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  • Yang, Fan (Helsingin yliopisto, 2014)
    Dependence logic is a new logic which incorporates the notion of dependence , as well as independence between variables into first-order logic. In this thesis, we study extensions and variants of dependence logic on the first-order, propositional and modal level. In particular, the role of intuitionistic connectives in this setting is emphasized. We obtain, among others, the following results: 1. First-order intuitionistic dependence logic is proved to have the same expressive power as the full second-order logic. 2. Complete axiomatizations for propositional dependence logic and its variants are obtained. 3. The complexity of model checking problem for modal intuitionistic dependence logic is analyzed.
  • Hirsikko, Anne (Helsingin yliopisto, 2011)
    Aerosol particles have effect on climate, visibility, air quality and human health. However, the strength of which aerosol particles affect our everyday life is not well described or entirely understood. Therefore, investigations of different processes and phenomena including e.g. primary particle sources, initial steps of secondary particle formation and growth, significance of charged particles in particle formation, as well as redistribution mechanisms in the atmosphere are required. In this work sources, sinks and concentrations of air ions (charged molecules, cluster and particles) were investigated directly by measuring air molecule ionising components (i.e. radon activity concentrations and external radiation dose rates) and charged particle size distributions, as well as based on literature review. The obtained results gave comprehensive and valuable picture of the spatial and temporal variation of the air ion sources, sinks and concentrations to use as input parameters in local and global scale climate models. Newly developed air ion spectrometers (Airel Ltd.) offered a possibility to investigate atmospheric (charged) particle formation and growth at sub-3 nm sizes. Therefore, new visual classification schemes for charged particle formation events were developed, and a newly developed particle growth rate method was tested with over one year dataset. These data analysis methods have been widely utilised by other researchers since introducing them. This thesis resulted interesting characteristics of atmospheric particle formation and growth: e.g. particle growth may sometimes be suppressed before detection limit (~ 3 nm) of traditional aerosol instruments, particle formation may take place during daytime as well as in the evening, growth rates of sub-3 nm particles were quite constant throughout the year while growth rates of larger particles (3-20 nm in diameter) were higher during summer compared to winter. These observations were thought to be a consequence of availability of condensing vapours. The observations of this thesis offered new understanding of the particle formation in the atmosphere. However, the role of ions in particle formation, which is not well understood with current knowledge, requires further research in future.
  • Pauna, Matti (Helsingin yliopisto, 2007)
    In this thesis we study a few games related to non-wellfounded and stationary sets. Games have turned out to be an important tool in mathematical logic ranging from semantic games defining the truth of a sentence in a given logic to for example games on real numbers whose determinacies have important effects on the consistency of certain large cardinal assumptions. The equality of non-wellfounded sets can be determined by a so called bisimulation game already used to identify processes in theoretical computer science and possible world models for modal logic. Here we present a game to classify non-wellfounded sets according to their branching structure. We also study games on stationary sets moving back to classical wellfounded set theory. We also describe a way to approximate non-wellfounded sets with hereditarily finite wellfounded sets. The framework used to do this is domain theory. In the Banach-Mazur game, also called the ideal game, the players play a descending sequence of stationary sets and the second player tries to keep their intersection stationary. The game is connected to precipitousness of the corresponding ideal. In the pressing down game first player plays regressive functions defined on stationary sets and the second player responds with a stationary set where the function is constant trying to keep the intersection stationary. This game has applications in model theory to the determinacy of the Ehrenfeucht-Fraisse game. We show that it is consistent that these games are not equivalent.
