4
ÚMV/FAN/06	Functional Analysis	6	3/1	Examination	1/2

Compulsory elective courses

ÚINF/ZNA1/06	Foundations of Knowledge Systems	4	2/-	Examination	1/1
ÚINF/MZK/06	Mathematical Foundations of Cryptography	6	3/2	Examination	1/1
ÚMV/PRA1/04	Law	3	2/-	Examination	1/1, 2/3
ÚMV/TGT/04	Graph Theory	4	2/-	Examination	1/1, 2/3
ÚMV/TIN1/03	Theory of Information	4	2/-	Examination	1/1, 2/3
ÚMV/TKO1/04	Theory of Codes	6	4/-	Examination	1/1, 2/3
ÚMV/TMT/04	Matroid Theory	5	3/-	Examination	1/1, 2/3
ÚINF/VYZ1/04	Computational Complexity	4	2/-	Examination	1/1, 2/3
ÚMV/GZ/04	Geometric Transformations	5	2/1	Examination	1/1, 2/3
ÚMV/FRG1/03	Fractal Geometry	4	2/-	Examination	1/2
ÚMV/TGR1/04	Theory of Groups	5	2/1	Examination	1/2
ÚMV/SKA1/99	Seminar on Combinatorial Algorithms	2	-/2	Assessment	1/2
ÚMV/APS1/99	Applied Statistics	6	3/2	Examination	1/2
ÚMV/ALA1/04	Applied Linear Algebra	5	2/1	Examination	1/2, 2/4
ÚMV/POT1/04	Polyhedral Theory	4	2/-	Examination	1/2, 2/4
ÚMV/UAL/04	Universal Algebra	5	3/-	Examination	1/2, 2/4

Recommended elective courses

ÚMV/SVK1/01

Student scientific conference

4

-/-

Assessment

1/2