Dirichlet problem at infinity for p-harmonic functions on negatively curved spaces

The object of this dissertation is to study globally defined bounded p-harmonic functions on Cartan-Hadamard manifolds and Gromov hyperbolic metric measure spaces. Such functions are constructed by solving the so called Dirichlet problem at infinity. This problem is to find a p-harmonic function on the space that extends continuously to the boundary at inifinity and obtains given boundary values there. The dissertation consists of an overview and three published research articles. In the first article the Dirichlet problem at infinity is considered for more general A-harmonic functions on Cartan-Hadamard manifolds. In the special case of two dimensions the Dirichlet problem at infinity is solved by only assuming that the sectional curvature has a certain upper bound. A sharpness result is proved for this upper bound. In the second article the Dirichlet problem at infinity is solved for p-harmonic functions on Cartan-Hadamard manifolds under the assumption that the sectional curvature is bounded outside a compact set from above and from below by functions that depend on the distance to a fixed point. The curvature bounds allow examples of quadratic decay and examples of exponential growth. In the final article a generalization of the Dirichlet problem at infinity for p-harmonic functions is considered on Gromov hyperbolic metric measure spaces. Existence and uniqueness results are proved and Cartan-Hadamard manifolds are considered as an application.

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.

Writing Otherwise than Seeing : Writing and Exteriority in Maurice Blanchot

Maurice Blanchot (1907-2003), the French writer and novelist, is one of the most important figures in post-war French literature and philosophy. The main intention of this study is to figure out his position and originality in the field of phenomenology. Since this thesis concentrates on the notion of vision in Blanchot s work, its primary context is the post-war discussion of the relation between seeing and thinking in France, and particularly the discussion of the conditions of non-violent vision and language. The focus will be on the philosophical conversation between Blanchot and his contemporary philosophers. The central premise is the following: Blanchot relates the criticism of vision to the criticism of the representative model of language. In this thesis, Blanchot s definition of literary language as the refusal to reveal anything is read as a reference pointing in two directions. First, to Hegel s idea of naming as negativity which reveals Being incrementally to man, and second, to Heidegger s idea of poetry as the simultaneity of revealing and withdrawal; the aim is to prove that eventually Blanchot opposes both Hegel s idea of naming as a gradual revelation of the totality of being and Heidegger s conception of poetry as a way of revealing the truth of Being. My other central hypothesis is that for Blanchot, the criticism of the privilege of vision is always related to the problematic of the exteriority. The principal intention is to trace how Blanchot s idea of language as infinity and exteriority challenges both the Hegelian idea of naming as conceptualizing things and Heidegger s concept of language as a way to truth (as aletheia). The intention is to show how Blanchot, with his concepts of fascination, resemblance and image, both affirms and challenges the central points of Heidegger s thinking on language. Blanchot s originality in, and contribution to, the discussion about the violence of vision and language is found in his answer to the question of how to approach the other by avoiding the worst violence . I claim that by criticizing the idea of language as naming both in Hegel and Heidegger, Blanchot generates an account of language which, since it neither negates nor creates Being, is beyond the metaphysical opposition between Being and non-Being.

Ajan sisäkkäiset tilat. Arkkitehtuuri, valokuva ja menneisyyden esittäminen W. G. Sebaldin romaanissa Austerlitz

In her thesis, Kaisa Kaakinen analyzes how the German emigrant author W. G. Sebald (1944-2001) uses architecture and photography in his last novel "Austerlitz" to represent time, history and remembering. Sebald describes time in spatial terms: it is like a building, the rooms and chambers of which are connected to each other. The poetics of spatial time manifests itself on multiple levels of the text. Kaakinen traces it in architectural representations, photographic images, intertextuality, as well as in the form of the text, using the concept of spatial form by Joseph Frank. Architectural and photographic representations serve as meeting points for different aspects and angles of the novel and illustrate the idea of a layered present that has multiple connections to the past. The novel tells a story of Jacques Austerlitz, who as a small child was sent from Prague to Britain in one of the so-called Kindertransports that saved children from Central Europe occupied by the National Socialists. Only gradually he remembers his Jewish parents, who have most likely perished in Nazi concentration camps. The novel brings the problematic of writing about another person's past to the fore by the fact that Austerlitz's story is told by an anonymous narrator, Austerlitz's interlocutor, who listens to and writes down Austerlitz's story. Kaakinen devotes the final part of her thesis to study the demands of representing a historical trauma, drawing on authors such as Dominick LaCapra and Michael Rothberg. Through the analysis of architectural and photographic representations in the novel, she demonstrates how Austerlitz highlights the sense of singularity and inaccessibility of memories of an individual, while also stressing the necessity - and therefore a certain kind of possibility - of passing these memories to another person. The coexistence of traumatic narrowness and of the infinity of history is reflected in ambivalent buildings. Some buildings in the novel resemble reversible figures: they can be perceived simultaneously as ruins and as construction sites. Buildings are also shown to be able to both cover and preserve memories - an idea that also is repeated in the use of photography, which tends to both replace memories and cause an experience of the presence of an absent thing. Commenting and critisizing some recent studies on Sebald, the author develops a reading which stresses the ambivalence inherent in Sebald's view on history and historiography. Austerlitz shows the need to recognize the inevitable absence of the past as well as the distance from the experiences of others. Equally important, however, is the refusal to give up narrating the past: Sebald's novel stresses the necessity to preserve the sites of the past, which carry silent traces of vanished life. The poetics of Austerlitz reflects the paradox of the simultaneous impossibility and indispensability of writing history.

