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Emergence of irreversible dynamics by the lack-of-fit reduction
Mladá, Kateřina ; Pavelka, Michal (advisor) ; Klika, Václav (referee)
The thesis studies theories of dimensional reduction on the example of the Kac- Zwanzig (heat bath) model. The studied methods are the Mori-Zwanzig projection for- malism and the lack-of-fit reduction, both applied for two sets of resolved variables. The methods give integro-differential and ordinary differential evolution equations re- spectively. For the Mori-Zwanzig formalism, a limit of the number of particles going to infinity is made, which leads to an exponential memory kernel and consequently to a set of stochastic differential equations. The evolution equations of the two methods are compared using numerical simulations. 1
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The Connection between Continuum Mechanics and Riemannian Geometry
Burýšek, Miroslav ; Pavelka, Michal (advisor) ; Klika, Václav (referee)
We investigate the systems of quasi-linear partial differential equations of hydrody- namic type. These equations occur mainly in hydrodynamics and continuum mechanics, but they arise in other various applications. In the study of such systems, one finds an intersection of Poisson and pseudo-Riemannian geometry. The Poisson bracket is deter- mined by functions that turn out to be metrics and Christoffel symbols. If the metric is non-degenerate, the existence of Poisson structure is equivalent to the existence of flat metric and Levi-Civita covariant derivative with zero curvature. Moreover, one can find special flat coordinates where the bracket is trivial. This result was found in the eighties by Dubrovin and Novikov for the one-dimensional case and later on extended to more dimensions. In this thesis we provide the proof of the Dubrovin-Novikov theorem, which was only sketched in the original paper. We also conducted an overview of current knowledge in the multi-dimensional case, where the theory gets much more complicated. In particular, the link between the compatible brackets and the possibility of finding flat coordinates is discussed. The Riemannian character of the Hamiltonian equations of hydrodynamic type can be used to prove their symmetric hyperbolicity, even when the equations are not in the...
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Vliv materiálových parametrů na stabilitu termální konvekce
Dostalík, Mark ; Matyska, Ctirad (advisor) ; Klika, Václav (referee)
The thesis is focused on the investigation of Rayleigh-Bénard problem in an extended setting approximating the conditions in the Earth's mantle. The aim is to evaluate the influence of depth- and temperature- dependent material parameters, dissipation, adiabatic heating/cooling and heat sources on the qualitative characteristics of thermal convection. We identify the critical values of dimensionless parameters that determine the onset of convection and characterize the dominating convection patterns in marginally supercritical states. These issues are addressed by the application of linear stability analysis and weakly non-linear analysis. It has been found that the character of convection differ substantially from the standard case of Rayleigh-Bénard convection. Powered by TCPDF (www.tcpdf.org)
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Hamiltonian and thermodynamic theory of solids and fluids
Sýkora, Martin ; Pavelka, Michal (advisor) ; Klika, Václav (referee)
The standard approach to modelling mechanics of continuum based on bal- ances of mass, momentum, angular momentum and energy is a very powerful tool. However, there is no connection between that and the Hamiltonian mechanics, that superbly describes kinematics of isolated particles. Thus, the two topics are rather isolated. Nevertheless, there is another approach to continuum mechan- ics - a one, whose reversible part is based on Hamiltonian mechanics, while the irreversible is generated by a dissipation potential. This framework, called GENERIC, is thus an interesting bridge between con- tinuous and discrete systems. In this thesis, we present the GENERIC framework applied to a continuous body, derive the governing equations and compare them to the standard theory. Both analytical and numerical solutions to a decent range of model examples are presented and analysed.
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Computer Modeling of Tissue Development
Bednář, Vojtěch ; Hedrlín, Zdeněk (advisor) ; Loebl, Martin (referee) ; Klika, Václav (referee)
Title: Computer Modeling of Tissue Development Author: Vojtěch Bednář Department: Department of applied mathematics Supervisor: Doc. RNDr. Zdeněk Hedrlín, CSc. Abstract: This thesis describes hybrid individual cell-based approach to modeling of systems of biological cells. In the first part reaction-diffusion model of environment is introduced together with vax equilibrium and model of a cell based on zygotic graph and cummulative states. Further, simulations modeling three biologically motivated situations are introduced: Lumen formation, tumor growth, and cellular migration in chronic inflammation. The first model shows a scenario of hollow structure formation based on directional division and cellular migration. The second model is concerned with the growth of a progeny of a slightly damaged cell. The resulting tumor exhibits three stages of malign transformation. Further, emergence of an aggressive tumor without detectable precursor is observed on one hand and a continual transformation of a benign neoplasm into a malign one is seen on the other hand. Each of these cases is a consequence of different parametrization of the model situation. The last model analyses the role of membrane enzymatic activity in migrating cells of the immune system in chronic inflammation. In this model it is observed that...
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Vliv materiálových parametrů na stabilitu termální konvekce
Dostalík, Mark ; Matyska, Ctirad (advisor) ; Klika, Václav (referee)
The thesis is focused on the investigation of Rayleigh-Bénard problem in an extended setting approximating the conditions in the Earth's mantle. The aim is to evaluate the influence of depth- and temperature- dependent material parameters, dissipation, adiabatic heating/cooling and heat sources on the qualitative characteristics of thermal convection. We identify the critical values of dimensionless parameters that determine the onset of convection and characterize the dominating convection patterns in marginally supercritical states. These issues are addressed by the application of linear stability analysis and weakly non-linear analysis. It has been found that the character of convection differ substantially from the standard case of Rayleigh-Bénard convection. Powered by TCPDF (www.tcpdf.org)
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Remodelace kostí po náhradě kyčelního kloubu. Numerické modelování a srovnání s klinickými pozorováními
Klika, Václav ; Maršík, František ; Landor, I.
The stress-strain changes in the surroundings of the hip-joint implant have dramatic impact on the response from the living bone tissue – by the remodeling process. This process is divided into three stages that can be described by five ordinary differential equations for the concentrations of five relevant chemical components. The driving force for the remodeling process is the dynamic loading. It is apparent that the joint implant creates a very new condition for natural remodeling process. In presented paper we simulate the consequences caused by remodeling phenomenon and compare it to clinical studies.
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