Renormalization methods in KAM theory

It is well known that an integrable (in the sense of Arnold-Jost) Hamiltonian system gives rise to quasi-periodic motion with trajectories running on invariant tori. These tori foliate the whole phase space. If we perturb an integrable system, the Kolmogorow-Arnold-Moser (KAM) theorem states that, provided some non-degeneracy condition and that the perturbation is sufficiently small, most of the invariant tori carrying quasi-periodic motion persist, getting only slightly deformed. The measure of the persisting invariant tori is large together with the inverse of the size of the perturbation. In the first part of the thesis we shall use a Renormalization Group (RG) scheme in order to prove the classical KAM result in the case of a non analytic perturbation (the latter will only be assumed to have continuous derivatives up to a sufficiently large order). We shall proceed by solving a sequence of problems in which theperturbations are analytic approximations of the original one. We will finally show that the approximate solutions will converge to a differentiable solution of our original problem. In the second part we will use an RG scheme using continuous scales, so that instead of solving an iterative equation as in the classical RG KAM, we will end up solving a partial differential equation. This will allow us to reduce the complications of treating a sequence of iterative equations to the use of the Banach fixed point theorem in a suitable Banach space.

Homogeneous model theory of metric structures

This thesis studies homogeneous classes of complete metric spaces. Over the past few decades model theory has been extended to cover a variety of nonelementary frameworks. Shelah introduced the abstact elementary classes (AEC) in the 1980s as a common framework for the study of nonelementary classes. Another direction of extension has been the development of model theory for metric structures. This thesis takes a step in the direction of combining these two by introducing an AEC-like setting for studying metric structures. To find balance between generality and the possibility to develop stability theoretic tools, we work in a homogeneous context, thus extending the usual compact approach. The homogeneous context enables the application of stability theoretic tools developed in discrete homogeneous model theory. Using these we prove categoricity transfer theorems for homogeneous metric structures with respect to isometric isomorphisms. We also show how generalized isomorphisms can be added to the class, giving a model theoretic approach to, e.g., Banach space isomorphisms or operator approximations. The novelty is the built-in treatment of these generalized isomorphisms making, e.g., stability up to perturbation the natural stability notion. With respect to these generalized isomorphisms we develop a notion of independence. It behaves well already for structures which are omega-stable up to perturbation and coincides with the one from classical homogeneous model theory over saturated enough models. We also introduce a notion of isolation and prove dominance for it.

On the Geometry of Infinite-Dimensional Grassmannian Manifolds and Gauge Theory

This PhD Thesis is about certain infinite-dimensional Grassmannian manifolds that arise naturally in geometry, representation theory and mathematical physics. From the physics point of view one encounters these infinite-dimensional manifolds when trying to understand the second quantization of fermions. The many particle Hilbert space of the second quantized fermions is called the fermionic Fock space. A typical element of the fermionic Fock space can be thought to be a linear combination of the configurations m particles and n anti-particles . Geometrically the fermionic Fock space can be constructed as holomorphic sections of a certain (dual)determinant line bundle lying over the so called restricted Grassmannian manifold, which is a typical example of an infinite-dimensional Grassmannian manifold one encounters in QFT. The construction should be compared with its well-known finite-dimensional analogue, where one realizes an exterior power of a finite-dimensional vector space as the space of holomorphic sections of a determinant line bundle lying over a finite-dimensional Grassmannian manifold. The connection with infinite-dimensional representation theory stems from the fact that the restricted Grassmannian manifold is an infinite-dimensional homogeneous (Kähler) manifold, i.e. it is of the form G/H where G is a certain infinite-dimensional Lie group and H its subgroup. A central extension of G acts on the total space of the dual determinant line bundle and also on the space its holomorphic sections; thus G admits a (projective) representation on the fermionic Fock space. This construction also induces the so called basic representation for loop groups (of compact groups), which in turn are vitally important in string theory / conformal field theory. The Thesis consists of three chapters: the first chapter is an introduction to the backround material and the other two chapters are individually written research articles. The first article deals in a new way with the well-known question in Yang-Mills theory, when can one lift the action of the gauge transformation group on the space of connection one forms to the total space of the Fock bundle in a compatible way with the second quantized Dirac operator. In general there is an obstruction to this (called the Mickelsson-Faddeev anomaly) and various geometric interpretations for this anomaly, using such things as group extensions and bundle gerbes, have been given earlier. In this work we give a new geometric interpretation for the Faddeev-Mickelsson anomaly in terms of differentiable gerbes (certain sheaves of categories) and central extensions of Lie groupoids. The second research article deals with the question how to define a Dirac-like operator on the restricted Grassmannian manifold, which is an infinite-dimensional space and hence not in the landscape of standard Dirac operator theory. The construction relies heavily on infinite-dimensional representation theory and one of the most technically demanding challenges is to be able to introduce proper normal orderings for certain infinite sums of operators in such a way that all divergences will disappear and the infinite sum will make sense as a well-defined operator acting on a suitable Hilbert space of spinors. This research article was motivated by a more extensive ongoing project to construct twisted K-theory classes in Yang-Mills theory via a Dirac-like operator on the restricted Grassmannian manifold.

