Boundedness and convergence of singular integrals on fractal type sets

The topic of this dissertation lies in the intersection of harmonic analysis and fractal geometry. We particulary consider singular integrals in Euclidean spaces with respect to general measures, and we study how the geometric structure of the measures affects certain analytic properties of the operators. The thesis consists of three research articles and an overview. In the first article we construct singular integral operators on lower dimensional Sierpinski gaskets associated with homogeneous Calderón-Zygmund kernels. While these operators are bounded their principal values fail to exist almost everywhere. Conformal iterated function systems generate a broad range of fractal sets. In the second article we prove that many of these limit sets are porous in a very strong sense, by showing that they contain holes spread in every direction. In the following we connect these results with singular integrals. We exploit the fractal structure of these limit sets, in order to establish that singular integrals associated with very general kernels converge weakly. Boundedness questions consist a central topic of investigation in the theory of singular integrals. In the third article we study singular integrals of different measures. We prove a very general boundedness result in the case where the two underlying measures are separated by a Lipshitz graph. As a consequence we show that a certain weak convergence holds for a large class of singular integrals.

On Games on Non-Wellfounded Sets and Stationary Sets

In this thesis we study a few games related to non-wellfounded and stationary sets. Games have turned out to be an important tool in mathematical logic ranging from semantic games defining the truth of a sentence in a given logic to for example games on real numbers whose determinacies have important effects on the consistency of certain large cardinal assumptions. The equality of non-wellfounded sets can be determined by a so called bisimulation game already used to identify processes in theoretical computer science and possible world models for modal logic. Here we present a game to classify non-wellfounded sets according to their branching structure. We also study games on stationary sets moving back to classical wellfounded set theory. We also describe a way to approximate non-wellfounded sets with hereditarily finite wellfounded sets. The framework used to do this is domain theory. In the Banach-Mazur game, also called the ideal game, the players play a descending sequence of stationary sets and the second player tries to keep their intersection stationary. The game is connected to precipitousness of the corresponding ideal. In the pressing down game first player plays regressive functions defined on stationary sets and the second player responds with a stationary set where the function is constant trying to keep the intersection stationary. This game has applications in model theory to the determinacy of the Ehrenfeucht-Fraisse game. We show that it is consistent that these games are not equivalent.

Explorations in the distributional and semantic similarity of words

A straightforward computation of the list of the words (the `tail words' of the list) that are distributionally most similar to a given word (the `head word' of the list) leads to the question: How semantically similar to the head word are the tail words; that is: how similar are their meanings to its meaning? And can we do better? The experiment was done on nearly 18,000 most frequent nouns in a Finnish newsgroup corpus. These nouns are considered to be distributionally similar to the extent that they occur in the same direct dependency relations with the same nouns, adjectives and verbs. The extent of the similarity of their computational representations is quantified with the information radius. The semantic classification of head-tail pairs is intuitive; some tail words seem to be semantically similar to the head word, some do not. Each such pair is also associated with a number of further distributional variables. Individually, their overlap for the semantic classes is large, but the trained classification-tree models have some success in using combinations to predict the semantic class. The training data consists of a random sample of 400 head-tail pairs with the tail word ranked among the 20 distributionally most similar to the head word, excluding names. The models are then tested on a random sample of another 100 such pairs. The best success rates range from 70% to 92% of the test pairs, where a success means that the model predicted my intuitive semantic class of the pair. This seems somewhat promising when distributional similarity is used to capture semantically similar words. This analysis also includes a general discussion of several different similarity formulas, arranged in three groups: those that apply to sets with graded membership, those that apply to the members of a vector space, and those that apply to probability mass functions.