  • Nikula, Miika (Helsingin yliopisto, 2014)
    Mandelbrot cascades and multiplicative chaos are two related natural constructions of random multifractal measures, or rather families of such measures indexed by a single real parameter. While these models were first defined decades ago, they have recently become topical again after they were applied in the field of planar random geometry. In that context, multiplicative chaos provides one rigorous way of defining the exponential of the Gaussian free field. Typically, for a Mandelbrot cascade or multiplicative chaos there exists a critical parameter value at which the behavior of the measure changes drastically. Away from criticality, the geometric properties of cascade and chaos measures have been studied intensively and much of the theory is by now classical. For small values of the parameter, the random measures considered here are natural examples of multifractality. The measures are exact dimensional with the dimension, depending deterministically on the parameter, strictly between 0 and the dimension of the ambient space in which the measure is constructed. For large values of the parameter the construction of the measures is a rather subtle issue, but it is known that in this case the measures are atomic. The main theme of this dissertation is the study of geometric properties of cascade and chaos measures at the critical parameter value. To mention one main result, it is shown that while the critical measures are almost surely supported on sets of Hausdorff dimension 0, they do not have atoms. Improving on the qualitative result, almost sure quantitative bounds are given for the mass carried by a set of a given size in the ambient geometry. One part of the analysis of the geometric properties of cascade and chaos measures is understanding the probability distributions of the mass of a given set. Since the measures are built to exhibit strong statistical self-similarity, this is a major part of understanding the full law of the measure, i.e. the joint laws arbitrary collections of sets. Results concerning the law of the mass of a fixed set are another main theme of this dissertation.
  • Järvinen, Riku (Helsingin yliopisto, 2011)
    This doctoral thesis is about the solar wind influence on the atmosphere of the planet Venus. A numerical plasma simulation model was developed for the interaction between Venus and the solar wind to study the erosion of charged particles from the Venus upper atmosphere. The developed model is a hybrid simulation where ions are treated as particles and electrons are modelled as a fluid. The simulation was used to study the solar wind induced ion escape from Venus as observed by the European Space Agency's Venus Express and NASA's Pioneer Venus Orbiter spacecraft. Especially, observations made by the ASPERA-4 particle instrument onboard Venus Express were studied. The thesis consists of an introductory part and four peer-reviewed articles published in scientific journals. In the introduction Venus is presented as one of the terrestrial planets in the Solar System and the main findings of the work are discussed within the wider context of planetary physics. Venus is the closest neighbouring planet to the Earth and the most earthlike planet in its size and mass orbiting the Sun. Whereas the atmosphere of the Earth consists mainly of nitrogen and oxygen, Venus has a hot carbon dioxide atmosphere, which is dominated by the greenhouse effect. Venus has all of its water in the atmosphere, which is only a fraction of the Earth's total water supply. Since planets developed presumably in similar conditions in the young Solar System, why Venus and Earth became so different in many respects? One important feature of Venus is that the planet does not have an intrinsic magnetic field. This makes it possible for the solar wind, a continuous stream of charged particles from the Sun, to flow close to Venus and to pick up ions from the planet's upper atmosphere. The strong intrinsic magnetic field of the Earth dominates the terrestrial magnetosphere and deflects the solar wind flow far away from the atmosphere. The region around Venus where the planet's atmosphere interacts with the solar wind is called the plasma environment or the induced magnetosphere. Main findings of the work include new knowledge about the movement of escaping planetary ions in the Venusian induced magnetosphere. Further, the developed simulation model was used to study how the solar wind conditions affect the ion escape from Venus. Especially, the global three-dimensional structure of the Venusian particle and magnetic environment was studied. The results help to interpret spacecraft observations around the planet. Finally, several remaining questions were identified, which could potentially improve our knowledge of the Venus ion escape and guide the future development of planetary plasma simulations.