Conrad Summenhart's theory of individual rights and its medieval background

This study focuses on the theory of individual rights that the German theologian Conrad Summenhart (1455-1502) explicated in his massive work Opus septipartitum de contractibus pro foro conscientiae et theologico. The central question to be studied is: How does Summenhart understand the concept of an individual right and its immediate implications? The basic premiss of this study is that in Opus septipartitum Summenhart composed a comprehensive theory of individual rights as a contribution to the on-going medieval discourse on rights. With this rationale, the first part of the study concentrates on earlier discussions on rights as the background for Summenhart s theory. Special attention is paid to language in which right was defined in terms of power . In the fourteenth century writers like Hervaeus Natalis and William Ockham maintained that right signifies power by which the right-holder can to use material things licitly. It will also be shown how the attempts to describe what is meant by the term right became more specified and cultivated. Gerson followed the implications that the term power had in natural philosophy and attributed rights to animals and other creatures. To secure right as a normative concept, Gerson utilized the ancient ius suum cuique-principle of justice and introduced a definition in which right was seen as derived from justice. The latter part of this study makes effort to reconstructing Summenhart s theory of individual rights in three sections. The first section clarifies Summenhart s discussion of the right of the individual or the concept of an individual right. Summenhart specified Gerson s description of right as power, taking further use of the language of natural philosophy. In this respect, Summenhart s theory managed to bring an end to a particular continuity of thought that was centered upon a view in which right was understood to signify power to licit action. Perhaps the most significant feature of Summenhart s discussion was the way he explicated the implication of liberty that was present in Gerson s language of rights. Summenhart assimilated libertas with the self-mastery or dominion that in the economic context of discussion took the form of (a moderate) self-ownership. Summenhart discussion also introduced two apparent extensions to Gerson s terminology. First, Summenhart classified right as relation, and second, he equated right with dominion. It is distinctive of Summenhart s view that he took action as the primary determinant of right: Everyone has as much rights or dominion in regard to a thing, as much actions it is licit for him to exercise in regard to the thing. The second section elaborates Summenhart s discussion of the species dominion, which delivered an answer to the question of what kind of rights exist, and clarified thereby the implications of the concept of an individual right. The central feature in Summenhart s discussion was his conscious effort to systematize Gerson s language by combining classifications of dominion into a coherent whole. In this respect, his treatement of the natural dominion is emblematic. Summenhart constructed the concept of natural dominion by making use of the concepts of foundation (founded on a natural gift) and law (according to the natural law). In defining natural dominion as dominion founded on a natural gift, Summenhart attributed natural dominion to animals and even to heavenly bodies. In discussing man s natural dominion, Summenhart pointed out that the natural dominion is not sufficiently identified by its foundation, but requires further specification, which Summenhart finds in the idea that natural dominion is appropriate to the subject according to the natural law. This characterization lead him to treat God s dominion as natural dominion. Partly, this was due to Summenhart s specific understanding of the natural law, which made reasonableness as the primary criterion for the natural dominion at the expense of any metaphysical considerations. The third section clarifies Summenhart s discussion of the property rights defined by the positive human law. By delivering an account on juridical property rights Summenhart connected his philosophical and theological theory on rights to the juridical language of his times, and demonstrated that his own language of rights was compatible with current juridical terminology. Summenhart prepared his discussion of property rights with an account of the justification for private property, which gave private property a direct and strong natural law-based justification. Summenhart s discussion of the four property rights usus, usufructus, proprietas, and possession aimed at delivering a detailed report of the usage of these concepts in juridical discourse. His discussion was characterized by extensive use of the juridical source texts, which was more direct and verbal the more his discussion became entangled with the details of juridical doctrine. At the same time he promoted his own language on rights, especially by applying the idea of right as relation. He also showed recognizable effort towards systematizing juridical language related to property rights.

