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.

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

2. 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.

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

3. Decomposition in mathematical programming

   Recent challenges in the solution of engineering optimization problems are related to their specialised and complicated distributed structure. It influences their properties and solvability. Hence, the key role is often played by decomposition approaches in model building, modifications of existing algorithms, and development of new solution techniques. Therefore, the proposed research goal in the area of mathematical programming and its applications is to develop, modify and integrate suitable decomposition techniques in both modeling and solution areas of studied problems.

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

4. Geometric Methods in Mechanics

   The topis is focued on the use of differential geometry methods in mechanics and in continuum mechanics. The scientific research of natural operators, differential invarinats and applications of commutative algebra methods for the description of mechanical systems and materials.

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

5. Geometrical methods of cloud of points processing

   The processing of cloud of points is an important problem in technical use. Huge amount of data is produced by modern measuring instruments (Lidar technology, 3D scanner, etc.) and it is necessary to process this set of data. In this area a lot of different approaches are possible (image recognition, filters, segmentation algorithms) and there is also the area of unsolved problems (building reconstruction in Lidar data, segmentation of objects). Then, the comparison of different type of visualisation will be shown with spline or subdivision surfaces or new proposal methods.

   Supervisor: Druckmüller Miloslav, prof. RNDr., CSc.

6. Jets and symplectic manifolds

   Symplectic manifolds are studied in differential geometry and they formalize the notion of phase space from physics. The topic is focused to a use of jets and jet groups in the theory of symplectic manifolds, a use of algebraic methods and applications of symplectic geometry.

   Supervisor: Kureš Miroslav, doc. RNDr., 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.

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

8. Stability of fractional differential equations and their discretizations

   Fractional differential equations (i.e. equations involving derivatives of an unknown function of non-integer orders) turn out to be a more suitable model of various problems than classical ordinary differential equations. Since these equations can be solved analytically only in special cases, qualitative and numerical methods of their solving become very important. The aim of the study is to perform the research of these methods, with a special attention paid to stability of investigated fractional differential equations and their basic numerical discretizations. Obtained results will be applied to real problems described via linear as well as nonlinear fractional differential equations.

   Supervisor: Čermák Jan, prof. RNDr., CSc.

9. The mathematic apparatus of the multi-axis manufacturing machines error analysis

   The multi--axis manufacturing machines analysis is a modern topic in the production technology. Together with the increased demands on the manufacturing machines accuracy, more affecting factors are included and the error measuring becomes more complex, too. This leads necessarily to the the employment of much more sophisticated mathematical tools than ever before. The topic of the doctoral studies is both research of the mathematical background, proposal of the algorithms specialized on the multi-axis machines error analysis and the appropriate tests on real data. At the beginning, we assume to use the tools of mathematical robotics such as Lie theory, geometric algebra and differential geometry together with advanced linear algebra and matrix theory such as pseudoinverse matrices. In further research, other mathematical disciplines such as statistics, graph theory, interpolation etc. are very likely to be included.

   Supervisor: Druckmüller Miloslav, prof. RNDr., CSc.