  • Jääskeläinen, Jarmo (Helsingin yliopisto, 2012)
    The dissertation consists of an introductory part and three articles. The starting point of the work is the consideration of quasiregular mappings in the plane. These are characterized by the Beltrami equation. One of the foundational results of the theory of quasiregular mappings is that the derivative of a nonconstant map does not vanish almost everywhere. The first paper (in collaboration with Kari Astala) shows that the same is true for so-called reduced quasiregular mappings when the function is one-to-one. The second paper studies linear classes of planar quasiregular mappings. In addition, it shows that the derivative of a nonconstant reduced quasiregular map is non-vanishing even in subdomains and without assuming injectivity. The principal tool of the proof is a reverse Hölder inequality for the adjoint equation of a 2nd order uniformly elliptic operator in the nondivergence form. The non-vanishing fact is used to show Wronsky-type theorem for general linear Beltrami systems. This result is applied to prove that the associated Beltrami equation of a linear quasiregular family is unique. The third article (joint work with Kari Astala, Albert Clop, Daniel Faraco, and László Székelyhidi Jr) tackles the problem of uniqueness of normalized solutions to the nonlinear Beltrami equation. It turns out that the uniqueness holds under explicit bounds in the ellipticity constant of the equation at infinity, but not in general. The fact is complemented with counterexamples.
  • Bissell-Siders, Ryan (Helsingin yliopisto, 2008)
    We solve the Dynamic Ehrenfeucht-Fra\"iss\'e Game on linear orders for both players, yielding a normal form for quantifier-rank equivalence classes of linear orders in first-order logic, infinitary logic, and generalized-infinitary logics with linearly ordered clocks. We show that Scott Sentences can be manipulated quickly, classified into local information, and consistency can be decided effectively in the length of the Scott Sentence. We describe a finite set of linked automata moving continuously on a linear order. Running them on ordinals, we compute the ordinal truth predicate and compute truth in the constructible universe of set-theory. Among the corollaries are a study of semi-models as efficient database of both model-theoretic and formulaic information, and a new proof of the atomicity of the Boolean algebra of sentences consistent with the theory of linear order -- i.e., that the finitely axiomatized theories of linear order are dense.
  • Petäjä, Tuukka (Helsingin yliopisto, 2006)
    Atmospheric aerosol particles affect the global climate as well as human health. In this thesis, formation of nanometer sized atmospheric aerosol particles and their subsequent growth was observed to occur all around the world. Typical formation rate of 3 nm particles at varied from 0.01 to 10 cm-3s-1. One order of magnitude higher formation rates were detected in urban environment. Highest formation rates up to 105 cm-3s-1 were detected in coastal areas and in industrial pollution plumes. Subsequent growth rates varied from 0.01 to 20 nm h-1. Smallest growth rates were observed in polar areas and the largest in the polluted urban environment. This was probably due to competition between growth by condensation and loss by coagulation. Observed growth rates were used in the calculation of a proxy condensable vapour concentration and its source rate in vastly different environments from pristine Antarctica to polluted India. Estimated concentrations varied only 2 orders of magnitude, but the source rates for the vapours varied up to 4 orders of magnitude. Highest source rates were in New Delhi and lowest were in the Antarctica. Indirect methods were applied to study the growth of freshly formed particles in the atmosphere. Also a newly developed Water Condensation Particle Counter, TSI 3785, was found to be a potential candidate to detect water solubility and thus indirectly composition of atmospheric ultra-fine particles. Based on indirect methods, the relative roles of sulphuric acid, non-volatile material and coagulation were investigated in rural Melpitz, Germany. Condensation of non-volatile material explained 20-40% and sulphuric acid the most of the remaining growth up to a point, when nucleation mode reached 10 to 20 nm in diameter. Coagulation contributed typically less than 5%. Furthermore, hygroscopicity measurements were applied to detect the contribution of water soluble and insoluble components in Athens. During more polluted days, the water soluble components contributed more to the growth. During less anthropogenic influence, non-soluble compounds explained a larger fraction of the growth. In addition, long range transport to a measurement station in Finland in a relatively polluted air mass was found to affect the hygroscopicity of the particles. This aging could have implications to cloud formation far away from the pollution sources.