Infinitiivi ja sen infiniittisyys : Tutkimus suomen kielen itsenäisistä A-infinitiivikonstruktioista

"Infinitive and its infinity" advocates an approach to infinitives that differs from most previous descriptions in several ways. Infinitives are generally considered to be an illustrative example of an inherently subordinated verb category. This is due to the fact that they are morphologically reduced and are allegedly not able to function as the only predicate of an independent clause. While former descriptions have thus treated infinitives as a linguistic category heavily dependent on the finite verb, my claim is that Finnish A-infinitives (e.g. juosta to run , olla to be ) can be used as independent grammatical units: they need not be either dependent or subordinated, but can have an equal status with finite constructions. In other words, they can be conceptually and interactionally non-dependent. Theoretically, the main objective of the thesis is to discover the nature of non-finite conceptualization and the ways in which it is utilized in everyday interactions. This is accomplished by contrasting finite and non-finite conceptualization with respect to the morphosyntactic marking of person, tense and modality. I argue that the morphologically reduced nature of infinitives can be used as an interactional resource. Independent A-infinitive constructions designate verbal processes that profile no participants, lack any connection with time, and present states of affairs as intensional, structural spaces. Consequently, they provide the interactants with a conceptual alternative in contrast to finite predications that are in Finnish always grammatically anchored to time, modality and person. The deictically unanchored character of A-infinitive constructions makes them highly affective and reflexive in nature. I discuss my findings primarily in the light of Cognitive Grammar. I have drawn insight from various other fields, too: among the theories that are touched upon are interactional linguistics, functional-typological linguistics, and studies on the poetic and metapragmatic use of language. The study is based on empirical data interpreted in qualitative terms. Analyses are based on 980 examples coming mainly from written language. Some 20 examples of spoken data are analyzed as well. In sum, the thesis presents a critical statement towards the finite-verb centred outlook on language and shows that analyzing non-finite elements as such reveals new aspects of grammar and interaction. This is to acknowledge the fact that infinitives, albeit prototypically participating in the coding of dependent events, can also be used outside of the context of the finite verb. Such a view poses several new research questions, as a linguistic category generally seen to code dependent, less prominent states of affairs , now is viewed on as possessing a full cognitive and pragmatic potential.

Inverse Problem for Fractional Brownian Motion with Discrete Data

The problem of recovering information from measurement data has already been studied for a long time. In the beginning, the methods were mostly empirical, but already towards the end of the sixties Backus and Gilbert started the development of mathematical methods for the interpretation of geophysical data. The problem of recovering information about a physical phenomenon from measurement data is an inverse problem. Throughout this work, the statistical inversion method is used to obtain a solution. Assuming that the measurement vector is a realization of fractional Brownian motion, the goal is to retrieve the amplitude and the Hurst parameter. We prove that under some conditions, the solution of the discretized problem coincides with the solution of the corresponding continuous problem as the number of observations tends to infinity. The measurement data is usually noisy, and we assume the data to be the sum of two vectors: the trend and the noise. Both vectors are supposed to be realizations of fractional Brownian motions, and the goal is to retrieve their parameters using the statistical inversion method. We prove a partial uniqueness of the solution. Moreover, with the support of numerical simulations, we show that in certain cases the solution is reliable and the reconstruction of the trend vector is quite accurate.

