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.

Minästä toiseen : Lacanilaisen subjektin syntymä Anaïs Ninin House of Incestissä ja Hélène Cixous'n Dedans'issa

The study approaches two modern novels using the conceptual frame of Lacanian psychoanalysis, especially the Lacanian notion of subject. The novels can be described as subversive “Bildungsromans” (development novels) highly influenced by psychoanalytic thought. Anaïs Nin’s (1903—1977) “poetic novel” House of Incest (1936) is a story of sexual and artistic awakening while Hélène Cixous’s (b. 1937) first novel Dedans (1969) depicts the growth of a little girl whose father dies. Both are first novels and first person narratives. Concentrating in the narrator’s internal life the novels writings break with the realistic conventions of narrative, bringing forth the themes of anguish, alienation from the world and escape into the prison like realm of the self. The study follows roughly the Lacanian process of becoming a subject. Each chapter opens up with a quick introduction to the Lacanian concepts used in the following part that analyses the novels. The study can thus also be used as a brief introduction to Lacanian theory in finnish. The psychoanalytic narrative/story of the birth of the subject and the novels stories can be seen as mirroring each other. The method of the study is thus based on a dialogue between the theoretical concepts and the analyses. Novels are being approached as texts that break with the Cartesian notion of an autonomous subject making room for a dialectics of self and other, for a movement in which the “I” builds an identity mirroring itself with others. While both of the novels recount the birth of a character called I, they also have a first person narrator apart from the character “I”. Having constituted the self’s identity, the narrator finds from inside of the self also an other or “you” – this discovery is the final clue to the coffin of the autonomous self. From the Lacanian perspective man’s great Other is the order of language, Symbolic, which constitutes the individual, the speaking subject. Using this perspective the novels are interpreted as describing the process of becoming a subject of the Symbolic; subjected to Symbolic order. This “birth process” happens in particular in the Imaginary register, where the self’s identity is built. In the Imaginary or Mirror phase the “I” mirrors himself with different others (e.g. with his mirror image and the family members, the surrounding others) learning to see his body and his selfhood both as familiar and strange, other. In the Imaginary phase the novels’ characters are also trying to deal with the opposite realm of the Symcolic, the Real. The Lacanian Real is not the reality “before words” but a reality left over from the Symbolic, aside of it but constituted by the Symbolic, to be deducted only from within it. In the novels the Real is experienced as a womblike state where the self is immersed in the other’s body. The process of coming a subject of the Symbolic is depicted also as a process of renouncing the “dream of the womb”, which, if realized, could only mean the non-existence of the subject, i.e. death. The study concentrates on analysing the novels’ writing, where meanings are constantly changing: “I” becomes you, the father becomes a mother, inside becomes outside. This technique enables also the deconstruction of certain opposing notions in the novels. The Lacanian point of view exposes language as a constantly moving universe where the subject has no more stability than the momentary meanings language creates. The self’s identity depicted in the novels is a Lacanian fixed identity, whose growth is necessary but opposes the flux imminent to the Symbolic. The anguish experienced in the novels, in the “house of incest” or “inside”, is due to clinging on the unchanging “I”. However, the writing of the novels shows how the meaning of the “I” changes constantly and the fixity thus becomes movement. This way House of Incest and Dedans, despite their pessimistic stories, manage to create an image of a new, moving subject.

Lausunta taiteena ja tapahtumana : Tekstikeskeisyys, esittäjäkeskeisyys ja lausunnan positiot lausuntaa koskevan kirjallisen aineiston ja teatterintutkimuksen viitekehyksessä

The research is analyzing the Finnish tradition on poetry elocution both from the discource analytical and theatrical and view point. The main questions are, whether there still is a fixed position for elocution in the field of art or are we dealing with just one form of thearte? -- The art of elocution has been considered as an independent art form, sometimes even in opposition to theatre, which has been regarded as a very physical and emotional art form by the elocutionists themselves. The self-image of the Finnish elocution art has been born and firmly sustained from the notion that elocution is linked to literature. Elocution as an art form has been seen as "pure" and humbly serving literature and poetry. The main function of an elocutionist has been to understand and vocally express the meanings found in a poem to larger audience. -- This function has changed over the time. There have been a transition from the traditonal text-centeredness to performer-centeredness, even to performance-centeredness as a new wave of theatrical elements and methods have reached the circle of elocution. New forms of poetry performances, such as poetry reading and poetry slam are new challenges to elocution as it must reconsider it´s traditional function in a new artistic and cultural context.

