Boundedness of weakly singular integral operators on domains

The monograph dissertation deals with kernel integral operators and their mapping properties on Euclidean domains. The associated kernels are weakly singular and examples of such are given by Green functions of certain elliptic partial differential equations. It is well known that mapping properties of the corresponding Green operators can be used to deduce a priori estimates for the solutions of these equations. In the dissertation, natural size- and cancellation conditions are quantified for kernels defined in domains. These kernels induce integral operators which are then composed with any partial differential operator of prescribed order, depending on the size of the kernel. The main object of study in this dissertation being the boundedness properties of such compositions, the main result is the characterization of their Lp-boundedness on suitably regular domains. In case the aforementioned kernels are defined in the whole Euclidean space, their partial derivatives of prescribed order turn out to be so called standard kernels that arise in connection with singular integral operators. The Lp-boundedness of singular integrals is characterized by the T1 theorem, which is originally due to David and Journé and was published in 1984 (Ann. of Math. 120). The main result in the dissertation can be interpreted as a T1 theorem for weakly singular integral operators. The dissertation deals also with special convolution type weakly singular integral operators that are defined on Euclidean spaces.

Numerical computation of the module of a quadrilateral

The module of a quadrilateral is a positive real number which divides quadrilaterals into conformal equivalence classes. This is an introductory text to the module of a quadrilateral with some historical background and some numerical aspects. This work discusses the following topics: 1. Preliminaries 2. The module of a quadrilateral 3. The Schwarz-Christoffel Mapping 4. Symmetry properties of the module 5. Computational results 6. Other numerical methods Appendices include: Numerical evaluation of the elliptic integrals of the first kind. Matlab programs and scripts and possible topics for future research. Numerical results section covers additive quadrilaterals and the module of a quadrilateral under the movement of one of its vertex.

A computer code for enclosed natural convection in the boundary layer regime

A computer code is developed for the numerical prediction of natural convection in rectangular two-dimensional cavities at high Rayleigh numbers. The governing equations are retained in the primitive variable form. The numerical method is based on finite differences and an ADI scheme. Convective terms may be approximated with either central or hybrid differencing for greater stability. A non-uniform grid distribution is possible for greater efficiency. The pressure is dealt with via a SIMPLE type algorithm and the use of a fast elliptic solver for the solenoidal velocity correction field significantly reduces computing times. Preliminary results indicate that the code is reasonably accurate, robust and fast compared with existing benchmarks and finite difference based codes, particularly at high Rayleigh numbers. Extension to three-dimensional problems and turbulence studies in similar geometries is readily possible and indicated.

On the definition of a small orifice

An attempt is made to draw a line of demarcation between small orifices and large orifices. It is proposed that an orifice can be considered 'small' if the discharge through it calculated on the small-orifice assumption differs from the exact discharge by less than half of one per cent. Using this criterion, it is shown that a circular or elliptic orifice can be deemed 'small' as long as the ratio of the depth of the orifice to the head causing the flow (measured from the center of the orifice to the liquid surface) is less than 0.8; a rectangular orifice can be deemed 'small' if the ratio is less than 0.7. A correction factor is suggested for the coefficient of discharge to account for the deviation from the exact discharge.

Effects of recreational use and fragmentation on the understorey vegetation and soil microbial communities of urban forests in southern Finland

