Quasiconformal mappings and inequalities involving special functions

This PhD thesis in Mathematics belongs to the field of Geometric Function Theory. The thesis consists of four original papers. The topic studied deals with quasiconformal mappings and their distortion theory in Euclidean n-dimensional spaces. This theory has its roots in the pioneering papers of F. W. Gehring and J. Väisälä published in the early 1960’s and it has been studied by many mathematicians thereafter. In the first paper we refine the known bounds for the so-called Mori constant and also estimate the distortion in the hyperbolic metric. The second paper deals with radial functions which are simple examples of quasiconformal mappings. These radial functions lead us to the study of the so-called p-angular distance which has been studied recently e.g. by L. Maligranda and S. Dragomir. In the third paper we study a class of functions of a real variable studied by P. Lindqvist in an influential paper. This leads one to study parametrized analogues of classical trigonometric and hyperbolic functions which for the parameter value p = 2 coincide with the classical functions. Gaussian hypergeometric functions have an important role in the study of these special functions. Several new inequalities and identities involving p-analogues of these functions are also given. In the fourth paper we study the generalized complete elliptic integrals, modular functions and some related functions. We find the upper and lower bounds of these functions, and those bounds are given in a simple form. This theory has a long history which goes back two centuries and includes names such as A. M. Legendre, C. Jacobi, C. F. Gauss. Modular functions also occur in the study of quasiconformal mappings. Conformal invariants, such as the modulus of a curve family, are often applied in quasiconformal mapping theory. The invariants can be sometimes expressed in terms of special conformal mappings. This fact explains why special functions often occur in this theory.

Exploring the role and transformative potential of human rights in development practice and food security : case study from Malawi

Avhandlingen är en analys av den roll som mänskliga rättigheter spelar i utvecklingssamarbete i allmänhet och tre matsäkerhetsprojekt i Malawi i synnerhet. Författaren undersöker huruvida mänskliga rättigheter kan bidra till samhälleliga förändringsprocesser. Undersökningen har en diskursiv syn på mänskliga rättigheter; rättigheter ses som sociala konstruktioner skapade av människor, konstruktioner som förändras i takt med att aktörerna ger dem ny mening. Människorättsbaserade utvecklingsstrategier riktar uppmärksamheten mot olika aktörers skyldigheter samt vikten av att mänskliga rättigheter inte kränks genom utvecklingspolitiska beslut. En analys som utgår från mänskliga rättigheter kan leda till att nya frågor lyfts fram: vem har skyldigheter, vem har rättigheter, och varför förverkligas inte dessa? Fokus flyttas till hur politiska, ekonomiska och juridiska strukturer kan förändras så att de främjar mänskliga rättigheter. Författaren använder empiriskt material från Malawi för att visa vilken roll mänskliga rättigheter och så kallade människorättsprinciper spelar inom tre projekt: (1) ett mathjälpsprojekt; (2) ett rättighetsbaserat projekt med fokus på att stärka strategier för livsuppehälle; (3) ett juridiskt projekt inom ramen för vilket man skapat ett lagförslag om rätten till föda. Analysen visar att det rättighetsbaserade projektet strävade efter den mest djupgående samhälleliga förändringen. Människorättsprinciperna ansvarighet och deltagande handlade inte endast om verksamheten inom projektet utan man strävade även efter att stärka dessa principer mellan medborgare och myndigheter. Rättigheter, som ett sätt att tala, tänka och handla, inte som abstrakta juridiska normer, påverkade de lokala aktörernas sätt att kräva bl.a. tjänster av lokala myndigheter. Teoretiskt kan man säga att det rättighetsbaserade projektet representerar ett aktörsperspektiv på mänskliga rättigheter där aktörernas egna uppfattningar om vad de är berättigade till samt krav på myndigheters ansvar formade vilken mening rättigheterna kom att få.

Palkanlaskentajärjestelmän kehittäminen

Työn tavoitteena oli selvittää palkanlaskennan nykytilaa ja sen uudistamisen mahdollisuuksia etenkin SAP HR-ohjelmiston valossa. Ensin selvitettiin palkanlaskentaan ja sovellushankintaan liittyviä vaatimuksia kirjallisuuden avulla. Tämän jälkeen tutustuttiin palkanlaskennan nykytilaan haastattelujen, prosessikuvausten ja sisäisten dokumenttien avulla. SAP HR-ohjelmistoon tutustuttiin siihen liittyneiden työtehtävien, erilaisten koulutusmateriaalien ja esittelyiden avulla. Yrityksessä on jo aiemmin yritetty korvata henkilöstö- ja palkanlaskentajärjestelmiä, mutta projekti keskeytettiin. SAP HR-ohjelmiston henkilöstöhallinto-osuutta ollaan ottamassa käyttöön koko yrityksen laajuudessa. Palkanlaskentajärjestelmä vaatii pikaista uusintaa vanhan teknologian ja suuren henkilöriippuvuuden takia. Nykyiset työajanhallinnan ja palkanlaskennan prosessit yrityksessä ovat tiukasti sidottuja nykyisten järjestelmien kanssa ja uuden järjestelmän käyttöönotto vaatii muutoksia koko ketjuun työvuorojen suunnittelusta palkanlaskentaan ja raportointiin asti. Johtopäätöksenä todetaan SAP HR:n palkanlaskennan soveltuvan yrityksen palkanlaskentaan ja tarjoavan monia uusia mahdollisuuksia. Suurimmat mahdollisuudet liittyvät kokonaisten prosessiketjujen läpinäkyvyyteen ja parempaan ohjailtavuuteen. Tämä kuitenkin vaatii nykyistä tarkempaa sisäisten prosessien selvittämistä, SAP HR-palkanlaskennan kyvykkyyden tarkempaa tutkimisia, vaihtoehtoisten ohjelmien vertailua sekä ennen kaikkea valmiutta muuttaa nykyisiä prosesseja.