Burnt area mapping in insular Southeast Asia using medium resolution satellite imagery

Burnt area mapping in humid tropical insular Southeast Asia using medium resolution (250-500m) satellite imagery is characterized by persisting cloud cover, wide range of land cover types, vast amount of wetland areas and highly varying fire regimes. The objective of this study was to deepen understanding of three major aspects affecting the implementation and limits of medium resolution burnt area mapping in insular Southeast Asia: 1) fire-induced spectral changes, 2) most suitable multitemporal compositing methods and 3) burn scars patterns and size distribution. The results revealed a high variation in fire-induced spectral changes depending on the pre-fire greenness of burnt area. It was concluded that this variation needs to be taken into account in change detection based burnt area mapping algorithms in order to maximize the potential of medium resolution satellite data. Minimum near infrared (MODIS band 2, 0.86μm) compositing method was found to be the most suitable for burnt area mapping purposes using Moderate Resolution Imaging Spectroradiometer (MODIS) data. In general, medium resolution burnt area mapping was found to be usable in the wetlands of insular Southeast Asia, whereas in other areas the usability was seriously jeopardized by the small size of burn scars. The suitability of medium resolution data for burnt area mapping in wetlands is important since recently Southeast Asian wetlands have become a major point of interest in many fields of science due to yearly occurring wild fires that not only degrade these unique ecosystems but also create regional haze problem and release globally significant amounts of carbon into the atmosphere due to burning peat. Finally, super-resolution MODIS images were tested but the test failed to improve the detection of small scars. Therefore, super-resolution technique was not considered to be applicable to regional level burnt area mapping in insular Southeast Asia.

Digitoitujen väri-infrailmakuvien numeerinen monikuvatulkinta metsäninventoinnissa

The use of remote sensing imagery as auxiliary data in forest inventory is based on the correlation between features extracted from the images and the ground truth. The bidirectional reflectance and radial displacement cause variation in image features located in different segments of the image but forest characteristics remaining the same. The variation has so far been diminished by different radiometric corrections. In this study the use of sun azimuth based converted image co-ordinates was examined to supplement auxiliary data extracted from digitised aerial photographs. The method was considered as an alternative for radiometric corrections. Additionally, the usefulness of multi-image interpretation of digitised aerial photographs in regression estimation of forest characteristics was studied. The state owned study area located in Leivonmäki, Central Finland and the study material consisted of five digitised and ortho-rectified colour-infrared (CIR) aerial photographs and field measurements of 388 plots, out of which 194 were relascope (Bitterlich) plots and 194 were concentric circular plots. Both the image data and the field measurements were from the year 1999. When examining the effect of the location of the image point on pixel values and texture features of Finnish forest plots in digitised CIR photographs the clearest differences were found between front-and back-lighted image halves. Inside the image half the differences between different blocks were clearly bigger on the front-lighted half than on the back-lighted half. The strength of the phenomenon varied by forest category. The differences between pixel values extracted from different image blocks were greatest in developed and mature stands and smallest in young stands. The differences between texture features were greatest in developing stands and smallest in young and mature stands. The logarithm of timber volume per hectare and the angular transformation of the proportion of broadleaved trees of the total volume were used as dependent variables in regression models. Five different converted image co-ordinates based trend surfaces were used in models in order to diminish the effect of the bidirectional reflectance. The reference model of total volume, in which the location of the image point had been ignored, resulted in RMSE of 1,268 calculated from test material. The best of the trend surfaces was the complete third order surface, which resulted in RMSE of 1,107. The reference model of the proportion of broadleaved trees resulted in RMSE of 0,4292 and the second order trend surface was the best, resulting in RMSE of 0,4270. The trend surface method is applicable, but it has to be applied by forest category and by variable. The usefulness of multi-image interpretation of digitised aerial photographs was studied by building comparable regression models using either the front-lighted image features, back-lighted image features or both. The two-image model turned out to be slightly better than the one-image models in total volume estimation. The best one-image model resulted in RMSE of 1,098 and the two-image model resulted in RMSE of 1,090. The homologous features did not improve the models of the proportion of broadleaved trees. The overall result gives motivation for further research of multi-image interpretation. The focus may be improving regression estimation and feature selection or examination of stratification used in two-phase sampling inventory techniques. Keywords: forest inventory, digitised aerial photograph, bidirectional reflectance, converted image co­ordinates, regression estimation, multi-image interpretation, pixel value, texture, trend surface

Composition operators, Aleksandrov measures and value distribution of analytic maps in the unit disc

