Composition operators on vector-valued BMOA and related function spaces

A composition operator is a linear operator between spaces of analytic or harmonic functions on the unit disk, which precomposes a function with a fixed self-map of the disk. A fundamental problem is to relate properties of a composition operator to the function-theoretic properties of the self-map. During the recent decades these operators have been very actively studied in connection with various function spaces. The study of composition operators lies in the intersection of two central fields of mathematical analysis; function theory and operator theory. This thesis consists of four research articles and an overview. In the first three articles the weak compactness of composition operators is studied on certain vector-valued function spaces. A vector-valued function takes its values in some complex Banach space. In the first and third article sufficient conditions are given for a composition operator to be weakly compact on different versions of vector-valued BMOA spaces. In the second article characterizations are given for the weak compactness of a composition operator on harmonic Hardy spaces and spaces of Cauchy transforms, provided the functions take values in a reflexive Banach space. Composition operators are also considered on certain weak versions of the above function spaces. In addition, the relationship of different vector-valued function spaces is analyzed. In the fourth article weighted composition operators are studied on the scalar-valued BMOA space and its subspace VMOA. A weighted composition operator is obtained by first applying a composition operator and then a pointwise multiplier. A complete characterization is given for the boundedness and compactness of a weighted composition operator on BMOA and VMOA. Moreover, the essential norm of a weighted composition operator on VMOA is estimated. These results generalize many previously known results about composition operators and pointwise multipliers on these spaces.

Rhetoric and Representation : Exploring the Cultural Meaning of the Natural Sciences in Contemporary Popular Science Writing and Literature

Kirjallisuuden- ja kulttuurintutkimus on viimeisten kolmen vuosikymmenen aikana tullut yhä enenevässä määrin tietoiseksi tieteen ja taiteen suhteen monimutkaisesta luonteesta. Nykyään näiden kahden kulttuurin tutkimus muodostaa oman kenttänsä, jolla niiden suhdetta tarkastellaan ennen kaikkea dynaamisena vuorovaikutuksena, joka heijastaa kulttuurimme kieltä, arvoja ja ideologisia sisältöjä. Toisin kuin aiemmat näkemykset, jotka pitävät tiedettä ja taidetta toisilleen enemmän tai vähemmän vastakkaisina pyrkimyksinä, nykytutkimus lähtee oletuksesta, jonka mukaan ne ovat kulttuurillisesti rakentuneita diskursseja, jotka kohtaavat usein samankaltaisia todellisuuden mallintamiseen liittyviä ongelmia, vaikka niiden käyttämät metodit eroavatkin toisistaan. Väitöskirjani keskittyy yllä mainitun suhteen osa-alueista popularisoidun tietokirjallisuuden (muun muassa Paul Davies, James Gleick ja Richard Dawkins) käyttämän kielen ja luonnontieteistä ideoita ammentavan kaunokirjallisuuden (muun muassa Jeanette Winterson, Tom Stoppard ja Richard Powers) hyödyntämien keinojen tarkasteluun nojautuen yli 30 teoksen kattavaa aineistoa koskevaan tyylin ja teemojen tekstianalyysiin. Populaarin tietokirjallisuuden osalta tarkoituksenani on osoittaa, että sen käyttämä kieli rakentuu huomattavassa määrin sellaisille rakenteille, jotka tarjoavat mahdollisuuden esittää todellisuutta koskevia argumentteja mahdollisimman vakuuttavalla tavalla. Tässä tehtävässä monilla klassisen retoriikan määrittelemillä kuvioilla on tärkeä rooli, koska ne auttavat liittämään sanotun sisällön ja muodon tiukasti toisiinsa: retoristen kuvioiden käyttö ei näin ollen edusta pelkkää tyylikeinoa, vaan se myös usein kiteyttää argumenttien taustalla olevat tieteenfilosofiset olettamukset ja auttaa vakiinnuttamaan argumentoinnin logiikan. Koska monet aikaisemmin ilmestyneistä tutkimuksista ovat keskittyneet pelkästään metaforan rooliin tieteellisissä argumenteissa, tämä väitöskirja pyrkii laajentamaan tutkimuskenttää analysoimalla myös toisenlaisten kuvioiden käyttöä. Osoitan myös, että retoristen kuvioiden käyttö muodostaa yhtymäkohdan tieteellisiä ideoita hyödyntävään kaunokirjallisuuteen. Siinä missä popularisoitu tiede käyttää retoriikkaa vahvistaakseen sekä argumentatiivisia että kaunokirjallisia ominaisuuksiaan, kuvaa tällainen sanataide tiedettä tavoilla, jotka usein heijastelevat tietokirjallisuuden kielellisiä rakenteita. Toisaalta on myös mahdollista nähdä, miten kaunokirjallisuuden keinot heijastuvat popularisoidun tieteen kerrontatapoihin ja kieleen todistaen kahden kulttuurin dynaamisesta vuorovaikutuksesta. Nykyaikaisen populaaritieteen retoristen elementtien ja kaunokirjallisuuden keinojen vertailu näyttää lisäksi, kuinka tiede ja taide osallistuvat keskusteluun kulttuurimme tiettyjen peruskäsitteiden kuten identiteetin, tiedon ja ajan merkityksestä. Tällä tavoin on mahdollista nähdä, että molemmat ovat perustavanlaatuisia osia merkityksenantoprosessissa, jonka kautta niin tieteelliset ideat kuin ihmiselämän suuret kysymyksetkin saavat kulttuurillisesti rakentuneen merkityksensä.