  • Zhang, Zhanhai (Helsingin yliopisto, 2000)
  • Tuomi, Laura (Helsingin yliopisto, 2014)
    The modelling of surface waves and vertical mixing in the northern Baltic Sea is a complicated task, in which taking into account the specific features of the area is essential. The seasonal ice conditions affect the wave climate of the northern Baltic Sea and the formulation of the wave statistics. Five different ways of formulating statistics in seasonally ice-covered seas were presented, and the differences between them in the mean values and the exceedance probabilities of significant wave height were evaluated based on six years of wave hindcasts. The severest wave climate was in the Baltic Proper, where the hindcast maximum value of significant wave height was 9.7 m. The highest values were reached in autumn and winter, spring and summer had considerably less severe wave climate. Due to the irregular shoreline and archipelago, the coastal areas of Finland are mostly well sheltered from the more severe wave conditions of the open sea. Modelling of wave conditions in these areas requires high-resolution grids with sufficiently accurate description of bathymetry and land-sea mask. The manual and automated methods developed for compiling representative model grids in archipelago areas were shown to improve the accuracy of wave modelling. Taking the sheltering effects of the coastal archipelago into account on a sub-grid scale by using additional grid obstructions was shown to result in sufficient accuracy in the modelled significant wave height even when a coarser resolution was applied. However, additional measures are needed to take into account wave refraction and depth-induced wave breaking on a sub-grid scale. The different factors affecting the accuracy of the wave and hydrodynamic modelling, such as the initial and boundary conditions, bathymetry, horizontal and vertical resolution and numerical solutions, make it difficult to distinguish a specific reason behind the modelling inaccuracies. It was shown, however, that the meteorological forcing plays a key role in all marine modelling applications in the Baltic Sea, and special emphasis should be given to the production of meteorological datasets with high quality and resolution. Increasing computational resources allow us to use more exact numerical solutions, e.g., in the nonlinear four-wave interaction source term of the wave model, thus improving the accuracy of the model results. To better describe the ocean surface layer dynamics there is an evident need for coupled models and inclusion of the effect of surface waves in the modelling of marine systems. The different parametrisations of vertical turbulence used in modelling 3D hydrodynamics in the Gulf of Finland were shown to underestimate the thermocline depth and to show less steep temperature gradients than those of the measured profiles. The role of surface waves in the vertical mixing in the Gulf of Finland was studied by calculating the turbulent Langmuir numbers based on wave hindcasts. In summer the hindcast values of the turbulent Langmuir number were within the interval where the Langmuir circulation is estimated to play a role in the turbulence production together with the wind-induced mean shear. In general, a coupled 3D hydrodynamic-wave-ice model might further improve our ability to simulate short and long-term changes in the marine systems of the Baltic Sea.
  • Lipponen, Henri (Helsingin yliopisto, 2013)
    We construct transgressive differential forms on manifolds with boundary, using the calculus of cusp pseudodifferential operators due to Melrose. Even and odd dimensional manifolds are considered. These differential forms are associated with the grading operators defined by Dirac operators coupled to vector potentials. In quantum field theory, these forms appear in the construction of the Dirac vacuum in the canonical quantization and are related to anomalies. Furthermore, we construct local representatives for the bulk-parts of these forms in BRST-cohomology. This gives the local representations for the associated boundary BRST-cocycles. Particularly, in odd dimensions we construct local representations for the Schwinger terms.
  • Vesalainen, Esa V. (Helsingin yliopisto, 2014)
    In this thesis, we extend the theory of non-scattering energies on two fronts. First, we shall consider the discreteness of non-scattering energies corresponding to non-compactly supported potentials using the approach via transmission eigenvalues and fourth-order operators. The method requires the support of the potential to exhibit certain compact Sobolev embedding and to be contained in a half-space and the potential to have controlled polynomial or exponential decay at infinity. Also, in order to connect the non-scattering energies to the fourth-order operators, a generalization of the classical Rellich theorem to unbounded domains is required. This is of independent interest, and we obtain several such results, including a discrete analogue. Our second contribution (joint work with L. Päivärinta and M. Salo) is extending a recent result on non-existence of non-scattering energies for potentials with rectangular corners to arbitrary corners of angle smaller than 180 degrees in two dimensions, and to prove in three dimensions that the set of strictly convex circular conical corners for which non-scattering energies might exist is at most countable.
