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. Closure operators in digital topology

   The topological approach to digital topology will be developed based on using closure operators. In particular, new closure operators on the digital space will be studied endowing convenient structure of the space for investigation of digital images. The results should provide new algorithms for processing these images..

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

2. Fuzzy Stochastic Models of Reliability

   Fuzzification of the probability distributions for reliability modeling of elements and systems by means of the fuzzy numerical characteristics of vague times between failures. Account of their properties, PC implementation of algorithms, and applications.

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

3. 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 exciting 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 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 nd optical cables, especially in the case of the nanosecond pulses.

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

4. Mathematical Programs for Dynamic Pricing - Demand Based Management

   The aim of the research is to study and develop suitable mathematical models and methods for dynamic pricing in decision making. Traditionally, most companies are organized by separating market and logistics decisions. Typically, the decisions (pricing, marketing strategies etc.) are made independently of knowledge of production and logistics constraints, possibly leading to sub-optimality of the decisions. The logistics field of dynamic pricing takes a different approach. Here, marketing decisions are seen together with logistic decisions aiming for improved company performance.Several distinguished researchers have pointed out the area of Demand Based Management involving Dynamic Pricing as among the most important in the near future; both application- as well as research-wise.

   Tutor: Haugen Kjetil Kare, prof., PhD

5. Models and Algorithms for Operational Resource based Project Scheduling

   Project based production has grown to be an important production mechanism. The early mass production facilities are faced out, and the rapidly changing competitive modern production environments rely in many instances on project based techniques. As such, this area is interesting in many dimensions. Our choice is related to the operational level focusing on time-cost and resource constrained models. In many cases, the strategical and tactical level is reasonably well taken care of, but models and computer systems to handle operational project scheduling are neither not present, nor very efficient. The main research idea is hence related to modelling and algorithmic development related to operational project scheduling. This could include deterministic as well as stochastic cases.

   Tutor: Haugen Kjetil Kare, prof., PhD

6. Models and Methods for rich Fleet Composition Problems

   The FCP includes strategic decisions about the composition of the vehicle fleet as an integral part of the problem definition; routing decisions may be modeled in various detail to capture the tactical and operational consequences of the strategic decisions. Thus, the objective is to minimize the total cost, including fixed costs for managing the vehicles in the fleet and variable routing costs. Applications range from strategic decisions on fleet composition, given expected demand, to tactical and operational decisions on fleet selection on a shorter term basis. We would like to focus on so called rich variants of this problem, getting closer to the real world applications, and methodologies for solving such problems will be the focus. These can encompass both exact and metaheuristic solvers, with possibilities for combined methods.

   Tutor: Lokketangen Arne, prof., Dr.Scient

7. Models and Methods for Rich Inventory Routing Problems

   The Inventory Routing Problem, often called Vendor-based Inventory Management, combines the problems of production, distribution and inventory. The idea is that the vendor (or producer) takes responsibility for maintaining a sufficient inventory at all times at each customer. This will enable the vendor to use the available transportation resources in a more optimal way, as well as coordinating his production with the planned distribution. We would like to focus on so called rich variants of this problem, getting closer to the real world applications, and methodologies for solving such problems will be the focus. There is also the possibility of making cooperating solvers, with each solver having responsibility for its own domain, but also trying to optimize some overall goal. Stochasticity can also be regarded as an inherent feature of these problem, and heuristic or exact solution approaches for stochastic extensions can be explored.

   Tutor: Lokketangen Arne, prof., Dr.Scient

8. Morphisms of Polynomial and Local Algebras in Geometry and Cryptography

   PhD topic is focused on a research in a field of rings of polynomials in more indereminates over fields or integral domains and local rings expressed as finite-dimensional factor rings with an emphasis to their homomorphisms, especially automorphisms. Mainly, the project follows up such properties of polynomial or local algebra automorphisms which have applications in geometry and cryptography. Hence, in particular, automorphisms of Weil algebras (with applications in the searching for natural oprators in differential geometry) and polynomial automorphisms over finite fields (with applications in multivariate cryptosystems) will be studied in more details. The topic continues both earlier research of the supervisor (especially classification results of special Weil algebras and Weil bundles) and highly up-to-date research of cryptographic community (e.g. Patarin’s attacks to TTM systems).

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

9. Numerical methods of solar corona image processing

   The aim of the work is to develop and implement new mathematical algorithms for solar corona image analysis especially for images acquired by space probes. The more important feature of these algorithms is visualization of information invisible for human vision without introducing of processing artifacts and enabling effective additive noise filtering. The new methods will be tested on SOHO a STEREO spacecrafts data at first and later we will articipate on ESA Proba 3 project as a part of FSI VUT team.

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

10. Random censoring and suvival analysis

    The goal of the work is the application of random censoring to survival analysis. The influence of the distribution of a time censor on a survival time will be studied. The supposed basic model will be the Green-Koziol model. Further the methods of parameter estimation od the basic model will be studied. Focal point of the estimation will consist in finding the estimates of parameters by maximum likelihood method and by bayesian method. The computational procedures will be studied and developed. Computer implementation of suggested algorithms will be an important part of the work. Then the suggested techniques of estimation will be compared theoretically and by computer simulations. The illustration of the properties of suggested methods will be demonstrated on real data and the analysis of reliability of real systems will be performed. According to the availability of data the application will be concentrated to the technical domain, to the financial mathematics, medicine and to the environmental and geological fields.

    Tutor: Michálek Jaroslav, doc. RNDr., CSc.

11. Statistical analysis of spatial data

    The first goal of the work is a description of different statistical methods for spatial data processing and their comparing. Among these methods the well known and basic method is the method called kriging. This method is mainly recommended for normal distributed studied random variables (gaussian kriging). Thus the second goal of the work is to try to generalize the kriging method to other distributions particularly to the distributions which are not symmetrical. The simulation study for comparing methods is welcomed. The statistical analysis of real date based on obtained results is expected.

    Tutor: Michálek Jaroslav, doc. RNDr., CSc.

12. Topological Structures on Categories

    The topic is focused on the study of topological structures on categories. In addition to the classical closure operators, particularly convergence structures and neighborhood systems will be investigated. The aim of the dissertation is to extend basic topological concepts like separation, compactness, connectedness, etc. onto categories with topological structures and then to describe the behaviour of the extended concepts.

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