On the Origin of Ideas : An Abductivist Approach to Discovery

The purpose of this study is to analyze and develop various forms of abduction as a means of conceptualizing processes of discovery. Abduction was originally presented by Charles S. Peirce (1839-1914) as a "weak", third main mode of inference -- besides deduction and induction -- one which, he proposed, is closely related to many kinds of cognitive processes, such as instincts, perception, practices and mediated activity in general. Both abduction and discovery are controversial issues in philosophy of science. It is often claimed that discovery cannot be a proper subject area for conceptual analysis and, accordingly, abduction cannot serve as a "logic of discovery". I argue, however, that abduction gives essential means for understanding processes of discovery although it cannot give rise to a manual or algorithm for making discoveries. In the first part of the study, I briefly present how the main trend in philosophy of science has, for a long time, been critical towards a systematic account of discovery. Various models have, however, been suggested. I outline a short history of abduction; first Peirce's evolving forms of his theory, and then later developments. Although abduction has not been a major area of research until quite recently, I review some critiques of it and look at the ways it has been analyzed, developed and used in various fields of research. Peirce's own writings and later developments, I argue, leave room for various subsequent interpretations of abduction. The second part of the study consists of six research articles. First I treat "classical" arguments against abduction as a logic of discovery. I show that by developing strategic aspects of abductive inference these arguments can be countered. Nowadays the term 'abduction' is often used as a synonym for the Inference to the Best Explanation (IBE) model. I argue, however, that it is useful to distinguish between IBE ("Harmanian abduction") and "Hansonian abduction"; the latter concentrating on analyzing processes of discovery. The distinctions between loveliness and likeliness, and between potential and actual explanations are more fruitful within Hansonian abduction. I clarify the nature of abduction by using Peirce's distinction between three areas of "semeiotic": grammar, critic, and methodeutic. Grammar (emphasizing "Firstnesses" and iconicity) and methodeutic (i.e., a processual approach) especially, give new means for understanding abduction. Peirce himself held a controversial view that new abductive ideas are products of an instinct and an inference at the same time. I maintain that it is beneficial to make a clear distinction between abductive inference and abductive instinct, on the basis of which both can be developed further. Besides these, I analyze abduction as a part of distributed cognition which emphasizes a long-term interaction with the material, social and cultural environment as a source for abductive ideas. This approach suggests a "trialogical" model in which inquirers are fundamentally connected both to other inquirers and to the objects of inquiry. As for the classical Meno paradox about discovery, I show that abduction provides more than one answer. As my main example of abductive methodology, I analyze the process of Ignaz Semmelweis' research on childbed fever. A central basis for abduction is the claim that discovery is not a sequence of events governed only by processes of chance. Abduction treats those processes which both constrain and instigate the search for new ideas; starting from the use of clues as a starting point for discovery, but continuing in considerations like elegance and 'loveliness'. The study then continues a Peircean-Hansonian research programme by developing abduction as a way of analyzing processes of discovery.

