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.

2. round (applications submitted from 01.07.2019 to 31.07.2019)

1. Aggregation operators in fuzzy logic

   Fuzzy logic is a form of many-valued logic or probabilistic logic. It has been applied to many fields, from control theory to artificial intelligence. The topic of the thesis is to study new constructions and properties of fuzzy logic connectives via aggreagtion operators.

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

2. Controllability problems for discrete equations with aftereffect

   The aim is to solve some controllabity problems on relative and trajectory controllability for systems of discrete equations with aftereffect. It is assumed that criteria of controllability will be derived and relevant algorithms for their solutions will be constructed (including constructions of controll functions). Starting literature – the book by M. Sami Fadali and Antonio Visioli, Digital Control Engineering, Analysis and Design, Elsewier, 2013 and paper by J. Diblík, Relative and trajectory controllability of linear discrete systems with constant coefficients and a single delay, IEEE Transactions on Automatic Control (Early Access), (https://ieeexplore-ieee-org.ezproxy.lib.vutbr.cz/stamp/stamp.jsp?tp=&arnumber=8443094), 1-8, 2018. During study a visit to Bialystok University, Poland, where similar problems are studied, is planned.

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

3. Semianalytical methods for solving fractional differential equations.

   The aim of the dissertation thesis is modification of the differential transformation method and the iteration method with a difference kernel to solving initial value problems for fractional differential equations. Convergence analysis of proposed methods will be investigated as well.

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

4. Stochastic differential equations in electrical engineering

   By adding some randomness to the coefficients of an ordinary differential equation we get stochastic differential equations. Such an equation describes the current in an RL circuit with stochastic source. Then the solution of the equation is a random process. The subject involves creating stochastic models, numerical solutions of stochastic differential equations and examinations of the statistical estimates of the solutions.

   Tutor: Kolářová Edita, doc. RNDr., Ph.D.

5. The study of the properties of the solution of matrix systems of differential and difference equations with delay.

   The aim of this work is to modify and expand the knowledge about the solution for the selected class of matrix systems of differential and difference equations with delay. Possible the applications will offer, inter alia, in the areas of optimization and control theory.

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

1. round (applications submitted from 01.04.2019 to 15.05.2019)

1. Algebraic, geometrical and topological methods with applications in information technologies

   The content of the dissertation will be focused on the study and development of mathematical methods or algorithms of continuous as well as  discrete nature, with possible applications in information technologies. The source of inspiration there will be especially the general and partial problems in robotics, artificial intelligence or processing of various kinds of information (for instance, including the image data). The main mathematical tools of the research there will be formal concept analysis in the sense of B. Ganter and R. Wille, the associated algebraic structures and their topological and geometrical properties. There will be also studied the generalized uniform and quasi-pseudometric properties of these structures in the sense of  H. P. Kunzi and S. Matthews, and, alternatively, the relationships of casual nature, whose foundations were developed by L. Crane and J. D. Christensen. We expect original scientific results of an interdisciplinary character in the mentioned above or wider range.

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

2. Applications of algebraic hyperstructure theory in electrical engineering

   The aim of the thesis is to describe relations between various types of hypergroups and hyperrings on one hand and concepts of the algebraic hyperstructure theory which make use of binary relations on the other. The relations will be applied for a construction of a mathematical model of a specific task of electrical engineering. Basic reading: P. Corsini and V. Leoreanu: Applications of Hyperstructure Theory, Kluwer Academi Publications, 2003, and selected chapters of B. Davvaz and V. Leoreanu-Fotea: Hyperring Theory and Applications, International Academic Press, 2007. The candidate is expected to have the knowledge of single-valued algebraic structures. In the course of the Ph.D. study the candidate will make a research stay at University of Nova Gorica, Slovenia, where the algebraic hyperstructure theory is studied.

   Tutor: Novák Michal, doc. RNDr., Ph.D.

3. Asymptotic properties of solutions to discrete Emden-Fowler equation

   The aim will be to study asymptotic behavior of solutions to discrete Emden-Fowler equation and derive new results. Special attention will be paid to the existence of solutions with different asymptotic behavior and their generalizations to Emden-Fowler equations on time scales. To derive results, it is assumed that, except others, the Wazewski topological principle for discrete equations will be used. Starting literature – the book by R. Bellman, Stability Theory of Differential Equations, New York, Toronto, London, 1953, recently published results for continuous case and discrete case, papers dealing with topological principle for discrete equations and equations on time scales. During study a visit to Bialystok University, Poland, where similar problems are studied, is planned.

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

4. General solutions to weakly delayed linear differential systems

   The aim will be to derive explicit formulas for general solutions to weakly delayed linear differential systems, to show if its reduction to linear systems of ordinary differential equations is possible, and prove results on conditional stability. To derive results, various mathematical tools will be used, one of them is the Laplace transform. Starting literature – the paper by D. Ya. Khusainov, D. B. Benditkis and J. Diblik, Weak delay in systems with an aftereffect, Functional Differential Equations, 9, 2002, No 3-4, 385-404 and recently published results. During study a visit to Bialystok University, Poland, where similar problems are studied, is planned.

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