Topological stability through tame retractions

A smooth map is said to be stable if small perturbations of the map only differ from the original one by a smooth change of coordinates. Smoothly stable maps are generic among the proper maps between given source and target manifolds when the source and target dimensions belong to the so-called nice dimensions, but outside this range of dimensions, smooth maps cannot generally be approximated by stable maps. This leads to the definition of topologically stable maps, where the smooth coordinate changes are replaced with homeomorphisms. The topologically stable maps are generic among proper maps for any dimensions of source and target. The purpose of this thesis is to investigate methods for proving topological stability by constructing extremely tame (E-tame) retractions onto the map in question from one of its smoothly stable unfoldings. In particular, we investigate how to use E-tame retractions from stable unfoldings to find topologically ministable unfoldings for certain weighted homogeneous maps or germs. Our first results are concerned with the construction of E-tame retractions and their relation to topological stability. We study how to construct the E-tame retractions from partial or local information, and these results form our toolbox for the main constructions. In the next chapter we study the group of right-left equivalences leaving a given multigerm f invariant, and show that when the multigerm is finitely determined, the group has a maximal compact subgroup and that the corresponding quotient is contractible. This means, essentially, that the group can be replaced with a compact Lie group of symmetries without much loss of information. We also show how to split the group into a product whose components only depend on the monogerm components of f. In the final chapter we investigate representatives of the E- and Z-series of singularities, discuss their instability and use our tools to construct E-tame retractions for some of them. The construction is based on describing the geometry of the set of points where the map is not smoothly stable, discovering that by using induction and our constructional tools, we already know how to construct local E-tame retractions along the set. The local solutions can then be glued together using our knowledge about the symmetry group of the local germs. We also discuss how to generalize our method to the whole E- and Z- series.

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.

Mediated music makers : Constructing author images in popular music

The subject of the thesis is the mediated construction of author images in popular music. In the study, the construction of images is treated as a process in which artists, the media and the members of the audience participate. The notions of presented, mediated and compiled author images are used in explaining the mediation process and the various authorial roles of the agents involved. In order to explore the issue more closely, I analyse the author images of a group of popular music artists representing the genres of rock, pop and electronic dance music. The analysed material consists mostly of written media texts through which the artists authorial roles and creative responsibilities are discussed. Theoretically speaking, the starting points for the examination lie in cultural studies and discourse analysis. Even though author images may be conceived as intertextual constructions, the artist is usually presented as a recognizable figure whose purpose is to give the music its public face. This study does not, then, deal with musical authors as such, but rather with their public images and mediated constructions. Because of the author-based functioning of popular music culture and the idea of the artist s individual creative power, the collective and social processes involved in the making of popular music are often superseded by the belief in a single, originating authorship. In addition to the collective practices of music making, the roles of the media and the marketing machinery complicate attempts to clarify the sharing of authorial contributions. As the case studies demonstrate, the differences between the examined author images are connected with a number of themes ranging from issues of auteurism and stardom to the use of masked imagery and the blending of authorial voices. Also the emergence of new music technologies has affected not only the ways in which music is made, but also how the artist s authorial status and artistic identity is understood. In the study at hand, the author images of auteurs, stars, DJs and sampling artists are discussed alongside such varied topics as collective authorship, evaluative hierarchies, visual promotion and generic conventions. Taken altogether, the examined case studies shed light on the functioning of popular music culture and the ways in which musical authorship is (re)defined.

Keitä huonokuuloiset oikein ovat? : Lähtökohtia huonokuuloisten sosiaalisten identiteettien tutkimukselle

To be hard-of-hearing (HOH) isn t only a hearing-related phenomenon. Some HOH people identify themselves with the majority population, where as others identify themselves with culturally Deaf or with other HOH people. This results in three different kinds of groups. The original aim of this study was to interview HOH people in order to discover the social identities of these three groups. It turned out, however, that there are only few studies exploring this subject. It was therefore first necessary to develop sustainable and suitable starting points for the examination of the information produced by the interviews. In other words, the construction of the theoretical and conceptual frame of reference became the objective of this study. It is, on one hand, the personal experiences of the researcher s mingling with different kinds of HOH people and with culturally Deaf and, on the other, the literature concerning identities that lie in the background of this study. The analysis concentrated on the studies of identities in general and on the previous works dealing with the identities of the HOH people as well as on other studies with reference to HOH people. The choice between mainstream education and special education is one of the main factors affecting the development of the identity. Yet, it became apparent that identifying with the relevant reference group (majority population, culturally Deaf or other HOH people) had an even greater influence than the school. The hard of hearing define themselves differently and take different views on being HOH, depending on their social relations, their values and on their communicative and cultural features. The conclusion is that HOH people should be examined as different groups, not as one group, since they differ clearly from each other.

