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.

Global, Local and Vector-Valued Tb Theorems on Non-Homogeneous Metric Spaces

Various Tb theorems play a key role in the modern harmonic analysis. They provide characterizations for the boundedness of Calderón-Zygmund type singular integral operators. The general philosophy is that to conclude the boundedness of an operator T on some function space, one needs only to test it on some suitable function b. The main object of this dissertation is to prove very general Tb theorems. The dissertation consists of four research articles and an introductory part. The framework is general with respect to the domain (a metric space), the measure (an upper doubling measure) and the range (a UMD Banach space). Moreover, the used testing conditions are weak. In the first article a (global) Tb theorem on non-homogeneous metric spaces is proved. One of the main technical components is the construction of a randomization procedure for the metric dyadic cubes. The difficulty lies in the fact that metric spaces do not, in general, have a translation group. Also, the measures considered are more general than in the existing literature. This generality is genuinely important for some applications, including the result of Volberg and Wick concerning the characterization of measures for which the analytic Besov-Sobolev space embeds continuously into the space of square integrable functions. In the second article a vector-valued extension of the main result of the first article is considered. This theorem is a new contribution to the vector-valued literature, since previously such general domains and measures were not allowed. The third article deals with local Tb theorems both in the homogeneous and non-homogeneous situations. A modified version of the general non-homogeneous proof technique of Nazarov, Treil and Volberg is extended to cover the case of upper doubling measures. This technique is also used in the homogeneous setting to prove local Tb theorems with weak testing conditions introduced by Auscher, Hofmann, Muscalu, Tao and Thiele. This gives a completely new and direct proof of such results utilizing the full force of non-homogeneous analysis. The final article has to do with sharp weighted theory for maximal truncations of Calderón-Zygmund operators. This includes a reduction to certain Sawyer-type testing conditions, which are in the spirit of Tb theorems and thus of the dissertation. The article extends the sharp bounds previously known only for untruncated operators, and also proves sharp weak type results, which are new even for untruncated operators. New techniques are introduced to overcome the difficulties introduced by the non-linearity of maximal truncations.

Liikennepolttoaineeksi tarkoitettu ohraetanoli, Suomen ohramarkkinat ja ilmastonmuutos

The aim of the thesis was to analyze the use of barley as an input for bioethanol production and the impacts the use has on the Finnish barley markets. Two main research questions were formulated. First, privately and socially optimal bioethanol production levels were examined. In the social optimum, the climate benefits of bioethanol production were considered. It was calculated that the production and use of bioethanol created smaller CO2 -emissions when compared with the production and use of gasoline. Second, the impacts of bioethanol production on farmland allocation and agricultural production were analyzed. In more detail, the second aim was to analyze the farmland allocation between wheat and barley cultivation and green set aside in the private and social optimum. An analytical model was produced to analyze the barley markets in Finland. To provide an empirical counterpart to this model, existing research data on bioethanol production and barley cultivation was used. The aim of the model was to analyze the supply and the demand as well as market equilibrium of barley. Furthermore, the model provided a framework for analyzing the differences between the private optimum and social optimum of bioethanol production in Finland. The demand for barley consists of animal feed demand and bioethanol demand. On the supply side, a heterogeneous model of farmland quality was used. With this framework, it is possible to analyze farmland allocation between barley and wheat cultivation and green set aside and how the climate benefits of bioethanol production affects the allocation. Moreover, the relative changes in barley price between the private and social optimum were analyzed. Based on the empirical analysis, the private optimum for barley based bioethanol production is 58 691 metric tons. However, the social optimum for barley based bioethanol production is 72 736 metric tons. The portion of farmland that is allocated to barley cultivation is increased if the climate benefits of bioethanol production are considered. In the private optimum, 1/19 of the total farmland is allocated to barley cultivation whereas in social optimum the share increases to 7/19. Furthermore, the increase in barley price between private and social optimum is rather modest. Total increase in price is only about 1,8 percent.

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.

