Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2003. / Includes bibliographical references (leaves 56-58). Also available in electronic version. Access restricted to campus users.

The diploma thesis included of informations of a Mathematical Olympiad, equations and systems of equations. The theoretical part contains a historical development and its organization, next it contains informations about equations and their teaching as it is given in the National Curriculum. The practical part consists of a collection of exercises on the topic of equations and systems of equations, which were included in the competition Mathematical Olympiad. Examples in the collection are divided into categories vhich appeard, and ranked by difficulty. Further practical part includes evaluation works heets completed by students of Pedagogicial faculty, University of South Bohemia. This thesis may serve as preparation pupils on Mathematical Olympiad or as a collection of examples for talented pupils

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Econometric Analysis of Financial Data Author: Matúš Baniar Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Jitka Zichová, Dr. Abstract: In some occasions, financial data can be represented as a combination of cross-sectional and time-series information. Hence it could be convenient to consider a system of econometric equations for modeling such data sets. At the beginning of this thesis, we describe general definitions and we talk about different types of variables from the perspective of exogeneity. Later, we describe some specific cases of these equations: SUR system, simultaneous equation models and the model of vector autoregression. For selected models, we also discuss estimation methods and their properties. In the final section, the described approach is applied to real financial data making use of appropriate software. Keywords: exogeneity, SUR system, simultaneous equations, VAR

The aim of this work is to get acquainted with the issue of solving word problems by means of equations, to state and to analyse various ways of solving them. The thesis also includes the comparison of strategies for solving word problems that are selected by students who have been already taught the given type of the equation, and students who have not yet mastered these procedures. The introductory part of the work deals with word problems in general. There the terms such as the word problems, procedures, methods and strategies used for solving word problems are specified. In the next part of the text, the basic types of equations are characterized. For each type of the equation, one typical word problem was selected and it was experimentally found out which solving strategies were used by students most often. Keywords: word problem, algebraic equation, non-algebraic equation, system of equations

The design process includes many different activities inwhich various computational mechanics tools are used forbehaviour modelling of mechanical systems and their buildingblocks, e.g. machine elements. These tools usually supportlarge and complex models and they produce large quantities ofdata with a high degree of complexity. In these situations,efficient data management and the ability to search and sharedata are important issues to achieve an efficient designprocess. Today, this ability is usually not supported by theindividual applications even though this probably would improveand facilitate the ability to search for data on a higher levelin the engineering information system. This work investigates the ability of searching andcomparing analysis data within behaviour models of technicalsystems as well as over the analysis results. This is done byinvestigating the potential benefits of integrating moderndatabase technology with a multibody system (MBS) analysissoftware in the same manner that has been successfully done forbusiness and administrative applications. This has resulted inan implemented pilot system, named MECHAMOS, that integratesthe main-memory resident object-relational database managementsystem (DBMS) AMOSwith the symbolic multibody system (MBS)software SOPHIA operating in MapleV. This provides MECHAMOSwith both symbolic and numeric mathematical capabilities forMBS analysis and data management capabilities to search andcompare engineering data in the database. The approach, making data managing tools available in acomputer aided engineering software, considerably improves theanalysis of technical systems. The analysis is brought to ahigher level through the available query language and thedesired data is specified, fairly intuitively, in a query. Whenthe query is processed, the DBMS knows how to retrieve andautomatically derive the required data. As shown in someexamples, the ability to search over stored and derived data inthe database is not restricted to a single MBS-model inMECHAMOS. Because of the implemented materialisation handling,it is also possible to search, combine, and compare data fromseveral simulation results which are based on several differentmodels in the database. This extends the ability to performoptimisation from a traditional parameter study to thepossibility to analyse and compare different technical conceptsthrough the query language and hereby retrieve those conceptsthat fulfil certain requirements. If submodel techniques aresupported, queries over a set of components in the databasewould automatically create, analyse and compare the possibleconcepts. This would assist the designer in choosing the bestcomponents for a design.

