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

A maximal operator, Carleson's embedding, and tent spaces for vector-valued functions

This thesis is concerned with the area of vector-valued Harmonic Analysis, where the central theme is to determine how results from classical Harmonic Analysis generalize to functions with values in an infinite dimensional Banach space. The work consists of three articles and an introduction. The first article studies the Rademacher maximal function that was originally defined by T. Hytönen, A. McIntosh and P. Portal in 2008 in order to prove a vector-valued version of Carleson's embedding theorem. The boundedness of the corresponding maximal operator on Lebesgue-(Bochner) -spaces defines the RMF-property of the range space. It is shown that the RMF-property is equivalent to a weak type inequality, which does not depend for instance on the integrability exponent, hence providing more flexibility for the RMF-property. The second article, which is written in collaboration with T. Hytönen, studies a vector-valued Carleson's embedding theorem with respect to filtrations. An earlier proof of the dyadic version assumed that the range space satisfies a certain geometric type condition, which this article shows to be also necessary. The third article deals with a vector-valued generalizations of tent spaces, originally defined by R. R. Coifman, Y. Meyer and E. M. Stein in the 80's, and concerns especially the ones related to square functions. A natural assumption on the range space is then the UMD-property. The main result is an atomic decomposition for tent spaces with integrability exponent one. In order to suit the stochastic integrals appearing in the vector-valued formulation, the proof is based on a geometric lemma for cones and differs essentially from the classical proof. Vector-valued tent spaces have also found applications in functional calculi for bisectorial operators. In the introduction these three themes come together when studying paraproduct operators for vector-valued functions. The Rademacher maximal function and Carleson's embedding theorem were applied already by Hytönen, McIntosh and Portal in order to prove boundedness for the dyadic paraproduct operator on Lebesgue-Bochner -spaces assuming that the range space satisfies both UMD- and RMF-properties. Whether UMD implies RMF is thus an interesting question. Tent spaces, on the other hand, provide a method to study continuous time paraproduct operators, although the RMF-property is not yet understood in the framework of tent spaces.

Discriminative learning of Bayesian networks via factorized conditional log-likelihood

We propose an efficient and parameter-free scoring criterion, the factorized conditional log-likelihood (ˆfCLL), for learning Bayesian network classifiers. The proposed score is an approximation of the conditional log-likelihood criterion. The approximation is devised in order to guarantee decomposability over the network structure, as well as efficient estimation of the optimal parameters, achieving the same time and space complexity as the traditional log-likelihood scoring criterion. The resulting criterion has an information-theoretic interpretation based on interaction information, which exhibits its discriminative nature. To evaluate the performance of the proposed criterion, we present an empirical comparison with state-of-the-art classifiers. Results on a large suite of benchmark data sets from the UCI repository show that ˆfCLL-trained classifiers achieve at least as good accuracy as the best compared classifiers, using significantly less computational resources.

Elämää keskustassa ja kaupunkiseudun reunoilla. Urbaani ja jälkiesikaupungillinen elämäntyyli asumisen valinnoissa ja arkiliikkumisessa Helsingin seudulla.

The lifestyles of people living in single-family housing areas on the outskirts of the Greater Helsinki Region (GHR) are different from those living in inner city area. The urban structure of the GHR is concentrated in the capital on the one hand, and spread out across the outskirts on the other. Socioeconomic spatial divisions are evident as well-paid and educated residents move to the inner city or the single-family house dominated suburban neighbourhoods depending on their housing preferences and life situations. The following thesis explores how these lifestyles have emerged through the housing choices and daily mobility of the residents living in the new single-family housing areas on the outskirts of the GHR and the inner city. The study shows that, when it comes to lifestyles, residents on the outskirts of the region have different housing preferences and daily mobility patterns when compared with their inner city counterparts. Based on five different case study areas my results show that these differences are related to residents values, preferences and attitudes towards the neighbourhood, on the one hand, and limited by urban structure on the other. This also confirms earlier theoretical analyses and findings from the GHR. Residents who moved to the outskirts of Greater Helsinki Region and the apartment buildings of the inner city were similar in the basic elements of their housing preferences: they sought a safe and peaceful neighbourhood close to the natural environment. However, where housing choices, daily mobility and activities vary different lifestyles develop in both the outskirts and the inner city. More specifically, lifestyles in the city apartment blocks were inherently urban. Liveliness and highest order facilities were appreciated and daily mobility patterns were supported by diverse modes of transportation for the purposes of work, shopping and leisure time. On the outskirts, by contrast, lifestyles were largely post-suburban and child-friendliness appreciated. Due to the heterachical urban structure, daily mobility was more car-dependent since work, shopping and free time activities of the residents are more spread around the region. The urban structure frames the daily mobility on the outskirts of the region, but this is not to say that short local trips replace longer regional ones. This comparative case study was carried out in the single-family housing areas of Sundsberg in Kirkkonummi, Landbo in Helsinki and Ylästö in Vantaa, as well as in the inner city apartment building areas of Punavuori and Katajanokka in Helsinki. The data is comprised of residential surveys, interviews, and statistics and GIS data sets that illustrate regional daily mobility, socio-economic structure and vis-à-vis housing stock.