The Nature, Origins, and Consequences of Finnish Shame-Proneness: A Grounded Theory Study

Although shame is a universal human emotion and is one of the most difficult emotions to overcome, its origins and nature as well as its effects on psychosocial functioning are not well understood or defined. While psychological and spiritual counselors are aware of the effects and consequences of shame for an individual s internal well-being and social life, shame is often still considered a taboo topic and is not given adequate attention. This study aims to explain the developmental process and effects of shame and shame-proneness for individuals and provide tools for practitioners to work more effectively with their clients who struggle with shame. This study presents the empirical foundation for a grounded theory that describes and explains the nature, origins, and consequences of shame-proneness. The study focused on Finnish participants childhood, adolescence and adulthood experiences and why they developed shame-proneness, what it meant for them as children and adolescents and what it meant for them as adults. The data collection phase of this study began in 2000. The participants were recruited through advertisements in local and country-wide newspapers and magazines. Altogether 325 people responded to the advertisements by sending an essay concerning their shame and guilt experiences. For the present study, 135 essays were selected and from those who sent an essay 19 were selected for in-depth interviews. In addition to essays and interviews, participants personal notebooks and childhood hospital and medical reports as well as their scores on the Internalized Shame Scale were analyzed. The development of shame-proneness and significant experiences and events during childhood and adolescence (e.g., health, parenting and parents behavior, humiliation, bullying, neglect, maltreatment and abuse) are discussed and the connections of shame-proneness to psychological concepts such as self-esteem, attachment, perfectionism, narcissism, submissiveness, pleasing others, heightened interpersonal subjectivity, and codependence are explained. Relationships and effects of shame-proneness on guilt, spirituality, temperament, coping strategies, defenses, personality formation and psychological health are also explicated. In addition, shame expressions and the development of shame triggers as well as internalized and externalized shame are clarified. These connections and developments are represented by the core category lack of gaining love, validation and protection as the authentic self. The conclusions drawn from the study include a categorization of shame-prone Finnish people according to their childhood and adolescent experiences and the characteristics of their shame-proneness and personality. Implications for psychological and spiritual counseling are also discussed. Key words: shame, internalized shame, external shame, shame development, shame triggers, guilt, self-esteem, attachment, narcissism, perfectionism, submissiveness, codependence, childhood neglect, childhood abuse, childhood maltreatment, emotional abuse, sexual abuse, spiritual abuse, psychological well-being