The impacts of fragmentation and recreational use on the hemiboreal urban forest understorey vegetation and the microbial community of the humus layer (the phospholipid fatty acid (PLFA) pattern, microbial biomass and microbial activity, measured as basal respiration) were examined in the greater Helsinki area, southern Finland. Trampling tolerance of 1) herb-rich OMT, 2) mesic MT, and 3) sub-xeric VT forests (in decreasing order of fertility) was studied by comparing relative understorey vegetation cover (urban/untrampled reference ratio) of the three forest types. The trampling tolerance of forest vegetation increased with the productivity of the site (sub-xeric < mesic < herb-rich). Wear of understorey vegetation correlated positively with the number of residents (i.e., recreational pressure) around the forest patch. An increase of 15000 residents within a radius of 1 km around a forest patch was associated with ca. 30% decrease in the relative understorey vegetation cover. The cover of dwarf shrub Vaccinium myrtillus in particular decreased with increasing levels of wear. The cover of mosses in urban forests was less than half of that in untrampled reference areas. Cover of tree saplings, mainly Sorbus aucuparia, and some resilient herbs was higher than in the reference areas. In small urban forest fragments, broad-leaved trees, grasses and herbs were more abundant and mosses were scarcer than in larger urban forest areas. Thus, due to trampling and edge effects, resilient herb and grass species are replacing sensitive dwarf shrubs, mosses and lichens in urban forests. Differences in the soil microbial community structure were found between paths and untrampled areas and the effects of paths extended more than one meter from the paths. Paths supported approximately 25-30% higher microbial biomass with a transition zone of at least 1 m from the path edge. However, microbial activity per unit of biomass was lower on paths than in untrampled areas. Furthermore, microbial biomass and activity were 30-45% lower at the first 20 m into the forest fragments, due to low moisture content of humus near the edge. The decreased microbial activity detected at forest edges and paths implies decreased litter decomposition rates, and thus, a change in nutrient cycling. Changes in the decomposition and nutrient supply may in turn affect the diversity and function of plant communities in urban forests. Keywords: boreal forest vegetation, edge effects, phospholipid fatty acids, trampling, urban woodlands, wear

Dark matter dominance at all radii in the superthin galaxy UGC 7321

We model the shape and density profile of the dark matter halo of the low surface brightness, superthin galaxy UGC 7321, using the observed rotation curve and the H i scale height data as simultaneous constraints. We treat the galaxy as a gravitationally coupled system of stars and gas, responding to the gravitational potential of the dark matter halo. An isothermal halo of spherical shape with a core density in the range of View the MathML source and a core radius between 2.5 and 2.9 kpc, gives the best fit to the observations for a range of realistic gas parameters assumed. We find that the best-fit core radius is only slightly higher than the stellar disc scale length (2.1 kpc), unlike the case of the high surface brightness galaxies where the halo core radius is typically 3–4 times the disc scale length of the stars. Thus our model shows that the dark matter halo dominates the dynamics of the low surface brightness, superthin galaxy UGC 7321 at all radii, including the inner parts of the galaxy.

The effects of habitat edges and trampling intensity on vegetation in urban forests

Although changes in urban forest vegetation have been documented in previous Finnish studies, the reasons for these changes have not been studied explicitly. Especially, the consequences of forest fragmentation, i.e. the fact that forest edges receive more solar radiation, wind and air-borne nutrients than interiors have been ignored. In order to limit the change in urban forest vegetation we need to know why it occurs. Therefore, the effects of edges and recreational use of urban forests on vegetation were investigated together in this thesis to reveal the relative strengths of these effects and to provide recommendations for forest management. Data were collected in the greater Helsinki area (in the cities of Helsinki, Vantaa and Espoo, and in the municipalities of Sipoo and Tuusula) and in the Lahti region (in the city of Lahti and in the municipality of Hollola) by means of systematic and randomized vegetation and soil sampling and tree measurements. Sample plots were placed from the forest edges to the interiors to investigate the effects of forest edges, and on paths of different levels of wear and off these paths to investigate the effects of trampling. The natural vegetation of mesic and sub-xeric forest site types studied was sensitive both to the effects of the edge and to trampling. The abundances of dwarf shrubs and bryophytes decreased, while light- and nitrogen-demanding herbs and grasses - and especially Sorbus aucuparia – were favoured at the edges and next to the paths. Results indicated that typical forest site types at the edges are changing toward more nitrophilic vegetation communities. Covers of the most abundant forest species decreased considerably – even tens of percentages – from interiors to the edges indicating strong edge effects. These effects penetrated at least up to 50 m from the forest edges into the interiors, especially at south to west facing open edges. The effects of trampling were pronounced on paths and even low levels of trampling decreased the abundances of certain species considerably. The effects of trampling extended up to 8 m from path edges. Results showed that the fragmentation of urban forest remnants into small and narrow patches should be avoided in order to maintain natural forest understorey vegetation in the urban setting. Thus, urban forest fragments left within urban development should be at least 3 ha in size, and as circular as possible. Where the preservation of representative original forest interior vegetation is a management aim, closed edges with conifers can act as an effective barrier against solar radiation, wind and urban load, thereby restricting the effects of the edge. Tree volume at the edge should be at least 225-250 m3 ha-1 and the proportion of conifers (especially spruce) 80% or more of the tree species composition. Closed, spruce-dominated edges may also prevent the excessive growth of S. aucuparia saplings at urban forest edges. In addition, closed edges may guide people’s movements to the maintained paths, thus preventing the spontaneous creation of dense path networks. In urban areas the effects of edges and trampling on biodiversity may be considerable, and are important to consider when the aim of management is to prevent the development of homogeneous herb-grass dominated vegetation communities, as was observed at the investigated edges.