This paper presents a study of the computer game ”Minesweeper”. The aim of the game is to search through a rectangular area of mined squares without hitting any mines. By using a strategy based on making every operation as safe as possible, series of the game have been simulated. The size of the playground is four times four squares. The si- mulations indicate how often the game will succeed and which square is the best one to start at. The strategy demands advanced mathematical calculations. The account of these is the ma- jor part of my work. My investigation shows that if there are three hidden mines the game will succeed about two times out of three. It also shows that the best startingpoint is a corner.

<p>The design process includes many different activities inwhich various computational mechanics tools are used forbehaviour modelling of mechanical systems and their buildingblocks, e.g. machine elements. These tools usually supportlarge and complex models and they produce large quantities ofdata with a high degree of complexity. In these situations,efficient data management and the ability to search and sharedata are important issues to achieve an efficient designprocess. Today, this ability is usually not supported by theindividual applications even though this probably would improveand facilitate the ability to search for data on a higher levelin the engineering information system.</p><p>This work investigates the ability of searching andcomparing analysis data within behaviour models of technicalsystems as well as over the analysis results. This is done byinvestigating the potential benefits of integrating moderndatabase technology with a multibody system (MBS) analysissoftware in the same manner that has been successfully done forbusiness and administrative applications. This has resulted inan implemented pilot system, named MECHAMOS, that integratesthe main-memory resident object-relational database managementsystem (DBMS) AMOSwith the symbolic multibody system (MBS)software SOPHIA operating in MapleV. This provides MECHAMOSwith both symbolic and numeric mathematical capabilities forMBS analysis and data management capabilities to search andcompare engineering data in the database.</p><p>The approach, making data managing tools available in acomputer aided engineering software, considerably improves theanalysis of technical systems. The analysis is brought to ahigher level through the available query language and thedesired data is specified, fairly intuitively, in a query. Whenthe query is processed, the DBMS knows how to retrieve andautomatically derive the required data. As shown in someexamples, the ability to search over stored and derived data inthe database is not restricted to a single MBS-model inMECHAMOS. Because of the implemented materialisation handling,it is also possible to search, combine, and compare data fromseveral simulation results which are based on several differentmodels in the database. This extends the ability to performoptimisation from a traditional parameter study to thepossibility to analyse and compare different technical conceptsthrough the query language and hereby retrieve those conceptsthat fulfil certain requirements. If submodel techniques aresupported, queries over a set of components in the databasewould automatically create, analyse and compare the possibleconcepts. This would assist the designer in choosing the bestcomponents for a design.</p>

Nesta dissertação tivemos como principal objetivo, conceber, implementar, analisar, reconstruir e validar uma proposta didática que agregasse um maior valor formativo ao ensino do tópico de sistemas de equações, através da geometria vetorial. No referencial teórico buscamos naturais aproximações entre álgebra e geometria, através de exemplos tomados da história e de exemplos que tratam de conteúdos presentes nos programas escolares. Nessas aproximações, também fazemos uma discussão que se apóia no trabalho de Douady (1998), onde é sugerido, sempre que possível, a utilização de pelo menos dois domínios de conhecimento (algébrico e geométrico) no processo de ensino e aprendizagem da Matemática. A metodologia de investigação escolhida, a Engenharia Didática (Artigue, 1996), nos permitiu um acurado trabalho de validação do material didático produzido. Este material assim se constitui: na fundamentação matemática de conteúdos relativos à geometria vetorial, de forma que estes pudessem ser trabalhados no Ensino Médio; na construção de uma seqüência didática, na forma de coletânea de atividades acompanhada de plano de execução; e na construção de um pequeno software “Vetores e Operações”, com o propósito de trazer um recurso que é facilitador da aprendizagem. Consideramos que nossa hipótese de investigação – através da geometria vetorial é possível desenvolver, na escola, o tópico de sistema de equações, de forma a ter-se nele agregado um maior valor formativo – se valida no contexto de nossa experiência. E com as devidas adaptações, de modo a atenderem as especificidades de cada turma de alunos, é com confiança que apostamos na sua viabilidade para outras situações de ensino e aprendizagem sobre sistemas de equações. / In this paper our main purpose is to conceive, implement, analyze, rebuild and validate an educational proposal that can provides a wide formative value to the teaching of the topic of system of equations through vectorial geometry. The theoretical frame is based on natural approach between algebra and geometry, through examples taken from History and examples related to subjects which are part of school programs. To support this approach, we have also made a discussion based on Douady’s work (1988), where it is stressed the importance of the interactions of different domains (geometrical and algebraic) in the student learning process. The methodology of investigation chosen, the Educational Engineering (Artigue, 1966), allowed us an accurate work of validation of the produced material. This material is in this way constituted: the mathematical foundation of the contents related to vectorial geometry, in a way to become possible to work in the high school level; an educational sequence, made of a collection of mathematical activities and an execution plan; a learning object (a small software) named “ Vectors and Operations“, with the purpose of bring a resource that can make easy the act of learning. We consider that our investigation hypothesis – through vectorial geometry it is possible to develop in the school, the topic of system of equation in a way that it carries a maximum formative value – was validate in the context of the experience. And, with the proper adaptation, in a way to attend the specificities of each group of students, it is with confidence that we believe in its applicability for other teaching and learning situations about these subject .

