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

Spectral Tuning and Adaptation to Different Light Environments of Mysid Visual Pigments

In the present thesis, questions of spectral tuning, the relation of spectral and thermal properties of visual pigments, and evolutionary adaptation to different light environments were addressed using a group of small crustaceans of the genus Mysis as a model. The study was based on microspectrophotometric measurements of visual pigment absorbance spectra, electrophysiological measurements of spectral sensitivities of dark-adapted eyes, and sequencing of the opsin gene retrieved through PCR. The spectral properties were related to the spectral transmission of the respective light environments, as well as to the phylogentic histories of the species. The photoactivation energy (Ea) was estimated from temperature effects on spectral sensitivity in the long-wavelength range, and calculations were made for optimal quantum catch and optimal signal-to-noise ratio in the different light environments. The opsin amino acid sequences of spectrally characterized individuals were compared to find candidate residues for spectral tuning. The general purpose was to clarify to what extent and on what time scale adaptive evolution has driven the functional properties of (mysid) visual pigments towards optimal performance in different light environments. An ultimate goal was to find the molecular mechanisms underlying the spectral tuning and to understand the balance between evolutionary adaptation and molecular constraints. The totally consistent segregation of absorption maxima (λmax) into (shorter-wavelength) marine and (longer-wavelength) freshwater populations suggests that truly adaptive evolution is involved in tuning the visual pigment for optimal performance, driven by selection for high absolute visual sensitivity. On the other hand, the similarity in λmax and opsin sequence between several populations of freshwater M. relicta in spectrally different lakes highlights the limits to adaptation set by evolutionary history and time. A strong inverse correlation between Ea and λmax was found among all visual pigments studied in these respects, including those of M. relicta and 10 species of vertebrate pigments, and this was used to infer thermal noise. The conceptual signal-to-noise ratios thus calculated for pigments with different λmax in the Baltic Sea and Lake Pääjärvi light environments supported the notion that spectral adaptation works towards maximizing the signal-to-noise ratio rather than quantum catch as such. Judged by the shape of absorbance spectra, the visual pigments of all populations of M. relicta and M. salemaai used exclusively the A2 chromophore (3, 4-dehydroretinal). A comparison of amino acid substitutions between M. relicta and M. salemaai indicated that mysid shrimps have a small number of readily available tuning sites to shift between a shorter - and a longer -wavelength opsin. However, phylogenetic history seems to have prevented marine M. relicta from converting back to the (presumably) ancestral opsin form, and thus the more recent reinvention of marine spectral sensitivity has been accomplished by some other novel mechanism, yet to be found

Spectral states and accretion geometries in Galactic black hole binaries

Black hole X-ray binaries, binary systems where matter from a companion star is accreted by a stellar mass black hole, thereby releasing enormous amounts of gravitational energy converted into radiation, are seen as strong X-ray sources in the sky. As a black hole can only be detected via its interaction with its surroundings, these binary systems provide important evidence for the existence of black holes. There are now at least twenty cases where the measured mass of the X-ray emitting compact object in a binary exceeds the upper limit for a neutron star, thus inferring the presence of a black hole. These binary systems serve as excellent laboratories not only to study the physics of accretion but also to test predictions of general relativity in strongly curved space time. An understanding of the accretion flow onto these, the most compact objects in our Universe, is therefore of great importance to physics. We are only now slowly beginning to understand the spectra and variability observed in these X-ray sources. During the last decade, a framework has developed that provides an interpretation of the spectral evolution as a function of changes in the physics and geometry of the accretion flow driven by a variable accretion rate. This doctoral thesis presents studies of two black hole binary systems, Cygnus~X-1 and GRS~1915+105, plus the possible black hole candidate Cygnus~X-3, and the results from an attempt to interpret their observed properties within this emerging framework. The main result presented in this thesis is an interpretation of the spectral variability in the enigmatic source Cygnus~X-3, including the nature and accretion geometry of its so-called hard spectral state. The results suggest that the compact object in this source, which has not been uniquely identified as a black hole on the basis of standard mass measurements, is most probably a massive, ~30 Msun, black hole, and thus the most massive black hole observed in a binary in our Galaxy so far. In addition, results concerning a possible observation of limit-cycle variability in the microquasar GRS~1915+105 are presented as well as evidence of `mini-hysteresis' in the extreme hard state of Cygnus X-1.