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  • 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.
  • Martin, Jussi (Helsingin yliopisto, 2014)
    In this thesis we study the spectrum of a certain boundary value problem in cuspidal domains. This boundary value problem originates from the linear theory of water-waves. One particular motivation for studying the problem in cuspidal domains, is that in some cases the cuspidal shape could act as a wave absorbing structure. Furthermore, the problem itself has also mathematical interest on its own, due to its unusual form where the spectral parameter appears in the boundary condition. The thesis consists of an introductory part and three articles. In the first article we study the problem in a domain that represents a lake or a pond, in which parameter describing the sharpness of the cusp is two. In the article we show that in this case the essential spectrum of the problem contains real numbers which are greater or equal to a certain positive number. In the second article the setting is similar, but the sharpness parameter is allowed to be any real number greater than two. We show that in this case the continuous spectrum of the problem is the set of all non-negative real numbers and this is also the whole spectrum. In addition, we improve the result of the first article by showing that values found in the essential spectrum also belong to the continuous spectrum. In the third article we consider a case where the domain is a canal, filled with two liquid layers of different densities. The cuspidal form is in one of the layers and its sharpness parameter is two. We show that in this case the continuous spectrum of the problem contains real numbers which are greater or equal to a certain positive number, and that the interval from zero to this number is contained in the discrete spectrum.
  • Niemelä, Sami (Helsingin yliopisto, 2007)
    Modern-day weather forecasting is highly dependent on Numerical Weather Prediction (NWP) models as the main data source. The evolving state of the atmosphere with time can be numerically predicted by solving a set of hydrodynamic equations, if the initial state is known. However, such a modelling approach always contains approximations that by and large depend on the purpose of use and resolution of the models. Present-day NWP systems operate with horizontal model resolutions in the range from about 40 km to 10 km. Recently, the aim has been to reach operationally to scales of 1 4 km. This requires less approximations in the model equations, more complex treatment of physical processes and, furthermore, more computing power. This thesis concentrates on the physical parameterization methods used in high-resolution NWP models. The main emphasis is on the validation of the grid-size-dependent convection parameterization in the High Resolution Limited Area Model (HIRLAM) and on a comprehensive intercomparison of radiative-flux parameterizations. In addition, the problems related to wind prediction near the coastline are addressed with high-resolution meso-scale models. The grid-size-dependent convection parameterization is clearly beneficial for NWP models operating with a dense grid. Results show that the current convection scheme in HIRLAM is still applicable down to a 5.6 km grid size. However, with further improved model resolution, the tendency of the model to overestimate strong precipitation intensities increases in all the experiment runs. For the clear-sky longwave radiation parameterization, schemes used in NWP-models provide much better results in comparison with simple empirical schemes. On the other hand, for the shortwave part of the spectrum, the empirical schemes are more competitive for producing fairly accurate surface fluxes. Overall, even the complex radiation parameterization schemes used in NWP-models seem to be slightly too transparent for both long- and shortwave radiation in clear-sky conditions. For cloudy conditions, simple cloud correction functions are tested. In case of longwave radiation, the empirical cloud correction methods provide rather accurate results, whereas for shortwave radiation the benefit is only marginal. Idealised high-resolution two-dimensional meso-scale model experiments suggest that the reason for the observed formation of the afternoon low level jet (LLJ) over the Gulf of Finland is an inertial oscillation mechanism, when the large-scale flow is from the south-east or west directions. The LLJ is further enhanced by the sea-breeze circulation. A three-dimensional HIRLAM experiment, with a 7.7 km grid size, is able to generate a similar LLJ flow structure as suggested by the 2D-experiments and observations. It is also pointed out that improved model resolution does not necessary lead to better wind forecasts in the statistical sense. In nested systems, the quality of the large-scale host model is really important, especially if the inner meso-scale model domain is small.
