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. 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.

3. 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.

4. Modern methods for solving nonlinear variationall differential equations

   Since intial boundary value problems for evolutionary mainly partial differential equations in technology often do not admit classical solution, various generalized formulations of these problems were proposed. The aim of the study will be comparison of these formulations and studying existence and uniqueness of their solutions. Then the theory will be applied to particular problems occuring in technology and alternatively to carry out numerical experiments.

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

5. Periodic boundary value problem for second order ODE.

   For certain classes of nonlinear nonautonomous ordinary differential equations, new sufficient conditions on the existence and uniqueness of periodic solutions with prescribed period will established.

   Tutor: Lomtatidze Aleksandre, prof., DrSc.