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Functional equations
Konopecký, František ; Hencl, Stanislav (advisor) ; Černý, Robert (referee)
Nazev pracc: Funkcionalni rovnice Autor: Frantisck Konopccky Katedra (ustav): Katcdra matcmaticke analyzy Vedouci bakalafske pracc: RNDr. Stanislav Hcncl, Ph.D. e-mail vedouciho: hencK(.i)karlni.mrT.cmri.cz Abstrakt: V pfcdlozcnc praci ye zabyvame mctodami fcscni funkcionalnich rovnic, triky, kterc fescni usnadnuji, vlastnostmi funkci, kterc pfi fescni pomahaji a myslenkovymi po- stupy pfi samotnem fescni. Nejprve za/Jvame pojem a praci s funkcionalni rovnici. Pote vysvetlnjeme dvc hlavni inetody - Substitucni-A Cauchyova. Naslcduji diilczite vlastnosti fnnkci a po nich ka]>itola o castych chybach pfi foscni. Ko konci se venujcme Cauchyove rovnici z vice stran a naslednje nckolik fesenych i nofcsenych pfikladu. Klicova slova: Fnnkcc, funkcionalni rovnice. Title: Functional Equations Author: Frantisek Konopccky Department; Department of Mathematical Analysis Supervisor: RNDr. Stanislav Hcncl, Ph.D. Supervisor's e-mail address: hencK(fikarlin.mfi.cnui.cz Abstract: In this thesis we study methods of solving strategies of functional equations, tricks, which make solving easier, special function properties, that help in solution and also ideas in solution itself. Firstly we get closer to term functional equation and show how to work with it. Then we explain two main methods Substitution method and Cauchy's...
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Neural crest biology with respect to diversity of vertebrates
Štundl, Jan ; Černý, Robert (advisor) ; Němec, Pavel (referee)
Neural crest is an extensively migrating population of cells that arise during early development of vertebrate embryos. It provides a huge variety of different cell types that generate new tissues which occur only in vertebrates. Neural crest cells together with the mesoderm participate on the formation of the head of vertebrates, which is viewed as one of the most important innovations in the evolution of vertebrates. Thanks to their skeletogenic potencial neural crest cells are percieved as a key factor causing massive craniofacial diversity. The aim of this thesis was to get acquainted with the population of neural crest cells and try to understand its importance for the evolution of vertebrates and especially for generating craniofacial diversity.
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Infinite products
Kudrnáč, Vojtěch ; Rokyta, Mirko (advisor) ; Černý, Robert (referee)
This paper offers a brief insight into the basic theory of convergence of the infinite products of real or complex sequences. Then it focuses mainly on the possibilities of developing some selected functions into the form of infinite product and on the corollaries and utilizations of being familiar with these. Purpose of the paper is not to prove the existence of infinite products for functions with certain characteristics in general, but rather to derive specific formulas and prove their validity. The attention is paid to those elementary functions which are derived from the exponential function, especially the sinus function, the nonelementary functions mentioned are the gamma and the zeta function. The text should be understandable even for a person, who has never came upon infinite products before.
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Developmental morphogenesis of attachment organs in lower vertebrates
Minařík, Martin ; Černý, Robert (advisor) ; Buchtová, Marcela (referee)
Adhesive organs are widespread structures among vertebrate larvae. They allow the larvae to attach to a substrate, so that the time for the development of mouth or motoric apparatus could be prolonged. Similar structures in ascidians, larvaceans and lancelets are known too. Thus, it might be hypothesized that the presence of some type of adhesive gland could indeed represent the ancestral state for chordate larvae. Interestingly, however, whilst in most species these glands take their developmental origin in ectodermal layer, in bichir, a member of a primitive actinopterygian lineage, their origin was suggested to be endodermal already at the beginning of 20th century. Since then, however, the former study has become almost forgotten and even recent analyses do not come with new findings on this topic. Because of the essential importance of study of bichir cement glands for understanding the relationship between these structures among chordates, I have decided to focus on this subject. To obtain appropriate comparative data Xenopus, Weather loach and Ribbed newt embryos were included in this study as well. By using combination of immunohistochemical and histological techniques the endodermal origin of cement glands in bichir was proven and their morphogenesis was described into considerable details. The...
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Models of default prediction of a client
Hezoučká, Šárka ; Černý, Rostislav (advisor) ; Hurt, Jan (referee)
The aim of the presented work is to investigate possible improvement of scor- ing models prediction power in retail credit segment by using structural models estimating the future development of behavioral score. These models contain the information about past development of the behavioral score by parameters which take into account the sensitivity of clients' probability of default on in- dividual market and life changes. These parameters are estimated with Markov Chain Monte Carlo methods based on score history. Eight different types of struc- tural models were applied on the real data. The diversification measure of indivi- dual models is compared using the Gini coefficient. These structural models were compared with each other and also with the existing scoring model of the credit institution which provided the underlying data. 1
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Mountain Pass Theorem and its applications
Příhoda, Vojtěch ; Černý, Robert (advisor) ; Vlasák, Václav (referee)
Cílem této práce je formulovat a dokázat Větu o horském sedle a ukázat příklady její aplikace tak, aby student třetího ročníku matematiky nebo fy- ziky, jenž absolvoval alespoň úvodní kurs funkcionální analýzy, byl schopen práci porozumět. D·kaz Věty o horském sedle provedeme s pomocí Ekelan- dova variačního principu, který si rovněž formulujeme a dokážeme. Následně si ukážeme dva příklady její aplikace na d·kaz existence netriviálního sla- bého řešení eliptické parciální diferenciální rovnice obsahující nelinearitu se subkritickým r·stem. 1
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