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.

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.

Pelon ja toivon välissä : Pätkätyöläisen urakerronta

This study analyzes the forming of the occupational identity of the well-educated fixed-term employees. Fixed-term employment contracts amongst the well-educated labour force are exceptionally common in Finland as compared to other European countries. Two groups of modern fixed-term employees are distinguished. The first comprises well-educated women employed in the public sector whose fixed-term employment often consists of successive periods as temporary substitutes. The other group comprises well-educated, upper white-collar men aged over 40, whose fixed-term employment careers often consist of jobs of project nature or posts that are filled for a fixed period only. Method of the study For the empirical data I interviewed 35 persons (26 women and 9 men) in 33 interviews, one of which was conducted by e-mail and one was a group interview. All the interviews were electronically recorded and coded. All the interviewees have two things in common: fixed-term employment and formal high education. Thirteen (13) of them are researchers, four nurses, four midwives, four journalists, and ten project experts. I used the snowball method to get in touch the interviewees. The first interviewees were those who were recommended by the trade unions and by my personal acquaintances. These interviewees, in turn, recommended other potential interviewees. In addition, announcements on the internet pages of the trade unions were used to reach other interviewees. In analysing process I read the research material several times to find the turning points in the narrative the interviewees told. I also searched for the most meaningful stories told and the meaning the interviewees gave to these stories and to the whole narrative. In addition to that I paid attention to co-production of the narrative with the interviewees and analyzed the narrative as performance to be able to search for the preferred identities the interviewees perform. (Riesman 2001, 698-701). I do not pay much attention to the question of truth of a narrative in the sense of its correspondence with facts; rather I think a working life narrative has two tasks: On the one hand one has to tell the facts and on the other hand, he/she has to describe the meaning of these facts to herself/himself. To emphasize the double nature of the narrative about one’s working life I analyzed the empirical data both by categorizing it according to the cultural models of storytelling (heroic story, comedy, irony and tragedy) and by studying the themes most of the interviewees talked about. Ethics of the study I chose to use narrative within qualitative interviews on the grounds that in my opinion is more ethical and more empowering than the more traditional structured interview methods. During the research process I carefully followed the ethical rules of a qualitative research. The purpose of the interviews and the research was told to the interviewees by giving them a written description of the study. Oral permission to use the interview in this research was obtained from the interviewees. The names and places, which are mentioned in the study, are changed to conceal the actual identity of the interviewees. I shared the analysis with the interviewees by sending each of them the first analysis of their personal interview. This way I asked them to make sure that the identity was hidden well enough and hoped to give interviewees a chance to look at their narratives, to instigate new actions and sustain the present one (Smith 2001, 721). Also I hoped to enjoy a new possibility of joint authorship. Main results As a result of the study I introduce six models of telling a story. The four typical western cultural models that guide the telling are: heroic story, comedy, tragedy and satirical story (Hänninen 1999). In addition to these models I found two ways of telling a career filled with fixed-term employments that differ significantly from traditional career story telling. However, the story models in which the interviewees pour their experience locates the fixed term employers work career in an imagined life trajectory and reveals the meaning they give to it. I analyze the many sided heroic story that Liisa tells as an example of the strength of the fear of failing or losing the job the fixed term employee feels. By this structure it is also possible to show that success is felt to be entirely a matter of chance. Tragedy, the failure in one’s trial to get something, is a model I introduce with the help of Vilppu’s story. This narrative gets its meaning both from the sorrow of the failure in the past and the rise of something new the teller has found. Aino tells her story as a comedy. By introducing her narrative, I suggest that the purpose of the comedy, a stronger social consensus, gets deeper and darker shade by fixed-term employment: one who works as a fixed term employee has to take his/her place in his/her work community by him/herself without the support the community gives to those in permanent position. By studying the satiric model Rauno uses, I argue that using irony both turns the power structures to a carnival and builds free space to the teller of the story and to the listener. Irony also helps in building a consensus, mutual understanding, between the teller and the listener and it shows the distance the teller tells to exist between him and others. Irony, however, demands some kind of success in one’s occupational career but also at least a minor disappointment in the progress of it. Helmi tells her story merely as a detective story. By introducing Helmi’s narrative, I argue that this story model strengthens the trust in fairness of the society the teller and the listener share. The analysis also emphasizes the central position of identity work, which is caused by fixed-term employment. Most of the interviewees talked about getting along in working life. I introduced Sari’s narrative as an example of this. In both of these latter narratives one’s personal character and habits are lifted as permanent parts of the actual professional expertise, which in turn varies according to different situations. By introducing these models, I reveal that the fixed-term employees have different strategies to cope with their job situations and these strategies vary according to their personal motives and situations and the actual purpose of the interview. However, I argue that they feel the space between their hopes and fears narrow and unsecure. In the research report I also introduce pieces of the stories – themes – that the interviewees use to build these survival strategies. They use their personal curriculum vitae or portfolio, their position in work community and their work morals to build their professional identity. Professional identity is flexible and varies in time and place, but even then it offers a tool to fix one’s identity work into something. It offers a viewpoint to society and a tool to measure one’s position in surrounding social nets. As one result of the study I analyze the position the fixed-term employees share on the edge of their job communities. I summarize the hopes and fears the interviewees have concerning employers, trade unions, educational institutions and the whole society. In their opinion, the solidarity between people has been weakened by the short-sighted power of the economy. The impact the fixed-term employment has on one’s professional identity and social capital is a many-sided and versatile process. Fixed-term employment both strengthens and weakens the professional identity, social capital and the building of trust. Fixed-term employment also affects one’s day-to-day life by excluding her/him from the norm and by one’s difficulty in making long-term plans (Jokinen 2005). Regardless of the nature of the job contract, the workers themselves are experts in making the best of their sometimes less than satisfying work life and they also build their professional identity by using creatively their education, work experiences and interpersonal relations. However, a long career of short fixed-term employments may seriously change the perception of employee about his/her role. He/she may start concentrating only in coping in his/her unsatisfactory situation and leaves the active improvement of the lousy working conditions to other people. Keywords: narrative, fixed-tem employment, occupational identity, work, story model, social capital, career  