On Orlicz-Sobolev capacities

This thesis consists of three articles on Orlicz-Sobolev capacities. Capacity is a set function which gives information of the size of sets. Capacity is useful concept in the study of partial differential equations, and generalizations of exponential-type inequalities and Lebesgue point theory, and other topics related to weakly differentiable functions such as functions belonging to some Sobolev space or Orlicz-Sobolev space. In this thesis it is assumed that the defining function of the Orlicz-Sobolev space, the Young function, satisfies certain growth conditions. In the first article, the null sets of two different versions of Orlicz-Sobolev capacity are studied. Sufficient conditions are given so that these two versions of capacity have the same null sets. The importance of having information about null sets lies in the fact that the sets of capacity zero play similar role in the Orlicz-Sobolev space setting as the sets of measure zero do in the Lebesgue space and Orlicz space setting. The second article continues the work of the first article. In this article, it is shown that if a Young function satisfies certain conditions, then two versions of Orlicz-Sobolev capacity have the same null sets for its complementary Young function. In the third article the metric properties of Orlicz-Sobolev capacities are studied. It is usually difficult or impossible to calculate a capacity of a set. In applications it is often useful to have estimates for the Orlicz-Sobolev capacities of balls. Such estimates are obtained in this paper, when the Young function satisfies some growth conditions.

Locality and order-invariant logics

Malli on logiikassa käytetty abstraktio monille matemaattisille objekteille. Esimerkiksi verkot, ryhmät ja metriset avaruudet ovat malleja. Äärellisten mallien teoria on logiikan osa-alue, jossa tarkastellaan logiikkojen, formaalien kielten, ilmaisuvoimaa malleissa, joiden alkioiden lukumäärä on äärellinen. Rajoittuminen äärellisiin malleihin mahdollistaa tulosten soveltamisen teoreettisessa tietojenkäsittelytieteessä, jonka näkökulmasta logiikan kaavoja voidaan ajatella ohjelmina ja äärellisiä malleja niiden syötteinä. Lokaalisuus tarkoittaa logiikan kyvyttömyyttä erottaa toisistaan malleja, joiden paikalliset piirteet vastaavat toisiaan. Väitöskirjassa tarkastellaan useita lokaalisuuden muotoja ja niiden säilymistä logiikkoja yhdistellessä. Kehitettyjä työkaluja apuna käyttäen osoitetaan, että Gaifman- ja Hanf-lokaalisuudeksi kutsuttujen varianttien välissä on lokaalisuuskäsitteiden hierarkia, jonka eri tasot voidaan erottaa toisistaan kasvavaa dimensiota olevissa hiloissa. Toisaalta osoitetaan, että lokaalisuuskäsitteet eivät eroa toisistaan, kun rajoitutaan tarkastelemaan äärellisiä puita. Järjestysinvariantit logiikat ovat kieliä, joissa on käytössä sisäänrakennettu järjestysrelaatio, mutta sitä on käytettävä siten, etteivät kaavojen ilmaisemat asiat riipu valitusta järjestyksestä. Määritelmää voi motivoida tietojenkäsittelyn näkökulmasta: vaikka ohjelman syötteen tietojen järjestyksellä ei olisi odotetun tuloksen kannalta merkitystä, on syöte tietokoneen muistissa aina jossakin järjestyksessä, jota ohjelma voi laskennassaan hyödyntää. Väitöskirjassa tutkitaan minkälaisia lokaalisuuden muotoja järjestysinvariantit ensimmäisen kertaluvun predikaattilogiikan laajennukset yksipaikkaisilla kvanttoreilla voivat toteuttaa. Tuloksia sovelletaan tarkastelemalla, milloin sisäänrakennettu järjestys lisää logiikan ilmaisuvoimaa äärellisissä puissa.