Christina Rossetti's "Goblin market" : Milton revised or revived?

The present study examines Christina Rossetti's (1830-94) best-known poem "Goblin Market" (1859) as a response to John Milton's (1608-74) Paradise Lost (1667) and Paradise Regained (1674). Rossetti's poem relates to Milton's twin epics with a surprising degree of complexity, something that the thesis demonstrates through a large number of hitherto unexplored textual parallels. In its recontextualisation of Rossetti's poetry, this study balances interpretative strategies evolved by feminism with more historicist readings. SinceGilbert and Gubar's landmark feminist work of 1979, it has become something of a critical commonplace that Milton was a figure of patriarchal oppression whom women writers have challenged through revision. As a consequence, intertextual Rossetti studies have tended to emphasise such a revisionist element in her achievement. But if we place "Goblin Market" within the larger context of Rossetti's thoughts on religion and poetry throughout her writing, and if we de-emphasise the dichotomy between Rossetti and her male precursor, alternative patterns of reading begin to emerge. This thesis argues that Rossetti's aim was not only to revise Milton but also to revive him. It develops this point in several ways: first, by introducing the subject and theoretically arguing that literary allusiveness is not necessarily a matter of oedipal or feminist envy only (Chapter One); second, by determining the nature and extent of Rossetti's Miltonic allusions in terms of theme, imagery and language (Chapter Two);and third, by examining their function in terms of argument (Chapters Three andFour). I n an age of increasing religious doubt, Rossetti's poetry provided a subtle challenge not so much to Milton as to the emerging scepticism and aestheticism in Victorian society and its art-world, and not least to the secular tendencies of the second generation of Pre-Raphaelites, including Dante Gabriel Rossetti. Christina Rossetti was ahead of her time (and perhaps ours) in seeing the feminist potential in both John Milton and the Christian message. In the soul's relation to Christ, gender does not matter. What matters is the individual's quest for salvation, a quest in which poetry is a sacramental act, with a transformative, unifying function. Rossetti's poems both depict and constitute such a quest for union with God, on the assumption that God's word offers a pathway that is open to men as well as women.

Resa i röstens landskap : en narrativ studie av hur lärare blir professionella röstanvändare

The theme of this thesis is the learning process that occurs when teachers become professional voice users. The aim is to explore what it (really) means to become a professional voice user in a teaching profession; thereby developing an understanding of how future education in this field can be arranged so as to both effectively prevent vocal problems and to develop oral didactical competence among teachers. The ambition is to describe, interpret, and understand the learning process through a combination of emic and ethic research perspectives. The theoretical frame of reference reflects the cross disciplinary character of the thesis. Voice problems are common among both student teachers and inservice teachers and adversely affect professional competence, identity and quality of life. Additionally, vocal problems are proven to have a negative impact on pupils´ learning. The individual elements of learning are explored in the light of experiential learning theory and transformative learning theory. The social elements of learning are explored in relation to the theory of situated learning. In addition, theories of teacher professionalisation in terms of competence and identity are outlined. The empirical study has a longitudinal and multi method character. It is anchored in a phenomenological hermeneutical tradition, more specifically in narrative inquiry. The point of departure is the learning experiences of ten student teachers, who attended a ten week long course on voice production as part of their teacher training at Åbo Akademi University, in the autumn of 2002 and the spring of 2003. Four interviews in the form of conversations were conducted with each participant. These were crystallised with a process diary, a Swedish Voice Handicap Index, a voice observation, and a video observation. A fifth interview was conducted with each participant five years post teacher training, in the spring of 2008. Participant observation was also conducted throughout the course. The research materials have been analysed and interpreted narratively using a phenomenological hermeneutical method. The results are presented descriptively as individual narratives, which are reflected in logopedic research materials. Learning is here understood as emergent awareness. This is followed by a meta narrative concerning learning as experiences in the four dimensions body, thought, feeling, and relation. Finally, interpretation is expressed with respect to the theory of relational education. Learning is here understood as a movement in the field between the actual and the possible voice. It is also viewed as fundamentally rooted in inter-human relationships, in moments of presence and coexistence. As a tentative answer to the call for an existential space for learning in order to be a professional voice user, I suggest the concept of a learning refuge as a locus for a learning process built on trust, mutuality and openness.

