
Space filling curves
Štěpánek, Adam ; Hencl, Stanislav (advisor) ; Pyrih, Pavel (referee)
Peano curves are continuous mappings from the unit interval [0, 1] onto the n dimensional square [0, 1]n , n ∈ N. There are many such curves and therefore we focuses especially on the Hilbert curve. We informally outline its geometrical interpretation and then we describe the construction in R2 by writing a number in a quaternary form. For such defined mapping we prove that it is a Peano curve and that it is 1/2  Hölder con tinuous. In conclusion, using the Haussdorf dimension, we show that there is no Peano curve in Rn that is also α  Hölder continuous for α > 1/n. 1


What has changed in the networks during the last 10 years
Slivka, Lukáš ; Pyrih, Pavel (advisor) ; Bulej, Lubomír (referee)
Bachelor thesis is focused on the evolution of computer networks during past 10 years. Part of it shortly describes development of individual technologies. It deals with problems caused by enormous interest of using Internet and other communication services. In the main, there is shortage of IP addresses and the changes awakened by customer's demand for new services. Specifically, we can mention interest in multimedia, higher bandwidth, comfort of using mobile devices and global roaming. Big section is devoted to the transport media and used technologies. I evaluate their advantages and disadvantages. There are also some specific projects and factors in the end, which had impact on general evolution of communication. Powered by TCPDF (www.tcpdf.org)


Managerial decision making about external relationships of high school
Rampír, Vojtěch ; Černý, Jan (advisor) ; Pyrih, Pavel (referee)
Diploma thesis acquants reader about decision problems at Střední odborné škola informatika a spojů a Střední odborné učiliště Kolín. Chosen problem are desribed and analysed, but also variants of solution are created, so tehy can be used as an support to decision making for school manager. These problems are about getting new students, usage of unused rooms of dorm and school cafeteria. For creating variant of solutions CobbDouglas utility function is used, moreover the thesis teaches reader how to work with it, so a manager of school can use it for his own decision making proces. Diploma thesis was created by request of a principal Ing. Miloš Hölzl as he is awaiting the results.


Construction of dendroids and their properties
Marciňa, Radek ; Pyrih, Pavel (advisor) ; Hušek, Miroslav (referee)
This thesis is about dendroids and their properties. Another example of a dendroid with two intersecting shore continua whose union is not a shore continuum is constructed. Moreover, a simplification of a proof that the union of finitely many pairwise disjoint shore continua is again a shore continuum has been made. But the main result is an affirmative answer to the question whether the union of finitely many closed shore sets is again a closed shore set in the case dendroids with only finitely many branch points. Powered by TCPDF (www.tcpdf.org)


Curves hidden in differential equations
Žalobínová, Petra ; Pyrih, Pavel (advisor) ; Bárta, Tomáš (referee)
Bachelor thesis Curves hidden in differential equations is engaged in solving and deriving differential equations, leading to the selected curve, which are cycloid curve and curves described by the hyperbolic functions . The core of the work is outlined into four transparent chapters, where the first of them gives a brief insight into the theory of curves and differential equations. In the following chapters, the work is specifically devoted to two historically significant problems, which are brachistochrone and catenary problems, but also deals with eigen, more modern problem of dynamics of symbiotic populations. The text explains the procedure of deriving differential equations from the considered problems and shows their solutions, by multiple methods. The contribution of this work, apart from the formulation and solution of the problem of symbiotic populations, is processing and making additions to solutions of these problems using a variety of methods from cited literature. The most important addition is relatively new and less known proof of the uniqueness of solutions of brachistochrone problem, which is enhanced by eigen intermediate steps and explanations.