A Decentralized Session Management Framework for Heterogeneous Ad-Hoc and Fixed Networks

Wireless technologies are continuously evolving. Second generation cellular networks have gained worldwide acceptance. Wireless LANs are commonly deployed in corporations or university campuses, and their diffusion in public hotspots is growing. Third generation cellular systems are yet to affirm everywhere; still, there is an impressive amount of research ongoing for deploying beyond 3G systems. These new wireless technologies combine the characteristics of WLAN based and cellular networks to provide increased bandwidth. The common direction where all the efforts in wireless technologies are headed is towards an IP-based communication. Telephony services have been the killer application for cellular systems; their evolution to packet-switched networks is a natural path. Effective IP telephony signaling protocols, such as the Session Initiation Protocol (SIP) and the H 323 protocol are needed to establish IP-based telephony sessions. However, IP telephony is just one service example of IP-based communication. IP-based multimedia sessions are expected to become popular and offer a wider range of communication capabilities than pure telephony. In order to conjoin the advances of the future wireless technologies with the potential of IP-based multimedia communication, the next step would be to obtain ubiquitous communication capabilities. According to this vision, people must be able to communicate also when no support from an infrastructured network is available, needed or desired. In order to achieve ubiquitous communication, end devices must integrate all the capabilities necessary for IP-based distributed and decentralized communication. Such capabilities are currently missing. For example, it is not possible to utilize native IP telephony signaling protocols in a totally decentralized way. This dissertation presents a solution for deploying the SIP protocol in a decentralized fashion without support of infrastructure servers. The proposed solution is mainly designed to fit the needs of decentralized mobile environments, and can be applied to small scale ad-hoc networks or also bigger networks with hundreds of nodes. A framework allowing discovery of SIP users in ad-hoc networks and the establishment of SIP sessions among them, in a fully distributed and secure way, is described and evaluated. Security support allows ad-hoc users to authenticate the sender of a message, and to verify the integrity of a received message. The distributed session management framework has been extended in order to achieve interoperability with the Internet, and the native Internet applications. With limited extensions to the SIP protocol, we have designed and experimentally validated a SIP gateway allowing SIP signaling between ad-hoc networks with private addressing space and native SIP applications in the Internet. The design is completed by an application level relay that permits instant messaging sessions to be established in heterogeneous environments. The resulting framework constitutes a flexible and effective approach for the pervasive deployment of real time applications.

Common past, divided memories: historical memory of the Polish minority members in Lithuania

The Master’s thesis examines historical memory of the Polish minority members in Lithuania with regard to how their interpretation of the common Polish-Lithuanian history reiterates or differs from the official Polish and Lithuanian narratives conveyed by the school textbooks. History teaching in high schools carries a crucial state-supported role of “identity building policies” – it maintains a national narrative of memory, which might be exclusive to minorities and their peculiar understanding of history. Lithuanians Poles, in this regard, represent a national minority, which is exposed to two conflicting national narratives of the common past – Polish and Lithuanian. As members of the Polish nation, their understanding of the common Polish-Lithuanian history is conditioned by the Polish historical narrative, acquired as part of the collective memory of the family and/or different minority organizations. On the other hand, they encounter Lithuanian historical narrative of the Polish-Lithuanian past throughout the secondary school history education, where the curriculum, even if taught in Polish, largely represents the Lithuanian point of view. The concept of collective memory is utilized to refer to collective representations of national memory (i.e. publicly articulated narratives and images of collective past in history textbooks) as well as to socially framed individual memories (i.e. historical memory of minority members, where individual remembering is framed by the social context of their identity). The thesis compares the official national historical narratives in Lithuania and Poland, as conveyed by the Polish and Lithuanian history textbooks. The consequent analysis of qualitative interviews with the Polish minority members in Lithuania offers insights into historical memory of Lithuanian Poles and its relation to the official Polish and Lithuanian national narratives of the common past. Qualitative content analysis is applied in both parts of the analysis. The narratives which emerge from the interview data could be broadly grouped into two segments. First, a more pronounced view on the past combines the following elements: i) emphasis on the value of multicultural and diverse past of Lithuania, ii) contestation of “Lithuanocentricity” of the Lithuanian narrative and iii) rejection of the term “occupation”, based on the cultural presuppositions – the dominant position of Polish culture and language in the Vilnius region, symbolic belonging and “Lithuanianness” of the local Poles. While the opposition to the term of “occupation” is in accord with the official Polish narrative conveyed by the textbooks, the former two elements do not neatly adhere to either Polish or Lithuanian textbook narratives. They should rather be considered as an expression of claims for inclusion of plural pasts into Lithuanian collective memory and hence as claims for symbolic enfranchisement into the Lithuanian “imagined community”. The second strand of views, on the other hand, does not exclude assertions about the historically dominant position of Polish culture in Lithuania, but at the same time places more emphasis on the political and historical continuity of the Lithuanian state and highlights a long-standing symbolic connectedness of Vilnius and Lithuania, thus, striking a middle way between the Polish and Lithuanian interpretations of the past.