The postgraduate study programme aims at preparing top scientific and research specialists in various areas of mathematics with applications in electrical engineering fields of study, especially in the area of stochastic processes, design of optimization and statistic methods for description of the systems studied, analysis of systems and multisystems using discrete and functional equations, digital topology application, AI mathematical background, transformation and representation of multistructures modelling automated processes, fuzzy preference structures application, multicriterial optimization, research into automata and multiautomata seen in the framework of discrete systems, stability and system controllability. The study programme will also focus on developing theoretical background of the above mentioned areas of mathematics.

The graduates of the postgraduate study programme Mathematics in Electrical Engineering will be prepared for future employment in the area of applied research and in technology research teams. Due to the comprehensive use of computer engineering throughout the study programme, the graduates will be well prepared for work in the area of scientific and technology software development and maintenance. The graduates will also be prepared for management and analytical positions in companies requiring good knowledge of mathematical modelling, statistics and optimization.

The graduates of the postgraduate study programme Mathematics in Electrical Engineering will be prepared for future employment in the area of applied research and in technology research teams. Due to the comprehensive use of computer engineering throughout the study programme, the graduates will be well prepared for work in the area of scientific and technology software development and maintenance. The graduates will also be prepared for management and analytical positions in companies requiring good knowledge of mathematical modelling, statistics and optimization.

doc. RNDr. Zdeněk Šmarda, CSc.

1. Analysis and reprezentation of discrete and continuous topological information

   The dissertation will be focused in the study of the spatio-relational discrete and continuous relationships and their proper, especially digital, representation (by suitable algebraic and topological structures), with respect to the later computer processing. The source of the studied spatio-relational properties it could be various data structures in computer science or even real-existing structures and objects of physical, biological or other nature. The theoretical background for the dissertation is provided especially by general and digital topology, formal concept analysis (FCA) and other selected parts of modern algebra and discrete mathematics.

   Supervisor: Kovár Martin, doc. RNDr., Ph.D.

2. Applications of structured systems in the sense of Mesarovic-Takahara and multiautomata within analysis of processes and signals.

   In the project there will be investigated input-output systems with structured input and output spaces endowed with algerbaic or analytical multistructures and corresponding compatible input-output relation. In the connection with multiautomata considered as actions of binary hyperstructures on suitable choosen state spaces, the constructes tools will be applied onto investigation of concrete systems and signals with continuous and discrete time.

   Supervisor: Chvalina Jan, prof. RNDr., DrSc.

3. Asymptotical properties of solutions of difference equations and systems of difference equations

   The project deals with asymptotical properties of solutions of difference equations and systems of difference equations. Next we will study sufficient and nessesary conditions which guarantee at least one solution with prescribed properties. Wazewki's topological method and its modification for difference equations and systems of difference equations will be the theoretical base for investigation.

   Supervisor: Baštinec Jaromír, doc. RNDr., CSc.

4. Estimates of solutions of differential delayed systems.

   In the project it is necessary to derive sufficient conditions for existence of bounded solutions of delayed differential systems. An attention will be paid especially to quasi-linear systems. Estimates of solutions will be, among others, constructed on the basis of properties of matrices of linear approximations. The project is related with the stability of solutions of differential delayed systems.

   Supervisor: Diblík Josef, prof. RNDr., DrSc.

5. General Choquet integral in multicriterial decision making

   In many decision problems a set of actions is evaluated with respect to a set of viewpoints, called criteria. In general, evaluations with respect to different criteria can be discordant with respect to preferences. One of the simplest aggregation procedures is the weighted sum of the evaluations with respect to considered criteria.Some more complex aggregation procedures have been proposed in order to take into account specific aspects in evaluating importance of criteria, such as interaction among criteria. The interaction of criteria has been considered through non-additive integrals such as Choquet and Sugeno integral. A useful tool is a generalization of the Choquet integral which takes into account the fact that the importance of criteria depends on the level of their evaluations. The properties and applications of Choquet integral will be investigated.

   Supervisor: Hliněná Dana, doc. RNDr., Ph.D.

6. Integral , integrodifferential inequalities and their applications in the theory of integrodifferential equations.

   The aim of the work is to devote new forms of integral and integrodifferential inequalities through which will be determined extremal solutions of some classes of integrodifferential equations and sufficient conditions of boudedness of solutions, as well.

   Supervisor: Šmarda Zdeněk, doc. RNDr., CSc.

7. Mathematical structures generated by differential equations and their applications

   Qualitative behaviour of differential equations. The study may be directed not only to analytic methods, but also to algebraic and geometric approaches. Investigation may be extended to functional differential equations, as well as to functional equations only. The situations when the studied objects are not suffciently smooth may also be under consideration. Especially in these cases it will be useful to find possible applications, e.g. in the theory of signal processing.

   Supervisor: Neuman František, prof. RNDr., DrSc.

8. Modification of the Wazewki's topological method for integrodifferential equations

   The project deals with a generalization of the Wazewski's qualitative method of the investigation of ordinary differential equations to integrodifferential equations. It will be aimed especially to modifications of notions as ingress, egress points , u,v-subset , the initial value problem for integrodifferential equations. The generalized topological method will be applied to the asymptotic investigation of continuous dynamic systems described by integrodifferential equations.

   Supervisor: Šmarda Zdeněk, doc. RNDr., CSc.

9. Representation of solutions of discrete and dynamic systems on time-scales.

   In the project will be considered the problem on a representation of solutions of discrete and dynamic linear systems on time scales. A special attention will be paid to some delaed systems. Properties of solutions induced from their representation will be studied.

   Supervisor: Diblík Josef, prof. RNDr., DrSc.

10. Topological analysis and digital reprezentation of microstructures

    The main contents of the dissertation will be analysis and representation of topological and geometrical properties of material structures with small dimensions. As examples of such structures there can be mentioned e.g. the cellular structure of DNA or the material structures on molecular or atomic level (alternatively, particle level). With respect to applicant's interests, the dissertation could be focused more in the study of the topological properties of the underlying structure, or more into the representation of these properties suitable for later digital processing. The theoretical background for the dissertation is provided especially by general, algebraic and digital topology, formal concept analysis (FCA) and other selected parts of modern mathematics but could be in close relationships with other scientific disciplines (molecular biology, quantum mechanics and other) with respect to the individual applicant's interests and abilities.

    Supervisor: Kovár Martin, doc. RNDr., Ph.D.