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. Airborne imaging spectroscopy in assesment of urban ecosystem‘s thermal properties

   Recently, significant fluctuations of climatological factors, mainly long-lasting hot seasons without precipitation, results in worsening living conditions from city themal regime point of view. Recent airborne imaging spectorscopy in reflective (VNIR, SWIR) and thermal (TIR) bands of electromagnetic spectrum together with LiDAR scanning offer sources for description of state and structural city quantities and their relations. The aims of the work are: 1/ a development of algorithms and mathematical models for urban thermal properties simulations using mentioned geoinformational data; 2/ assessment of impact of urban greenery on mitigation of weveheat at different spatial scales. The suggested topic is in the focus of long-term research activities of the RS group, Global Change Research Institute CAS and CzechGlobe varanties both data/soft. availability and part time job..

   Tutor: Zemek František, doc. Ing. Mgr., Ph.D.

2. Algebraic Control Theory

   Using the general ring theory we will analyze the feedback control systems. Our goal is to design a stable minimum control elements.

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

3. Algebraic principles of volumetric accuracy

   We will use methods of linear algebra, general algebra, and discrete geometry to solve general problems arising with the analysis of volumetric precision multi-axis production machines

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

4. Algebraic-geometric methods in continuum mechanics and in materials with microstructure

   The theme is focused on the application of the theory of jets and Weil algebras for materials corresponding with Cosserat continuum and generalizations. It is a new use of methods of commutative algebra and modern differential geometry in applications.

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

5. Analysis of dynamical systems exhibiting a chaotic behavior

   Some dynamical systems exhibit a complex behavior known as deterministic chaos. The topic is focused on analysis of suitable chaotic models (with respect to a widest set of system's parameters). This analysis can be extended on models of non-integer (fractional) order as well.

   Tutor: Nechvátal Luděk, doc. Ing., Ph.D.

6. Asymptotics and oscillation of dynamic equations

   We shall study qualitative properties of various second order and higher order nonlinear differential equations, which arise from applications (including, e.g., the equations with a (generalized) Laplacian). The research will be focused, for example, on obtaining asymptotic formulae for solutions or establishing new oscillation criteria. We shall deal not only with differential equations but also with their discrete (or time scale) analogues. This will enable us, among others, to compare (and explain) similarities and/or discrepancies between a continuous case and some of its discretization. We plan to utilize various (modifications of and combinations of) standard tools, but also to develop new techniques.

   Tutor: Řehák Pavel, prof. Mgr., Ph.D.

7. Fractional Calculus Applications in Control Theory

   Due to utilization of differential and integral terms of non-integer order, fractional calculus allows generalization of the classical models of control theory with preserved linearity. The goals of the thesis is theoretical and practical comparison of properties of classical and fractional control terms, in particular in aerospace industry.

   Tutor: Kisela Tomáš, Ing., Ph.D.

8. Generalized spatial models

   The topic aims at spatial models. These models allow us to describe a distribution of random variables which can be spatially dependent and their parameters can be a function of available explanatory variables. Often the normal distribution of the explained variables is assumed. The goal of the study is to describe available methods of spatial models for variables with distribution other than the normal distribution, examine their properties and compare them at least by simulations in suitable software. Moreover, an application of the models in real data analysis will be of interest.

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

9. Mathematical modelling of dynamical systems

   Dynamical systems theory provides a useful tool for description and qualitative investigation of many engineering problems. There is a need of profound problem analysis for a construction of adequate mathematical model. Considering too many details is generally leading to complications in the model investigation whereas a negligence of fundamental factors can depreciate obtained analysis. Therefore it is necessary to compare the model analysis with real data (if it is possible). The work consists in applying mathematical and numerical analysis in engineering problems modelling and proper interpretation of obtained results.

   Tutor: Tomášek Petr, doc. Ing., Ph.D.

10. 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.

11. Optimization of atmospheric correction of airborne thermal imagery data and improvement of algorithms for separation of emissivity from temperature

    The theoretical part will focus on existing atmospheric correction of airborne hyperspectral thermal data and separation of emissivity from temperature. The improvements of following algorithms are expected: 1/ an algorithm for atmospheric correction of the HS thermal data, including specification and data source for proper parametrization; 2/ algorithm for separation of emissivity from temperature. The practical aspect of the thesis will be payed to implementation of the improved algorithms in the form of software modules which would became an important part of the whole processing chain for the airborne data acquired by TASI hyperspectral sensor. All data, software will be available at the Global change research institute, ASCR (CzechGlobe) where the Ph.D. candidate will be trained and will get a part time job.

    Tutor: Zemek František, doc. Ing. Mgr., Ph.D.

12. 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.

13. Spatial model of nongaussian data

    Frequently applied models based on kriging can be used for spatial modelling of dependencies between random variables with normal distribution. However, in some cases random variables with a nongaussian distribution are to be modeled. The aim of the study is to compare available methods and their application in real data analysis.

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

14. Stability and stabilization of dynamic systems with a time delay

    The study will be focused on analysis of continuous and discrete dynamic systems involving factor of a time delay. These systems appear in many engineering and natural science branches (control theory, neural networks, population dynamics) and model equations are supposed to by analyzed with respect to these applications. The studied topics are: stabilizing (or destabilizing) effect of delays on dynamics of a system, asymptotics of solutions and relationship between continuous and discrete models. These models can involve also modern types of dynamic systems, including chaotic and fractional ones.

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