Reconsidering Relativism and Intentionalism in Interpretation : Donald Davidson, Hermeneutics, and Pragmatism

This study focuses on two philosophical issues related to the interpretation of art. Firstly, it considers the role of authorial intentions in interpretation. Secondly, the study raises the issue of relativism in interpretation through a discussion of the relativistic tendencies apparent in the views of three major figures of contemporary philosophy: Joseph Margolis, Hans-Georg Gadamer, and Richard Rorty. The major goal of the thesis is to develop a theory of interpretation supporting the role of authorial intentions in interpretation on the basis of Donald Davidson s late philosophy of language and the holistic account of interpretation that underlies different parts of his philosophy. It is my belief that an intentionalist view of interpretation built on Davidsonian elements manages to form the most convincing defense of that interpretive position against the skepticism present in the views of Margolis, Gadamer, and Rorty. The theoretical issues addressed in the thesis are illuminated by discussions of case-examples, most importantly Richard Wagner s The Valkyrie, Thomas Adés America: A Prophecy, and some symphonies by Dimitri Shostakovich. In chapter one, I present a critical discussion of Margolis robust relativism. While finding Margolis criticism of the self-refutive argument plausible, I, nevertheless, argue that the relativistic logic Margolis offers should not be favored in interpretation. The first parts of chapter two outline Davidsonian intentionalism by presenting a reading of Davidson s later work in philosophy of language and mind, and by indicating its relationship to Davidson s views of literature. Then, I shall compare Davidson s ideas with some recent modest forms of intentionalism found in analytic aesthetics, and argue that Davidsonian intentionalism is in many respects more satisfactory compared to them. Chapter three engages Gadamer s hermeneutics by defending E.D. Hirsch s criticism of Gadamer. Uncovering the shortcomings in the replies of Gadamer s followers to Hirsch s criticism serves as a basis for the defense of intentionalism in interpretation carried out in the chapter. That defense is then extended with a discussion of some recent hermeneutic readings of Davidson s views. Chapter four deals with the standing of intentionalism through Rorty s pragmatist approach to literature. By indicating the position of pragmatist notions of aesthetic experience and imagination in Davidsonian intentionalism, it is shown that an intentionalist approach need not be as impoverished with regard to the value Rorty attributes to literature as he assumes. The concluding chapter outlines some ways in which one can be a pluralist with regard to art and interpretation without falling into relativism.

Logic as the Universal Science : Bertrand Russell's Early Conception of Logic and Its Philosophical Context

Bertrand Russell (1872 1970) introduced the English-speaking philosophical world to modern, mathematical logic and foundational study of mathematics. The present study concerns the conception of logic that underlies his early logicist philosophy of mathematics, formulated in The Principles of Mathematics (1903). In 1967, Jean van Heijenoort published a paper, Logic as Language and Logic as Calculus, in which he argued that the early development of modern logic (roughly the period 1879 1930) can be understood, when considered in the light of a distinction between two essentially different perspectives on logic. According to the view of logic as language, logic constitutes the general framework for all rational discourse, or meaningful use of language, whereas the conception of logic as calculus regards logic more as a symbolism which is subject to reinterpretation. The calculus-view paves the way for systematic metatheory, where logic itself becomes a subject of mathematical study (model-theory). Several scholars have interpreted Russell s views on logic with the help of the interpretative tool introduced by van Heijenoort,. They have commonly argued that Russell s is a clear-cut case of the view of logic as language. In the present study a detailed reconstruction of the view and its implications is provided, and it is argued that the interpretation is seriously misleading as to what he really thought about logic. I argue that Russell s conception is best understood by setting it in its proper philosophical context. This is constituted by Immanuel Kant s theory of mathematics. Kant had argued that purely conceptual thought basically, the logical forms recognised in Aristotelian logic cannot capture the content of mathematical judgments and reasonings. Mathematical cognition is not grounded in logic but in space and time as the pure forms of intuition. As against this view, Russell argued that once logic is developed into a proper tool which can be applied to mathematical theories, Kant s views turn out to be completely wrong. In the present work the view is defended that Russell s logicist philosophy of mathematics, or the view that mathematics is really only logic, is based on what I term the Bolzanian account of logic . According to this conception, (i) the distinction between form and content is not explanatory in logic; (ii) the propositions of logic have genuine content; (iii) this content is conferred upon them by special entities, logical constants . The Bolzanian account, it is argued, is both historically important and throws genuine light on Russell s conception of logic.