  • Gagné, Stéphanie (Helsingin yliopisto, 2011)
    Floating in the air that surrounds us is a number of small particles, invisible to the human eye. The mixture of air and particles, liquid or solid, is called an aerosol. Aerosols have significant effects on air quality, visibility and health, and on the Earth's climate. Their effect on the Earth's climate is the least understood of climatically relevant effects. They can scatter the incoming radiation from the Sun, or they can act as seeds onto which cloud droplets are formed. Aerosol particles are created directly, by human activity or natural reasons such as breaking ocean waves or sandstorms. They can also be created indirectly as vapors or very small particles are emitted into the atmosphere and they combine to form small particles that later grow to reach climatically or health relevant sizes. The mechanisms through which those particles are formed is still under scientific discussion, even though this knowledge is crucial to make air quality or climate predictions, or to understand how aerosols will influence and will be influenced by the climate's feedback loops. One of the proposed mechanisms responsible for new particle formation is ion-induced nucleation. This mechanism is based on the idea that newly formed particles were ultimately formed around an electric charge. The amount of available charges in the atmosphere varies depending on radon concentrations in the soil and in the air, as well as incoming ionizing radiation from outer space. In this thesis, ion-induced nucleation is investigated through long-term measurements in two different environments: in the background site of Hyytiälä and in the urban site that is Helsinki. The main conclusion of this thesis is that ion-induced nucleation generally plays a minor role in new particle formation. The fraction of particles formed varies from day to day and from place to place. The relative importance of ion-induced nucleation, i.e. the fraction of particles formed through ion-induced nucleation, is bigger in cleaner areas where the absolute number of particles formed is smaller. Moreover, ion-induced nucleation contributes to a bigger fraction of particles on warmer days, when the sulfuric acid and water vapor saturation ratios are lower. This analysis will help to understand the feedbacks associated with climate change.
  • Mattsson, Maria (Helsingin yliopisto, 2012)
    During the last decade, the cosmological observations have indicated that the homogeneous and isotropic Friedmann models with linear perturbations fail to describe our universe at late times unless a dominant energy component with negative pressure called dark energy is introduced. In this thesis, we study the implications of the nonlinear nature of general relativity on the cosmological model building beyond the standard Friedmann models. Despite the well established observational status of cosmic structures, their effects have gained more attention only along with the dark energy debate. In particular, the fact that the start of the supposed dark energy domination coincides with the time the nonlinear inhomogeneities started to form on larger scales, motivates the study of the dynamics of the cosmic structures. In cosmology, the implication of the nonlinearity of gravity is that averages of inhomogeneous quantities do not evolve in time like the corresponding homogeneous quantities - a phenomenon referred to as the backreaction. Due to the new precision observations during the recent years, the evaluation of the backreaction in our universe is a topical, but complex task. In this thesis, rather than trying to fully quantify the backreaction, the emphasis is on the model building. We explicitly demonstrate the importance of the exact matching conditions in the solutions representing cosmic structures in the context of backreaction evaluation. Indeed, the cosmic web of structures is made of very differently behaving regions and the shear on the interface between the different regions seems to play an important role. The backreaction term emerging from averaging the Einstein equation is not the only effect that cosmic structures can have on the observations. Indeed, we also demonstrate that even though the backreaction would remain small, large effects can arise from the choice of the smoothing scale and, perhaps surprisingly, from perturbative models as well. As we find, at least the supernova data can be explained within a linearly perturbed Friedmann model - without dark energy. The key point is to take into account the effects of structures on the observable distance measures, ignored in the standard cosmological perturbation theory. Further inspection shows that the model is actually equivalent to a nonperturbative inhomogeneous solution, confirming that the supernova data does not necessarily imply additional nonperturbative corrections. Considering physical quantities such as the expansion rate of space and the matter density, there are large local variations in the cosmic web. The main question to answer is whether (and to what extent) the effects of the local variations average out or accumulate in the observables. It appears likely that when combining all the cosmological data, more sophisticated models than the perturbed Friedmann or the simplest spherically symmetric exact inhomogeneous solutions are required to fully quantify the effects of the structures on the cosmological observations.