A composition operator is a linear operator that precomposes any given function with another function, which is held fixed and called the symbol of the composition operator. This dissertation studies such operators and questions related to their theory in the case when the functions to be composed are analytic in the unit disc of the complex plane. Thus the subject of the dissertation lies at the intersection of analytic function theory and operator theory. The work contains three research articles. The first article is concerned with the value distribution of analytic functions. In the literature there are two different conditions which characterize when a composition operator is compact on the Hardy spaces of the unit disc. One condition is in terms of the classical Nevanlinna counting function, defined inside the disc, and the other condition involves a family of certain measures called the Aleksandrov (or Clark) measures and supported on the boundary of the disc. The article explains the connection between these two approaches from a function-theoretic point of view. It is shown that the Aleksandrov measures can be interpreted as kinds of boundary limits of the Nevanlinna counting function as one approaches the boundary from within the disc. The other two articles investigate the compactness properties of the difference of two composition operators, which is beneficial for understanding the structure of the set of all composition operators. The second article considers this question on the Hardy and related spaces of the disc, and employs Aleksandrov measures as its main tool. The results obtained generalize those existing for the case of a single composition operator. However, there are some peculiarities which do not occur in the theory of a single operator. The third article studies the compactness of the difference operator on the Bloch and Lipschitz spaces, improving and extending results given in the previous literature. Moreover, in this connection one obtains a general result which characterizes the compactness and weak compactness of the difference of two weighted composition operators on certain weighted Hardy-type spaces.

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.

A Floor Control Server in a Distributed Conference Service

The conferencing systems in IP Multimedia (IM) networks are going through restructuring, accomplished in the near future. One of the changes introduced is the concept of floors and floor control in its current form with matching entity roles. The Binary Floor Control Protocol (BFCP) is a novelty to be exploited in distributed tightly coupled conferencing services. The protocol defines the floor control server (FCS), which implements floor control giving access to shared resources. As the newest tendency is to distribute the conferencing services, the locations of different functionality units play an important role in developing the standards. The floor control server location is not yet single-mindedly fixed in different standardization bodies, and the debate goes on where to place it within the media server, providing the conferencing service. The thesis main objective is to evaluate two distinctive alternatives in respect the Mp interface protocol between the respective nodes, as the interface in relation to floor control is under standardization work at the moment. The thesis gives a straightforward preamble in IMS network, nodes of interest including floor control server and conferencing. Knowledge on several protocols – BFCP, SDP, SIP and H.248 provides an important background for understanding the functionality changes introduced in the Mp interface and therefore introductions on those protocols and how they are connected to the full picture is given. The actual analysis on the impact of the floor control server into the Mp reference point is concluded in relation to the locations, giving basic flows, requirements analysis including a limited implementation proposal on supporting protocol parameters. The overall conclusion of the thesis is that even if both choices are seemingly useful, not one of the locations is clearly the most suitable in the light of this work. The thesis suggests a solution having both possibilities available to be chosen from in separate circumstances, realized with consistent standardization. It is evident, that if the preliminary assumption for the analysis is kept regarding to only one right place for the floor control server, more work is to be done in connected areas to discover the one most appropriate location.

