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.

Boundedness of weakly singular integral operators on domains

The monograph dissertation deals with kernel integral operators and their mapping properties on Euclidean domains. The associated kernels are weakly singular and examples of such are given by Green functions of certain elliptic partial differential equations. It is well known that mapping properties of the corresponding Green operators can be used to deduce a priori estimates for the solutions of these equations. In the dissertation, natural size- and cancellation conditions are quantified for kernels defined in domains. These kernels induce integral operators which are then composed with any partial differential operator of prescribed order, depending on the size of the kernel. The main object of study in this dissertation being the boundedness properties of such compositions, the main result is the characterization of their Lp-boundedness on suitably regular domains. In case the aforementioned kernels are defined in the whole Euclidean space, their partial derivatives of prescribed order turn out to be so called standard kernels that arise in connection with singular integral operators. The Lp-boundedness of singular integrals is characterized by the T1 theorem, which is originally due to David and Journé and was published in 1984 (Ann. of Math. 120). The main result in the dissertation can be interpreted as a T1 theorem for weakly singular integral operators. The dissertation deals also with special convolution type weakly singular integral operators that are defined on Euclidean spaces.

Uso del pronombre personal sujeto de la primera persona del singular en español y portugués hablados: factores semánticos y pragmáticos

Tutkimuksen aihe on subjektipronominin ei-pakollinen käyttö finiittisten verbimuotojen yhteydessä espanjan ja portugalin kielessä. Tutkimuskohteena ovat yksikön ensimmäisen persoonan verbimuodot Espanjassa ja Portugalissa kerätyissä puhekielen korpuksissa. Tutkimuksen tarkoitus on selvittää, mitkä semanttiset ja pragmaattiset tekijät vaikuttavat subjektipronominin ei-pakollisen käytön yleisyyteen ja mitä systemaattisia eroja subjektipronominin käytössä on espanjan ja portugalin välillä. Tutkimus kuuluu korpuslingvistiikan alaan ja ensisijaisena tutkimusmetodina on kvantitatiivinen vertailu. Tutkimus osoittaa, että yksikön ensimmäisen persoonan subjektipronominin ei-pakollinen käyttö on käytännössä kaikissa konteksteissa yleisempää portugalissa kuin espanjassa. Tätä eroa voidaan selittää kielten konstituenttirakenteen typologisella erilaisuudella. Subjektin semanttinen rooli on tutkimuksen perusteella sidoksissa subjektipronominin käyttöön enemmän espanjassa kuin portugalissa, mutta kummassakaan kielessä subjektipronominin käyttöä ei voida selittää pelkästään subjektin semanttisella roolilla. Molemmissa kielissä samanviitteisyys edellisen subjektin kanssa vähentää subjektipronominin käyttöä, kun taas subjektipronominin ei-referentiaalinen käyttö ja toisaalta verbin ilmaiseman toiminnan irreaalisuus lisäävät sitä. Tutkimustulokset antavat aihetta lisätutkimukseen pronominien ja verbien ei-referentiaalisesta ja irreaalisesta käytöstä espanjassa ja portugalissa sekä typologi-seen tutkimukseen subjektipronominien käyttöön vaikuttavista tekijöistä eri kielissä.

Interactions of natural products with β-lactoglobulins, members of the lipocalin family