Constructive Perspectives on Inductive Logic

Constructive (intuitionist, anti-realist) semantics has thus far been lacking an adequate concept of truth in infinity concerning factual (i.e., empirical, non-mathematical) sentences. One consequence of this problem is the difficulty of incorporating inductive reasoning in constructive semantics. It is not possible to formulate a notion for probable truth in infinity if there is no adequate notion of what truth in infinity is. One needs a notion of a constructive possible world based on sensory experience. Moreover, a constructive probability measure must be defined over these constructively possible empirical worlds. This study defines a particular kind of approach to the concept of truth in infinity for Rudolf Carnap's inductive logic. The new approach is based on truth in the consecutive finite domains of individuals. This concept will be given a constructive interpretation. What can be verifiably said about an empirical statement with respect to this concept of truth, will be explained, for which purpose a constructive notion of epistemic probability will be introduced. The aim of this study is also to improve Carnap's inductive logic. The study addresses the problem of justifying the use of an "inductivist" method in Carnap's lambda-continuum. A correction rule for adjusting the inductive method itself in the course of obtaining evidence will be introduced. Together with the constructive interpretation of probability, the correction rule yields positive prior probabilities for universal generalizations in infinite domains.

Leikin asia : Näkökulmia varhaisnuorten romanttiseen seurustelukulttuuriin

The subject of my research is the romantic dating culture, the practice of 'going with', among preadolescents ('tweens') in Finland during the 1990s. Preadolescence is a cultural construction of the post-industrial period, experienced by school students between the ages of 7 to 13. Deemed by researchers as a shallow, unchallenging and uninteresting period, it has been shadowed in previous studies by early childhood and puberty. This study combines paradigms of the folkloristic research of children's lore, which began in the 1970s, with those of later, turn-of-the-century girls study. The phenomena of romantic girl culture are studied in several ways, through ample and varied subject materials collected in different places at different times. The research material was collected directly from schoolchildren through interviews, questionnaires and the observations of preadolescents' behavior in discos, among other methods. Part of the material consists of reminiscent thematic writings and parts have been quoted from tween message boards. A general picture of romantic preadolescent dating culture is formed in this study from five previously published articles and a summary. The influence of western culture, with its respect for relationships, is evident in tween dating culture. Seven- to thirteen-year olds use the elements of the society around them to construct an appropriate way for themselves to 'go out' with someone. Many expressions in preadolescent dating culture are contrary to the models of adult relationships. For example, a couple isn't necessarily expected to meet each other even once, or the other party, the boy, doesn't even need to know he's dating someone. Girls organize and experience relationships by playing card fortune-telling, calculating 'Love Percentages', and other methods. Categorizing tween dating culture and its related emotional qualities from an adult point of view as simply a play is one example of the hierarchical system of generations where childhood emotions, actions and conceptions of reality aren't valued as highly as the 'real life' of adults. Lowest on the totem pole are little girls, who in this study get their voices backed up by the researcher's adulthood and research-based sisterhood. Keywords: childhood, children's lore, dating culture, girls and boys, girls study, fortune-telling games, preadolescence/tweens

