Direct and Inverse Scattering for Beltrami Fields

We consider an obstacle scattering problem for linear Beltrami fields. A vector field is a linear Beltrami field if the curl of the field is a constant times itself. We study the obstacles that are of Neumann type, that is, the normal component of the total field vanishes on the boundary of the obstacle. We prove the unique solvability for the corresponding exterior boundary value problem, in other words, the direct obstacle scattering model. For the inverse obstacle scattering problem, we deduce the formulas that are needed to apply the singular sources method. The numerical examples are computed for the direct scattering problem and for the inverse scattering problem.

Direct and Inverse Scattering for Beltrami Fields

We consider an obstacle scattering problem for linear Beltrami fields. A vector field is a linear Beltrami field if the curl of the field is a constant times itself. We study the obstacles that are of Neumann type, that is, the normal component of the total field vanishes on the boundary of the obstacle. We prove the unique solvability for the corresponding exterior boundary value problem, in other words, the direct obstacle scattering model. For the inverse obstacle scattering problem, we deduce the formulas that are needed to apply the singular sources method. The numerical examples are computed for the direct scattering problem and for the inverse scattering problem.

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.

Passive vector turbulence

This thesis consists of three articles on passive vector fields in turbulence. The vector fields interact with a turbulent velocity field, which is described by the Kraichnan model. The effect of the Kraichnan model on the passive vectors is studied via an equation for the pair correlation function and its solutions. The first paper is concerned with the passive magnetohydrodynamic equations. Emphasis is placed on the so called "dynamo effect", which in the present context is understood as an unbounded growth of the pair correlation function. The exact analytical conditions for such growth are found in the cases of zero and infinite Prandtl numbers. The second paper contains an extensive study of a number of passive vector models. Emphasis is now on the properties of the (assumed) steady state, namely anomalous scaling, anisotropy and small and large scale behavior with different types of forcing or stirring. The third paper is in many ways a completion to the previous one in its study of the steady state existence problem. Conditions for the existence of the steady state are found in terms of the spatial roughness parameter of the turbulent velocity field.

The ozone transfer between atmosphere and vegetation. A study on Scots pine in the field

Ozone (O3) is a reactive gas present in the troposphere in the range of parts per billion (ppb), i.e. molecules of O3 in 109 molecules of air. Its strong oxidative capacity makes it a key element in tropospheric chemistry and a threat to the integrity of materials, including living organisms. Knowledge and control of O3 levels are an issue in relation to indoor air quality, building material endurance, respiratory human disorders, and plant performance. Ozone is also a greenhouse gas and its abundance is relevant to global warming. The interaction of the lower troposphere with vegetated landscapes results in O3 being removed from the atmosphere by reactions that lead to the oxidation of plant-related components. Details on the rate and pattern of removal on different landscapes as well as the ultimate mechanisms by which this occurs are not fully resolved. This thesis analysed the controlling processes of the transfer of ozone at the air-plant interface. Improvement in the knowledge of these processes benefits the prediction of both atmospheric removal of O3 and its impact on vegetation. This study was based on the measurement and analysis of multi-year field measurements of O3 flux to Scots pine (Pinus sylvestris L.) foliage with a shoot-scale gas-exchange enclosure system. In addition, the analyses made use of simultaneous CO2 and H2O exchange, canopy-scale O3, CO2 and H2O exchange, foliage surface wetness, and environmental variables. All data was gathered at the SMEAR measuring station (southern Finland). Enclosure gas-exchange techniques such as those commonly used for the measure of CO2 and water vapour can be applied to the measure of ozone gas-exchange in the field. Through analysis of the system dynamics the occurring disturbances and noise can be identified. In the system used in this study, the possible artefacts arising from the ozone reactivity towards the system materials in combination with low background concentrations need to be taken into account. The main artefact was the loss of ozone towards the chamber walls, which was found to be very variable. The level of wall-loss was obtained from simultaneous and continuous measurements, and was included in the formulation of the mass balance of O3 concentration inside the chamber. The analysis of the field measurements in this study show that the flux of ozone to the Scots pine foliage is generated in about equal proportions by stomatal and non-stomatal controlled processes. Deposition towards foliage and forest is sustained also during night and winter when stomatal gas-exchange is low or absent. The non-stomatal portion of the flux was analysed further. The pattern of flux in time was found to be an overlap of the patterns of biological activity and presence of wetness in the environment. This was seen to occur both at the shoot and canopy scale. The presence of wetness enhanced the flux not only in the presence of liquid droplets but also during existence of a moisture film on the plant surfaces. The existence of these films and their relation to the ozone sinks was determined by simultaneous measurements of leaf surface wetness and ozone flux. The results seem to suggest ozone would be reacting at the foliage surface and the reaction rate would be mediated by the presence of surface wetness. Alternative mechanisms were discussed, including nocturnal stomatal aperture and emission of reactive volatile compounds. The prediction of the total flux could thus be based on a combination of a model of stomatal behaviour and a model of water absorption on the foliage surfaces. The concepts behind the division of stomatal and non-stomatal sinks were reconsidered. This study showed that it is theoretically possible that a sink located before or near the stomatal aperture prevents or diminishes the diffusion of ozone towards the intercellular air space of the mesophyll. This obstacle to stomatal diffusion happens only under certain conditions, which include a very low presence of reaction sites in the mesophyll, an extremely strong sink located on the outer surfaces or stomatal pore. The relevance, or existence, of this process in natural conditions would need to be assessed further. Potentially strong reactions were considered, including dissolved sulphate, volatile organic compounds, and apoplastic ascorbic acid. Information on the location and the relative abundance of these compounds would be valuable. The highest total flux towards the foliage and forest happens when both the plant activity and ambient moisture are high. The highest uptake into the interior of the foliage happens at large stomatal apertures, provided that scavenging reactions located near the stomatal pore are weak or non-existent. The discussion covers the methodological developments of this study, the relevance of the different controlling factors of ozone flux, the partition amongst its component, and the possible mechanisms of non-stomatal uptake.