Polymerization of Ethene and Branched α-olefins with Group IV Metal Complexes Bearing Nitrogen- and Oxygen-Based Ligands

The commodity plastics that are used in our everyday lives are based on polyolefin resins and they find wide variety of applications in several areas. Most of the production is carried out in catalyzed low pressure processes. As a consequence polymerization of ethene and α-olefins has been one of the focus areas for catalyst research both in industry and academia. Enormous amount of effort have been dedicated to fine tune the processes and to obtain better control of the polymerization and to produce tailored polymer structures The literature review of the thesis concentrates on the use of Group IV metal complexes as catalysts for polymerization of ethene and branched α-olefins. More precisely the review is focused on the use of complexes bearing [O,O] and [O,N] type ligands which have gained considerable interest. Effects of the ligand framework as well as mechanical and fluxional behaviour of the complexes are discussed. The experimental part consists mainly of development of new Group IV metal complexes bearing [O,O] and [O,N] ligands and their use as catalysts precursors in ethene polymerization. Part of the experimental work deals with usage of high-throughput techniques in tailoring properties of new polymer materials which are synthesized using Group IV complexes as catalysts. It is known that the by changing the steric and electronic properties of the ligand framework it is possible to fine tune the catalyst and to gain control over the polymerization reaction. This is why in this thesis the complex structures were designed so that the ligand frameworks could be fairly easily modified. All together 14 complexes were synthesised and used as catalysts in ethene polymerizations. It was found that the ligand framework did have an impact within the studied catalyst families. The activities of the catalysts were affected by the changes in complex structure and also effects on the produced polymers were observed: molecular weights and molecular weight distributions were depended on the used catalyst structure. Some catalysts also produced bi- or multi-modal polymers. During last decade high-throughput techniques developed in pharmaceutical industries have been adopted into polyolefin research in order to speed-up and optimize the catalyst candidates. These methods can now be regarded as established method suitable for both academia and industry alike. These high-throughput techniques were used in tailoring poly(4-methyl-1-pentene) polymers which were synthesized using Group IV metal complexes as catalysts. This work done in this thesis represents the first successful example where the high-throughput synthesis techniques are combined with high-throughput mechanical testing techniques to speed-up the discovery process for new polymer materials.

Phylogeny of a taxonomically difficult group and evolution of host location mechanism

The parasitic wasps are one of the largest insect groups and their life histories are remarkably variable. Common to all parasitic wasps is that they kill their hosts, which are usually beetles, butterflies and sometimes spiders. Hosts are often at a larval or pupal stage and live in concealed conditions, such as in plant tissue. Parasitic wasps have two main ways of finding their host. 1) They can detect chemical compounds emitted by damaged plant material or released by larvae living in plant tissue, and 2) detect the larvae by sound vibrations. Even though pupae are immobile and silent, and therefore do not cause vibration, parasitoids have, however, adapted to find passive developmental stages by producing vibration themselves by knocking the substrate with their antennae, and then detecting the echoes with their legs. This echolocation allows a parasitoid to locate its potential hosts that are deeply buried in wood. This study focuses on the relationships of the subfamily Cryptinae (Hymenoptera: Ichneumonidae) and related taxa, and the evolution of host location mechanism. There are no earlier studies of the phylogeny of the Cryptinae, and the position of related taxa are unclear. According to the earlier classification, which is entirely intuitional, the Cryptinae is divided into three tribes: Cryptini, Hemigasterini and Phygadeuontini. Further, these tribes are subdiveded into numerous subtribes. This work, based on molecular characters, shows that the cryptine tribes Cryptini, Phygadeuon¬tini and Hemigasterini come out largely as monophyletic groups, thus agreeing with the earlier classification. The earlier subtribal classification had no support. In addition, it is shown that modified antennal structures are associated with host usage of wood-boring coleopteran hosts. The cryptines have a clear modification series on their antennal tips from a simply tip to a hammer-like structure. The species with strongly modified antennae belong mostly to the tribe Cryptini and they utilise wood-boring beetles as hosts. Also, field observations on insect behaviour support this result.

A Historical Approach to Children s Physics Education : Modelling of DC-circuit Phenomena in a Small Group