Numbers and Languages

The thesis presents results obtained during the authors PhD-studies. First systems of language equations of a simple form consisting of just two equations are proved to be computationally universal. These are systems over unary alphabet, that are seen as systems of equations over natural numbers. The systems contain only an equation X+A=B and an equation X+X+C=X+X+D, where A, B, C and D are eventually periodic constants. It is proved that for every recursive set S there exists natural numbers p and d, and eventually periodic sets A, B, C and D such that a number n is in S if and only if np+d is in the unique solution of the abovementioned system of two equations, so all recursive sets can be represented in an encoded form. It is also proved that all recursive sets cannot be represented as they are, so the encoding is really needed. Furthermore, it is proved that the family of languages generated by Boolean grammars is closed under injective gsm-mappings and inverse gsm-mappings. The arguments apply also for the families of unambiguous Boolean languages, conjunctive languages and unambiguous languages. Finally, characterizations for morphisims preserving subfamilies of context-free languages are presented. It is shown that the families of deterministic and LL context-free languages are closed under codes if and only if they are of bounded deciphering delay. These families are also closed under non-codes, if they map every letter into a submonoid generated by a single word. The family of unambiguous context-free languages is closed under all codes and under the same non-codes as the families of deterministic and LL context-free languages.

Hyperbolic type metrics and distortion of quasiconformal map pings

This Ph.D. thesis consists of four original papers. The papers cover several topics from geometric function theory, more specifically, hyperbolic type metrics, conformal invariants, and the distortion properties of quasiconformal mappings. The first paper deals mostly with the quasihyperbolic metric. The main result gives the optimal bilipschitz constant with respect to the quasihyperbolic metric for the M¨obius self-mappings of the unit ball. A quasiinvariance property, sharp in a local sense, of the quasihyperbolic metric under quasiconformal mappings is also proved. The second paper studies some distortion estimates for the class of quasiconformal self-mappings fixing the boundary values of the unit ball or convex domains. The distortion is measured by the hyperbolic metric or hyperbolic type metrics. The results provide explicit, asymptotically sharp inequalities when the maximal dilatation of quasiconformal mappings tends to 1. These explicit estimates involve special functions which have a crucial role in this study. In the third paper, we investigate the notion of the quasihyperbolic volume and find the growth estimates for the quasihyperbolic volume of balls in a domain in terms of the radius. It turns out that in the case of domains with Ahlfors regular boundaries, the rate of growth depends not merely on the radius but also on the metric structure of the boundary. The topic of the fourth paper is complete elliptic integrals and inequalities. We derive some functional inequalities and elementary estimates for these special functions. As applications, some functional inequalities and the growth of the exterior modulus of a rectangle are studied.

Paul and death : a study in psychological coping

The present investigation looks into the attitudes toward death in Paul’s authentic letters, and puts them in relation to modern theories of psychological coping. Drawing on psychologically-oriented hermeneutic theory, and theories about psychological coping in particular, I argue that each case of psychological coping must be understood in its historical situation as strategies emanating from a specific person’s subjective appraisal (cf. Pargament, Lazarus and Folkman). Paul’s letters frequently refer to persecution and violent death. To aid in psychological coping is often integral to the purpose of the letters, which makes the perspective of psychological coping akin to their genre. In the course of a tentatively assumed chronological order of 1 Thessalonians, Galatians, 1 Corinthians, 2 Corinthians, Romans, Philippians, and Philemon, Paul moves from the perception of Jesus dying for the faithful to the understanding of dying with Jesus. His coping strategies concerning death are gradually transformed from conservative and deferring coping styles, to a more self-directing coping style, to collaborative and transformative coping styles, and finally to a new sense of deferring coping style in prison. The last case of deferring coping carries the traits of generosity and flexibility even in the face of death, which is in contrast to his previous letters. Through his correspondence, we see Paul’s attitude toward death transformed from denial to reaction, to processing, to acceptance (cf. Lindemann, Kübler-Ross, Bowlby, Parkes, among others). His strategies also shift in accordance with these understandings. Denial is accompanied by diversion, threat by aggression, processing by rumination, and acceptance by joy. The study shows the hermeneutic benefits of reading Paul’s letters as the rhetorically framed expressions of a person in a particular historical situation. The letters open small windows through which we can glimpse the coping process of a person of antiquity. In adopting the method of psychological exegesis, the study shows that the variety of attitudes toward death in Paul’s letters makes sense from the perspective of psychological coping. The psychological aspect of these letters is an underexamined richness that can extend into areas of contemporary individual and group identity, and from there to public policy and ethics.