Methods for Answer Extraction in Textual Question Answering

In this thesis we present and evaluate two pattern matching based methods for answer extraction in textual question answering systems. A textual question answering system is a system that seeks answers to natural language questions from unstructured text. Textual question answering systems are an important research problem because as the amount of natural language text in digital format grows all the time, the need for novel methods for pinpointing important knowledge from the vast textual databases becomes more and more urgent. We concentrate on developing methods for the automatic creation of answer extraction patterns. A new type of extraction pattern is developed also. The pattern matching based approach chosen is interesting because of its language and application independence. The answer extraction methods are developed in the framework of our own question answering system. Publicly available datasets in English are used as training and evaluation data for the methods. The techniques developed are based on the well known methods of sequence alignment and hierarchical clustering. The similarity metric used is based on edit distance. The main conclusions of the research are that answer extraction patterns consisting of the most important words of the question and of the following information extracted from the answer context: plain words, part-of-speech tags, punctuation marks and capitalization patterns, can be used in the answer extraction module of a question answering system. This type of patterns and the two new methods for generating answer extraction patterns provide average results when compared to those produced by other systems using the same dataset. However, most answer extraction methods in the question answering systems tested with the same dataset are both hand crafted and based on a system-specific and fine-grained question classification. The the new methods developed in this thesis require no manual creation of answer extraction patterns. As a source of knowledge, they require a dataset of sample questions and answers, as well as a set of text documents that contain answers to most of the questions. The question classification used in the training data is a standard one and provided already in the publicly available data.

Measuring and tracking quality factors in Free and Open Source Software projects

Free and Open Source Software (FOSS) has gained increased interest in the computer software industry, but assessing its quality remains a challenge. FOSS development is frequently carried out by globally distributed development teams, and all stages of development are publicly visible. Several product and process-level quality factors can be measured using the public data. This thesis presents a theoretical background for software quality and metrics and their application in a FOSS environment. Information available from FOSS projects in three information spaces are presented, and a quality model suitable for use in a FOSS context is constructed. The model includes both process and product quality metrics, and takes into account the tools and working methods commonly used in FOSS projects. A subset of the constructed quality model is applied to three FOSS projects, highlighting both theoretical and practical concerns in implementing automatic metric collection and analysis. The experiment shows that useful quality information can be extracted from the vast amount of data available. In particular, projects vary in their growth rate, complexity, modularity and team structure.

Total hip arthroplasty in young patients : with special references to patients under 55 years of age and to patients with developmental dysplasia of the hip