This three-phase design research describes the modelling processes for DC-circuit phenomena. The first phase presents an analysis of the development of the DC-circuit historical models in the context of constructing Volta s pile at the turn of the 18th century. The second phase involves the designing of a teaching experiment for comprehensive school third graders. Among other considerations, the design work utilises the results of the first phase and research literature of pupils mental models for DC-circuit phenomena. The third phase of the research was concerned with the realisation of the planned teaching experiment. The aim of this phase was to study the development of the external representations of DC-circuit phenomena in a small group of third graders. The aim of the study has been to search for new ways to guide pupils to learn DC-circuit phenomena while emphasing understanding at the qualitative level. Thus, electricity, which has been perceived as a difficult and abstract subject, could be learnt more comprehensively. Especially, the research of younger pupils learning of electricity concepts has not been of great interest at the international level, although DC-circuit phenomena are also taught in the lower classes of comprehensive schools. The results of this study are important, because there has tended to be more teaching of natural sciences in the lower classes of comprehensive schools, and attempts are being made to develop this trend in Finland. In the theoretical part of the research an Experimental-centred representation approach, which emphasises the role of experimentalism in the development of pupil s representations, is created. According to this approach learning at the qualitative level consists of empirical operations like experimenting, observations, perception, and prequantification of nature phenomena, and modelling operations like explaining and reasoning. Besides planning teaching, the new approach can be used as an analysis tool in describing both historical modelling and the development of pupils representations. In the first phase of the study, the research question was: How did the historical models of DC-circuit phenomena develop in Volta s time? The analysis uncovered three qualitative historical models associated with the historical concept formation process. The models include conceptions of the electric circuit as a scene in the DC-circuit phenomena, the comparative electric-current phenomenon as a cause of different observable effect phenomena, and the strength of the battery as a cause of the electric-current phenomenon. These models describe the concept formation process and its phases in Volta s time. The models are portrayed in the analysis using fragments of the models, where observation-based fragments and theoretical fragements are distinguished from each other. The results emphasise the significance of the qualitative concept formation and the meaning of language in the historical modelling of DC-circuit phenomena. For this reason these viewpoints are stressed in planning the teaching experiment in the second phase of the research. In addition, the design process utilised the experimentation behind the historical models of DC-circuit phenomena In the third phase of the study the research question is as follows: How will the small group s external representations of DC-circuit phenomena develop during the teaching experiment? The main question is divided into the following two sub questions: What kind of talk exists in the small group s learning? What kinds of external representations for DC-circuit phenomena exist in the small group discourse during the teaching experiment? The analysis revealed that the teaching experiment of the small group succeeded in its aim to activate talk in the small group. The designed connection cards proved especially successful in activating talk. The connection cards are cards that represent the components of the electric circuit. In the teaching experiment the pupils constructed different connections with the connection cards and discussed, what kinds of DC-circuit phenomena would take place in the corresponding real connections. The talk of the small group was analysed by comparing two situations, firstly, when the small group discussed using connections made with the connection cards and secondly with the same connections using real components. According to the results the talk of the small group included more higher-order thinking when using the connection cards than with similar real components. In order to answer the second sub question concerning the small group s external representations that appeared in the talk during the teaching experiment; student talk was visualised by the fragment maps which incorporate the electric circuit, the electric current and the source voltage. The fragment maps represent the gradual development of the external representations of DC-circuit phenomena in the small group during the teaching experiment. The results of the study challenge the results of previous research into the abstractness and difficulty of electricity concepts. According to this research, the external representations of DC-circuit phenomena clearly developed in the small group of third graders. Furthermore, the fragment maps uncover that although the theoretical explanations of DC-circuit phenomena, which have been obtained as results of typical mental model studies, remain undeveloped, learning at the qualitative level of understanding does take place.

Group Beliefs : Studies on the Nature and Logic of Collective Doxastic Attitudes

This thesis studies the nature and logic of collective doxastic attitudes, or what is referred to in ordinary language as "group beliefs". Beliefs and other intentional attitudes are attributed to groups and collections of people, and such attributions are used to explain and predict the actions of groups. The thesis develops an understanding of group beliefs as voluntarily adopted views or acceptances rather than as ordinary beliefs. Such an understanding can provide new answers to questions concerning collective knowledge and justification of group beliefs, and it allows developing modal logics with collective doxastic and epistemic notions. The thesis consists of six articles. The first three articles are philosophical studies concerned with the nature of group beliefs. The last three articles are logical studies that aim at developing proof-theoretical calculi for reasoning about collective doxastic attitudes.

Struggling over subjectivity: A discursive analysis of strategic development in a group

We have seen growing interest in discursive perspectives on strategy. This perspective holds great promise for development of an understanding on how strategy discourse and subjectivity are intertwined. We wish to add to this existing research by outlining a discursive struggle approach to subjectivity. To understand the complex subjectification and empowering/disempowering effects of organizational strategy discourse, this approach focuses on organization-specific discourse mobilizations an various ways of resistance. Drawing on an analysis of the discourses and practices of ‘strategic development’ in an engineering and consulting group we provide an empirical illustration of such struggles over subjectivity. In particular, we report three examples of competing ways of making sense of and giving sense to strategic development, with specific subjectification tendencies. First, we show how corporate management can mobilize and appropriate a specific kind of discourse to attempt to gain control of the organization, which tends to reproduce managerial hegemony, but also trigger discursive and other forms of resistance. Second, we will illustrate how middle managers resist this hegemony by initiating a strategy discourse of their own to create room for manoeuvre in controversial situations. Third, we show how project engineers can distance themselves from managerial-initiated strategy discourses to maintain a viable identity despite all kinds of pressures. Although our examples are case-specific, we believe that similar discursive dynamics also characterize strategizing in other organizations.

Influence of forest owners as an interest group in achieving the forest policy goals in Finland.

XVIII IUFRO World Congress, Ljubljana 1986.