The One and The Many : In the works of Heraclitus, Parmenides and Lao Zi

This paper concentrates on Heraclitus, Parmenides and Lao Zi. The focus is on their ideas on change and whether the world is essentially One or if it is composed of many entities. In the first chapter I go over some general tendences in Greek and Chinese philosophy. The differences in the cultural background have an influence in the ways philosophy is made, but the paper aims to show that two questions can be brought up when comparing the philosophies of Heraclitus, Parmenides and Lao Zi. The questions are; is the world essentially One or Many? Is change real and if it is, what is the nature of it and how does it take place? For Heraclitus change is real, and as will be shown later in the chapter, quite essential for the sustainability of the world-order (kosmos). The key-concept in the case of Heraclitus is Logos. Heraclitus uses Logos in several senses, most well known relating to his element-theory. But another important feature of the Logos, the content of real wisdom, is to be able to regard everything as one. This does not mean that world is essentially one for Heraclitus in the ontological sense, but that we should see the underlying unity of multiple phenomena. Heraclitus regards this as hen panta: All from One, One from All. I characterize Heraclitus as epistemic monist and an ontological pluralist. It is plausible that the views of Heraclitus on change were the focus of Parmenides’ severe criticism. Parmenides held the view that the world is essentially one and that to see it as consisting of many entities was the error of mortals, i.e. the common man and his philosophical predecessors. For Parmenides what-is, can be approached by two routes; The Way of Truth (Aletheia) and The Way of Seeming (Doxa). Aletheia essentially sees the world as one, where even time is an illusion. In Doxa Parmenides is giving an explanation of the world seen as consisting of many entities and this is his contribution to the line of thought of his predecessors. It should be noted that a strong emphasis is given to the Aletheia, whereas the world-view given is in Doxa is only probable. I go on to describe Parmenides as ontological monist, who gives some plausibility to pluralistic views. In the work of Lao Zi world can be seen as One or as consisting of Many entities. In my interpretation, Lao Zi uses Dao in two different senses; Dao is the totality of things or the order in change. The wu-aspect (seeing-without-form) attends the world as one, whereas the you-aspect attends the world of many entities. In wu-aspect, Dao refers to the totality of things, when in you-aspect Dao is the order or law in change. There are two insights in Lao Zi regarding the relationship between wu- and- you-apects; in ch.1 it is stated that they are two separate aspects in seeing the world, the other chapters regarding that you comes from wu. This naturally brings in the question whether the One is the peak of seeing the world as many. In other words, is there a way from pluralism to monism. All these considerations make it probable that the work attributed to Lao Zi has been added new material or is a compilation of oral sayings. In the end of the paper I will go on to give some insights on how Logos and Dao can be compared in a relevant manner. I also compare Parmenides holistic monism to Lao Zi’s Dao as nameless totality (i.e. in its wu-aspect). I briefly touch the issues of Heidegger and the future of comparative philosophy.

Tacit Pedagogical Knowing: At the Core of Teacher's Professionality

The purpose of this research was to examine class teachers' interactive pedagogical thinking and action, in other words their tacit pedagogical knowing. Tacit pedagogical knowing was defined as a process in interactive teaching situation, through which a teacher finds solutions to surprising and challenging situations, pedagogical moments, so that the lesson continues. Teachers are able to describe their tacit pedagogical knowing afterwards and also find some reasons for it as well. More specifically, the aim was to study, 1) how does a class teacher's tacit pedagogical knowing appear in teacher's actions, and 2) what kinds of contents include in class teacher's tacit pedagogical knowing. The research material was gathered from four class teachers by videotaping their lessons and by stimulated recall interviews. In addition to this, the researcher spent a relatively long time in the research participants' classrooms. She conducted initial interviews and orientating observations by means of participant observation in order to get to know the participants and their contexts better. A phenomenologically oriented approach, which proceeded by following abductive logic, was used in the analysis procedures of the videotaped and stimulated recall data. In addition to this, correlation examinations were used in the validation of stimulated recall data analyses. The appearance of the tacit pedagogical knowing was observed in the videotaped data. The contents of tacit pedagogical knowing were defined by the analyses of stimulated recall data. According to the research results, a class teacher's tacit pedagogical knowing appears in the maintenance of the pedagogical relation, the teacher's relation to content, and the didactical relation. The contents of class teacher's tacit pedagogical knowing were many sided. The maintenance of the pedagogical relation, the teacher's relation to content, and the didactical relation were elements of the contents as well. In addition to these, the maintenance of teacher's pedagogical authority, the maintenance of the student's role or pedagogical authority, and the awareness of the nature of the content of instruction are included in the contents of teacher's tacit pedagogical knowing. The phenomenon of tacit pedagogical knowing was observed to be clearly a process-like and relational phenomenon. Based on the research results, a model of teacher's tacit pedagogical knowing was developed. Using the model, it is possible to illustrate the factors that are at the core of teacher's professionality. This model could be used in the context of teacher education, supervision, or in-service training.

