On Random Planar Curves and Their Scaling Limits

Planar curves arise naturally as interfaces between two regions of the plane. An important part of statistical physics is the study of lattice models. This thesis is about the interfaces of 2D lattice models. The scaling limit is an infinite system limit which is taken by letting the lattice mesh decrease to zero. At criticality, the scaling limit of an interface is one of the SLE curves (Schramm-Loewner evolution), introduced by Oded Schramm. This family of random curves is parametrized by a real variable, which determines the universality class of the model. The first and the second paper of this thesis study properties of SLEs. They contain two different methods to study the whole SLE curve, which is, in fact, the most interesting object from the statistical physics point of view. These methods are applied to study two symmetries of SLE: reversibility and duality. The first paper uses an algebraic method and a representation of the Virasoro algebra to find common martingales to different processes, and that way, to confirm the symmetries for polynomial expected values of natural SLE data. In the second paper, a recursion is obtained for the same kind of expected values. The recursion is based on stationarity of the law of the whole SLE curve under a SLE induced flow. The third paper deals with one of the most central questions of the field and provides a framework of estimates for describing 2D scaling limits by SLE curves. In particular, it is shown that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical physics model will have scaling limits and those will be well-described by Loewner evolutions with random driving forces.

Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

The effect of interrogative function on intonation in spontaneous and read Finnish

Goals This study aims to map the effect of interrogative function on the intonation of spontaneous and read Finnish. Earlier research shows that the most prominent feature in Finnish question intonation is an appeal to the listener. Question word questions typically start with a high peak which is followed by falling intonation. In yes/no questions, F0 remains on a high level until the word carrying sentence stress and then falls. Final rises are mainly found in intonation clichés such as "Ai mitä?" ("What?") These earlier results are based on read speech and enacted dialogues. In this study, questions and statements found in spontaneous dialogues were compared. These utterances were also compared with read versions of the same utterances. Fundamental frequency values were compared using a mixed model. Contours were also grouped using auditory and visual inspection. Thus it was possible to compare frequencies of contour types according to utterance type and speech style. The position of questions in the F0 distribution of the whole material was also investigated in this study. Method The material consisted of four spontaneous dialogues and their read versions. The speakers were young adults from the Helsinki metropolitan area, four females and four males. The whole material was first divided into broad dialogue function categories arising from the material and F0 curves were calculated for each category. After this, 277 questions and 244 statements were selected for closer inspection. Values reflecting F0 distribution and contour shape were measured from the F0 contours of these utterances. A mixed model was used to analyse the differences. Utterance type, question type, speech style and speaker gender were used as fixed effects. The frequencies of F0 contour types were compared using a Chi square test. Additional material in this study came from eight young female speakers in central Finland. Results and conclusions In the mixed model analysis, significant differences were found both between questions and statements and between spontaneous and read speech. Generally, utterance type affected the variables reflecting contour type while speech style affected the variables reflecting F0 distribution. The effect of question type was not clearly visible. In read speech the contours resembled earlier results more closely. Speakers had different strategies in differentiating between questions and statements. In the whole material, F0 was slightly higher in questions than in statements. The effect of dialectal background could be seen in the contour types. The results show that interrogative function affects intonation in both spontaneous and read Finnish.

Aritmetiikasta algebraan : Muutoksia osaamisessa peruskoulun päättöluokalla 20 vuoden aikana

From Arithmetic to Algebra. Changes in the skills in comprehensive school over 20 years. In recent decades we have emphasized the understanding of calculation in mathematics teaching. Many studies have found that better understanding helps to apply skills in new conditions and that the ability to think on an abstract level increases the transfer to new contexts. In my research I take into consideration competence as a matrix where content is in a horizontal line and levels of thinking are in a vertical line. The know-how is intellectual and strategic flexibility and understanding. The resources and limitations of memory have their effects on learning in different ways in different phases. Therefore both flexible conceptual thinking and automatization must be considered in learning. The research questions that I examine are what kind of changes have occurred in mathematical skills in comprehensive school over the last 20 years and what kind of conceptual thinking is demonstrated by students in this decade. The study consists of two parts. The first part is a statistical analysis of the mathematical skills and their changes over the last 20 years in comprehensive school. In the test the pupils did not use calculators. The second part is a qualitative analysis of the conceptual thinking of pupils in comprehensive school in this decade. The study shows significant differences in algebra and in some parts of arithmetic. The largest differences were detected in the calculation skills of fractions. In the 1980s two out of three pupils were able to complete tasks with fractions, but in the 2000s only one out of three pupils were able to do the same tasks. Also remarkable is that out of the students who could complete the tasks with fractions, only one out of three pupils was on the conceptual level in his/her thinking. This means that about 10% of pupils are able to understand the algebraic expression, which has the same isomorphic structure as the arithmetical expression. This finding is important because the ability to think innovatively is created when learning the basic concepts. Keywords: arithmetic, algebra, competence