There is an ongoing controversy as to which methods in total hip arthroplasty (THA) could provide young patients with best long-term results. THA is an especially demanding operation in patients with severely dysplastic hips. The optimal surgical treatment for these patients also remains controversial. The aim of this study was to evaluate the long-term survival of THA in young patients (<55 years at the time of the primary operation) on a nation-wide level, and to analyze the long-term clinical and radio-graphical outcome of uncemented THA in patients with severely dysplastic joints. Survival of 4661 primary THAs performed for primary osteoarthritis (OA), 2557 primary THAs per-formed for rheumatoid arthritis (RA), and modern uncemented THA designs performed for primary OA in young patients, were analysed from the Finnish Arthroplasty Register. A total of 68 THAs were per-formed in 56 consecutive patients with high congenital hip dislocation between 1989-1994, and 68 THAs were performed in 59 consecutive patients with severely dysplastic hips and a previous Schanz osteotomy of the femur between 1988-1995 at the Orton Orthopaedic Hospital, Helsinki, Finland. These patients underwent a detailed physical and radiographical evaluation at a mean of 12.3 years and 13.0 years postoperatively, respectively. The risk of stem revision due to aseptic loosening in young patients with primary OA was higher for cemented stems than for proximally porous-coated or HA-coated uncemented stems implanted over the 1991-2001 period. There was no difference in the risk of revision between all-poly cemented-cups and press-fit porous-coated uncemented cups implanted during the same period, when the end point was defined as any revision (including exchange of liner). All uncemented stem designs studied in young patients with primary OA had >90% survival rates at 10 years. The Biomet Bi-Metric stem had a 95% (95% CI 93-97) survival rate even at 15 years. When the end point was defined as any revision, 10 year survival rates of all uncemented cup designs except the Harris-Galante II decreased to <80%. In young patients with RA, the risk of stem revision due to aseptic loosening was higher with cemented stems than with proximally porous-coated uncemented stems. In contrast, the risk of cup revision was higher for all uncemented cup concepts than for all-poly cemented cups with any type of cup revision as the end point. The Harris hip score increased significantly (p<0.001) both in patients with high con-genital hip dislocation and in patients with severely dysplastic hips and a previous Schanz osteotomy, treated with uncemented THA. There was a negative Trendelenburg sign in 92% and in 88% of hips, respectively. There were 12 (18%) and 15 (22%) perioperative complications. The rate of survival for the CDH femoral components, with revision due to aseptic loosening as the end point, was 98% (95% CI 97-100) at 10 years in patients with high hip dislocation and 92% (95% CI, 86-99) at 14 years in patients with a previous Schanz osteotomy. The rate of survival for press-fit, porous-coated acetabular components, with revision due to aseptic loosening as the end point, was 95% (95% CI 89-100) at 10 years in patients with high hip dislocation, and 98% (95% CI 89-100) in patients with a previous Schanz osteotomy. When revision of the cup for any reason was defined as the end point, 10 year sur-vival rates declined to 88% (95% CI 81-95) and to 69% (95% CI, 56-82), respectively. For young patients with primary OA, uncemented proximally circumferentially porous- and HA-coated stems are the implants of choice. However, survival rates of modern uncemented cups are no better than that of all-poly cemented cups. Uncemented proximally circumferentially porous-coated stems and cemented all-poly cups are currently the implants of choice for young patients with RA. Uncemented THA, with placement of the cup at the level of the true acetabulum, distal advancement of the greater trochanter and femoral shortening osteotomy provided patients with high congenital hip dislocation good long-term outcomes. Most of the patients with severely dysplastic hips and a previous Schanz osteotomy can be successfully treated with the same method. However, the subtrochanteric segmental shortening with angular correction gives better leg length correction for the patients with a previous low-seated unilateral Schanz osteotomy.

Juvenile diversity of tree species in Theobroma cacao L. agroforestry and primary forest – Central Sulawesi, Indonesia

This thesis studies the tree species’ juvenile diversity in cacao (Theobroma cacao L.) based agroforestry and in primary forest in a natural conservation forest environment of Lore Lindu National Park, Sulawesi, Indonesia. Species’ adult composition in Lore Lindu National Park is relatively well studied, less is known about tree species’ diversity in seedling communities particularly in frequently disturbed cacao agroforestry field environment. Cacao production forms a potentially serious thread for maintaining the conservation areas pristine and forested in Sulawesi. The impacts of cacao production on natural environment are directly linked to the diversity and abundance of shade tree usage. The study aims at comparing differences between cacao agroforestry and natural forest in the surrounding area in their species composition in seedling and sapling size categories. The study was carried out in two parts. Biodiversity inventory of seedlings and saplings was combined with social survey with farmer interviews. Aim of the survey was to gain knowledge of the cacao fields, and farmers’ observations and choices regarding tree species associated with cacao. Data was collected in summer 2008. The assessment of the impact of environmental factors of solar radiation, weeding frequency, cacao tree planting density, distance to forest and distance to main park road, and type of habitat on seedling and sapling compositions was done with Non-metric Multidimensional Scaling (NMS). Outlier analysis was used to assess distorting variables for NMS, and Multi-Response Permutation Procedures (MRPP) analysis to differentiate the impact of categorical variables. Sampling success was estimated with rarefaction curves and jackknife estimate of species richness. In the inventory 135 species of trees and shrubs were found. Only some agroforestry related species were dominating. The most species rich were sapling communities in forest habitat. NMS was showing generally low linear correlation between variation of species composition and environmental variables. Solar radiation was having most significance as explaining variable. The most clearly separated in ordination were cacao and forest habitats. The results of seedling and sapling inventory were only partly coinciding with farmers’ knowledge of the tree species occurring on their fields. More research with frequent assessment of seedling cohorts is needed due to natural variability of cohorts and high mortality rate of seedlings.

