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Stability of the Zero Solution of Stochastic Differential System
Klimešová, Marie
Stability of stochastic differential equations (SDEs) has become a very popular theme of recent research in mathematics and its applications. The method of Lyapunov functions for the analysis of qualitative behavior of SDEs provide some very powerful instruments in the study of stability properties for concrete stochastic dynamical systems, conditions of existence the stationary solutions of SDEs and related problems.
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Microrheology modeling with COMSOL Multiphysics package
Koláček, Jakub ; Sedláček, Petr (referee) ; Pekař, Miloslav (advisor)
This bachelor thesis focuses on modeling Brownian motion using the COMSOL Multiphysics package and its Particle Tracing module. The aim of the work is to design and create elementary models that will be able to suitably simulate the movement of microparticles in viscous and viscoelastic environments, which can later be used for modeling passive microrheology. Within this work, Matlab scripts were created for the calculation of MSD from the simulation results, validation of the viscous model was performed on experimental data and elementary models for the simulation of the viscoelastic environment were also designed. Two different approaches were chosen for the design of these models, namely the use of rigid obstacles under the assumption of a discrete environment and a mathematical model assuming continuous environment. Data from the viscous model showed good agreement with the experimental results. The results of viscoelastic simulations are presented, and further possible development of these models is discussed. The continuous mathematical model is considered closest to modeling viscoelastic behavior because of a characteristic curvature that was observed in the evaluation of MSD.
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Modelling of particle movement in liquid and hydrogel media
Orság, Miroslav ; Sedláček, Petr (referee) ; Pekař, Miloslav (advisor)
Main goal of this thesis was to became familiar with general mathematical description of particle motion in fluid and hydrogel media in COMSOL software. Simulate a basic Brownian motion of particles with various diameter in media with different density. According to software options, simulate motion in polymer structure or in viscoelastic media. Consider an options of the model for use in microrheology and suggest next process in this field.
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Fractals in Computer Graphics
Šelepa, Jan ; Venera, Jiří (referee) ; Sumec, Stanislav (advisor)
The goal of this work is to give introduction and specification of fractals. The first chapter presents the basics of fractal geometry. The second chapter maps the history of fractals and points out the most important people of fractal science. The third chapter presents fractal classification based on several criteria and gives basic examples. The fourth chapter is a summary of widely used applications for fractal creation. The last chapter describes the application that was made to demonstrate given algorithms and fractals mentioned in this thesis.
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Stochastic ordinary differential equations
Bahník, Michal ; Kolářová, Edita (referee) ; Franců, Jan (advisor)
Diplomová práce se zabývá problematikou obyčejných stochastických diferenciálních rovnic. Po souhrnu teorie stochastických procesů, zejména tzv. Brownova pohybu je zaveden stochastický Itôův integrál, diferenciál a tzv. Itôova formule. Poté je definováno řešení počáteční úlohy stochastické diferenciální rovnice a uvedena věta o existenci a jednoznačnosti řešení. Pro případ lineární rovnice je odvozen tvar řešení a rovnice pro jeho střední hodnotu a rozptzyl. Závěr tvoří rozbor vybraných rovnic.
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Simulations of synthetic diffusion MRI data based on Brownian motion
Valla, Radek ; Mikl, Michal (referee) ; Labounek, René (advisor)
This master thesis focuses on dMRI (diffusion magnetic resonance imaging) and its dependance on diffusion in human brain tissue. It is described how to retrieve an image from gained data and its properties, advantages and disadvantages. It mentions problem in detecting kissing fibres due to its similarity with crossing fibres. Design of mathematical models of axons is decribed and suggested measurement to detect difference in signals for kissing and crossing fibres. It describes new simulator of diffusion-weighted MRI (dMRI) data which is able to generate it based on random walk algorithm with geometrical constraints not only for crossing fiber geometry, but also as o novelty for bending and kissing fiber geometries. This study contains results of simulations and disscusion about their usefulness with suggestions for simulator improvement. Simulated dMRI data shows significant difference in certain gradients. Data reconstruction shows, that these reults cannot be reconstructed into the same geometry as it was simulated for.
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Correlation analysis of optically trapped particle motion
Koudelníček, Karel ; Jákl,, Petr (referee) ; Harabiš, Vratislav (advisor)
This bachelor’s thesis deals with the description of random (Brownian) motion of microscopic particle trapped by optical trap. This thesis provides overview of possible usage in biology. The major part of the thesis is dedicated to description of methods that allow characterize optical trap with etc. stiffness thanks to the random motion. First, the estimation methods are tested on the simulated data. Before using these methods for real data, it was necessary to modify them, specifically to eliminate crosstalk and frequency noise. The differences in the estimation of the stiffness from the autocorrelation function and the power spectrum were described in the conclusion.
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Design of Hydrogen Peroxide Disinfector
Makarová, Sylvia ; Rubínová, Dana (referee) ; Fridrichová, Eva (advisor)
The aim of this bachelor thesis is to design a hydrogen peroxide disinfector with regard to all its functional and technical parameters. Emphasis is placed simultaneously on the aesthetics of the product. The work is based on the analysis of accesible products and detailed technical analysis. The final device is intended primarily for medical facilities, but it can be used in any other areas, where is disinfection needed
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