Aluesuunnittelu ja kestävä kehitys Luoteis-Venäjällä : Ympäristösuunnittelun mahdollisuus paikallisissa hallintatavoissa

In this Ph.D. thesis I have studied how the objectives of sustainable development have been integrated into Northwest Russian urban and regional planning, and how the Russian planning discourse has changed after the collapse of the Soviet Union. By analysing the planning discussion, processes, and strategic documents I have also investigated the use of power and governmentality in urban and regional planning. As a methodological foundation I have used an approach that I call geographical constructivism . It was possible to answer in a relevant manner the question of how sustainable development has become a part of planning in Northwest Russia through a discourse analysis of the planning discussion. During the last decades, the aim of sustainable development has become globally one of the most central societal challenges. Urban and regional planning has a central role to play in promoting this process, since many meta-level objectives actually take shape within its sphere. An ever more actual challenge brought by sustainable development is to plan regions and places while balancing the conflicts of the pressures of safeguarding a good environment and of taking into consideration social and economic needs. I have given these unavoidable conflicts of sustainable development a central place in my work. In my view, complementing instrumental and communicative rationality with conflict rationality gives environmental planning a well-equipped toolbox. Sustainable development can be enhanced in urban and regional planning by seeking open, and especially hidden, potential conflicts. Thus, the expressed thinking (mentality) and actions taken by power regimes in and around conflicts open an interesting viewpoint into Northwest Russian governmentality. I examine the significance of sustainable development in planning through Northwest Russian geography, and also through recent planning legislation and four case studies. In addition, I project my analysis of empirical material onto the latest discussion of planning theory. My four case studies, which are based on independent and separate empirical material (42 thematic interviews and planning documents), consider the republics of Karelia and Komi, Leningrad oblast and the city of Saint Petersburg. In the dissertation I argue how sustainable development is, in the local governmentalities of Northwest Russia, understood as a concept where solving environmental problems is central, and that they can be solved through planning carried out by the planning professionals. Despite this idealism, environmental improvements have been overlooked by appealing to difficult economic factors. This is what I consider environmental racism, which I think is the most central barrier to sustainable development in Northwest Russia. The situation concerning the social dimension of sustainable development is even more difficult, since, for example, the development of local democracy is not highly valued. In the planning discourse this democracy racism is explained by a short history of democracy in Russia. However, precisely through planning conflicts, for example in St. Petersburg, planning has become socially more sustainable: protests by local inhabitants have bypassed the poorly functioning representational democracy, when the governmentality has changed from a mute use of power to one that adopts a stand on a conflicting issue. Keywords: Russia, urban and regional planning, sustainable development, environmental planning, power and conflicts in planning, governmentality, rationalities.

Dependence Logic: Investigations into Higher-Order Semantics Defined on Teams

This thesis is a study of a rather new logic called dependence logic and its closure under classical negation, team logic. In this thesis, dependence logic is investigated from several aspects. Some rules are presented for quantifier swapping in dependence logic and team logic. Such rules are among the basic tools one must be familiar with in order to gain the required intuition for using the logic for practical purposes. The thesis compares Ehrenfeucht-Fraïssé (EF) games of first order logic and dependence logic and defines a third EF game that characterises a mixed case where first order formulas are measured in the formula rank of dependence logic. The thesis contains detailed proofs of several translations between dependence logic, team logic, second order logic and its existential fragment. Translations are useful for showing relationships between the expressive powers of logics. Also, by inspecting the form of the translated formulas, one can see how an aspect of one logic can be expressed in the other logic. The thesis makes preliminary investigations into proof theory of dependence logic. Attempts focus on finding a complete proof system for a modest yet nontrivial fragment of dependence logic. A key problem is identified and addressed in adapting a known proof system of classical propositional logic to become a proof system for the fragment, namely that the rule of contraction is needed but is unsound in its unrestricted form. A proof system is suggested for the fragment and its completeness conjectured. Finally, the thesis investigates the very foundation of dependence logic. An alternative semantics called 1-semantics is suggested for the syntax of dependence logic. There are several key differences between 1-semantics and other semantics of dependence logic. 1-semantics is derived from first order semantics by a natural type shift. Therefore 1-semantics reflects an established semantics in a coherent manner. Negation in 1-semantics is a semantic operation and satisfies the law of excluded middle. A translation is provided from unrestricted formulas of existential second order logic into 1-semantics. Also game theoretic semantics are considerd in the light of 1-semantics.