Metsikön rakenteen ja hakkuiden vaikutus pintakasvillisuuteen Etelä-Suomen yleisimmillä kasvupaikkatyypeillä

Taman tutkielman tarkoituksena oli selvittaa metsikon rakenteen seka hakkuiden vaikutuksia pintakasvillisuuden lajikoostumukseen ja biomassaan Etela-Suomen lehtomaisilla, tuoreilla ja kuivahkoilla kankailla. Aineistona tassa tyossa on 8. valtakunnan metsien inventoinnin yhteydessa vuosina 1985–86 metsaluonnon ja ympariston tilan seurantaa varten perustetuista noin 3 000 pysyvasta koealasta poimittu otos. Pintakasvillisuuden lajisto muuttuu metsikon kehitysvaiheen mukaan. Hakkuu on huomattava hairio, joka aiheuttaa nopeita ja suuria muutoksia pintakasvillisuudessa. Pintakasvillisuutta on tarkasteltu lahinna lajiryhmittain (heinat, ruohot, varvut, sammalet seka jakalat). Kunkin lajiryhman peittavyyden eroavaisuuksia testattiin varianssianalyysilla kun selittavana muuttujana ovat luokittain metsikon ika ja edellisesta hakkuusta kulunut aika. Lajikohtaisia tarkasteluja on sen sijaan tehty kasvillisuuden ordinaatioanalyyseilla. Tassa kaytetty ordinaatiomenetelma on epametrinen moniulotteinen skaalaus (Non-metric multidimensional scaling, NMDS), jonka avulla voidaan tehda paatelmia kasvillisuuden rakenteen ekologisesta vaihtelusta ymparistomuuttujien suhteen. Harvennus- ja avohakkuiden vaikutuksia pintakasvillisuuteen myos mallinnettiin lajiryhmittain kayttaen yleistettyja lineaarisia malleja (Generalized linear models). Lajiryhmien peittavyyksien kehitysta mallinnettiin puuston pohjapinta-alan funktiona. Metsikon ian kasvaessa heinien ja ruohojen osuus pienenee, kun taas varpujen ja sammalten osuus lisaantyy. Harvennushakkuiden vaikutukset ovat lievempia kuin avohakkuiden eivatka ne useimmiten aiheuttaneet tilastollisesti merkittavia muutoksia pintakasvillisuuden peittavyyksissa. Avohakkuu sen sijaan on voimakkaampi ja aiheuttaa merkittavia muutoksia. Heinia ja ruohoja esiintyy hakkuun jalkeen enemman ja vastaavasti sammalet ja varvut taantuvat. Kasvillisuuden kokonaispeittavyys ja biomassa ovat suurimmillaan hakkaamattomissa metsikoissa. Harvennushakkuun jalkeen peittavyys ja biomassa voi kuitenkin hetkellisesti olla suurimmillaan kun harvennuksesta on kulunut muutama vuosi. Yleistetyt lineaariset mallit kuvasivat pintakasvillisuuden kehitysta metsikon pohjapinta-alan funktiona luotettavasti. Malleja voidaan kayttaa myos ennustamaan miten pintakasvillisuus kehittyy avohakkuun jalkeen. Malleja voidaan soveltaa esimerkiksi laskettaessa pintakasvillisuuden sitoman hiilen maaraa eriikaisissa metsissa. Niiden avulla voidaan myos arvioida esimerkiksi avohakkuuta voimaperaisemman energiapuun korjuun vaikutuksia pintakasvillisuuden runsauteen.

Bounded turning circles are weak-quasicircles

We show that a metric Jordan curve $\Gamma$ is \emph{bounded turning} if and only if there exists a \emph{weak-quasisymmetric} homeomorphism $\phi\colon \mathsf{S}^1 \to \Gamma$.