Spirituality of Finnish peacekeepers

The previous academic research on Finnish peacekeeping has clarified the operative and historical aspects of Finnish peacekeeping lacking the view of an individual who does the actual peacekeeping work. This research is based on the underlying theoretical assumption of human beings possessing different kinds of talents and intelligences creating a holistic entity. In this broad perspective spirituality was explored as an umbrella concept, as a holistic ability or talent, that can be explored as the deepest aspect of defining what it means to be human. The theoretical framework incorporated the concept of an intelligence, which is defined in Gardner's theory of multiple intelligences as the ability to solve problems, or to create products, that are valued within one or more cultural settings (Gardner, 1993, x). The viability of this theory was studied in the sample of Finnish peacekeepers. Spirituality in the theoretical and conceptual horizon was viewed as an extension of Gardner's theory of intelligences as one potential Gardnerian intelligence candidate. In addition to Gardner's theory, spirituality was explored as sensitivity which includes capacities such as sensing awareness, sensing mystery and sensing value (Hay, 1998). Also the practical aspects of spirituality were taken in account as shown in our everyday lives giving us the direction and influencing our social responsibilities and concerns (Bradford, 1995). Spirituality was explored also involving the element of the peacekeepers' community, the element of personal moral orientations and in the domain of religion and coping. The purpose of this research aimed in two dimensions. First, the aim was to outline the intelligence profile and the spiritual sensitivity profile of peacekeepers. Second, the aim was to understand qualitatively the nature of peacekeepers' spirituality The research interests were studied with different kinds of peacekeepers. Applying the mixed methods approach the research was conducted in two phases: first the former SFOR peacekeepers (N=6) were interviewed and the data was analysed. Inspired by the primary findings of these interviews, the data for the case-study of one peacekeeper was collected in co-operation with one former SFOR peacekeeper (N=1). In the second phase the data was collected from KFOR peacekeepers through the quantitative MI-Survey and the spiritual sensitivity survey (N=195). The quantitative method was used to outline the intelligence profile and the spiritual sensitivity profile of peacekeepers (N=195). In the mixed methods approach this method highlighted the general overview of intelligence traits and spiritual sensitivity of peacekeepers. In the mixed methods approach the qualitative method including interviews (N=6) and a case-study of one peacekeeper (N=1) increased subjective, qualitative information of spirituality of peacekeepers. The intelligence profile of peacekeepers highlighted the bodily-kinesthetic and interpersonal dimensions as the practical and social aspects of peacekeepers. Strong inter-item dependencies in the intrapersonal intelligence profile meant that peacekeepers possess a self-reflection and self-knowledge component and they reflect on deep psychological and philosophical issues. Regarding the spiritual sensitivity, peacekeepers found awareness-sensing, mystery-sensing, value-sensing and community-sensing important. The community-sensing emphasised a strong will to advance peace and to help people who are in need: things that are close to the heart of the peacekeepers. These results depicted practicality, being socially capable, and reflecting one's inner world as essential to peacekeepers. Moreover, spirituality as peacekeepers' moral endeavour became clearer because the sub-model of their community-sensing described morally charged destinations: advancing peace and helping people in need. In the qualitative findings peacekeepers articulated justice orientation and rule-following characterising the nature of peacekeepers' moral attitude and moral call (Kohlberg, 1969). An ethic of care (Gilligan, 1982) describes mainly female moral orientation, but the findings revealed that an ethic of care is also an important agent supporting strongly male peacekeepers in their aim to carry out qualitatively good peacekeeping work. The moral endeavour was voiced, when the role of religion in coping meant the assessment of the a way of life, a way of conduct, a way of being truthful to one's own values in confusing surroundings. The practical level of spiritual and religious contemplation was voiced as morally charged inner motivation to fulfil one's duties and at the same time to cope with various peacekeeping challenges. The results of different data sets were combined and interpreted as the moral endeavour, which characterises peacekeepers' spirituality. As the combining result, the perspective of peacekeepers' spirituality is considered moral or at least morally charged.

