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Alternative K-functions for stationary point processes
Koňasová, Kateřina ; Dvořák, Jiří (advisor) ; Prokešová, Michaela (referee)
The main theme of this thesis is the theory of stationary point processes, in particular the directional K-function. In the first chapter we explain the essentials of planar point process theory including the classical definition of K-function and its estimator. The second chapter introduces two types of the directional K-function: cylindrical K-function whose structural element is a cylinder and directional K-function using double spherical cones. The third chapter presents the comparison of directional K-function and its estimator on an anisotropic version of Thomas process. We also illustrate the major contribution of directional K-function in orientation analysis of point patterns. We introduce a heuristic method for detecting anisotropies in clustered or regular data. 1
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Mixing of Markov chains - lower bounds for mixing
Ditrich, Jakub ; Prokešová, Michaela (advisor) ; Swart, Jan (referee)
The focus of the thesis is the convergence of irreducible aperiodic homoge- neous Markov chains with a finite and discrete set of states. Specifically, lower bounds on the time needed for the chain's marginal probability distribution to be sufficiently close to the stationary distribution, so called mixing time. Multiple methods are introdu- ced, properly motivated and proven. Finally, each method is demonstrated on a suitable example. The result is an overview of three methods that can be used to derive lower bounds for mixing time. 1
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Random walks on networks, hitting times and cover times
Havránek, Jiří ; Prokešová, Michaela (advisor) ; Večeř, Jan (referee)
This thesis studies the cover time of random walks on finite connected graphs. Work contains the derivation of upper and lower estimates of cover time which allows us to focus on hitting time instead of cover time. We show that in some cases the problem of searching for the hitting time can be further simplified with the usage of the electrical networks, which can provide a different model for considered random walks and can help finding the hitting time through the effective resistance between some vertices. This procedure is used to find the upper and lower bounds for specific families of structures, which ilustrates that in some cases the bounds are asymptotically very tight and in other cases they give poor results. 1
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The rheological response to a history of knee joint loading
Prokešová, Michaela ; Otáhal, Stanislav (advisor) ; Jelen, Karel (referee) ; Vilímek, Miloslav (referee)
Title: The rheological response to a history of knee joint loading Aim: The purpose of this work is to illustrate the possibilities of rheological interpretation of passive resistance in the knee joint during simple forced movement of the knee flexion and extension in the sagittal plane. Methods: With the help of biorheometrical measurements, we identify the effect that a weight bearing history has on the actual rheological properties of the knee joint. The methods are based on an experiment carried out in vivo, passive momentum (resistance) of the knee joint during forced flexion and extension are measured and expressed as the specific rheological response of the knee joint. The dependence of passive momentum M on the angle of flexion φ is hereby characterized as the total rheological characteristic of passive forces on the whole knee and its surroundings. The graphical interpretation of the duration of the momentum force is named - biorheogram. Result: Results of experimental study presented in this dissertation distinctly shows that rheologic interpretation of passive resistance in the knee joint during simple forced movement of the knee joint in the sagittal plane is possible in flexion and extension. Conclusion: Rheologic description of hysteretic response of passive resistance {moment of...
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The rheological response to a history of knee joint loading
Prokešová, Michaela ; Otáhal, Stanislav (advisor) ; Jelen, Karel (referee) ; Vilímek, Miloslav (referee)
Title: The rheological response to a history of knee joint loading Aim: The purpose of this work is to illustrate the possibilities of rheological interpretation of passive resistance in the knee joint during simple forced movement of the knee flexion and extension in the sagittal plane. Methods: With the help of biorheometrical measurements, we identify the effect that a weight bearing history has on the actual rheological properties of the knee joint. The methods are based on an experiment carried out in vivo, passive momentum (resistance) of the knee joint during forced flexion and extension are measured and expressed as the specific rheological response of the knee joint. The dependence of passive momentum M on the angle of flexion φ is hereby characterized as the total rheological characteristic of passive forces on the whole knee and its surroundings. The graphical interpretation of the duration of the momentum force is named - biorheogram. Result: Results of experimental study presented in this dissertation distinctly shows that rheologic interpretation of passive resistance in the knee joint during simple forced movement of the knee joint in the sagittal plane is possible in flexion and extension. Conclusion: Rheologic description of hysteretic response of passive resistance {moment of...
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Random walks on networks and mixing of Markov chains
Gemrotová, Kateřina ; Prokešová, Michaela (advisor) ; Pawlas, Zbyněk (referee)
The thesis presents the study of deriving upper bounds of the speed of convergence of reversible Markov chains with discrete time and discrete finite space state to their stationary distributions. We express the derived upper bound in terms of several variables and we make use of the theory of electrical networks, which will help us to represent random walks on a graph. The result of this thesis will be simply obtainable upper bound of mixing time of random walks on connected graphs with an arbitrary number of vertices and edges. Partial results will be demonstrated on simple examples and counterexamples. 1
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Separability of the intensity function of a Poisson point process
Petráková, Martina ; Dvořák, Jiří (advisor) ; Prokešová, Michaela (referee)
Our main interest in the thesis is Poisson point process and one of its charac- teristics - intensity function. Whenever Poisson process has intensity function, its distribution is uniquely determined by it. Our main goal is to determine how to deduce from observed data whether intensity function is separable. We present a formal test of this hypothesis assuming exponential model of the in- tensity function depending on finite number of parameters. Properties of this test are then examined in a simulation study. 1
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