
A characterization of sets with positive reach
Fryš, Filip ; Rataj, Jan (advisor) ; Pokorný, Dušan (referee)
The main source of this thesis is the article Sets with positive reach by the German mathematician Prof. Dr. Victor Bangert from 1982. In this paper, Victor Bangert gives a characterization of sets with positive reach as subsets of connected Riemannian manifolds using weakly regular sublevel sets of functions, the class of which he introduces in his earlier article Analytische Eigenschaften konvexer Funtionen auf Riemannschen Mannigfaltigkeiten from 1979. The aim of this thesis is to study the above mentioned article from 1982 from Bangert and to give a detailed proof for the special case of the Riemannian manifold Rn . After the introductory chapter, where we shall get acquainted with Bangert's article and the aim of the thesis, the first chapter follows, in which we will introduce the basic notation and introduce some necessary knowledge and definitions. In the second chapter we shall deal with the sets with positive reach themselves, give some examples and their basic properties. In the third chapter we will take a closer look at the Bangert's class of functions, and in the fourth chapter we will characterize the sets with positive reach in Rn . 1


Integralgeometric measure
Penkov, Deyvid ; Rataj, Jan (advisor) ; Campbell, Daniel Cameron (referee)
We define a class of lower dimensional measures by averaging the Lebesgue measure from the set projection over all orthogonal projections. We show some relations with the Hausdorff measure and construct sets on which this measure differs from the Hausdorff measure. We define a class of lower dimensional measures by averaging the Lebesgue measure from the set projection over all orthogonal projections. We show some relations with the Hausdorff measure and construct sets on which this measure differs from the Hausdorff measure. 1


JUDr Jiří Branžovský. A political portrait of a modern Czech nationalist
Beroun, Zdeněk ; Šebek, Jaroslav (advisor) ; Čechurová, Jana (referee) ; Rataj, Jan (referee)
The dissertation submitted aims at forming a political portrait of JUDr Jiří Branžovský (18981955), an important representative of the extreme rightwing political scene during the so called First Czechoslovak Republic. We can trace back his public activities already from the outset of 1920s when he engaged in the students' academic fellowships. He belonged to the founders of the nationalist club of Reds andWhites. What inscribed him to the historical awareness, was his prominent membership of The National Fascists' Community. He operated as their lawyer in the fellow position of their president, Radola Gajda. He sat on the highest Party bodies, he participated in the editorial board of the Fascist press, as well. Standing as a candidate of The National Fascists' Community, he was elected a member of the National Parliament, in 1935. Side by side with his mandate performance, there were protracted quarrels with the Party leaders, though. During the Nazi occupation, he enter into contact with the resistance movement. In April 1943, he was arrested by Gestapo and imprisoned till the end of War; at first in the Little Fortress Terezín, later in the Buchenwald and Dachau concentration camps. By the liberation, he returned to his civil profession, soon after the February 1948 he fell into disgrace. He...


Functional receptor alterations in urinary bladder and urethra in an animal model of chronic prostatitischronic pelvic pain syndrome
Rataj, Jan ; Štaud, František (advisor) ; Vokřál, Ivan (referee)
University of Gothenburg; Charles University Sahlgrenska Academy; Faculty of Pharmacy in Hradec Králové Institute of Neuroscience and Physiology; Department of Pharmacology & Toxicology Student: Jan Rataj Supervisor: Michael Winder, Ph.D.; Özgü Aydogdu, M.D.; Prof. PharmDr. František Štaud, Ph.D. Title of diploma thesis: Functional receptor alterations in urinary bladder and urethra in an animal model of chronic prostatitischronic pelvic pain syndrome Chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS) is a common urological disorder. Current available pharmacological treatment options are not effective enough, mostly due to the etiology of CP/CPPS not being fully understood. My thesis aimed to examine potential urethral changes, using immunohistochemistry methods and an organ bath setup, in a rat model of CP/CPPS. The thesis also examined changes in the expression of muscarinic receptor subtype 3 (M3), β3 adrenoceptors, and purinergic P2X1 and P2X3 receptors in the urinary bladder using immunohistochemistry. Specifically, it was examined how these differ when celecoxib, BAY 602770 and a combination of celecoxib and BAY 602770 had been administered during the development of prostate inflammation in the rat model of CP/CPPS. The data show an upregulation of M3 receptors in animals with...


Sets with positive reach and their intersections
Komárek, Daniel ; Rataj, Jan (advisor) ; Pokorný, Dušan (referee)
The goal of this thesis is to collect various properties of sets with positive reach and to describe generalization of the directional curvatures in R3 as the intersection of a plane and a set with positive reach. Firstly, we define sets with positive reach, their Tangent and Normal cones, show basic properties accompanied by some characterizations of sets with positive reach. Then, we generalize principal curvatures for sets with positive reach and describe generalization of Euler's identity about normal curvature in R3 . 1


Asymptotic inference for stochastic geometry models
Flimmel, Daniela ; Pawlas, Zbyněk (advisor) ; Schulte, Matthias (referee) ; Rataj, Jan (referee)
We compare three methods used in stochastic geometry in order to investigate asymp totic behaviour of random geometrical structures in large domains or in a large intensity regime. Namely, we describe in detail the MalliavinStein method, the method of sta bilization and the method of cumulants. Then, we discuss some of its possible variants, combinations or extensions. Each method is supplemented with numerous examples con cerning limit behaviour of different kinds of point processes, random tessellations and graphs or particle processes. Specially, for a geometric characteristic of the typical cell in a weighted Voronoi tessellation, we use the minussampling technique to construct an unbiased estimator of the average value of this characteristic and using the method of stabilization, we establish variance asymptotic and the asymptotic normality of such es timator. Next, we study asymptotic properties of a cylinder process in the plane derived by a Brillingertype mixing point process. We prove a weak law of large numbers as well as a formula of the asymptotic variance for the area of the process. Under comparatively stronger assumptions, we also derive a central limit theorem for the cylinder process using the method of cumulants. 1

 
 

Nonstationary particle processes
Jirsák, Čeněk ; Rataj, Jan (advisor) ; Beneš, Viktor (referee)
Title: Nonstacionary particle processes Author: Čeněk Jirsák Department: Department of Probability and Mathematical Statistics Supervisor: Doc. RNDr. Jan Rataj, CSc., Mathematical Institute, Charles University Supervisor's email address: rataj@karlin.mff.cuni.cz Abstract: Many real phenomena can be modeled as random closed sets of different Hausdorff dimension in Rd . One of the main characteristics of such random set is its expected Hausdorff measure. In case that this measure has a density, the density is called intensity function. In present paper we define a nonparametric kernel estimation of the intensity function. The concept of Hk rectifiable set has a key role here. Properties of kernel estimation such as unbiasness or convergence behavior are studied. As the esti mation may be difficult to compute precisely numerical approximations are derived for practical use. Parametric models are also briefly mentioned and the kernel estimation is used with the minimum contrast method to estimate the parameters of the model. At last the suggested methods are tested on simulated data. Keywords: stochastic geometry, intensity measure, random closed set, kernel estimation 1

 