Pell's equation, continued fractions and Diophantine approximations of irrational numbers

This bachelor's thesis deals with Pell's equation, while clearly presenting structured information from studied domestic and foreign books, articles, and other sources. The goal of this thesis is to create study material primarily for university students but also for inquisitive high school students, and thus explain as intuitively as possible what Pell's equation is, how to find its solutions, and how it is related, for example, to continued fractions, approximations of irrational numbers, and invertible elements in Z[√n ]. The main motivation for solving Pell's equation throughout the work is specifically that its solutions give best approximations of irrational square roots. Pell's equation is presented in a brief historical context. Further, it is proved that there is a non-trivial integer solution for every Pell equation, and the theory of continued fractions is used to find it. To make the creation of continued fractions easier, the so-called Tenner's algorithm is introduced. Specifically, the search for a solution to Pell's equation is derived using convergents and the periodicity of continued fractions of irrational roots. Subsequently, the structure of the solution is described: it is proved that there is a so-called minimal solution that generates all positive solutions, and a set of...Katedra matematiky a didaktiky matematikyFaculty of EducationPedagogická fakult

Ústní tradice cantiones v českých zemích a její otisk v pozdně středověkých rukopisech

Ústní tradice cantiones v českých zemích a její otisk v pozdně středověkých rukopisech Abstrakt Prameny dochované v českých zemích nabízejí rozsáhlý soubor cantiones, tj. latinských jednohlasých nebo vícehlasých duchovních písní, které poskytují bohatou základnu pro výzkum jejich tradice a transmise v pozdním středověku. Ačkoli jsou předmětem vědeckého studia již půldruhého století, velkou část literatury limituje použitý přístup, ať už jde o filologickou metodu, která v dochovaných pramenech cantia spatřovala prostředek k nalezení jeho archetypu a ve variantách zápisu pouze chyby k emendaci (Mužík, Černý), nebo o nacionalistickou předpojatost (Nejedlý). V případě chorálu badatelé ukázalo, že tradice středověké hudby byla silně ovlivněna ústní transmisí (Treitler, Hucke), a proto by se na ni mělo nahlížet z perspektivy jiné, blízké etnomuzikologii (Jeffery). Ačkoli některé práce z poslední doby tento přístup reflektují (Gancarczyk), nebyl dosud ověřen na větším souboru písní ani nebyly systematicky nastíněny jeho důsledky. Má studie několika desítek písní zaznamenaných kolem roku 1410 v rukopisu CZ- VB 42, které, jak dokládám na vybraných pramenech, přežily a vzkvétaly i v následujícím století, ukazuje rozmanitost žánru a také to, že navzdory nástupu polyfonie nadále převládaly jednohlasé skladby. A co je...Oral tradition of cantiones in Czech lands and its imprint in late medieval manuscripts Abstract The sources extant in the Czech lands transmit a vast body of cantiones, i.e., Latin spiritual monophonic or polyphonic songs that provide a rich basis for research into their tradition and transmission during the late Middle Ages. Though they have been subject of scholarly study for a century and a half, much of the literature is limited by the approach employed, be it the philological method, which saw a cantio's extant sources a tool for deriving its archetype and in its variants merely errors to be emended (Mužík, Černý), or a nationalistic bias (Nejedlý). In the case of chant, scholars have suggested that the tradition of medieval music was strongly impacted by oral transmission (Treitler, Hucke) and hence should be looked at from a different perspective, one close to that of ethnomusicology (Jeffery). Though some recent papers reflect this approach (Gancarczyk), it has not yet been tested on a larger body of songs, nor its implications systematically outlined. My study of several dozen songs recorded around 1410 in CZ-VB 42 that survived and thrived-as evidenced by a selection of sources-well into the following century demonstrates the diversity of the genre and the continuing dominance of monophonic...Institute of MusicologyÚstav hudební vědyFaculty of ArtsFilozofická fakult

Application and installation of solar panels on the roof of Citelis city bus

V práci je řešen návrh montáže fotovoltaických panelů na střechu nízkopodlažního autobusu CITELIS 12 m za účelem udržování kapacity akumulátoru vozidla. Dále je zde ukázán vliv dodávky elektrické energie na spotřebu vozidla. Nakonec je provedena ekonomická rozvaha obsahující hrubý odhad nákladů na pořízení fotovoltaického systému a výpočet úspory paliva při použití fotovoltaických panelů.Katedra dopravních prostředkůDokončená práce s úspěšnou obhajobo