Natural products constitute an important source of new drugs. The bioavailability of the drugs depends on their absorption, distribution, metabolism and elimination. To achieve good bioavailability, the drug must be soluble in water, stable in the gastrointestinal tract and palatable. Binding proteins may improve the solubility of drug compounds, masking unwanted properties, such as bad taste, bitterness or toxicity, transporting or protecting these compounds during processing and storage. The focus of this thesis was to study the interactions, including ligand binding and the effect of pH and temperature, of bovine and reindeer β-lactoglobulin (βLG) with such compounds as retinoids, phenolic compounds as well as with compounds from plant extracts, and to investigate the transport properties of the βLG-ligand complex. To examine the binding interactions of different ligands to βLG, new methods were developed. The fluorescence binding method for the evaluation of ligand binding to βLG was miniaturized from a quartz cell to a 96-well plate. A method of ultrafiltration sampling combined with high-performance liquid chromatography was developed to assess the binding of compounds from extracts. The interactions of phenolic compounds or retinoids and βLG were investigated using the 96-well plate method. The majority of flavones, flavonols, flavanones and isoflavones and all of the retinoids included were shown to bind to bovine and reindeer βLG. Phenolic compounds, contrary to retinol, were not released at acidic pH. Those results suggest that βLG may have more binding sites, probably also on the surface of βLG. An extract from Camellia sinensis (L.) O. Kunze (black tea), Urtica dioica L. (nettle) and Piper nigrum (black pepper) were used to evaluate whether βLG could bind compounds from plant extracts. Piperine from P. nigrum was found to bind tightly and rutin from U. dioica weakly to βLG. No components from C. sinensis bound to βLG in our experiment. The uptake and membrane permeation of bovine and reindeer βLG, free and bound with retinol, palmitic acid and cholesterol, were investigated using Caco-2 cell monolayers. Both bovine and reindeer βLG were able to cross the Caco-2 cell membrane. Free and βLG-bound retinol and palmitic acid were transported equally, whereas cholesterol could not cross the Caco-2 cell monolayer free or bound to βLG. Our results showed that βLG can bind different natural product compounds, but cannot enhance transport of retinol, palmitic acid or cholesterol through Caco-2 cells. Despite this, βLG, as a water-soluble binding protein, may improve the solubility of natural compounds, possibly protecting them from early degradation and transporting some of them through the stomach. Furthermore, it may decrease their bad or bitter taste during oral administration of drugs or in food preparations. βLG can also enhance or decrease the health benefits of herbal teas and food preparations by binding compounds from extracts.

Agreement Patterns in English : Diachronic Corpus Studies on Common-Number Pronouns

This study reports a diachronic corpus investigation of common-number pronouns used to convey unknown or otherwise unspecified reference. The study charts agreement patterns in these pronouns in various diachronic and synchronic corpora. The objective is to provide base-line data on variant frequencies and distributions in the history of English, as there are no previous systematic corpus-based observations on this topic. This study seeks to answer the questions of how pronoun use is linked with the overall typological development in English and how their diachronic evolution is embedded in the linguistic and social structures in which they are used. The theoretical framework draws on corpus linguistics and historical sociolinguistics, grammaticalisation, diachronic typology, and multivariate analysis of modelling sociolinguistic variation. The method employs quantitative corpus analyses from two main electronic corpora, one from Modern English and the other from Present-day English. The Modern English material is the Corpus of Early English Correspondence, and the time frame covered is 1500-1800. The written component of the British National Corpus is used in the Present-day English investigations. In addition, the study draws supplementary data from other electronic corpora. The material is used to compare the frequencies and distributions of common-number pronouns between these two time periods. The study limits the common-number uses to two subsystems, one anaphoric to grammatically singular antecedents and one cataphoric, in which the pronoun is followed by a relative clause. Various statistical tools are used to process the data, ranging from cross-tabulations to multivariate VARBRUL analyses in which the effects of sociolinguistic and systemic parameters are assessed to model their impact on the dependent variable. This study shows how one pronoun type has extended its uses in both subsystems, an increase linked with grammaticalisation and the changes in other pronouns in English through the centuries. The variationist sociolinguistic analysis charts how grammaticalisation in the subsystems is embedded in the linguistic and social structures in which the pronouns are used. The study suggests a scale of two statistical generalisations of various sociolinguistic factors which contribute to grammaticalisation and its embedding at various stages of the process.