Subject Teacher Training and Teaching in English

The purpose of this Master s thesis is on one hand to find out how CLIL (Content and Language Integrated Learning) teachers and English teachers perceive English and its use in teaching, and on the other hand, what they consider important in subject teacher education in English that is being planned and piloted in STEP Project at the University of Helsinki Department of Teacher Education. One research question is also what kind of language requirements teachers think CLIL teachers should have. The research results are viewed in light of previous research and literature on CLIL education. Six teachers participate in this study. Two of them are English teachers in the comprehensive school, two are class teachers in bilingual elementary education, and two are subject teachers in bilingual education, one of whom teaches in a lower secondary school and the other in an upper secondary school. One English teacher and one bilingual class teacher have graduated from a pilot class teacher program in English that started at the University of Helsinki in the middle of the 1990 s. The bilingual subject teachers are not trained in English but they have learned English elsewhere, which is a particular focus of interest in this study because it is expected that a great number of CLIL teachers in Finland do not have actual studies in English philology. The research method is interview and this is a qualitative case study. The interviews are recorded and transcribed for the ease of analysis. The English teachers do not always use English in their lessons and they would not feel confident in teaching another subject completely in English. All of the CLIL teachers trust their English skills in teaching, but the bilingual class teachers also use Finnish during lessons either because some teaching material is in Finnish, or they feel that rules and instructions are understood better in mother tongue or students English skills are not strong enough. One of the bilingual subject teachers is the only one who consciously uses only English in teaching and in discussions with students. Although teachers good English skills are generally considered important, only the teachers who have graduated from the class teacher education in English consider it important that CLIL teachers would have studies in English philology. Regarding the subject teacher education program in English, the respondents hope that its teachers will have strong enough English skills and that it will deliver what it promises. Having student teachers of different subjects studying together is considered beneficial. The results of the study show that acquiring teaching material in English continues to be the teachers own responsibility and a huge burden for the teachers, and there has, in fact, not been much progress in the matter since the beginning of CLIL education. The bilingual subject teachers think, however, that using one s own material can give new inspiration to teaching and enable the use of various pedagogical methods. Although it is questionable if the language competence requirements set for CLIL teachers by the Finnish Ministry of Education are not adhered to, it becomes apparent in the study that studies in English philology do not necessarily guarantee strong enough language skills for CLIL teaching, but teachers own personality and self-confidence have significance. Keywords: CLIL, bilingual education, English, subject teacher training, subject teacher education in English, STEP

Whose ETSIP? National and local ownership of Namibia's education and training sector improvement programme

Tämä työ tarkastelee kansallista ja paikallista omistajuutta Namibian opetussektorin kehittämisohjelmassa. Opetussektorin kehittämisohjelma ETSIP on 15-vuotinen sektoriohjelma vuosille 2005-2015 ja sen tavoitteena on edesauttaa Namibian kehittymistä tietoyhteiskunnaksi. Tutkimuksen tarkoituksena on selvittää miten kansallinen ja paikallinen omistajuus on toteutunut ETSIP prosessin aikana. Erityisesti pyritään selvittämään paikallistason opetussektorin virkamiesten näkemyksiä ETSIP prosessista, heidän roolistaan siinä ja siitä millaisia vaikuttamisen ja hallinnan mahdollisuuksia heillä on ollut prosessin aikana. Tutkimuksen lähtökohta on laadullinen ja lähestymistapa konstruktionistinen: tutkimus tarkastelee todellisuutta ihmisten kokemusten, näkemysten ja toiminnan kautta. Tutkimusaineisto koostuu haastatteluista, epävirallisista keskusteluista, lehtiartikkeleista ja ETSIP dokumenteista. Tutkimus osoittaa että kansallinen omistajuus on epämääräinen käsite sillä kansallisia toimijoita ja näkemyksiä on useita. Tutkimus vahvistaa Castel-Brancon huomion siitä, että omistajuutta on tarkasteltava kontekstissaan: muuttuvana ja kilpailtuna. ETSIPin rinnalle ollaan valmistelemassa uutta strategista ohjelmaa opetusministeriölle mikä saattaa muuttaa omistajuutta ETSIPiin. ETSIP dokumenttien omistajuusretoriikka myötäilee kansainvälisiä sitoumuksia avun vaikuttavuuden parantamiseksi mutta niistä puuttuu syvällisempi analyysi siitä, miten kansallinen ja paikallinen omistajuus toteutuisi käytännössä. Avunantajien näkemys omistajuudesta on suppea: omistajuus nähdään lähinnä sitoutumisena ennalta määrättyyn politiikkaohjelmaan. Haastatteluaineistosta nousee esiin Whitfieldin ja Frazerin jaottelu suppeista ja laajoista omistajuuskäsityksistä. Sitoutumista ETSIP ohjelmaan pidetään tärkeänä mutta riittämättömänä määritteenä omistajuudelle. Paikallisella tasolla sitoutuminen ETSIP ohjelman periaatteisiin ja tavoitteisiin on toteutunut melko hyvin mutta jos omistajuutta tarkastellaan laajemmin vaikutusvallan ja hallinnan käsitteiden kautta voidaan todeta että omistajuus on ollut heikkoa. Paikallisella tasolla ei ole ollut juurikaan vaikutusvaltaa ETSIP ohjelman sisältöön eikä mahdollisuutta hallita ohjelman toteutusta ja päättää siitä mitä hankkeita ohjelman kautta rahoitetaan. Tujanin demokraattisen omistajuuden käsite kuvaa tarvetta muuttaa ja laajentaa omistajuusajattelua huomioiden paikallisen tason paremmin. Tämä tutkimus viittaa siihen että omistajuuden toteutuminen paikallisella tasolla edellyttäisi institutionaalisen kulttuurin muutosta ja institutionaalisen legitimiteetin vahvistamista. Omistajuuden mahdollistamiseksi paikallisella tasolla tarvittaisiin poliittista johtajuutta, luottamusta, vastuullisuuden kulttuurin kehittämistä, tehokkaampaa tiedonjakoa, laajaa osallistumista, vuoropuhelua ja yhteistyötä. Ennen kaikkea tarvittaisiin paikallisen tason vaikutusvaltaa päätöksenteossa ja kontrollia resurssien käytöstä. Tälle muutokselle on selvä tarve ja tilaus.