Návrh automatizované vyhřívané pece pro testování FDM výtisků

This bachelor thesis deals with the analysis, design, construction and optimization of a special furnace for heat treatment and testing of products manufactured by additive 3D printing technology. The first part of the thesis deals with a literature search of currently available industrial and DYI solutions. Based on the conducted research, a custom solution is proposed based on DIY (Do It Yourself) reflow oven projects, which are mainly used in the electronics sector for industrial soldering of components. The control of the furnace is implemented using ESP32 microcontroller and FreeRTOS operating system. All the necessary mechanical components were modelled in SolidWorks software and then manufactured either by 3D printing or conventional machining methods. The work also includes an oven control system that runs on the mentioned microcontroller, runing in access point mode and can therefore be remotely connected to it. The described control system also allows for the creation of custom predefined temperature profiles that can be reused. The proposed solution was subsequently tested with satisfactory results

Pell's equation, continued fractions and Diophantine approximations of irrational numbers

This bachelor's thesis deals with Pell's equation, while clearly presenting structured information from studied domestic and foreign books, articles, and other sources. The goal of this thesis is to create study material primarily for university students but also for inquisitive high school students, and thus explain as intuitively as possible what Pell's equation is, how to find its solutions, and how it is related, for example, to continued fractions, approximations of irrational numbers, and invertible elements in Z[√n ]. The main motivation for solving Pell's equation throughout the work is specifically that its solutions give best approximations of irrational square roots. Pell's equation is presented in a brief historical context. Further, it is proved that there is a non-trivial integer solution for every Pell equation, and the theory of continued fractions is used to find it. To make the creation of continued fractions easier, the so-called Tenner's algorithm is introduced. Specifically, the search for a solution to Pell's equation is derived using convergents and the periodicity of continued fractions of irrational roots. Subsequently, the structure of the solution is described: it is proved that there is a so-called minimal solution that generates all positive solutions, and a set of..

Year-round project for preschool children - Water changes

The bachelor thesis deals with the creation and implementation of a year-round project for preschool children. The project focuses on water transformation during the year. Water becomes known to children through environmental education. The theoretical part explains the term environmental education in the context of preschool education. By defining examples of the relationships between the components of living and inanimate nature, the work deals in detail with information on the topic of water, its various forms, changes and uses. The practical part contains suggestions of suitable activities and activities used to understand the relationship to nature, especially water during the ten thematic weeks, when each month in a given school year children discuss one of the thematic weeks (solid state of water, liquid state of water, gaseous state of water, utility water, waste water, water origin, water cycle). During the weeks, the children gained basic knowledge about nature, especially about water, and their positive attitude towards nature in their surroundings was deepened. The children understood that there would be no life without water

Důsledky nepřesného finančního plánování poradenských společností pro klienty v ČR

Kodýtek, P., Consequences of Inaccurate Financial Planning of Consulting Companies for Clients in the Czech Republic. Brno: Mendel university in Brno, 2022. The bachelor's thesis deals with the possible negative consequences in the process of personal financial planning caused by inaccurately determined pension insurance benefits. The thesis presents a theoretical description of personal financial planning and the pension system in the Czech Republic. The aim of this work is to identify and quantify the differences in the outputs of personal financial plans prepared with precisely determined pensions benefits compared to estimated pensions benefits. The research took the form of a structured interview and processing of results for 2 selected households. The results of the work are discussed, and factors are determined that may affect the importance of determining the exact benefits of pension insurance for the purposes of developing a personal financial plan

Zkoumání kulturních a politických významů Mé vlasti v první Československé republice