Problems of Proto-Slavic Historical Nominal Morphology : On the Basis of Old Church Slavic

In this study I offer a diachronic solution for a number of difficult inflectional endings in Old Church Slavic nominal declensions. In this context I address the perhaps most disputed and the most important question of the Slavic nominal inflectional morphology: whether there was in Proto-Slavic an Auslautgesetz (ALG), a law of final syllables, that narrowed the Proto-Indo-European vowel */o/ to */u/ in closed word-final syllables. In addition, the work contains an exhaustive morphological classification of the nouns and adjectives that occur in canonical Old Church Slavic. I argue that Proto-Indo-European */o/ became Proto-Slavic */u/ before word-final */s/ and */N/. This conclusion is based on the impossibility of finding credible analogical (as opposed to phonological) explanations for the forms supporting the ALG hypothesis, and on the survival of the neuter gender in Slavic. It is not likely that the */o/-stem nominative singular ending */-u/ was borrowed from the accusative singular, because the latter would have been the only paradigmatic form with the stem vowel */-u-/. It is equally unlikely that the ending */-u/ was borrowed from the */u/-stems, because the latter constituted a moribund class. The usually stated motivation for such an analogical borrowing, i.e. a need to prevent the merger of */o/-stem masculines with neuters of the same class, is not tenable. Extra-Slavic, as well as intra-Slavic evidence suggests that phonologically-triggered mergers between two semantically opaque genders do not tend to be prevented, but rather that such mergers lead to the loss of the gender opposition in question. On the other hand, if */-os/ had not become */-us/, most nouns and, most importantly, all adjectives and pronouns would have lost the formal distinction between masculines and neuters. This would have necessarily resulted in the loss of the neuter gender. A new explanation is given for the most apparent piece of evidence against the ALG hypothesis, the nominative-accusative singular of the */es/-stem neuters, e.g. nebo 'sky'. I argue that it arose in late Proto-Slavic dialects, replacing regular nebe, under the influence of the */o/- and */yo/-stems where a correlation had emerged between a hard root-final consonant and the termination -o, on the one hand, and a soft root-final consonant and the termination -e, on the other.

Vyra enie obob čёnno-ličnogo značenija v russkom jazyke

Expressing generalized-personal meaning in Russian Based on data from Russian, this doctoral dissertation examines generalized-personal meaning that is, generic expressions referring to all human beings, people in general, each or any person (e.g. S vozrastom načinae cenit prostye ve či With age you start to appreciate simple things ). The study shares its basic theoretical orientation with functional approaches going from meaning to form . The objective of the thesis is to determine and describe the various linguistic means which can be used by the speaker to express generalized-personal meaning. The main material of the study consists of 2,000 examples collected from modern Russian literature, newspapers, and magazines. The linguistic means of expressing generalized-personal meaning are divided into three main classes. Morphological and lexico-grammatical means (22% of the material) include the use of personal pronouns and personal verbal endings. In Russian, all personal forms except the 3rd person singular can be used in a generalized-personal meaning. Lexical means (14% of the material) involve, above all, pronouns like vse all , ka dyj everyone , nikto no one , as well as the nouns čelovek man and ljudi people . In emotional speech, generalized-personal meaning can also be conveyed lexically by using utterances like da e idiot znaet even an idiot knows . In rhetorical questions the pronoun kto who can appear in this meaning (cf. Kto ne ljubit moro enoe?! Who doesn t like ice cream?! ). The third main class, syntactic means (64% of the material), consists of constructions in which the generic person is not expressed at the surface level. This class mainly includes two-component structures in which the infinitive relates to a modal predicative adverb (e.g. mo no can, be allowed to , nado must ), modal verb (e.g. stoit be worth(while) , sleduet must, be obliged to ), or predicative adverb ending in -о (e.g. trudno it is hard to , neprilično is not appropriate ). Other syntactic means are: one-component infinitive structures, so-called embedded structures, structures with a processual noun, passive constructions, and gerund constructions. The different forms of expression available in Russian are not interchangeable in all contexts. Even if a given context tolerates the substitution of one construction for another, the two expressions are never entirely synonymous. In addition to determining the range of forms which can express generalized-personal meaning, the study aims to compare these forms and to specify the conditions and possible restrictions (contextual, semantic, syntactic, stylistic, etc.) associated with the use of each construction. In Russian linguistics, the generalized-personal meaning has not been extensively studied from a functional perspective. The advantage of a meaning-based functional approach is that it gives a comprehensive picture of the diversity and distribution of the phenomenon.

