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. Advanced stochastic programming models

   Modern applied mathematical programs often involve uncertain parameters. Their presence significantly reduces usefulness of classical deterministic optimization techniques. So, stochastic programming belongs among the fundamental modelling approaches for decision making under risk and uncertainty. In model building phase, the important role is played by advanced deterministic reformulations and their transformations. The goal of the proposed research is to modify existing and develop original deterministic reformulations and to study their theoretical properties, solvability, applicability and connectivity.

   Tutor: Popela Pavel, RNDr., Ph.D.

2. Applications of differential geometry in the control of nonholonomic systems in bionics.

   The goal is to analyze and control nonholonomic mechanisms (particularly robotic snakes) by methods of modern differential geometry and advanced linear algebra. More precisely, the theory of Lie groups and algebras, Pfaff systems, geometric (Clifford) algebras and representation theory will be involved. The geometric algebra starts to play (due to good computational complexity) an important role in the control of autonomous robots and one of our goals is to translate entire description of a control machinery to the language of geometric algebra for specific mechanisms.

   Tutor: Hrdina Jaroslav, doc. Mgr., Ph.D.

3. Computation methods of geometric algebras with applications in bionics

   The goal is the systematic analysis, mathematic model and creation of software tools to control bionic mechanisms (robotic snakes) by menas of geometric (Clifford) algebras. In the implementation of selected algorithms, our goal is to use the computational complexity based on algebraic properties of Clifford algebras. It is the situation analogous to the use of quaternionic algebras for orthogonal transformation preserving the orientation.

   Tutor: Hrdina Jaroslav, doc. Mgr., Ph.D.

4. Deconvolution and visualization of digital images in physics aplications

   The main goal of the work is to develop a deconvolution algorithm suitable for processing of digital high dynamic range images with very complicated point spread function. Data obtained in extreme ultraviolet part of the spectrum (10-40 nm) by means of spacecrafts observing plasma in solar corona (SDO, TRACE, SOHO) are a good example of such images and will be used for algorithm testing. The second goal is to find a suitable algorithm for resulting images artifact-free visualization.

   Tutor: Štarha Pavel, doc. Ing., Ph.D.

5. Distributed and hierarchical mathematical programs

   Recent challenges in the solution of some advanced engineering optimization problems are related to their specialised hierarchical and distributed structure. Such a structure influences theoretical properties of models and also efficiency of classical approaches to modelling and solving. Thus, the important role is played by suitable modelling techniques, which further influence development of new and modification of existing algorithms. So, the proposed research task in the area of mathematical programming is to enhance existing and develop new hierarchical and distributed apporaches to modelling and solution of mathematical programs together with their implementation and application.

   Tutor: Popela Pavel, RNDr., Ph.D.

6. Non-homogeneous ideals in polynomial and local R-algebras with regard to applications in differential geometry

   The PhD theme is focused on research in algebras of polynomials in more indeterminates over the real field, particularly on their factor algebra with emphasis on the factorization by non-homogeneous ideals. These algebras are used in several situations in differential geometry: first, as Weil algebras serving as an effective tool for describing of generalized velocities and contact elements, second, for the constructions of manifolds as spectra of such algebras over which the differential calculus is then being built. A role of non-homogeneous ideals has not been systematically investigated up to now, and research in this area is a new and challenging scientific research.

   Tutor: Kureš Miroslav, doc. RNDr., Ph.D.

7. Numerical methods of spatial objects analysis

   The main goal of the work is to develop a numerical methods for analyzing a hollow fibers distribution in a heat exchangers. Spatial distribution, orientation and interaction of the fibers influences a heat exchanger efficacy. The next task is to find and describe a good fibers distribution with respect to the heat exchanger efficacy. It is necessary to create special software application for this problem solving.

   Tutor: Štarha Pavel, doc. Ing., Ph.D.