  • Mangs, Johan (Helsingin yliopisto, 2004)
  • Dannenberg, Alia (Helsingin yliopisto, 2011)
    In the thesis I study various quantum coherence phenomena and create some of the foundations for a systematic coherence theory. So far, the approach to quantum coherence in science has been purely phenomenological. In my thesis I try to answer the question what quantum coherence is and how it should be approached within the framework of physics, the metatheory of physics and the terminology related to them. It is worth noticing that quantum coherence is a conserved quantity that can be exactly defined. I propose a way to define quantum coherence mathematically from the density matrix of the system. Degenerate quantum gases, i.e., Bose condensates and ultracold Fermi systems, form a good laboratory to study coherence, since their entropy is small and coherence is large, and thus they possess strong coherence phenomena. Concerning coherence phenomena in degenerate quantum gases, I concentrate in my thesis mainly on collective association from atoms to molecules, Rabi oscillations and decoherence. It appears that collective association and oscillations do not depend on the spin-statistics of particles. Moreover, I study the logical features of decoherence in closed systems via a simple spin-model. I argue that decoherence is a valid concept also in systems with a possibility to experience recoherence, i.e., Poincaré recurrences. Metatheoretically this is a remarkable result, since it justifies quantum cosmology: to study the whole universe (i.e., physical reality) purely quantum physically is meaningful and valid science, in which decoherence explains why the quantum physical universe appears to cosmologists and other scientists very classical-like. The study of the logical structure of closed systems also reveals that complex enough closed (physical) systems obey a principle that is similar to Gödel's incompleteness theorem of logic. According to the theorem it is impossible to describe completely a closed system within the system, and the inside and outside descriptions of the system can be remarkably different. Via understanding this feature it may be possible to comprehend coarse-graining better and to define uniquely the mutual entanglement of quantum systems.
  • Blåsten, Eemeli (Helsingin yliopisto, 2013)
    We prove uniqueness and stability for the inverse boundary value problem of the two dimensional Schrödinger equation. We do not assume the potentials to be continuous or even bounded. Instead, we assume that some of their positive fractional derivatives are in a specific Lorentz space. These spaces are a natural generalization to the usual fractional Sobolev spaces. The thesis consists of two parts. In the first part, we define the generalized fractional Sobolev spaces and prove some of their properties including embeddings and interpolation identities. In particular we sharpen the usual Sobolev embedding into the space of Hölder-continuous functions, by showing that a particular kind of space embeds into the space of continuous functions without any modulus of continuity. The inverse problem is considered in the second part of the thesis. We prove a new Carleman estimate for ∂. This estimate has a fast decay rate, which will allow us to consider potentials with very low regularity. After that we use Bukhgeim s oscillating exponential solutions, Alessandrini s identity and stationary phase to get information about the difference of the potentials from the difference of the Cauchy data. The stability estimate will be of logarithmic type, but works with potentials of low regularity.