Word Sense Discovery and Disambiguation

The work is based on the assumption that words with similar syntactic usage have similar meaning, which was proposed by Zellig S. Harris (1954,1968). We study his assumption from two aspects: Firstly, different meanings (word senses) of a word should manifest themselves in different usages (contexts), and secondly, similar usages (contexts) should lead to similar meanings (word senses). If we start with the different meanings of a word, we should be able to find distinct contexts for the meanings in text corpora. We separate the meanings by grouping and labeling contexts in an unsupervised or weakly supervised manner (Publication 1, 2 and 3). We are confronted with the question of how best to represent contexts in order to induce effective classifiers of contexts, because differences in context are the only means we have to separate word senses. If we start with words in similar contexts, we should be able to discover similarities in meaning. We can do this monolingually or multilingually. In the monolingual material, we find synonyms and other related words in an unsupervised way (Publication 4). In the multilingual material, we ?nd translations by supervised learning of transliterations (Publication 5). In both the monolingual and multilingual case, we first discover words with similar contexts, i.e., synonym or translation lists. In the monolingual case we also aim at finding structure in the lists by discovering groups of similar words, e.g., synonym sets. In this introduction to the publications of the thesis, we consider the larger background issues of how meaning arises, how it is quantized into word senses, and how it is modeled. We also consider how to define, collect and represent contexts. We discuss how to evaluate the trained context classi?ers and discovered word sense classifications, and ?nally we present the word sense discovery and disambiguation methods of the publications. This work supports Harris' hypothesis by implementing three new methods modeled on his hypothesis. The methods have practical consequences for creating thesauruses and translation dictionaries, e.g., for information retrieval and machine translation purposes. Keywords: Word senses, Context, Evaluation, Word sense disambiguation, Word sense discovery.

At the Limits of Presentation : Coming-into-Presence and its Aesthetic Relevance in Jean-Luc Nancy's Philosophy

This study investigates the significance of art in Jean-Luc Nancy s philosophy. I argue that the notion of art contributes to some of Nancy s central ontological ideas. Therefore, I consider art s importance in its own right whether art does have ontological significance, and if so, how one should describe this with respect to the theme of presentation. According to my central argument, with his thinking on art Nancy attempts to give one viewpoint to what is called the metaphysics of presence and to its deconstruction. On which grounds, as I propose, may one say that art is not reducible to philosophy? The thesis is divided into two main parts. The first part, Presentation as a Philosophical Theme, is a historical genesis of the central concepts associated with the birth of presentation in Nancy s philosophy. I examine this from the viewpoint of the differentiation between the ontological notions of presentation and representation by concentrating on the influence of Martin Heidegger and Jacques Derrida, as well as of Hegel and Kant. I give an overview of the way in which being or sense for Nancy is to be described as a coming-into-presence or presentation . Therefore, being takes place in its singular plurality. I argue that Nancy redevelops Heidegger s account of being in two principal ways: first, in rethinking the ontico-ontological difference, and secondly, by striving to radicalize the Heideggerian concept of Mitsein, being-with . I equally wish to show the importance of Derrida s notion of différance and its inherence in Nancy s questioning of being that rests on the unfoundedness of existence. The second part, From Ontology to Art, draws on the importance of art and the aesthetic. If, in Nancy, the question of art touches upon its own limit as the limit of nothingness, how is art able to open its own strangeness and our exposure to this strangeness? My aim is to investigate how Nancy s thinking on art finds its place within the conceptual realm of its inherent difference and interval. My central concern is the thought of originary ungroundedness and the plurality of art and of the arts. As for the question of the difference between art and philosophy, I wish to show that what differentiates art from thought is the fact that art exposes what is obvious but not apparent, if apparent is understood in the sense of givenness. As for art s ability to deconstruct Nancy s ontological notions, I suggest that in question in art is its original heterogeneity and diversity. Art is a matter of differing art occurs singularly, as a local difference. With this in mind, I point out that in reflecting on art in terms of spacing and interval, as a thinker of difference Nancy comes closer to Derrida and his idea of différance than to the structure of Heidegger s ontological difference.