Locality and order-invariant logics

Malli on logiikassa käytetty abstraktio monille matemaattisille objekteille. Esimerkiksi verkot, ryhmät ja metriset avaruudet ovat malleja. Äärellisten mallien teoria on logiikan osa-alue, jossa tarkastellaan logiikkojen, formaalien kielten, ilmaisuvoimaa malleissa, joiden alkioiden lukumäärä on äärellinen. Rajoittuminen äärellisiin malleihin mahdollistaa tulosten soveltamisen teoreettisessa tietojenkäsittelytieteessä, jonka näkökulmasta logiikan kaavoja voidaan ajatella ohjelmina ja äärellisiä malleja niiden syötteinä. Lokaalisuus tarkoittaa logiikan kyvyttömyyttä erottaa toisistaan malleja, joiden paikalliset piirteet vastaavat toisiaan. Väitöskirjassa tarkastellaan useita lokaalisuuden muotoja ja niiden säilymistä logiikkoja yhdistellessä. Kehitettyjä työkaluja apuna käyttäen osoitetaan, että Gaifman- ja Hanf-lokaalisuudeksi kutsuttujen varianttien välissä on lokaalisuuskäsitteiden hierarkia, jonka eri tasot voidaan erottaa toisistaan kasvavaa dimensiota olevissa hiloissa. Toisaalta osoitetaan, että lokaalisuuskäsitteet eivät eroa toisistaan, kun rajoitutaan tarkastelemaan äärellisiä puita. Järjestysinvariantit logiikat ovat kieliä, joissa on käytössä sisäänrakennettu järjestysrelaatio, mutta sitä on käytettävä siten, etteivät kaavojen ilmaisemat asiat riipu valitusta järjestyksestä. Määritelmää voi motivoida tietojenkäsittelyn näkökulmasta: vaikka ohjelman syötteen tietojen järjestyksellä ei olisi odotetun tuloksen kannalta merkitystä, on syöte tietokoneen muistissa aina jossakin järjestyksessä, jota ohjelma voi laskennassaan hyödyntää. Väitöskirjassa tutkitaan minkälaisia lokaalisuuden muotoja järjestysinvariantit ensimmäisen kertaluvun predikaattilogiikan laajennukset yksipaikkaisilla kvanttoreilla voivat toteuttaa. Tuloksia sovelletaan tarkastelemalla, milloin sisäänrakennettu järjestys lisää logiikan ilmaisuvoimaa äärellisissä puissa.

On the Rademacher maximal function

Tools known as maximal functions are frequently used in harmonic analysis when studying local behaviour of functions. Typically they measure the suprema of local averages of non-negative functions. It is essential that the size (more precisely, the L^p-norm) of the maximal function is comparable to the size of the original function. When dealing with families of operators between Banach spaces we are often forced to replace the uniform bound with the larger R-bound. Hence such a replacement is also needed in the maximal function for functions taking values in spaces of operators. More specifically, the suprema of norms of local averages (i.e. their uniform bound in the operator norm) has to be replaced by their R-bound. This procedure gives us the Rademacher maximal function, which was introduced by Hytönen, McIntosh and Portal in order to prove a certain vector-valued Carleson's embedding theorem. They noticed that the sizes of an operator-valued function and its Rademacher maximal function are comparable for many common range spaces, but not for all. Certain requirements on the type and cotype of the spaces involved are necessary for this comparability, henceforth referred to as the “RMF-property”. It was shown, that other objects and parameters appearing in the definition, such as the domain of functions and the exponent p of the norm, make no difference to this. After a short introduction to randomized norms and geometry in Banach spaces we study the Rademacher maximal function on Euclidean spaces. The requirements on the type and cotype are considered, providing examples of spaces without RMF. L^p-spaces are shown to have RMF not only for p greater or equal to 2 (when it is trivial) but also for 1 < p < 2. A dyadic version of Carleson's embedding theorem is proven for scalar- and operator-valued functions. As the analysis with dyadic cubes can be generalized to filtrations on sigma-finite measure spaces, we consider the Rademacher maximal function in this case as well. It turns out that the RMF-property is independent of the filtration and the underlying measure space and that it is enough to consider very simple ones known as Haar filtrations. Scalar- and operator-valued analogues of Carleson's embedding theorem are also provided. With the RMF-property proven independent of the underlying measure space, we can use probabilistic notions and formulate it for martingales. Following a similar result for UMD-spaces, a weak type inequality is shown to be (necessary and) sufficient for the RMF-property. The RMF-property is also studied using concave functions giving yet another proof of its independence from various parameters.