  • Tähtinen, Vesa (Helsingin yliopisto, 2010)
    This PhD Thesis is about certain infinite-dimensional Grassmannian manifolds that arise naturally in geometry, representation theory and mathematical physics. From the physics point of view one encounters these infinite-dimensional manifolds when trying to understand the second quantization of fermions. The many particle Hilbert space of the second quantized fermions is called the fermionic Fock space. A typical element of the fermionic Fock space can be thought to be a linear combination of the configurations m particles and n anti-particles . Geometrically the fermionic Fock space can be constructed as holomorphic sections of a certain (dual)determinant line bundle lying over the so called restricted Grassmannian manifold, which is a typical example of an infinite-dimensional Grassmannian manifold one encounters in QFT. The construction should be compared with its well-known finite-dimensional analogue, where one realizes an exterior power of a finite-dimensional vector space as the space of holomorphic sections of a determinant line bundle lying over a finite-dimensional Grassmannian manifold. The connection with infinite-dimensional representation theory stems from the fact that the restricted Grassmannian manifold is an infinite-dimensional homogeneous (Kähler) manifold, i.e. it is of the form G/H where G is a certain infinite-dimensional Lie group and H its subgroup. A central extension of G acts on the total space of the dual determinant line bundle and also on the space its holomorphic sections; thus G admits a (projective) representation on the fermionic Fock space. This construction also induces the so called basic representation for loop groups (of compact groups), which in turn are vitally important in string theory / conformal field theory. The Thesis consists of three chapters: the first chapter is an introduction to the backround material and the other two chapters are individually written research articles. The first article deals in a new way with the well-known question in Yang-Mills theory, when can one lift the action of the gauge transformation group on the space of connection one forms to the total space of the Fock bundle in a compatible way with the second quantized Dirac operator. In general there is an obstruction to this (called the Mickelsson-Faddeev anomaly) and various geometric interpretations for this anomaly, using such things as group extensions and bundle gerbes, have been given earlier. In this work we give a new geometric interpretation for the Faddeev-Mickelsson anomaly in terms of differentiable gerbes (certain sheaves of categories) and central extensions of Lie groupoids. The second research article deals with the question how to define a Dirac-like operator on the restricted Grassmannian manifold, which is an infinite-dimensional space and hence not in the landscape of standard Dirac operator theory. The construction relies heavily on infinite-dimensional representation theory and one of the most technically demanding challenges is to be able to introduce proper normal orderings for certain infinite sums of operators in such a way that all divergences will disappear and the infinite sum will make sense as a well-defined operator acting on a suitable Hilbert space of spinors. This research article was motivated by a more extensive ongoing project to construct twisted K-theory classes in Yang-Mills theory via a Dirac-like operator on the restricted Grassmannian manifold.
  • Yli-Juuti, Taina (Helsingin yliopisto, 2013)
    Atmospheric aerosol particles affect the visibility, damage human health and influence the Earth's climate by scattering and absorbing radiation and acting as cloud condensation nuclei (CCN). Considerable uncertainties are associated with the estimates of aerosol climatic effects and the extent of these effects depends on the particles size, composition, concentration and location in the atmosphere. Improved knowledge on the processes affecting these properties is of great importance in predicting future climate. Significant fraction of the atmospheric aerosol particles are formed in the atmosphere from trace gases through a phase change, i.e. nucleation. The freshly nucleated secondary aerosol particles are about a nanometer in diameter, and they need to grow tens of nanometers by condensation of vapors before they affect the climate. During the growth, the nanoparticles are subject to coagulational losses, and their survival to CCN sizes is greatly dependent on their growth rate. Therefore, capturing the nanoparticle growth correctly is crucial for representing aerosol effects in climate models. A large fraction of nanoparticle growth in many environments is expected to be due to organic compounds. However a full identification of the compounds and processes involved in the growth is lacking to date. In this thesis the variability in atmospheric nanoparticle growth rates with particle size and ambient conditions was studied based on observations at two locations, a boreal forest and a Central European rural site. The importance of various organic vapor uptake mechanisms and particle phase processes was evaluated, and two nanoparticle growth models were developed to study the effect of acid-base chemistry in the uptake of organic compounds by nanoparticles. Further, the effect of inorganic solutes on the partitioning of organic aerosol constituents between gas and particle phase was studied based on laboratory experiments. Observations of the atmospheric nanoparticle growth rates supported the hypothesis of organic compounds controlling the particle growth. The growth rates of particles with diameter PIENEMPI 20 nm vary with particle size, and the processes covering the uptake of organic vapors and limiting the nanoparticle growth were concluded to be size dependent. Formation of organic salts in the particle phase is likely to play a role in nanoparticle growth, however, according to the model predictions, it does not explain the uptake of semi-volatile organic compounds entirely. Small amount of inorganic salt does not seem to affect the volatility of organic acids, however with an increased inorganic content the case is not as clear.
