Master Degree Exam Requirements – Applied
Informatics
Subject:
Mathematical Methods
2013/2014
1.
Measuring
uncertainty. Probability, basic definitions and rules
(addition rule for elementary events, conditional probability, multiplication
rule, Bayes’ Theorem).
2.
Random
variable, discrete and continuous probability distributions. Most common models of
distribution (for example: alternative, binomial, Poisson, uniform, normal,
exponential).
3.
Statistical
inference. Population and random sample, methods of sampling, sampling distribution,
sampling error. Statistical point estimate, the quality of an estimate,
confidence interval for mean and proportion.
4.
Principles
of testing of statistical hypothesis,
5.
Description
of relationships between variables. Regression
and correlation methods. Tests of independence for qualitative data,
contingency table.
6.
Stochastic
and deterministic models. Stochastic processes.
Applications of stochastic models. Renewal models: renewal table and solution
with the use of regular Markov chains. Markov
chains and their description, regular chains, absorption chains, long – run
properties of Markov chains.
7.
Modelling
and simulation. Random numbers, congruent generators.
Transformation random numbers to random numbers from specific distributions.
The assessment of the quality of random number generators (statistical
properties).
8.
Principles
of Numerical mathematics and approximation of functions. Arithmetic
operations and errors in numerical computations. Interpolation by polynomials,
interpolation by spline functions, least square method.
9.
Solution of
nonlinear equations and numerical optimization. Bracketing
method for locating roots, methods for finding roots of equations, estimation
of error bounds, conditions of convergence. Minimization of function.
10. Numerical solutions of systems
of linear algebraic equations – direct and indirect methods. Gaussian
elimination method, partial pivoting, triangular factorization, ill
conditioning. Iterative methods.
11. Numerical differentiation and
integration, solution of ordinary differential equations. Numerical
differentiation, basic formulas. Numerical Integrations, basic and composite
rules. Euler’s method and Runge-Kutta methods.
12. Graph coloring
and its application. Graph coloring,
chromatic number, independent set, independence number, relation between χ(G) a α(G). Heuristic algorithms determining
chromatic number and their practical applications.
13. Searching of labyrinths and Eulerian graphs. Trémaux, Tarry, Edmonds-Johnson algorithms, mutual
relations among these algorithms. Searching of Eulerian
trail in Eulerian graph. Searching of the minimum
number of trails containing all edges of the given graph. Chinese postman
problem. Practical applications of mentioned algorithms.
References:
Arltová M. a kol.: Sbírka příkladů ze statistiky (Statistika A). VŠE Praha, 1997
Čermák L., Hlavička
R.: Numerické metody, Akademické nakladatelství CERM,
s.r.o., Brno, 2006
Demel, J.: GRAFY
a jejich aplikace,
Academia, Praha, 2002
Hebák P., Kahounová J.: Počet pravděpodobnosti v příkladech. Informatorium, Praha 2005.
Hindls R., Hronová S., Seger J.: Statistika pro ekonomy. Professional
Publishing, Praha 2003
Kučera, L.: Kombinatorické
algoritmy. SNTL, Praha,
1989
Míka, S., Brandner, M.:
Skalská H.: Aplikovaná statistika. Sbírka elektronických textů, UHK 2003
Skalská H.: Stochastické modelování. Gaudeamus, Hradec Králové, 2006
English
References Equivalents:
Gentle E.J.: Random Number Generation and
Grinstead
Groebner D.F.,
Hillier F.S., Lieberman G.J.: Introduction to operations research.
McGraw-Hill, 2004
McClave J.T.,
Benson P.G., Sincich T.: Statistics for Business and Economics.
Prentice Hall, Inc., 2003
Roberts, F. S., Tesman, B.: Applied Combinatorics,
Second Edition, Pearson Prentice Hall,