Effects of seed origin latitude on the timing of height growth cessation and field performance of silver birch

The aim of this thesis was to increase our knowledge about the effects of seed origin on the timing of height growth cessation and field performance of silver birch from different latitudes, with special attention paid to the browsing damage by moose in young birch plantations. The effect of seed origin latitude and sowing time on timing of height growth cessation of first-year seedlings was studied in a greenhouse experiment with seven seed origins (lat. 58º - 67ºN). Variation in critical night length (CNL) for 50 % bud set within two latitudinally distant stands (60º and 67ºN) was studied in three phytotron experiments. Browsing by moose on 5-11 -year-old silver birch saplings from latitudinally different seed origins (53º - 67ºN) was studied in a field experiment in southern Finland. Yield and stem quality of 22-year-old silver birch trees of Baltic, Finnish and Russian origin (54º - 63ºN) and the effect of latitudinal seed transfers were studied in two provenance trials at Tuusula, southern and Viitasaari, central Finland. The timing of height growth cessation depended systematically on latitude of seed origin and sowing date. The more northern the seed origin, the earlier the growth cessation and the shorter the growth period. Later sowing dates delayed growth cessation but also shortened the growth period. The mean CNL of the southern ecotype was longer, 6.3 ± 0.2 h (95 % confidence interval), than that of the northern ecotype, 3.1 ± 0.3 h. Within-ecotype variance of the CNL was higher in the northern ecotype (0.484 h2) than in the southern ecotype (0.150 h2). Browsing by moose decreased with increasing latitude of seed origin and sapling height. Origins transferred from more southern latitudes were more heavily browsed than the more northern native ones. Southern Finnish seed origins produced the highest volume per unit area in central Finland (lat. 63º11'N). Estonian and north Latvian stand seed origins, and the southern Finnish plus tree origins, were the most productive ones in southern Finland (lat. 60º21'N). Latitudinal seed transfer distance had a significant effect on survival, stem volume/ha and proportion of trees with a stem defect. The relationship of both survival and stem volume/ha to the latitudinal seed transfer distance was curvilinear. Volume was increased by transferring seed from ca. 2 degrees of latitude from the south. A longer transfer from the south, and transfer from the north, decreased the yield. The proportion of trees with a stem defect increased linearly in relation to the latitudinal seed transfer distance from the south.