  • Hienola, Anca (Helsingin yliopisto, 2008)
    The conversion of a metastable phase into a thermodynamically stable phase takes place via the formation of clusters. Clusters of different sizes are formed spontaneously within the metastable mother phase, but only those larger than a certain size, called the critical size, will end up growing into a new phase. There are two types of nucleation: homogeneous, where the clusters appear in a uniform phase, and heterogeneous, when pre-existing surfaces are available and clusters form on them. The nucleation of aerosol particles from gas-phase molecules is connected not only with inorganic compounds, but also with nonvolatile organic substances found in atmosphere. The question is which ones of the myriad of organic species have the right properties and are able to participate in nucleation phenomena. This thesis discusses both homogeneous and heterogeneous nucleation, having as theoretical tool the classical nucleation theory (CNT) based on thermodynamics. Different classes of organics are investigated. The members of the first class are four dicarboxylic acids (succinic, glutaric, malonic and adipic). They can be found in both the gas and particulate phases, and represent good candidates for the aerosol formation due to their low vapor pressure and solubility. Their influence on the nucleation process has not been largely investigated in the literature and it is not fully established. The accuracy of the CNT predictions for binary water-dicarboxylic acid systems depends significantly on the good knowledge of the thermophysical properties of the organics and their aqueous solutions. A large part of the thesis is dedicated to this issue. We have shown that homogeneous and heterogeneous nucleation of succinic, glutaric and malonic acids in combination with water is unlikely to happen in atmospheric conditions. However, it seems that adipic acid could participate in the nucleation process in conditions occurring in the upper troposphere. The second class of organics is represented by n-nonane and n-propanol. Their thermophysical properties are well established, and experiments on these substances have been performed. The experimental data of binary homogeneous and heterogeneous nucleation have been compared with the theoretical predictions. Although the n-nonane - n-propanol mixture is far from being ideal, CNT seems to behave fairly well, especially when calculating the cluster composition. In the case of heterogeneous nucleation, it has been found that better characterization of the substrate - liquid interaction by means of line tension and microscopic contact angle leads to a significant improvement of the CNT prediction. Unfortunately, this can not be achieved without well defined experimental data.
  • Lindberg, Sauli (Helsingin yliopisto, 2015)
    The dissertation deals with the Jacobian equation in the plane. R.R. Coifman, J.-P. Lions, Y. Meyer and S. Semmes proved in their seminal paper from 1993 that when a mapping from the n-space to the n-space belongs to a suitable homogeneous Sobolev space, its Jacobian determinant belongs to a real-variable Hardy space. Coifman, Lions, Meyer and Semmes proceeded to ask the following famous open problem: can every function in the Hardy space be written as the Jacobian of some Sobolev mapping? It follows from the work of G. Cupini, B. Dacorogna and O. Kneuss that the range of the Jacobian operator is dense in the Hardy space. As a consequence of this, solving the Jacobian equation reduces to proving that every so-called energy-minimal solution satisfies certain natural a priori estimate. In the dissertation we use Lagrange multipliers in Banach spaces to prove the sought after a priori estimate for a large class of energy-minimal solutions. It remains unclear whether the class is large enough to imply the surjectivity of the Jacobian operator, but we present many partial results on the properties of the class. To cite an example, when the Hardy space is endowed with a particular norm that is well suited to the study of the Jacobian equation, all the extreme points of the unit ball are Jacobians. Furthermore, the energy-minimal solutions for the extreme points satisfy the wanted a priori estimate. As one of the main results of the dissertation we reduce solving the Jacobian equation to a fairly concrete finite-dimensional problem. As the main tools of the dissertation we use Banach space geometry, harmonic analysis in the plane and methods from the theory of incompressible elasticity.