Nesta dissertação tivemos como principal objetivo, conceber, implementar, analisar, reconstruir e validar uma proposta didática que agregasse um maior valor formativo ao ensino do tópico de sistemas de equações, através da geometria vetorial. No referencial teórico buscamos naturais aproximações entre álgebra e geometria, através de exemplos tomados da história e de exemplos que tratam de conteúdos presentes nos programas escolares. Nessas aproximações, também fazemos uma discussão que se apóia no trabalho de Douady (1998), onde é sugerido, sempre que possível, a utilização de pelo menos dois domínios de conhecimento (algébrico e geométrico) no processo de ensino e aprendizagem da Matemática. A metodologia de investigação escolhida, a Engenharia Didática (Artigue, 1996), nos permitiu um acurado trabalho de validação do material didático produzido. Este material assim se constitui: na fundamentação matemática de conteúdos relativos à geometria vetorial, de forma que estes pudessem ser trabalhados no Ensino Médio; na construção de uma seqüência didática, na forma de coletânea de atividades acompanhada de plano de execução; e na construção de um pequeno software “Vetores e Operações”, com o propósito de trazer um recurso que é facilitador da aprendizagem. Consideramos que nossa hipótese de investigação – através da geometria vetorial é possível desenvolver, na escola, o tópico de sistema de equações, de forma a ter-se nele agregado um maior valor formativo – se valida no contexto de nossa experiência. E com as devidas adaptações, de modo a atenderem as especificidades de cada turma de alunos, é com confiança que apostamos na sua viabilidade para outras situações de ensino e aprendizagem sobre sistemas de equações. / In this paper our main purpose is to conceive, implement, analyze, rebuild and validate an educational proposal that can provides a wide formative value to the teaching of the topic of system of equations through vectorial geometry. The theoretical frame is based on natural approach between algebra and geometry, through examples taken from History and examples related to subjects which are part of school programs. To support this approach, we have also made a discussion based on Douady’s work (1988), where it is stressed the importance of the interactions of different domains (geometrical and algebraic) in the student learning process. The methodology of investigation chosen, the Educational Engineering (Artigue, 1966), allowed us an accurate work of validation of the produced material. This material is in this way constituted: the mathematical foundation of the contents related to vectorial geometry, in a way to become possible to work in the high school level; an educational sequence, made of a collection of mathematical activities and an execution plan; a learning object (a small software) named “ Vectors and Operations“, with the purpose of bring a resource that can make easy the act of learning. We consider that our investigation hypothesis – through vectorial geometry it is possible to develop in the school, the topic of system of equation in a way that it carries a maximum formative value – was validate in the context of the experience. And, with the proper adaptation, in a way to attend the specificities of each group of students, it is with confidence that we believe in its applicability for other teaching and learning situations about these subject .