Branch Details

# Applied Mathematics

Original title in Czech: **Aplikovaná matematika**FSIAbbreviation: **D-APM**Acad. year: **2020/2021**

Programme: **Applied Natural Sciences**

Length of Study: **4 years**

Accredited from: **1.1.1999**Accredited until: **31.12.2024**

Profile

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.

Supervisor

Issued topics of Doctoral Study Program

**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.

**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.

**Asymptotics and oscillation of dynamic equations**We shall study qualitative properties of various second order and higher order nonlinear differential equations, which arise from aplications (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 to compare and explain similaritities between the continuous case and some of its discretization, to get an extension to new time scales, or to obtain new results e.g. in the classical discrete case through a suitable transformation to other time scale.

**Distributed stochastic optimization**Many recent engineering optimization problems are related to large-scale stochastic programs. They involve random elements and they are often specific by a specialized structure and a huge set of input data. Thus, their modelling and solution algorithms need the use of decomposition ideas and so called distributed programs that are based on distributed computing principles. Hence, the proposed research goal is to develop new and modify existing approaches in the modelling and solving of advanced stochastic programs together with their implementation and application. Student will develop his original research in the area of hierarchical and distributed optimization models. He will collaborate with faculty institutes on real-world applications, e.g., IPE (TIRSM project) and EI (COMSI project) in the area of modelling of complex energy systems. International collaboration will be realized with Norwegian partners (NTNU Trondheim, NHH Bergen, MUC Molde).

Tutor: Popela Pavel, RNDr., Ph.D.

**Functional differential equations**Functional differential equations are a generalisation of ordinary differential equations. One of their further specification leads to equations with delayed argument which has been a widely studied topic recently. The advantage lies in the description of real-world situations better than ordinary differential equations. Apart from delayed equations we will also handle advanced differential equations because this has not been considered seriously so far. We shall mainly focus on qualitative analysis of particular functional differential equations which are derived from real models. More precisely, we shall study oscillatory properties of solutions to the considered equations.

**Geometric algebras in elements of autonomous control**Autonomous control consists among others of the external feedback information analysis. This includes visual, sound or other sensoric data (signal in general) which should be processed, analysed and evaluated providing the basis for further control. In every one of the mentioned disciplines geometric algebra may be used in an efficient way.

Tutor: Vašík Petr, doc. Mgr., Ph.D.

**Geometrical structures, invariants and their applications in the continuum mechanics**The student will study structures of modern differential geometry, namely bundle functors, Lie groups and invariants. Besides the theoretical results the attention will be focused on the applications of the theory, eventually the own results in the continuum mechanics, including thermodynamics.

Tutor: Tomáš Jiří, doc. RNDr., Dr.

**Homogenous spaces, foliations and their physical applications**A student will devote his studies to the theory of homogenous spaces in connection with other geometrical structures like Lie groups, connections and some kinds of bundle functors. As for applications, besides continuum mechanics the attention can be focused on selected branch of physics.

Tutor: Tomáš Jiří, doc. RNDr., Dr.

**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.

**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 chaotic 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.

**Numerical processing methods of experimental data for imaging spectroscopic reflectometry within the framework of the optical characterization of thin solid films**The content of the dissertation thesis is to find effective algorithms for numerical processing of big sets of experimental data obtained by means of imaging spectroscopic Reflectometer (built in The Coherence Optics Laboratory of IPE FME BUT) from non-uniform thin films for the determination of the optical parameters of these films. The goal is to realize aforementioned algorithms in the form of a software.

**Periodic solutions to non-linear second-order differential equations**We shall study existence of periodic solutions to non-linear second-order differential equations. We will focus on differential equations appearing in mathematical modelling, in particular ordinary differential equations in mechanics. Typical example of such equation is the so-called Duffing differential equation, which is derived, for instance, when aproximating a non-linearity in the equation of motion of certain forced oscillators.

Tutor: Šremr Jiří, doc. Ing., Ph.D.

**Qualitative properties of discrete dynamical systems**Many technical problems are need to be modelled by a discrete dynamical system since the independent variable has to be considered as discrete one instead of a continuous one. These systems have many different properties with respect to its continuous counterparts. The analysis of qualitative properties is significant from considered model behaviour prediction point of view (or dealing with its control).

## Course structure diagram with ECTS credits

Abbreviation | Title | L. | Cr. | Com. | Compl. | Hr. range | Gr. | Op. |
---|---|---|---|---|---|---|---|---|

9EMM | Empiric Models | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9FMS | Fuzzy Models of Technical Processes and Systems | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9GTR | Geometric Control Theory | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9MKP | FEM in Engineering Computations | cs | 0 | Recommended | DrEx | P - 20 | yes | |

9STH | Structure of Matter | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9SLT | Sturm-Lieouville Theory | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9TTD | Theory of Measurements, Measurement Techniques and Technical Diagnostics | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9TKD | Basics of Category Theory | cs, en | 0 | Recommended | DrEx | P - 20 | yes |

Abbreviation | Title | L. | Cr. | Com. | Compl. | Hr. range | Gr. | Op. |
---|---|---|---|---|---|---|---|---|

9ARA | Algebras of rotations and their applications | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9AMK | Analytical Mechanics and Mechanics of Continuum | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9AHA | Applied Harmonic Analysis | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9APT | Applied Topology | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9DVM | Dynamic and Multivariate Stochastic Models | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9FKP | Functions of a Complex Variable | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9FAP | Functional Analysis and Function Spaces | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9FZM | Physical Base of Materials Fracture | cs | 0 | Recommended | DrEx | P - 20 | yes | |

9ISY | Invariants and Symmetry | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9MOR | Mathematical Methods Of Optimal Control | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9MPK | Mathematical Principles of Cryptographic Algorithms | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9NMT | Nonlinear Mechanics and FEM | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9PVP | Programming in Python | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9UMS | Ordered Sets and Lattices | cs, en | 0 | Recommended | DrEx | P - 20 | yes |

Abbreviation | Title | L. | Cr. | Com. | Compl. | Hr. range | Gr. | Op. |
---|---|---|---|---|---|---|---|---|

9AJ | English for Doctoral Degree Study | en | 0 | Compulsory | DrEx | Cj - 60 | yes | |

9APH | Applied Hydrodynamics | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9ARV | Automation and Control of Manufacturing Systems | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9FLI | Fluid Engineering | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9GRA | Graph Algorithms | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9MBO | Mathematical Modeling of Machine Mechanisms | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9IDS | Modelling and Control of Dynamic Systems | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9PAR | Control Equipments | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9VIN | Computational Intelligence | cs, en | 0 | Recommended | DrEx | P - 20 | yes | |

9VMT | Computational Modeling of the Turbulent Flow | cs, en | 0 | Recommended | DrEx | P - 20 | yes |