Playing Games on Sets and Models

The most prominent objective of the thesis is the development of the generalized descriptive set theory, as we call it. There, we study the space of all functions from a fixed uncountable cardinal to itself, or to a finite set of size two. These correspond to generalized notions of the universal Baire space (functions from natural numbers to themselves with the product topology) and the Cantor space (functions from natural numbers to the {0,1}-set) respectively. We generalize the notion of Borel sets in three different ways and study the corresponding Borel structures with the aims of generalizing classical theorems of descriptive set theory or providing counter examples. In particular we are interested in equivalence relations on these spaces and their Borel reducibility to each other. The last chapter shows, using game-theoretic techniques, that the order of Borel equivalence relations under Borel reduciblity has very high complexity. The techniques in the above described set theoretical side of the thesis include forcing, general topological notions such as meager sets and combinatorial games of infinite length. By coding uncountable models to functions, we are able to apply the understanding of the generalized descriptive set theory to the model theory of uncountable models. The links between the theorems of model theory (including Shelah's classification theory) and the theorems in pure set theory are provided using game theoretic techniques from Ehrenfeucht-Fraïssé games in model theory to cub-games in set theory. The bottom line of the research declairs that the descriptive (set theoretic) complexity of an isomorphism relation of a first-order definable model class goes in synch with the stability theoretical complexity of the corresponding first-order theory. The first chapter of the thesis has slightly different focus and is purely concerned with a certain modification of the well known Ehrenfeucht-Fraïssé games. There we (me and my supervisor Tapani Hyttinen) answer some natural questions about that game mainly concerning determinacy and its relation to the standard EF-game

Distributed algorithms for edge dominating sets

An edge dominating set for a graph G is a set D of edges such that each edge of G is in D or adjacent to at least one edge in D. This work studies deterministic distributed approximation algorithms for finding minimum-size edge dominating sets. The focus is on anonymous port-numbered networks: there are no unique identifiers, but a node of degree d can refer to its neighbours by integers 1, 2, ..., d. The present work shows that in the port-numbering model, edge dominating sets can be approximated as follows: in d-regular graphs, to within 4 − 6/(d + 1) for an odd d and to within 4 − 2/d for an even d; and in graphs with maximum degree Δ, to within 4 − 2/(Δ − 1) for an odd Δ and to within 4 − 2/Δ for an even Δ. These approximation ratios are tight for all values of d and Δ: there are matching lower bounds.