Branch Details

# Applied Mathematics

Original title in Czech: **Aplikovaná matematika**FSIAbbreviation: **D-APM**Acad. year: **2019/2020**

Programme: **Applied Natural Sciences**

Length of Study: **4 years**

Accredited from: **1.1.1999**Accredited until: **31.12.2020**

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

**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..

**Algebraic methods in signal processing**The applicant will study algebraic structures to be applied in signal processing. Particularly, quaternions and Clifford algebras (maily geometric algebras) will be involved. Main tools for signal processing will be fomed of quaternionic Fourier transform and Clifford wavelets.

Tutor: Vašík Petr, doc. Mgr., Ph.D.

**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.

**Algebras of dual numbers and their generalizations, applications in screw calculus and mechanics**The topic is focused on the application of a screw calculus in the sense of F. M. Dimentberg in mechanics and generalizations for other local algebras.

**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.

**Autonomous vehicle control based on visual information**Analysis of 3D clusters of points using higher order Clifford algebras in combination with neural network. Specifically, the use of geometric algebra of quadrics for edge detection and the design of a neural network allowing manipulation of Clifford algebras. Geometric algebra of quadrices linearises quadratic surfaces in three dimensional spaces. Different obstacles of an autonomous vehicle can then be found and approximated as quadrics of the appropriate type or specific parameters. The assembled linear model is then the input of a suitable neural network.

**Cooperative theory of games in engineering**Using the Cooperative Game Theory apparatus to find optimal strategies on graphs and networks. In particular, we will use the cooperative aparatus to set the parameters of individual players. One of the goals is to choose the core of cooperative game. We will discuse its axiomatic definition tosolve the problem.

**Feedback in the geometric control theory**Using feedback in autonomous control of both nonholonomic and holonomic mechanisms. The addition of the purely algebraic apparatus to the appropriate methods of control theory. In paricular , the reconstructing of a 3D scene is usually solved by Binocular vision. I.e. we get an analytical solution. The error of this solution appears by control of the mechanisms. This gives a natural feedback control.

**Functional differential equations**Functional differential equations generalize ordinary differential equations. The research of theirs properties is much more difficult than it is for the ordinary differnetial equations. We shall analyze qualitative properties of the particular functional differential equations, which may appear in real models. In particular, we deal with oscillatory properties of considered equations.

**Geometric control models**The applicant will study various geometric-based control models, among all let us mention those using sub-Riemannian geometry or geometric algebra. Specific mechanisms with outstanding geometric properties will be determined and new results regarding the appropriate controlling structure will be delivered. Resulting algorithms will be validated and verified.

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.

**Hyperspectral airborne data quality assessment**Assessment of data quality is a crucial and meaningful stage in the processing of hyperspectral airborne data. In practice, existing methodologies used for quality assessment are moreover only providing rough quality tags or recommendations without proposing practial methodologies. The objective of the work is to develop a methodology (in an automatic and generic fashion) on hyperspectral airborne data quality assessment, which covers the complete data processing chain. The quality of each step has to be investigated: ranging from sensor calibration up to mapping results. All data, software will be available at the Global Change Research Institute, CAS (CzehGlobe) where Ph.D. candidate will be trained and will get a part time job.

**Infinitesimal automorphisms in geometric control theory**Study of the role of infinitesimal automorphisms of filtration determined by the nonholonomic constrains of the particalr mechanisms for optimal control analysis. Application of the advanced differential geometry in the mathematics control theory. In particular, we will discuss the extensiona of the control algebras and the geometric structures underliying to our control distibution determined by the non-homonymous system. The aim is to design new control algorithms for the selected class of mechanisms.

**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 exiting 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 the 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 and optical cables, especially in the case of the nanosecond pulses.

**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.

**Modern methods for solving nonlinear evolutionary differential equations**Since intial boundary value problems for evolutionary partial differential equations in technology often do not admit classical solution, various generalized formulations of these problems were proposed. The aim of the study will be comparison of these formulations and studying existence and uniqueness of their solutions. Then the theory will be applied to particular problems occuring in technology and alternatively to carry out numerical experiments.

**Numerical image processing methods for space-based externally occulter coronagraf**The aim of the work is to create a new methods for visualization of K-corona in digital images made with space-based coronagraphs. Three fundamental problems must be solved - filtration of impulse noise which is caused by cosmic rays, suppression of F-corona and low contrast features visualization in HDR images.

**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.

**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.

**Spectral and radiometric calibration of spectroradiometers**In order to facilitate proper functioning of spectroradiometers is necessary regular spectral and radiometric calibration of instruments. Calibrations are necessary for numerical as well as imaging spectroradiometers, The objective of the work is to develop a methodology for calibration of multispectral and hyperspectral spectroradiometers. Alternatively, to verify methodology recommended by producer of instrument. Expected outcome of work is calibration methodology including scripts for practical use. Example of instruments for calibration: FieldSpec-4, CASI-1500. All data, software will be available at the Global Change Research Institute, CAS (CzechGlobe) where Ph.D. candidate will be trained and will get a part time job.

**Stability and stabilization of discrete dynamic systems**The study will be focused on analysis of discrete dynamic systems. These systems appear in many engineering and natural science branches and model equations are supposed to by analyzed with respect to these applications. The other studied topics are stabilization of unstable dynamic systems by use of a suitable feedback control, stabilizing (or destabilizing) effects of time delays on dynamics of a system, synchronization of solutions of discrete dynamic systems and relationship between continuous and discrete models. These models can involve also modern types of dynamic systems, including chaotic and fractional ones.

**Stochastic Programming Related Statistical Techniques**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 solution algorithms need the use of approximating scenario-based programs designed by suitable statistical techniques during a preprocessing phase. Therefore, the proposed research goal is to develop new and modify existing methods in the creation of approximating stochastic programs and their solution techniques together with their implementation and application.

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

**3D reconstruction of objects in confocal microscopy using lens deliberate aberration**The aim of the work is development of new numerical method for object 3D reconstruction in confocal microscopy. This method is using lens deliberate aberration which enables to increase resolution within one optical cut. The method will be applied to a twin-line confocal microscope based on a rotating Nipkow disc.

## Course structure diagram with ECTS credits

Study plan wasn't generated yet for this year.