  • Wang, Keguang (Helsingin yliopisto, 2007)
    Pack ice is an aggregate of ice floes drifting on the sea surface. The forces controlling the motion and deformation of pack ice are air and water drag forces, sea surface tilt, Coriolis force and the internal force due to the interaction between ice floes. In this thesis, the mechanical behavior of compacted pack ice is investigated using theoretical and numerical methods, focusing on the three basic material properties: compressive strength, yield curve and flow rule. A high-resolution three-category sea ice model is applied to investigate the sea ice dynamics in two small basins, the whole Gulf Riga and the inside Pärnu Bay, focusing on the calibration of the compressive strength for thin ice. These two basins are on the scales of 100 km and 20 km, respectively, with typical ice thickness of 10-30 cm. The model is found capable of capturing the main characteristics of the ice dynamics. The compressive strength is calibrated to be about 30 kPa, consistent with the values from most large-scale sea ice dynamic studies. In addition, the numerical study in Pärnu Bay suggests that the shear strength drops significantly when the ice-floe size markedly decreases. A characteristic inversion method is developed to probe the yield curve of compacted pack ice. The basis of this method is the relationship between the intersection angle of linear kinematic features (LKFs) in sea ice and the slope of the yield curve. A summary of the observed LKFs shows that they can be basically divided into three groups: intersecting leads, uniaxial opening leads and uniaxial pressure ridges. Based on the available observed angles, the yield curve is determined to be a curved diamond. Comparisons of this yield curve with those from other methods show that it possesses almost all the advantages identified by the other methods. A new constitutive law is proposed, where the yield curve is a diamond and the flow rule is a combination of the normal and co-axial flow rule. The non-normal co-axial flow rule is necessary for the Coulombic yield constraint. This constitutive law not only captures the main features of forming LKFs but also takes the advantage of avoiding overestimating divergence during shear deformation. Moreover, this study provides a method for observing the flow rule for pack ice during deformation.
  • Vakkari, Ville (Helsingin yliopisto, 2013)
    Aerosol is defined as solid or liquid particles suspended in a gas lighter than the particles, which means that the atmosphere we live in is an aerosol in itself. Although aerosol particles are only a trace component of the atmosphere they affect our lives in several ways. The aerosol particles can cause adverse health effects and deteriorate visibility, but they affect also the Earth s climate directly by scattering and absorbing solar radiation and indirectly by modulating the properties of the clouds. Anthropogenic aerosol particles have a net cooling effect on the climate, but the uncertainty in the amount of cooling is presently as large as the heating effect of carbon dioxide. To reduce the uncertainty in the aerosol climate effects, spatially representative reference data of high quality are needed for the global climate models. To be able to capture the diurnal and seasonal variability the data have to be collected continuously over time periods that cover at least one full seasonal cycle. Until recently such data have been nearly non-existing for continental Africa and hence one aim of this work was to establish a permanent measurement station measuring the key aerosol particle properties in a continental location in southern Africa. In close collaboration with the North-West University in South Africa this aim has now been achieved at the Welgegund measurement station. The other aims of this work were to determine the aerosol particle concentrations including their seasonal and diurnal variation and to study the most important aerosol particle sources in continental southern Africa. In this thesis the aerosol size distribution and its seasonal and diurnal variation is reported for different environments ranging from a clean rural background to an anthropogenically heavily influenced mining region in continental southern Africa. Atmospheric regional scale new particle formation has been observed at a world record high frequency and it dominates the diurnal variation except in the vicinity of low-income residential areas, where domestic heating and cooking are a stronger source. The concentration of aerosol particles in sizes that can act as cloud condensation nuclei was found to increase during the dry season because of reduced wet removal and increased aerosol production from incomplete combustion, which can be either domestic heating or savannah and grassland fires depending on location. During the wet season new particle formation was shown to be an important source of particles in the size range of cloud condensation nuclei.