Tila tilassa : Liturgian ja tilan dialogi alttarin äärellä

This study examines the organisation and transformation of altar space in the modern Evangelical Lutheran Church of Finland in liturgical and architectural perspective. The research data consists of 65 altar spaces in The Finnish Evangelical Lutheran church buildings. All of these were characterised in Church Government records as churches , built 1962 1999 and had been consecrated. The main data was collected by means of observation, photographing, and drawing sketches of altar spaces. The focus of this study concerns the organisation of modern Finnish Evangelical Lutheran altar spaces and, in particular, their changes also in relation to the liturgical movement. The challenge of this approach was especially in discovering the spatial identity of an altar space in terms of unequivocal boundaries. The analysis was realised in three stages. Interiors, the organisation of altar space, as well as architectonic qualities of altar spaces in terms of floor elevations, shapes of ceilings, lighting, and openings in the altar space were analysed. Moreover, attention was focused on furnishing and fixed versus movable pieces of furniture (such as the altar, altar rail, the pulpit, the baptismal font, and lectern). Finally, the potential qualitative and quantitative changes in altar space were examined. All in all, the majority of churches in the data featured elongated church halls with an altar at the end of the nave. To look at the data in chronological perspective, increasingly wide church halls had been built since the 1980s (yet there was only one central hall in which the altar was placed at the middle point of the church). Every third church altar was movable. As for the focal point of this study and the altar in particular, it was my aim to pay attention to the versus populum altar and its development in relation to the (Lutheran) liturgy. Hence, it was meaningful to determine, in terms of interior design, whether liturgists were able to celebrate facing the people attending the service. In the 1960s and 70s, a versus orientem altar featured in more than half of all new Finnish Lutheran churches, yet in 2000 two out of three churches featured a versus populum altar. For architectural and esthetic reasons (and not primarily due to liturgical ideas), also altars standing freely off the walls had been constructed. In terms of the liturgy, versus populum altars had been realised in expectation of increased communication between liturgist and worshippers. However, the analysis indicated that the altar could also become a divider of space. This aspect is a novel finding in relation to earlier and concurrent discussions concerning the liturgical movement. This study concluded, all in all, that altars had been increasingly constructed closer and closer to the worshiping parish and, accordingly, used increasingly often in the versus populum manner. Lecterns were often movable until the millennium this was the case in most altar spaces. Baptismal fonts did not have a permanent place in this data, and the data even included altar spaces with no baptismal fonts in the choir, nor the church hall. The position and status of fonts was generally weakened even if baptism in the Lutheran Church was regarded as one of the two sacraments together with the eucharist. The study concluded that even if baptism is regarded as a sacrament in the church, the position and status of baptismal fonts had weakened overall in newer church architecture. In other words, the tendency of the liturgical movement to emphasise the service and its celebration had obviously had its effect on the placement of baptismal fonts in the church hall. This research indicated that the pieces of furniture that mostly involved (many kinds of) visual and spatial changes included the altar and the lectern. In certain instances, fixed furnishings had been substituted by movable pieces or, moreover, new pieces of furniture and paraphernalia such as music instruments, pieces of art, tables, chairs and plants were brought in. In the Evangelical Lutheran Church of Finland, liturgical changes were principally inspired by the Catholic Church, in which liturgical changes are essentially based on Canon Law. Unlike Finnish Lutheranism, Catholicism provides detailed rules and principles even regarding the design of an altar space. According to this study, in the Finnish Lutheran Church, the primarily functional nature of given guidelines and instructions characterises several practical solutions in furnishing.

Strategic Information Acquisition: Applications to Herding and Auctions

The information that the economic agents have and regard relevant to their decision making is often assumed to be exogenous in economics. It is assumed that the agents either poses or can observe the payoff relevant information without having to exert any effort to acquire it. In this thesis we relax the assumption of ex-ante fixed information structure and study what happens to the equilibrium behavior when the agents must also decide what information to acquire and when to acquire it. This thesis addresses this question in the two essays on herding and two essays on auction theory. In the first two essays, that are joint work with Klaus Kultti, we study herding models where it is costly to acquire information on the actions that the preceding agents have taken. In our model the agents have to decide both the action that they take and additionally the information that they want to acquire by observing their predecessors. We characterize the equilibrium behavior when the decision to observe preceding agents' actions is endogenous and show how the equilibrium outcome may differ from the standard model, where all preceding agents actions are assumed to be observable. In the latter part of this thesis we study two dynamic auctions: the English and the Dutch auction. We consider a situation where bidder(s) are uninformed about their valuations for the object that is put up for sale and they may acquire this information for a small cost at any point during the auction. We study the case of independent private valuations. In the third essay of the thesis we characterize the equilibrium behavior in an English auction when there are informed and uninformed bidders. We show that the informed bidder may jump bid and signal to the uninformed that he has a high valuation, thus deterring the uninformed from acquiring information and staying in the auction. The uninformed optimally acquires information once the price has passed a particular threshold and the informed has not signalled that his valuation is high. In addition, we provide an example of an information structure where the informed bidder initially waits and then makes multiple jumps. In the fourth essay of this thesis we study the Dutch auction. We consider two cases where all bidders are all initially uninformed. In the first case the information acquisition cost is the same across all bidders and in the second also the cost of information acquisition is independently distributed and private information to the bidders. We characterize a mixed strategy equilibrium in the first and a pure strategy equilibrium in the second case. In addition we provide a conjecture of an equilibrium in an asymmetric situation where there is one informed and one uninformed bidder. We compare the revenues that the first price auction and the Dutch auction generate and we find that under some circumstances the Dutch auction outperforms the first price sealed bid auction. The usual first price sealed bid auction and the Dutch auction are strategically equivalent. However, this equivalence breaks down in case information is acquired during the auction.

