Creative research conducted in a field of applied mathematics and co-operation with experts in other engineering and scientific fields are major focuses of the study that the students should master. They are encouraged in a maximum degree possible to engage in research projects of the Institute of Mathematics at the BUT Faculty of Mechanical Engineering. Computers are available to students as standard equipment.

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. Computational 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. Connectedness and Jordan curves in the digital plane

   The topic is oriented on finding and studying convenient structures on the digital plane by using tools of the graph theory and general topology. We will be interested in structures providing definitions of connectedness and possessing analogues of the Jordan curve theorem. The research is motivated by applications of the obtained results for solving problems of digital image processing.

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

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. Evaluation of models of radiation characteristics of optical elements

   New trends in industrial applications dealing with lighting, using optical elements combining the principles of diffraction and refraction of light. Design of such optical element requires monitoring of factors affecting resulting radiation characteristic. Therefore modelling of radiation characteristics of such elements is essential for their production. Goal of this topic is to study and statistical evaluate effects of various factors on optical properties of these elements.

   Tutor: Bednář Josef, Ing., Ph.D.

7. Mathematical Description of Electromagnetic Pulse Energy Center Velocity in the Case of Pulse Transfer of Informations in Dispersive Medium.

   Applications of tools of informatics, computer science and numerical mathematics for the description of motion of an electromagnetic pulse in dispersive medium. This approach shall be exiting from the solution of an equation describing these sorts of waving, which is identical, from the mathematical point of view, with the relativistic wave equation. It is possible to make an effort to apply the Vainshtein generalized definition of the group velocity of a pulse, eventually another definitions of this velocity, to various types of dispersive media and to different types of input pulses. The applications is expected in the pulse transfer of informations for example in waveguides, optical fibres and optical cables, especially in the case of the nanosecond pulses.

   Tutor: Klapka Jindřich, doc. RNDr., CSc.

8. 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.

9. Spatiotemporal discrete data modelling

   There are frequently applied methods of kriging available for spatiotemporal description a dependencies of random variables with normal distribution. However, in some cases the modelled variables can be discrete. The aim of the study is to review available models for spatiotemporal models of random variables with discrete distribution, their properties and their comparison at least by simulations in suitable software.

   Tutor: Hübnerová Zuzana, doc. Mgr., Ph.D.