From a tree to a stand in Finnish boreal forests: biomass estimation and comparison of methods

There is an increasing need to compare the results obtained with different methods of estimation of tree biomass in order to reduce the uncertainty in the assessment of forest biomass carbon. In this study, tree biomass was investigated in a 30-year-old Scots pine (Pinus sylvestris) (Young-Stand) and a 130-year-old mixed Norway spruce (Picea abies)-Scots pine stand (Mature-Stand) located in southern Finland (61º50' N, 24º22' E). In particular, a comparison of the results of different estimation methods was conducted to assess the reliability and suitability of their applications. For the trees in Mature-Stand, annual stem biomass increment fluctuated following a sigmoid equation, and the fitting curves reached a maximum level (from about 1 kg/yr for understorey spruce to 7 kg/yr for dominant pine) when the trees were 100 years old. Tree biomass was estimated to be about 70 Mg/ha in Young-Stand and about 220 Mg/ha in Mature-Stand. In the region (58.00-62.13 ºN, 14-34 ºE, ≤ 300 m a.s.l.) surrounding the study stands, the tree biomass accumulation in Norway spruce and Scots pine stands followed a sigmoid equation with stand age, with a maximum of 230 Mg/ha at the age of 140 years. In Mature-Stand, lichen biomass on the trees was 1.63 Mg/ha with more than half of the biomass occurring on dead branches, and the standing crop of litter lichen on the ground was about 0.09 Mg/ha. There were substantial differences among the results estimated by different methods in the stands. These results imply that a possible estimation error should be taken into account when calculating tree biomass in a stand with an indirect approach.

Structural analyses of (1->3),(1->4)-β-D-glucan of oats and barley

The structures of (1→3),(1→4)-β-D-glucans of oat bran, whole-grain oats and barley and processed foods were analysed. Various methods of hydrolysis of β-glucan, the content of insoluble fibre of whole grains of oats and barley and the solution behaviour of oat and barley β-glucans were studied. The isolated soluble β-glucans of oat bran and whole-grain oats and barley were hydrolysed with lichenase, an enzyme specific for (1→3),(1→4)-β-D-β-glucans. The amounts of oligosaccharides produced from bran were analysed with capillary electrophoresis and those from whole-grains with high-performance anion-exchange chromatography with pulse-amperometric detection. The main products were 3-O-β-cellobiosyl-D-glucose and 3-O-β-cellotriosyl-D-glucose, the oligosaccharides which have a degree of polymerisation denoted by DP3 and DP4. Small differences were detected between soluble and insoluble β-glucans and also between β-glucans of oats and barley. These differences can only be seen in the DP3:DP4 ratio which was higher for barley than for oat and also higher for insoluble than for soluble β-glucan. A greater proportion of barley β-glucan remained insoluble than of oat β-glucan. The molar masses of soluble β-glucans of oats and barley were the same as were those of insoluble β-glucans of oats and barley. To analyse the effects of cooking, baking, fermentation and drying, β-glucan was isolated from porridge, bread and fermentate and also from their starting materials. More β-glucan was released after cooking and less after baking. Drying decreased the extractability for bread and fermentate but increased it for porridge. Different hydrolysis methods of β-glucan were compared. Acid hydrolysis and the modified AOAC method gave similar results. The results of hydrolysis with lichenase gave higher recoveries than the other two. The combination of lichenase hydrolysis and high-performance anion-exchange chromatography with pulse-amperometric detection was found best for the analysis of β-glucan content. The content of insoluble fibre was higher for barley than for oats and the amount of β-glucan in the insoluble fibre fraction was higher for oats than for barley. The flow properties of both water and aqueous cuoxam solutions of oat and barley β-glucans were studied. Shear thinning was stronger for the water solutions of oat β-glucan than for barley β-glucan. In aqueous cuoxam shear thinning was not observed at the same concentration as in water but only with high concentration solutions. Then the viscosity of barley β-glucan was slightly higher than that of oat β-glucan. The oscillatory measurements showed that the crossover point of the G´ and G´´ curves was much lower for barley β-glucan than for oat β-glucan indicating a higher tendency towards solid-like behaviour for barley β-glucan than for oat β-glucan.