Carbon balance and component CO2 fluxes in boreal Scots pine stands

The forest vegetation takes up atmospheric carbon dioxide (CO2) in photosynthesis. Part of the fixed carbon is released back into the atmosphere during plant respiration but a substantial part is stored as plant biomass, especially in the stems of trees. Carbon also accumulates in the soil as litter and via the roots. CO2 is released into the atmosphere from these carbon stocks in the decomposition of dead biomass. Carbon balance of a forest stand is the difference between the CO2 uptake and CO2 efflux. This study quantifies and analyses the dynamics of carbon balance and component CO2 fluxes in four Southern Finnish Scots pine stands that covered the typical economic rotation time of 80 years. The study was based on direct flux measurements with chambers and eddy covariance (EC), and modelling of component CO2 fluxes. The net CO2 exchange of the stand was partitioned into component fluxes: photosynthesis of trees and ground vegetation, respiration of tree foliage and stems, and CO2 efflux from the soil. The relationships between the component fluxes and the environmental factors (light, temperature, atmospheric CO2, air humidity and soil moisture) were studied with mathematical modelling. The annual CO2 balance varied from a source of about 400 g C/m2 at a recently clearcut site to net CO2 uptake of 200 300 g C/m2 in a middle-aged (40-year-old) and a mature (75-year-old) stand. A 12-year-old sapling site was at the turning point from source to a sink of CO2. In the middle-aged stand, photosynthetic production was dominated by trees. Under closed pine canopies, ground vegetation accounted for 10 20% of stand photosynthesis whereas at the open sites the proportion and also the absolute photosynthesis of ground vegetation was much higher. The aboveground respiration was dominated by tree foliage which accounted for one third of the ecosystem respiration. Rate of wood respiration was in the order of 10% of total ecosystem respiration. CO2 efflux from the soil dominated the ecosystem respiratory fluxes in all phases of stand development. Instantaneous and delayed responses to the environmental driving factors could predict well within-year variability in photosynthetic production: In the short term and during the growing season photosynthesis follows primarily light while the seasonal variation is more strongly connected to temperature. The temperature relationship of the annual cycle of photosynthesis was found to be almost equal in the southern boreal zone and at the timberline in the northern boreal zone. The respiratory fluxes showed instantaneous and seasonal temperature relationships but they could also be connected to photosynthesis at an annual timescale.

The effects of low-intensity ultrasound in bioabsorbable self-reinforced poly-L-lactide -fixed cancellous bone fracture

The purpose of the present study was to investigate the effects of low-intensity ultrasound on bioabsorbable self-reinforced poly-L-lactide (SR-PLLA) screws and on fracture healing after SR-PLLA device fixation in experimental and clinical cancellous bone fracture. In the first experimental study, the assessment of the mechanical strengths of the SR-PLLA screws was performed after 12 weeks of daily 20-minute ultrasound exposure in vitro. In the second experimental study, 32 male Wistar rats with an experimental distal femur osteotomy fixed with an SR-PLLA rod were exposed for daily low-intensity ultrasound treatment for 21 days. The effects on the healing bone were assessed. The clinical studies consist of three prospective, randomized, and placebo-controlled series of dislocated lateral malleolar fractures fixed with one SR-PLLA screw. The total number of the patients in these series was 52. Half of the patients were provided randomly with a sham ultrasound device. The patients underwent ultrasound therapy 20 minutes daily for six weeks. Radiological bone healing was assessed both by radiographs at two, six, nine, and 12 weeks and by multidetector computed tomography (MDCT) scans at two weeks, nine weeks, and 18 months. Bone mineral density was assessed by dual-energy X-ray absorptiometry (DXA). The clinical outcome was assessed by both Olerud-Molander scoring and clinical examination of the ankle. Low-intensity ultrasound had no effects on the mechanical properties and degradation behaviour of the SR-PLLA screws in vitro. There were no obvious signs of low-intensity ultrasound-induced enhancement in the bone healing in SR-PLLA-rod-fixed metaphyseal distal femur osteotomy in rats. The biocompatibility of low-intensity ultrasound treatment and SR-PLLA was found to be good. In the clinical series low-intensity ultrasound was observed to have no obvious effects on the bone mineral density of the fractured lateral malleolus. There were no obvious differences in the radiological bone healing times of the SR-PLLA-screw-fixed lateral malleolar fractures after low-intensity ultrasound treatment. Low-intensity ultrasound did not have any effects on radiological bone morphology, bone mineral density or clinical outcome 18 months after the injury. There were no obvious findings in the present study to support the hypothesis that low-intensity pulsed ultrasound enhances bone healing in SR-PLLA-rod-fixed experimental metaphyseal distal femur osteotomy in rats or in clinical SR-PLLA-screw-fixed lateral malleolar fractures. It is important to limit the conclusions of the present set of studies only to lateral malleolar fractures fixed with an SR-PLLA screw.