Ammonia emissions from pig and cattle slurry in the field and utilization of slurry nitrogen in crop production

Volatilization of ammonia (NH3) from animal manure is a major pathway for nitrogen (N) losses that cause eutrophication, acidification, and other environmental hazards. In this study, the effect of alternative techniques of manure treatment (aeration, separation, addition of peat) and application (broadcast spreading, band spreading, injection, incorporation by harrowing) on ammonia emissions in the field and on nitrogen uptake by ley or cereals was studied. The effect of a mixture of slurry and peat on soil properties was also investigated. The aim of this study was to find ways to improve the utilization of manure nitrogen and reduce its release to the environment. Injection into the soil or incorporation by harrowing clearly reduced ammonia volatilization from slurry more than did the surface application onto a smaller area by band spreading or reduction of the dry matter of slurry by aeration or separation. Surface application showed low ammonia volatilization, when pig slurry was applied to tilled bare clay soil or to spring wheat stands in early growth stages. Apparently, the properties of both slurry and soil enabled the rapid infiltration and absorption of slurry and its ammoniacal nitrogen by the soil. On ley, however, surface-applied cattle slurry lost about half of its ammoniacal nitrogen. The volatilization of ammonia from surface-applied peat manure was slow, but proceeded over a long period of time. After rain or irrigation, the peat manure layer on the soil surface retarded evaporation. Incorporation was less important for the fertilizer effect of peat manure than for pig slurry, but both manures were more effective when incorporated. Peat manure applications increase soil organic matter content and aggregate stability. Stubble mulch tillage hastens the effect in surface soil compared with ploughing. The apparent recovery of ammoniacal manure nitrogen in crop yield was higher with injection and incorporation than with surface applications. This was the case for leys as well as for spring cereals, even though ammonia losses from manures applied to cereals were relatively low with surface applications as well. The ammoniacal nitrogen of surface-applied slurry was obviously adsorbed by the very surface soil and remained mostly unavailable to plant roots in the dry soil. Supplementing manures with inorganic fertilizer nitrogen, which adds plant-available nitrogen to the soil at the start of growth, increased the overall recovery of applied nitrogen in crop yields.

Noncommutative Quantum Field Theories and UV/IR Mixing

Our present-day understanding of fundamental constituents of matter and their interactions is based on the Standard Model of particle physics, which relies on quantum gauge field theories. On the other hand, the large scale dynamical behaviour of spacetime is understood via the general theory of relativity of Einstein. The merging of these two complementary aspects of nature, quantum and gravity, is one of the greatest goals of modern fundamental physics, the achievement of which would help us understand the short-distance structure of spacetime, thus shedding light on the events in the singular states of general relativity, such as black holes and the Big Bang, where our current models of nature break down. The formulation of quantum field theories in noncommutative spacetime is an attempt to realize the idea of nonlocality at short distances, which our present understanding of these different aspects of Nature suggests, and consequently to find testable hints of the underlying quantum behaviour of spacetime. The formulation of noncommutative theories encounters various unprecedented problems, which derive from their peculiar inherent nonlocality. Arguably the most serious of these is the so-called UV/IR mixing, which makes the derivation of observable predictions especially hard by causing new tedious divergencies, to which our previous well-developed renormalization methods for quantum field theories do not apply. In the thesis I review the basic mathematical concepts of noncommutative spacetime, different formulations of quantum field theories in the context, and the theoretical understanding of UV/IR mixing. In particular, I put forward new results to be published, which show that also the theory of quantum electrodynamics in noncommutative spacetime defined via Seiberg-Witten map suffers from UV/IR mixing. Finally, I review some of the most promising ways to overcome the problem. The final solution remains a challenge for the future.