Poetics of Subjectivity : Existence and Expressivity in Simone de Beauvoir's Philosophy

In the last thirty years, primarily feminist scholars have drawn attention to and re-evaluated the philosophy of Simone de Beauvoir (1908 1986). Her philosophical practice has been described as non-systematic, and her literary writing has been viewed as part of her non-systematic mode of philosophising. This dissertation radically deepens the question concerning Beauvoir s philosophical motivations for turning to literature as a mode to express subjectivity. It explicates the central concepts of Beauvoir s philosophy of existence, which are subjectivity, ambiguity, paradox and temporality, and their background in the modern traditions of existential philosophy and phenomenology. It also clarifies Beauvoir s main reason to turn to literature in order to express subjectivity as both singular and universal: as a specific mode of communication, literature is able to make the universality of existence manifest in the concrete, singular and temporal texture of life. In addition, the thesis gives examples of how Beauvoir s literary works contribute to an understanding of the complexity of subjectivity. I use the expression poetics of subjectivity to refer to the systematic relation between Beauvoir s existential and phenomenological notion of subjectivity and her literary works, and to her articulations of a creative mode of using language, especially in the novel. The thesis is divided into five chapters, of which the first three investigate Beauvoir s philosophy of existence at the intersection of the modern traditions of thought that began with René Descartes and Søren Kierkegaard s intuitions about subjectivity. Chapter 1 interprets Beauvoir s notion of ambiguity, as compared to paradox, and argues that both determine her notion of existence. Chapters 2 and 3 investigate the phenomenological side of Beauvoir s philosophy through a study of her response to early French interpretations of transcendental subjectivity, especially in the works of Jean-Paul Sartre and Maurice Merleau-Ponty. My analysis shows that Edmund Husserl s distinction between different levels of subjective experience is central to Beauvoir s understanding of subjectivity and to the different ego concepts she uses. Chapter 4 is a study of Beauvoir s reflections on the expression of subjective thought, and, more specifically, her philosophical conceptions of the metaphysical novel and the autobiography as two modes of indirect communication. Chapter 5, finally, compares two modes of investigating concrete subjectivity; Beauvoir s conceptual study of femininity in Le deuxième sexe and her literary expression of subjectivity in the novel L Invitée. My analysis reveals and explicates Beauvoir s original contribution to a comprehensive understanding of the becoming and paradox of human existence: the fundamental insight that these phenomena are sexed, historically as well as imaginatively.

Complicated Presence : The Unity of Being in Parmenides and Heidegger

The attempt to refer meaningful reality as a whole to a unifying ultimate principle - the quest for the unity of Being - was one of the basic tendencies of Western philosophy from its beginnings in ancient Greece up to Hegel's absolute idealism. However, the different trends of contemporary philosophy tend to regard such a speculative metaphysical quest for unity as obsolete. This study addresses this contemporary situation on the basis of the work of Martin Heidegger (1889-1976). Its methodological framework is Heidegger's phenomenological and hermeneutical approach to the history of philosophy. It seeks to understand, in terms of the metaphysical quest for unity, Heidegger's contrast between the first (Greek) beginning or "onset" (Anfang) of philosophy and another onset of thinking. This other onset is a possibility inherent in the contemporary situation in which, according to Heidegger, the metaphysical tradition has developed to its utmost limits and thereby come to an end. Part I is a detailed interpretation of the surviving fragments of the Poem of Parmenides of Elea (fl. c. 500 BC), an outstanding representative of the first philosophical beginning in Heidegger's sense. It is argued that the Poem is not a simple denial of apparent plurality and difference ("mortal acceptances," doxai) in favor of an extreme monism. Parmenides' point is rather to show in what sense the different instances of Being can be reduced to an absolute level of truth or evidence (aletheia), which is the unity of Being as such (to eon). What in prephilosophical human experience is accepted as being is referred to the source of its acceptability: intelligibility as such, the simple and undifferentiated presence to thinking that ultimately excludes unpresence and otherness. Part II interprets selected key texts from different stages in Heidegger's thinking in terms of the unity of Being. It argues that one aspect of Heidegger's sustained and gradually deepening philosophical quest was to think the unity of Being as singularity, as the instantaneous, context-specific, and differential unity of a temporally meaningful situation. In Being and Time (1927) Heidegger articulates the temporal situatedness of the human awareness of meaningful presence. His later work moves on to study the situational correlation between presence and the human awareness. Heidegger's "postmetaphysical" articulation seeks to show how presence becomes meaningful precisely as situated, in an event of differentiation from a multidimensional context of unpresence. In resigning itself to this irreducibly complicated and singular character of meaningful presence, philosophy also faces its own historically situated finitude. This resignation is an essential feature of Heidegger's "other onset" of thinking.

