The study of mathematical engineering focuses on developing mathematical disciplines used as theoretical foundations of engineering fields especially mechanical engineering. These are above all modern parts of approximate and numerical methods, stochastic methods, fuzzy and qualitative models, discrete mathematics, modern computer methods, parts of modern mathematical analysis. Doctoral dissertations are devoted both to developing the respective mathematical disciplines and to specific theoretical and experimental engineering problems.

prof. RNDr. Miloslav Druckmüller, CSc.

1. Arithmetic of the generalized polynomials

   First mention the motivation of introduction of the notion of generalized polynomial by means of derivatives and integrals of fractional orders. Introduce the definition of generalized polynomials over a commutative field whose exponents are rational, real or complex numbers. For the ring of generalized polynomials describe known larithmetical laws (existence of gcd, Noether property etc.). Attempt to extend this notion to polynomials whose exponents belong to other algebraic structures and attempt also to extend the known arithmetical laws on generalized polynomials. Some problems on the arithmetic of these polynomials solve using a computer.

   Tutor: Skula Ladislav, prof. RNDr., DrSc.

2. Closure operators in digital topology

   The topological approach to digital topology will be developed based on using closure operators. The results should provide new algorithms for digital image processing.

   Tutor: Šlapal Josef, prof. RNDr., CSc.

3. Fuzzy Stochastic Models of Reliability

   Fuzzification of the probability distributions for reliability modeling of elements and systems by means of the fuzzy numerical characteristics of vague times between failures. Account of their properties, PC implementation of algorithms, and applications.

   Tutor: Karpíšek Zdeněk, doc. RNDr., CSc.

4. Numerical methods for stochastic differential equations

   Stochastic differential equations enable to model situations with random data, in the equation coefficients and solutions are random processes. For applications of the equations in technical praxis numerical methods for solving these equations are important. The aim of the project is to learn theoretical base of the stochastic differential equations and compile a survey of methods for their numerical solution. The selected methods will be tested by numerical experiments, eventually will be applied to solution of particular problem of practise.

   Tutor: Franců Jan, prof. RNDr., CSc.

5. Reduction of square matrices over an integral domain for search of elementarity of matrices

   The aim of this theme is the investigation of integral domains such that there are invertible matrices with coefficients from these domains which are not elementary.To this purpose it is possible to use the integral domains with discrete norm. By means of this norm in many cases the square matrices can be reducated to matrices with less norm and to obtain in a such way the"standard form" of a matrix. From the standard form it is possible to find out whether an invertible matrix is elementary. The topic of the PhD theses to directed to the integral domain of the integral quaternions. For this purpose it is possible to make use of a computer.

   Tutor: Skula Ladislav, prof. RNDr., DrSc.

6. Statistical analysis of censored samples

   The goal of the work is the determination of suitable estimators for censored samples for type I and type II censoring and for random censoring. Focal point of the work will consist in finding the parameter estimates for censored samples from extreme value distributions. The algorithm development and computer implementation of suggested algorithms will be an important part of the work. The illustration of the properties of suggested methods will be demonstrated by simulations and on real data as well. The environmental data (rainfall data and air pollution data) and geological data will be preferred.

   Tutor: Michálek Jaroslav, doc. RNDr., CSc.

7. The method of the lest squares and its application

   Using the method of the least squares introduce the notions of various generalized inverses of a matrix. Particularly, describe the Moore-Penrose inverse. Use this M-P inverse to the least squares solution of the sytem of linear equations. Describe various methods of a numerical solution of this problem which are using a computer, and perform their evaluation.

   Tutor: Skula Ladislav, prof. RNDr., DrSc.

8. Topological Structures on categories

   The topic is focused on the study of topological structures on categories. In addition to the classical closure operators, particularly convergence structures and neighborhood systems will be investigated.

   Tutor: Šlapal Josef, prof. RNDr., CSc.