Aspects of atomic decompositions and Bergman projections

The concept of an atomic decomposition was introduced by Coifman and Rochberg (1980) for weighted Bergman spaces on the unit disk. By the Riemann mapping theorem, functions in every simply connected domain in the complex plane have an atomic decomposition. However, a decomposition resulting from a conformal mapping of the unit disk tends to be very implicit and often lacks a clear connection to the geometry of the domain that it has been mapped into. The lattice of points, where the atoms of the decomposition are evaluated, usually follows the geometry of the original domain, but after mapping the domain into another this connection is easily lost and the layout of points becomes seemingly random. In the first article we construct an atomic decomposition directly on a weighted Bergman space on a class of regulated, simply connected domains. The construction uses the geometric properties of the regulated domain, but does not explicitly involve any conformal Riemann map from the unit disk. It is known that the Bergman projection is not bounded on the space L-infinity of bounded measurable functions. Taskinen (2004) introduced the locally convex spaces LV-infinity consisting of measurable and HV-infinity of analytic functions on the unit disk with the latter being a closed subspace of the former. They have the property that the Bergman projection is continuous from LV-infinity onto HV-infinity and, in some sense, the space HV-infinity is the smallest possible substitute to the space H-infinity of analytic functions. In the second article we extend the above result to a smoothly bounded strictly pseudoconvex domain. Here the related reproducing kernels are usually not known explicitly, and thus the proof of continuity of the Bergman projection is based on generalised Forelli-Rudin estimates instead of integral representations. The minimality of the space LV-infinity is shown by using peaking functions first constructed by Bell (1981). Taskinen (2003) showed that on the unit disk the space HV-infinity admits an atomic decomposition. This result is generalised in the third article by constructing an atomic decomposition for the space HV-infinity on a smoothly bounded strictly pseudoconvex domain. In this case every function can be presented as a linear combination of atoms such that the coefficient sequence belongs to a suitable Köthe co-echelon space.

Ethics and Aesthetics : Intersections in Iris Murdoch's Philosophy

The study addresses the question concerning the relationship between ethics and aesthetics in the philosophy of Iris Murdoch. The main argument is that Murdoch s philosophy cannot be accurately understood without an understanding of the relationship she sees between the aesthetic experience and morality. Reading Murdoch s philosophy with this relationship in mind shows that it must be considered as a relevant alternative to the main forms of aesthetic-ethical theories. The study consists of seven previously published articles and a summary. It shows that Murdoch belongs to a tradition of philosophers who seek to broaden the scope of ethics by reference to aesthetic value and aesthetic experience. She sees an attitude responsible for aesthetic experiences as relevant for morality. However, she does not collapse morality into aesthetic experience. The two meet on the level of the subject s attitude towards its object, but there is a distinction between the experiences that accompany the attitudes. Aesthetic experiences can function as a clue to morals in that they present in a pleasing manner moral truths which otherwise might be psychologically too difficult to face. Murdoch equates the aesthetic attitude with virtuous love characterized by unselfish attention to its object. The primary object of such love is in Murdoch s account another human individual in her particularity. She compares the recognition of the other person as a particular existence to the experience of the Kantian sublime and offers her own version of the true sublime which is the experience of awe in the face of the infinity of the task of understanding others. One of the most central claims in Murdoch s philosophy is that human consciousness is evaluatively structured. This claim challenges the distinction between facts and values which has had an immense influence on modern moral philosophy. One argument with which Murdoch supports her claim is the nature of great literature. According to her, the standard of greatness in literature is the authors awareness of the independent existence of individuals in the particularity of their evaluative consciousnesses. The analysis of the standard of greatness in literature is also Murdoch s only argument for the claim that the primary object of the loving unselfish attention is the other particular individual. She is convinced that great literature reveals a deep truth about the human condition with its capacity to capture the particular. Abstract philo¬sophical discourse cannot compete with this capacity but it should take truths revealed by literature seriously in its theorising. Recognising this as Murdoch s stand on the question of the relation between philosophy and literature as forms of human discourse settles whether she is part of what has been called philosophy s turn to literature. The answer is yes.

The Role of Abduction in Self-Similarity

Based on the Aristotelian criterion referred to as 'abductio', Peirce suggests a method of hypothetical inference, which operates in a different way than the deductive and inductive methods. “Abduction is nothing but guessing” (Peirce, 7.219). This principle is of extreme value for the study of our understanding of mathematical self-similarity in both of its typical presentations: relative or absolute. For the first case, abduction incarnates the quantitative/qualitative relationships of a self-similar object or process; for the second case, abduction makes understandable the statistical treatment of self-similarity, 'guessing' the continuity of geometric features to the infinity through the use of a systematic stereotype (for instance, the assumption that the general shape of the Sierpiński triangle continuates identically into its particular shapes). The metaphor coined by Peirce, of an exact map containig itself the same exact map (a map of itself), is not only the most important precedent of Mandelbrot’s problem of measuring the boundaries of a continuous irregular surface with a logarithmic ruler, but also still being a useful abstraction for the conceptualisation of relative and absolute self-similarity, and its mechanisms of implementation. It is useful, also, for explaining some of the most basic geometric ontologies as mental constructions: in the notion of infinite convergence of points in the corners of a triangle, or the intuition for defining two parallel straight lines as two lines in a plane that 'never' intersect.