Essays on Labor Market Frictions, Technological Change and Macroeconomic Fluctuations

The dissertation consists of an introductory chapter and three essays that apply search-matching theory to study the interaction of labor market frictions, technological change and macroeconomic fluctuations. The first essay studies the impact of capital-embodied growth on equilibrium unemployment by extending a vintage capital/search model to incorporate vintage human capital. In addition to the capital obsolescence (or creative destruction) effect that tends to raise unemployment, vintage human capital introduces a skill obsolescence effect of faster growth that has the opposite sign. Faster skill obsolescence reduces the value of unemployment, hence wages and leads to more job creation and less job destruction, unambiguously reducing unemployment. The second essay studies the effect of skill biased technological change on skill mismatch and the allocation of workers and firms in the labor market. By allowing workers to invest in education, we extend a matching model with two-sided heterogeneity to incorporate an endogenous distribution of high and low skill workers. We consider various possibilities for the cost of acquiring skills and show that while unemployment increases in most scenarios, the effect on the distribution of vacancy and worker types varies according to the structure of skill costs. When the model is extended to incorporate endogenous labor market participation, we show that the unemployment rate becomes less informative of the state of the labor market as the participation margin absorbs employment effects. The third essay studies the effects of labor taxes on equilibrium labor market outcomes and macroeconomic dynamics in a New Keynesian model with matching frictions. Three policy instruments are considered: a marginal tax and a tax subsidy to produce tax progression schemes, and a replacement ratio to account for variability in outside options. In equilibrium, the marginal tax rate and replacement ratio dampen economic activity whereas tax subsidies boost the economy. The marginal tax rate and replacement ratio amplify shock responses whereas employment subsidies weaken them. The tax instruments affect the degree to which the wage absorbs shocks. We show that increasing tax progression when taxation is initially progressive is harmful for steady state employment and output, and amplifies the sensitivity of macroeconomic variables to shocks. When taxation is initially proportional, increasing progression is beneficial for output and employment and dampens shock responses.

Stress and developmental responses in Arabidopsis thaliana: regulation through the transcription factor-interacting protein RCD1

Plants constantly face adverse environmental conditions, such as drought or extreme temperatures that threaten their survival. They demonstrate astonishing metabolic flexibility in overcoming these challenges and one of the key responses to stresses is changes in gene expression leading to alterations in cellular functions. This is brought about by an intricate network of transcription factors and associated regulatory proteins. Protein-protein interactions and post-translational modifications are important steps in this control system along with carefully regulated degradation of signaling proteins. This work concentrates on the RADICAL-INDUCED CELL DEATH1 (RCD1) protein which is an important regulator of abiotic stress-related and developmental responses in Arabidopsis thaliana. Plants lacking this protein function display pleiotropic phenotypes including sensitivity to apoplastic reactive oxygen species (ROS) and salt, ultraviolet B (UV-B) and paraquat tolerance, early flowering and senescence. Additionally, the mutant plants overproduce nitric oxide, have alterations in their responses to several plant hormones and perturbations in gene expression profiles. The RCD1 gene is transcriptionally unresponsive to environmental signals and the regulation of the protein function is likely to happen post-translationally. RCD1 belongs to a small protein family and, together with its closest homolog SRO1, contains three distinguishable domains: In the N-terminus, there is a WWE domain followed by a poly(ADP-ribose) polymerase-like domain which, despite sequence conservation, does not seem to be functional. The C-terminus of RCD1 contains a novel domain called RST. It is present in RCD1-like proteins throughout the plant kingdom and is able to mediate physical interactions with multiple transcription factors. In conclusion, RCD1 is a key point of signal integration that links ROS-mediated cues to transcriptional regulation by yet unidentified means, which are likely to include post-translational mechanisms. The identification of RCD1-interacting transcription factors, most of whose functions are still unknown, opens new avenues for studies on plant stress as well as developmental responses.