3 Abstract Smetana's Má vlast (My Country) has long held a unique place in Czech national culture and is considered one of the constitutive elements of the Smetana myth in popular consciousness. Over forty years ago, in musicological discourse, Vladimír Lébl and Jitka Ludvová (1981) advocated for "removing the layers" of the Smetana myth, challenging the prevailing notion of Smetana as the sole genius by mapping the contemporary roots of Má vlast. Only recently, however, have musicologists begun to uncover the layers of the myth to a more serious degree, among them Brian S. Locke (2006), Christopher Campo-Bowen (2016), and Kelly St. Pierre (2017). This thesis contributes to this discussion by exploring the meanings of Má vlast and the Smetana myth in the context of the 1924 Smetana centenary celebrations. It reveals how the myth was repurposed by various stakeholders to justify diverse cultural policies and demonstrates that the different, overtly political interpretations were primarily a product of the ideologies and interests of these protagonists. The study also highlights the crucial role of the Czechoslovak government in enabling the celebrations and its use of them to achieve particular political objectives. Drawing on period sources, including archival materials, journal, and newspaper articles, as...4 Abstrakt Smetanova Má vlast zaujímá v české národní kultuře dlouhodobě jedinečné místo a je považována za jeden z konstitutivních prvků smetanovského mýtu zakotveného v obecném povědomí. Před více než čtyřiceti lety se v muzikologickém diskurzu Vladimír Lébl a Jitka Ludvová (1981) vyslovili pro "odstraňování nánosů" smetanovského mýtu a zpochybnili převládající představu o Smetanovi jako jediném géniovi zmapováním dobových kořenů Mé vlasti. Teprve v posledních letech však začali muzikologové vrstvy mýtu ve větší míře odkrývat, mezi nimi Brian S. Locke (2006), Christopher Campo- Bowen (2016) a Kelly St. Pierre (2017). Tato práce přispívá do této diskuse zkoumáním významů Mé vlasti a Smetanova mýtu v kontextu oslav stého výročí Smetanova narození v roce 1924. Odhaluje, jak byl mýtus různými aktéry přetvářen k ospravedlnění různých kulturních politik, a ukazuje, že různé, veskrze politické interpretace byly především výsledkem ideologií a zájmů těchto protagonistů. Studie rovněž zdůrazňuje klíčovou roli československého státu při realizaci oslav a to, jak je využil k dosažení konkrétních politických cílů. Na základě dobových pramenů, včetně archivních materiálů, časopiseckých a novinových článků a monografií, odhaluje tato studie dynamiku vyjednávání mezi různými zájmovými skupinami a státem při formování...Institute of MusicologyÚstav hudební vědyFaculty of ArtsFilozofická fakult

Oral tradition of cantiones in Czech lands and its imprint in late medieval manuscripts

Oral tradition of cantiones in Czech lands and its imprint in late medieval manuscripts Abstract The sources extant in the Czech lands transmit a vast body of cantiones, i.e., Latin spiritual monophonic or polyphonic songs that provide a rich basis for research into their tradition and transmission during the late Middle Ages. Though they have been subject of scholarly study for a century and a half, much of the literature is limited by the approach employed, be it the philological method, which saw a cantio's extant sources a tool for deriving its archetype and in its variants merely errors to be emended (Mužík, Černý), or a nationalistic bias (Nejedlý). In the case of chant, scholars have suggested that the tradition of medieval music was strongly impacted by oral transmission (Treitler, Hucke) and hence should be looked at from a different perspective, one close to that of ethnomusicology (Jeffery). Though some recent papers reflect this approach (Gancarczyk), it has not yet been tested on a larger body of songs, nor its implications systematically outlined. My study of several dozen songs recorded around 1410 in CZ-VB 42 that survived and thrived-as evidenced by a selection of sources-well into the following century demonstrates the diversity of the genre and the continuing dominance of monophonic..

Friedman's Excess energy and the McMillan-Mayer theory of solutions:Thermodynamics

In his version of the theory of multicomponent systems, Friedman used the analogy which exists between the virial expansion for the osmotic pressure obtained from the McMillan-Mayer (MM) theory of solutions in the grand canonical ensemble and the virial expansion for the pressure of a real gas. For the calculation of the thermodynamic properties of the solution, Friedman proposed a definition for the"excess free energy" that is a reminder of the ancient idea for the"osmotic work". However, the precise meaning to be attached to his free energy is, within other reasons, not well defined because in osmotic equilibrium the solution is not a closed system and for a given process the total amount of solvent in the solution varies. In this paper, an analysis based on thermodynamics is presented in order to obtain the exact and precise definition for Friedman"s excess free energy and its use in the comparison with the experimental data