Ryhmäidentiteetin mittaaminen verkkoroolipelissä World of Warcraft

The objective of the study was to explore the dimensions of group identity in the guilds of World of Warcraft. Previous research shows that social interaction has an important role in playing games for many players. Social identities are an important aspect of self-concept and since group related cues are more salient than personal clues in computer-mediated communication, the social gaming experience was approached through group identity. In the study a new scale will be developed to measure the group identity in games. Secondary goal is to study how different guild attributes affect the group identity and third goal is to explore the connection between group identity and gaming experience and amount of play. Subjects were 1203 guild members and 106 players not in a guild. The data was gathered by an Internet survey which measured group identity with nine scales, gaming experience with three scales and guild attributes with four scales. Also various background data was gathered. The construct of group identity was analyzed with explorative factor analysis. The typical experiences of group identity was analyzed with cluster analysis and effects of guild attributes with multivariate analysis of covariance. As a result of the study a new scale was developed which measured group identity on six dimensions: self-stereotyping, public and private evaluation, importance, interconnection of self and others and awareness of content. Group identity was experienced strongest in elder middle-sized guilds that had formal rules and that emphasized social interaction. The players with strong group identity had more positive gaming experience and played World of Warcraft more per week than the players who were not in a guild or identified to guild weakly. This result encourages game developers to produce environments that enhance group identity as it seems to increase the enjoyment in games. As a whole this study proposes that group identity in guilds is constructed from the same elements as in traditional groups. If this is truly the case, guild membership may have similar positive effects on individual’s mental well-being as traditional positively evaluated group memberships have.

Noncommutative Quantum Field Theories and UV/IR Mixing

Our present-day understanding of fundamental constituents of matter and their interactions is based on the Standard Model of particle physics, which relies on quantum gauge field theories. On the other hand, the large scale dynamical behaviour of spacetime is understood via the general theory of relativity of Einstein. The merging of these two complementary aspects of nature, quantum and gravity, is one of the greatest goals of modern fundamental physics, the achievement of which would help us understand the short-distance structure of spacetime, thus shedding light on the events in the singular states of general relativity, such as black holes and the Big Bang, where our current models of nature break down. The formulation of quantum field theories in noncommutative spacetime is an attempt to realize the idea of nonlocality at short distances, which our present understanding of these different aspects of Nature suggests, and consequently to find testable hints of the underlying quantum behaviour of spacetime. The formulation of noncommutative theories encounters various unprecedented problems, which derive from their peculiar inherent nonlocality. Arguably the most serious of these is the so-called UV/IR mixing, which makes the derivation of observable predictions especially hard by causing new tedious divergencies, to which our previous well-developed renormalization methods for quantum field theories do not apply. In the thesis I review the basic mathematical concepts of noncommutative spacetime, different formulations of quantum field theories in the context, and the theoretical understanding of UV/IR mixing. In particular, I put forward new results to be published, which show that also the theory of quantum electrodynamics in noncommutative spacetime defined via Seiberg-Witten map suffers from UV/IR mixing. Finally, I review some of the most promising ways to overcome the problem. The final solution remains a challenge for the future.