Future predicting and the axiom of choice
Jarosil, Lukáš ; Pyrih, Pavel (advisor) ; Kalenda, Ondřej (referee)
Given arbitrary function f : R → R it seems practically impossible to predict its future values based on our knowledge of its previous values. Nevertheless, axiom of choice surprisingly implies the existence of strategy that from values of the function f on some interval (s, t) correctly predicts its values on interval [t, t+ ) for every t of real line except for countable set. This result of Christopher Hardin and Alan Taylor is presented along with its generalization to mappings from topological space in the context of hat guessing games, mathematical games in which the players are supposed to guess color of their own hat while knowing only colors of other's hats. 1


Homeomorphisms in topological structures
Vejnar, Benjamin ; Pyrih, Pavel (advisor) ; Charatonik, Włodzimierz (referee) ; Illanes, Alejandro (referee)
In this thesis, we present solutions to several problems concerning onedimensi onal continua. We give an inductive description of graphs with a given disconnec tion number, this answers a question of S. B. Nadler. Further, we state a topo logical characterization of the Sierpi'nski triangle. In the study of shore sets in dendroids and λdendroids we obtain several positive results and we also provide some counterexamples. By doing this, we continue in the recent work of several authors. We are also dealing with the notion of 1 2 homogeneity and we prove that up to homeomorphism there are only two 1 2 homogeneous chainable continua with just two end points. We present also a new elegant proof of a classical theorem of Waraszkiewicz. 1


Monotonicity of functions which can be expressed using elementary functions
Peltan, Libor ; Bárta, Tomáš (advisor) ; Pyrih, Pavel (referee)
For certain types of functions expressible with formula (equivalently: functions from classes closed to arithmetic operations) under stated assumptions, we prove monotonicity at some neighbourhood of +∞. They are: formulas containing exp, log, sin, arctan, etc. with constrainted domain of these functions; power series with cofinite many coefficients positive; various classes of functions expressible with formulas with the requirement of preserving monotony in summation, or multiplication, or the monotony resulting from having a finite number of zero points; and finally formulas with square root. 1


What is a curve?
Koudela, Libor ; Veselý, Jiří (advisor) ; Pyrih, Pavel (referee) ; Bobok, Jozef (referee)
The notion of a curve played important role in the history of mathematical thought. This dissertation is focused on the conception of a curve in analysis, point set theory and topology. The rectification of curves and the notion of arc length are considered in connection with the history of analysis from antiquity to the beginning of the 20th century. "Measurement of curves" is also discussed from the measuretheoretic viewpoint and various definitions of linear measure and fractional dimension are described. Historically, there are two main approaches to understanding curves. Jordan defined a curve as a continuous image of a closed interval. However, his definition appeared to be too wide, since it was met by objects such as the Peano curve. In the point set theory, a curve is considered to be a onedimensional continuum. The development of the dimension theory and the continuum theory, starting with the pioneering work of Bolzano, was motivated by the search for rigorous topological definition of a curve, a surface etc. Among "pathological" curves, that were often introduced as counterexamples in the development of modern analysis, we can find early examples of fractals. The fractal theory motivated further study of mathematical properties of these curves in the late 20th century, such as selfsimilarity and...


Hausdirff metric and its application in fractals
Roháľ, Branislav Ján ; Hušek, Miroslav (advisor) ; Pyrih, Pavel (referee)
Title: Hausdorff metric and its application in fractals Author: Branislav Ján Roháľ Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Miroslav Hušek, DrSc., Department of Mathematical Analysis Abstract: In this thesis we focus on the themes naturally connected with the con cept of a fractal. In the first part of the thesis we pay attention to Banach fixed point theorem and to the Hausdorff metric which are later used when studying selfsimilar sets. There are included parts on the Hausdorff, similarity, and box counting dimension, too. In the second part of the thesis the new approaches to fractal dimension and some their properties are refered. We introduce generaliza tion of this concept for any space admitting a fractal structure and for a distance space where also the "size" of sets on each level of fractal structure is considered. In the last chapter the contribution of new approache is demonstrated,  this enables defining the notion needed and counting fractal dimension where it was not possible under the classical approaches, too. Application to the domain of words and counting of dimensions of a language generated by a regular expresion are presented. Keywords: Hausdorff metric, Banach fixed point theorem, selfsimilar set, Hausdorff dimension, fractal dimension
