Homogeneous model theory of metric structures

This thesis studies homogeneous classes of complete metric spaces. Over the past few decades model theory has been extended to cover a variety of nonelementary frameworks. Shelah introduced the abstact elementary classes (AEC) in the 1980s as a common framework for the study of nonelementary classes. Another direction of extension has been the development of model theory for metric structures. This thesis takes a step in the direction of combining these two by introducing an AEC-like setting for studying metric structures. To find balance between generality and the possibility to develop stability theoretic tools, we work in a homogeneous context, thus extending the usual compact approach. The homogeneous context enables the application of stability theoretic tools developed in discrete homogeneous model theory. Using these we prove categoricity transfer theorems for homogeneous metric structures with respect to isometric isomorphisms. We also show how generalized isomorphisms can be added to the class, giving a model theoretic approach to, e.g., Banach space isomorphisms or operator approximations. The novelty is the built-in treatment of these generalized isomorphisms making, e.g., stability up to perturbation the natural stability notion. With respect to these generalized isomorphisms we develop a notion of independence. It behaves well already for structures which are omega-stable up to perturbation and coincides with the one from classical homogeneous model theory over saturated enough models. We also introduce a notion of isolation and prove dominance for it.

Finitary Abstract Elementary Classes

The research in model theory has extended from the study of elementary classes to non-elementary classes, i.e. to classes which are not completely axiomatizable in elementary logic. The main theme has been the attempt to generalize tools from elementary stability theory to cover more applications arising in other branches of mathematics. In this doctoral thesis we introduce finitary abstract elementary classes, a non-elementary framework of model theory. These classes are a special case of abstract elementary classes (AEC), introduced by Saharon Shelah in the 1980's. We have collected a set of properties for classes of structures, which enable us to develop a 'geometric' approach to stability theory, including an independence calculus, in a very general framework. The thesis studies AEC's with amalgamation, joint embedding, arbitrarily large models, countable Löwenheim-Skolem number and finite character. The novel idea is the property of finite character, which enables the use of a notion of a weak type instead of the usual Galois type. Notions of simplicity, superstability, Lascar strong type, primary model and U-rank are inroduced for finitary classes. A categoricity transfer result is proved for simple, tame finitary classes: categoricity in any uncountable cardinal transfers upwards and to all cardinals above the Hanf number. Unlike the previous categoricity transfer results of equal generality the theorem does not assume the categoricity cardinal being a successor. The thesis consists of three independent papers. All three papers are joint work with Tapani Hyttinen.

Kehittämistutkimus: Tieto- ja viestintätekniikkaa kemian opetukseen

The main aim of the present study was to develop information and communication technology (ICT) based chemistry education. The goals for the study were to support meaningful chemistry learning, research-based teaching and diffusion of ICT innovations. These goals were used as guidelines that form the theoretical framework for this study. This Doctoral Dissertation is based on eight-stage research project that included three design researches. These three design researches were scrutinized as separate case studies in which the different cases were formed according to different design teams: i) one researcher was in charge of the design and teachers were involved in the research process, ii) a research group was in charge of the design and students were involved in the research process, and iii) the design was done by student teams, the research was done collaboratively, and the design process was coordinated by a researcher. The research projects were conducted using mixed method approach, which enabled a comprehensive view on education design. In addition, the three central areas of design research: problem analysis, design solution and design process were included in the research, which was guided by the main research questions formed according to these central areas: 1) design solution: what kind of elements are included in ICT-based learning environments that support meaningful chemistry learning and diffusion of innovation, 2) problem analysis: what kind of new possibilities the designed learning environments offer for the support of meaningful chemistry learning, and 3) design process: what kind of opportunities and challenges does collaboration bring to the design of ICT-based learning environments? The main research questions were answered according to the analysis of the survey and observation data, six designed learning environments and ten design narratives from the three case studies. Altogether 139 chemistry teachers and teacher students were involved in the design processes. The data was mainly analysed by methods of qualitative content analysis. The first main result from the study give new information on the meaningful chemistry learning and the elements of ICT-based learning environment that support the diffusion of innovation, which can help in the development of future ICT-education design. When the designed learning environment was examined in the context of chemistry education, it was evident that an ICT-based chemistry learning environment supporting the meaningful learning of chemistry motivates the students and makes the teacher s work easier. In addition, it should enable the simultaneous fulfilment of several pedagogical goals and activate higher-level cognitive processes. The learning environment supporting the diffusion of ICT innovation is suitable for Finnish school environment, based on open source code, and easy to use with quality chemistry content. According to the second main result, new information was acquired about the possibilities of ICT-based learning environments in supporting meaningful chemistry learning. This will help in setting the goals for future ICT education. After the analysis of design solutions and their evaluations, it can be said that ICT enables the recognition of all elements that define learning environments (i.e. didactic, physical, technological and social elements). The research particularly demonstrates the significance of ICT in supporting students motivation and higher-level cognitive processes as well as versatile visualization resources for chemistry that ICT makes possible. In addition, research-based teaching method supports well the diffusion of studied innovation on individual level. The third main result brought out new information on the significance of collaboration in design research, which guides the design of ICT education development. According to the analysis of design narratives, it can be said that collaboration is important in the execution of scientifically reliable design research. It enables comprehensive requirement analysis and multifaceted development, which improves the reliability and validity of the research. At the same time, it sets reliability challenges by complicating documenting and coordination, for example. In addition, a new method for design research was developed. Its aim is to support the execution of complicated collaborative design projects. To increase the reliability and validity of the research, a model theory was used. It enables time-pound documenting and visualization of design decisions that clarify the process. This improves the reliability of the research. The validity of the research is improved by requirement definition through models. This way learning environments that meet the design goals can be constructed. The designed method can be used in education development from comprehensive to higher level. It can be used to recognize the needs of different interest groups and individuals with regard to processes, technology and substance knowledge as well as interfaces and relations between them. The developed method has also commercial potential. It is used to design learning environments for national and international market.