Magmatic evolution of topaz-bearing granite stocks within the Wiborg rapakivi granite batholith

This study brings new insights into the magmatic evolution of natural F-enriched peraluminous granitic systems. The Artjärvi, Sääskjärvi and Kymi granite stocks within the 1.64 Ga Wiborg rapakivi granite batholith have been investigated by petrographic, geochemical, experimental and melt inclusion methods. These stocks represent late-stage leucocratic and weakly peraluminous intrusive phases typical of rapakivi granites worldwide. The Artjärvi and Sääskjärvi stocks are multiphase intrusions in which the most evolved phase is topaz granite. The Kymi stock contains topaz throughout and has a well-developed zoned structure, from the rim to the center: stockscheider pegmatite equigranular topaz granite porphyritic topaz granite. Geochemically the topaz granites are enriched in F, Li, Be, Ga, Rb, Sn and Nb and depleted in Mg, Fe, Ti, Ba, Sr, Zr and Eu. The anomalous geochemistry and mineralogy of the topaz granites are essentially magmatic in origin; postmagmatic reactions have only slightly modified the compositions. The Kymi equigranular topaz granite shows the most evolved character, and the topaz granites at Artjärvi and Sääskjärvi resemble the less evolved porphyritic topaz granite of the Kymi stock. Stockscheiders are found at the roof contacts of the Artjärvi and Kymi stocks. The stockscheider at Artjärvi is composed of biotite-rich schlieren and pegmatite layers parallel to the contact. The schlieren layering is considered to have formed by velocity-gradient sorting mechanism parallel to the flow, which led to the accumulation of mafic minerals along the upper contact of the topaz granite. Cooling and contraction of the topaz granite formed fractures parallel to the roof contact and residual pegmatite magmas were injected along the fractures and formed the pegmatite layers. The zoned structure of the Kymi stock is the result of intrusion of highly evolved residual melt from deeper parts of the magma chamber along the fractured contact between the porphyritic granite crystal mush and country rock. The equigranular topaz granite and marginal pegmatite (stockscheider) crystallized from this evolved melt. Phase relations of the Kymi equigranular topaz granite have been investigated utilizing crystallization experiments at 100 to 500 MPa as a function of water activity and F content. Fluorite and topaz can crystallize as liquidus phases in F-rich peraluminous systems, but the F content of the melt should exceed 2.5 - 3.0 wt % to facilitate crystallization of topaz. In peraluminous F-bearing melts containing more than 1 wt % F, topaz and muscovite are expected to be the first F-bearing phases to crystallize at high pressure, whereas fluorite and topaz should crystallize first at low pressure. Overall, the saturation of fluorite and topaz follows the reaction: CaAl2Si2O8 (plagioclase) + 2[AlF3]melt = CaF2 (fluorite) + 2Al2SiO4F2 (topaz). The obtained partition coefficient for F between biotite and glass D(F)Bt/glass is 1.89 to 0.80 (average 1.29) and can be used as an empirical fluormeter to determine the F content of coexisting melts. In order to study the magmatic evolution of the Kymi stock, crystallized melt inclusions in quartz and topaz grains in the porphyritic and the equigranular topaz granites and the marginal pegmatite were rehomogenized and analyzed. The homogenization conditions for the melt inclusions from the granites were 700 °C, 300 MPa, and 24 h, and for melt inclusions from the pegmatite, 700 °C, 100 MPa, and 24/96 h. The majority of the melt inclusions is chemically similar to the bulk rocks (excluding H2O content), but a few melt inclusions in the equigranular granite show clearly higher F and low K2O contents (on average 11.6 wt % F, 0.65 wt % K2O). The melt inclusion compositions indicate coexistence of two melt fractions, a prevailing peraluminous and a very volatile-rich, possibly peralkaline. Combined petrological, experimental and melt inclusion studies of the Kymi equigranular topaz granite indicate that plagioclase was the liquidus phase at nearly water-saturated (fluid-saturated) conditions and that the F content of the melt was at least 2 wt %. The early crystallization of biotite and the presence of muscovite in crystallization experiments at 200 MPa contrasts with the late-stage crystallization of biotite and the absence of muscovite in the equigranular granite, indicating that crystallization pressure may have been lower than 200 MPa for the granite.

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.