The Spin-Statistics Relation and Noncommutative Quantum Field Theory

The efforts of combining quantum theory with general relativity have been great and marked by several successes. One field where progress has lately been made is the study of noncommutative quantum field theories that arise as a low energy limit in certain string theories. The idea of noncommutativity comes naturally when combining these two extremes and has profound implications on results widely accepted in traditional, commutative, theories. In this work I review the status of one of the most important connections in physics, the spin-statistics relation. The relation is deeply ingrained in our reality in that it gives us the structure for the periodic table and is of crucial importance for the stability of all matter. The dramatic effects of noncommutativity of space-time coordinates, mainly the loss of Lorentz invariance, call the spin-statistics relation into question. The spin-statistics theorem is first presented in its traditional setting, giving a clarifying proof starting from minimal requirements. Next the notion of noncommutativity is introduced and its implications studied. The discussion is essentially based on twisted Poincaré symmetry, the space-time symmetry of noncommutative quantum field theory. The controversial issue of microcausality in noncommutative quantum field theory is settled by showing for the first time that the light wedge microcausality condition is compatible with the twisted Poincaré symmetry. The spin-statistics relation is considered both from the point of view of braided statistics, and in the traditional Lagrangian formulation of Pauli, with the conclusion that Pauli's age-old theorem stands even this test so dramatic for the whole structure of space-time.

Asymmetrical flow field-flow fractionation in the study of water-soluble macromolecules

Asymmetrical flow field-flow fractionation (AsFlFFF) was constructed, and its applicability to industrial, biochemical, and pharmaceutical applications was studied. The effect of several parameters, such as pH, ionic strength, temperature and the reactants mixing ratios on the particle sizes, molar masses, and the formation of aggregates of macromolecules was determined by AsFlFFF. In the case of industrial application AsFlFFF proved to be a valuable tool in the characterization of the hydrodynamic particle sizes, molar masses and phase transition behavior of various poly(N-isopropylacrylamide) (PNIPAM) polymers as a function of viscosity and phase transition temperatures. The effect of sodium chloride salt and the molar ratio of cationic and anionic polyelectrolytes on the hydrodynamic particle sizes of poly (methacryloxyethyl trimethylammonium chloride) and poly (ethylene oxide)-block-poly (sodium methacrylate) and their complexes were studied. The particle sizes of PNIPAM polymers, and polyelectrolyte complexes measured by AsFlFFF were in agreement with those obtained by dynamic light scattering. The molar masses of PNIPAM polymers obtained by AsFlFFF and size exclusion chromatography agreed also well. In addition, AsFlFFF proved to be a practical technique in thermo responsive behavior studies of polymers at temperatures up to about 50 oC. The suitability of AsFlFFF for biological, biomedical, and pharmaceutical applications was proved, upon studying the lipid-protein/peptide interactions, and the stability of liposomes at different temperatures. AsFlFFF was applied to the studies on the hydrophobic and electrostatic interactions between cytochrome c (a basic peripheral protein) and anionic lipid, and oleic acid, and sodium dodecyl sulphate surfactant. A miniaturized AsFlFFF constructed in this study was exploited in the elucidation of the effect of copper (II), pH, ionic strength, and vortexing on the particle sizes of low-density lipoproteins.

Composition operators on vector-valued BMOA and related function spaces

A composition operator is a linear operator between spaces of analytic or harmonic functions on the unit disk, which precomposes a function with a fixed self-map of the disk. A fundamental problem is to relate properties of a composition operator to the function-theoretic properties of the self-map. During the recent decades these operators have been very actively studied in connection with various function spaces. The study of composition operators lies in the intersection of two central fields of mathematical analysis; function theory and operator theory. This thesis consists of four research articles and an overview. In the first three articles the weak compactness of composition operators is studied on certain vector-valued function spaces. A vector-valued function takes its values in some complex Banach space. In the first and third article sufficient conditions are given for a composition operator to be weakly compact on different versions of vector-valued BMOA spaces. In the second article characterizations are given for the weak compactness of a composition operator on harmonic Hardy spaces and spaces of Cauchy transforms, provided the functions take values in a reflexive Banach space. Composition operators are also considered on certain weak versions of the above function spaces. In addition, the relationship of different vector-valued function spaces is analyzed. In the fourth article weighted composition operators are studied on the scalar-valued BMOA space and its subspace VMOA. A weighted composition operator is obtained by first applying a composition operator and then a pointwise multiplier. A complete characterization is given for the boundedness and compactness of a weighted composition operator on BMOA and VMOA. Moreover, the essential norm of a weighted composition operator on VMOA is estimated. These results generalize many previously known results about composition operators and pointwise multipliers on these spaces.

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.