  • Uusikivi, Jari (Helsingin yliopisto, 2013)
    Sea ice has been recognized as one of the key elements of polar and sub-polar seas, including Baltic Sea. The existence of sea ice cover and its properties have influence to many aspects of marine biology, climate and seafaring. This thesis is concentrated on describing physical and optical properties of landfast ice, and also pack ice, in the Baltic Sea. The aim of the thesis is to use measurements to study the interactions between optical and physical properties of sea ice and how these can affect the biology in sea ice. Decade long observations of ice properties were used to construct a statistical model of properties of landfast ice. Temperature was found to be the most important factor determining ice thickness and contribution of snow ice to the ice thickness was determined by the amount of winter time precipitation. Stratigraphy of ice and growth history had influence to the vertical distribution of organisms in the ice cover as snow ice layers and columnar ice layers were found to favor different types of organisms. Thickness of meteoric ice layer, including snow ice and superimposed ice, controlled the albedo of ice cover when no snow cover was on the ice. Based on the observations of fast ice conditions and albedo, the effects of snow thickness and meteoric ice thickness to the albedo of sea ice were formulated as albedo parameterization equations. The optical properties of sea ice with spectral resolution were studied on the landfast sea ice. Emphasis in these studies was given to optical properties in the ultraviolet and visible wavelengths. Organic matter, dissolved and particulate, was the most important factor determining the ultraviolet properties of sea ice cover. The optical properties in the ultraviolet were also actively modified by the living organisms in the ice cover by producing mycosporine like amino acids (MAAs) in relatively high amounts. MAAs are a family of photoprotective compounds that absorb UV radiation efficiently. At the visible part of spectrum the ice by itself and the thickness of meteoric ice layer were the most important determinants. Salinity and the initial salt entrapment during ice growth in the Baltic Sea were measured to be less than in the oceans with equal ice growth rates. The turbulent fluxes of heat and salinity under the landfast sea ice were measured to be small.
  • Joensuu, Jani (Helsingin yliopisto, 2009)
    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.
  • Saarela, Olli (Helsingin yliopisto, 2010)
    Whether a statistician wants to complement a probability model for observed data with a prior distribution and carry out fully probabilistic inference, or base the inference only on the likelihood function, may be a fundamental question in theory, but in practice it may well be of less importance if the likelihood contains much more information than the prior. Maximum likelihood inference can be justified as a Gaussian approximation at the posterior mode, using flat priors. However, in situations where parametric assumptions in standard statistical models would be too rigid, more flexible model formulation, combined with fully probabilistic inference, can be achieved using hierarchical Bayesian parametrization. This work includes five articles, all of which apply probability modeling under various problems involving incomplete observation. Three of the papers apply maximum likelihood estimation and two of them hierarchical Bayesian modeling. Because maximum likelihood may be presented as a special case of Bayesian inference, but not the other way round, in the introductory part of this work we present a framework for probability-based inference using only Bayesian concepts. We also re-derive some results presented in the original articles using the toolbox equipped herein, to show that they are also justifiable under this more general framework. Here the assumption of exchangeability and de Finetti's representation theorem are applied repeatedly for justifying the use of standard parametric probability models with conditionally independent likelihood contributions. It is argued that this same reasoning can be applied also under sampling from a finite population. The main emphasis here is in probability-based inference under incomplete observation due to study design. This is illustrated using a generic two-phase cohort sampling design as an example. The alternative approaches presented for analysis of such a design are full likelihood, which utilizes all observed information, and conditional likelihood, which is restricted to a completely observed set, conditioning on the rule that generated that set. Conditional likelihood inference is also applied for a joint analysis of prevalence and incidence data, a situation subject to both left censoring and left truncation. Other topics covered are model uncertainty and causal inference using posterior predictive distributions. We formulate a non-parametric monotonic regression model for one or more covariates and a Bayesian estimation procedure, and apply the model in the context of optimal sequential treatment regimes, demonstrating that inference based on posterior predictive distributions is feasible also in this case.
  • Kemppainen, Antti (Helsingin yliopisto, 2009)
    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.