ESPOOLAISTEN KOETTU TERVEYS - Pienalue-estimoinnin soveltaminen aluetasoisen tiedon tuottamiseen

Koettu terveys on subjektiivinen mittari, jota voidaan käyttää objektiivisten mittareiden ohella kunnan sosiaali- ja terveyspolitiikan onnistumisen arviointiin sekä ohjaamaan palveluiden järjestämistä. Tutkimuksessa selvitettiin mahdollisuuksia tuottaa pienalue-estimointimenetelmillä tietoa Espoon eri alueiden 20–64 -vuotiaan väestön kokemasta terveydestä. Erityisesti työ keskittyi selvittämään; kuinka pienille Espoon alueille voidaan tuottaa luotettavaa tietoa käytettävissä olevasta otosaineistosta ja miten käytetty mallitaso sekä otoskoon kasvattaminen muiden pääkaupunkiseudun asukkaiden vastauksilla vaikuttaa estimointitulokseen? Tutkimusaineistona käytettiin vuoden 2008 aikana Helsingin sosiaalialan osaamiskeskuksen keräämän Pääkaupunkiseudun hyvinvointitutkimus -aineiston lisäksi Aluesarjat-tilastotietokannasta sekä Tilastokeskuksen Väestötilastopalvelusta saatavaa tietoa. Pienalue-estimointimenetelminä käytettiin malliavusteista GREG-estimointia sekä malliperusteista EBLUP-estimointia. Sekä Espoon että koko pääkaupunkiseudun otosaineistosta muodostettujen yksikkö- ja aluetason mallien parametrien ja Espoon eri alueiden 20–64 -vuotiaaseen väestöön liittyvän tiedon avulla tuotettiin alue-estimaatteja Espoon pien-, tilasto- ja suuralueille. Koetun terveyden aluekeskiarvon estimointi onnistui kaikilla aluetasoilla kyseisen aluetason malliin perustuvalla EBLUP-estimaattorilla. GREG-estimaattori onnistui vain suuraluetason estimoinnissa, muilla aluetasoilla alueiden pienet otoskoot huononsivat GREG-estimaatin tarkkuutta. Yksikkötason sekamallin huono selitysvoima ja mallista puuttuva selittäjä huononsivat siihen perustuvan EBLUP-estimaattorin tarkkuutta. Estimoinnin kannalta mallitasoa tärkeämmäksi osoittautui mallin hyvyyden toteutuminen. Voiman lainaaminen kohdejoukon ulkopuoliselta otokselta heikensi satunnaisvaikutuksen merkitsevyyttä ja alue-estimaattien välistä vaihtelua sekä lisäsi estimaattien tarkkuutta. Pienaluetiedon tuottaminen onnistuu EBLUP-estimaattoreilla jopa 85 pienalueelle noin 800 havainnon otosaineistosta, mikäli käytössä on luotettavaa lisäinformaatiota ja hyvä malli. GREG-estimaattori sallii huonomman mallin käytön, mutta edellyttää suurempia pienalueittaisia otoskokoja kuin EBLUP-estimaattorit. EBLUP-estimaattoreiden etuna on alueittaisen otoskoon lisäksi mahdollisuus perustaa estimointi sekä yksikkötasoiseen että aluetasoiseen malliin. Pienalueestimointimenetelmät lisäävät otosaineistojen hyödyntämismahdollisuuksia. Onnistumisen takaa menetelmiin sisältyvien, aineistoon ja malliin kohdistuvien vaatimusten huomiointi tiedonkeruun suunnitteluvaiheessa mm. kysymysten asettelussa. Pienalue-estimointimenetelmien käyttö edellyttää tilastollista osaamista, kriittisyyttä saatuja tuloksia kohtaan ja vastuullisuutta tuloksia julkaistaessa. Laatuvaatimukset täyttävät pienalue-estimaatit soveltuvat hyvin päätöksenteon tueksi, kun halutaan vertailla alueita ja kohdentaa resursseja tarvelähtöisesti.