Molecular Tuning of Rhodopsins for High Visual Sensitivity in Different Light Environments : Variation in Absorbance Spectrum and Opsin Sequence within and between Species

Visual pigments of different animal species must have evolved at some stage to match the prevailing light environments, since all visual functions depend on their ability to absorb available photons and transduce the event into a reliable neural signal. There is a large literature on correlation between the light environment and spectral sensitivity between different fish species. However, little work has been done on evolutionary adaptation between separated populations within species. More generally, little is known about the rate of evolutionary adaptation to changing spectral environments. The objective of this thesis is to illuminate the constraints under which the evolutionary tuning of visual pigments works as evident in: scope, tempo, available molecular routes, and signal/noise trade-offs. Aquatic environments offer Nature s own laboratories for research on visual pigment properties, as naturally occurring light environments offer an enormous range of variation in both spectral composition and intensity. The present thesis focuses on the visual pigments that serve dim-light vision in two groups of model species, teleost fishes and mysid crustaceans. The geographical emphasis is in the brackish Baltic Sea area with its well-known postglacial isolation history and its aquatic fauna of both marine and fresh-water origin. The absorbance spectrum of the (single) dim-light visual pigment were recorded by microspectrophotometry (MSP) in single rods of 26 fish species and single rhabdoms of 8 opossum shrimp populations of the genus Mysis inhabiting marine, brackish or freshwater environments. Additionally, spectral sensitivity was determined from six Mysis populations by electroretinogram (ERG) recording. The rod opsin gene was sequenced in individuals of four allopatric populations of the sand goby (Pomatoschistus minutus). Rod opsins of two other goby species were investigated as outgroups for comparison. Rod absorbance spectra of the Baltic subspecies or populations of the primarily marine species herring (Clupea harengus membras), sand goby (P. minutus), and flounder (Platichthys flesus) were long-wavelength-shifted compared to their marine populations. The spectral shifts are consistent with adaptation for improved quantum catch (QC) as well as improved signal-to-noise ratio (SNR) of vision in the Baltic light environment. Since the chromophore of the pigment was pure A1 in all cases, this has apparently been achieved by evolutionary tuning of the opsin visual pigment. By contrast, no opsin-based differences were evident between lake and sea populations of species of fresh-water origin, which can tune their pigment by varying chromophore ratios. A more detailed analysis of differences in absorbance spectra and opsin sequence between and within populations was conducted using the sand goby as model species. Four allopatric populations from the Baltic Sea (B), Swedish west coast (S), English Channel (E), and Adriatic Sea (A) were examined. Rod absorbance spectra, characterized by the wavelength of maximum absorbance (λmax), differed between populations and correlated with differences in the spectral light transmission of the respective water bodies. The greatest λmax shift as well as the greatest opsin sequence difference was between the Baltic and the Adriatic populations. The significant within-population variation of the Baltic λmax values (506-511 nm) was analyzed on the level of individuals and was shown to correlate well with opsin sequence substitutions. The sequences of individuals with λmax at shorter wavelengths were identical to that of the Swedish population, whereas those with λmax at longer wavelengths additionally had substitution F261F/Y in the sixth transmembrane helix of the protein. This substitution (Y261) was also present in the Baltic common gobies and is known to redshift spectra. The tuning mechanism of the long-wavelength type Baltic sand gobies is assumed to be the co-expression of F261 and Y261 in all rods to produce ≈ 5 nm redshift. The polymorphism of the Baltic sand goby population possibly indicates ambiguous selection pressures in the Baltic Sea. The visual pigments of all lake populations of the opossum shrimp (Mysis relicta) were red-shifted by 25 nm compared with all Baltic Sea populations. This is calculated to confer a significant advantage in both QC and SNR in many humus-rich lakes with reddish water. Since only A2 chromophore was present, the differences obviously reflect evolutionary tuning of the visual protein, the opsin. The changes have occurred within the ca. 9000 years that the lakes have been isolated from the Sea after the most recent glaciation. At present, it seems that the mechanism explaining the spectral differences between lake and sea populations is not an amino acid substitution at any other conventional tuning site, but the mechanism is yet to be found.