Contribution from unresolved discrete sources to the extragalactic gamma-ray background (EGRB)

The origin of the extragalactic gamma-ray background (EGRB) is still an open question, even nearly forty years after its discovery. The emission could originate either from truly diffuse processes or from unresolved point sources. Although the majority of the 271 point sources detected by EGRET (Energetic Gamma Ray Experiment Telescope) are unidentified, of the identified sources, blazars are the dominant candidates. Therefore, unresolved blazars may be considered the main contributor to the EGRB, and many studies have been carried out to understand their distribution, evolution and contribution to the EGRB. Considering that gamma-ray emission comes mostly from jets of blazars and that the jet emission decreases rapidly with increasing jet to line-of-sight angle, it is not surprising that EGRET was not able to detect many large inclination angle active galactic nuclei (AGNs). Though Fermi could only detect a few large inclination angle AGNs during the first three months of its survey, it is expected to detect many such sources in the near future. Since non-blazar AGNs are expected to have higher density as compared to blazars, these could also contribute significantly to the EGRB. In this paper, we discuss contributions from unresolved discrete sources including normal galaxies, starburst galaxies, blazars and off-axis AGNs to the EGRB.

Third post-Newtonian angular momentum flux and the secular evolution of orbital elements for inspiralling compact binaries in quasi-elliptical orbits

The angular-momentum flux from an inspiralling binary system of compact objects moving in quasi-elliptical orbits is computed at the third post-Newtonian (3PN) order using the multipolar post-Minkowskian wave generation formalism. The 3PN angular-momentum flux involves the instantaneous, tail, and tail-of-tails contributions as for the 3PN energy flux, and in addition a contribution due to nonlinear memory. We average the angular-momentum flux over the binary's orbit using the 3PN quasi-Keplerian representation of elliptical orbits. The averaged angular-momentum flux provides the final input needed for gravitational-wave phasing of binaries moving in quasi-elliptical orbits. We obtain the evolution of orbital elements under 3PN gravitational radiation reaction in the quasi-elliptic case. For small eccentricities, we give simpler limiting expressions relevant for phasing up to order e(2). This work is important for the construction of templates for quasi-eccentric binaries, and for the comparison of post-Newtonian results with the numerical relativity simulations of the plunge and merger of eccentric binaries.

Global lopsided instability in a purely stellar galactic disc

It is shown that pure exponential discs in spiral galaxies are capable of supporting slowly varying discrete global lopsided modes, which can explain the observed features of lopsidedness in the stellar discs. Using linearized fluid dynamical equations with the softened self-gravity and pressure of the perturbation as the collective effect, we derive self-consistently a quadratic eigenvalue equation for the lopsided perturbation in the galactic disc. On solving this, we find that the ground-state mode shows the observed characteristics of the lopsidedness in a galactic disc, namely the fractional Fourier amplitude A(1), increases smoothly with the radius. These lopsided patterns precess in the disc with a very slow pattern speed with no preferred sense of precession. We show that the lopsided modes in the stellar disc are long-lived because of a substantial reduction (approximately a factor of 10 compared to the local free precession rate) in the differential precession. The numerical solution of the equations shows that the groundstate lopsided modes are either very slowly precessing stationary normal mode oscillations of the disc or growing modes with a slow growth rate depending on the relative importance of the collective effect of the self-gravity. N-body simulations are performed to test the spontaneous growth of lopsidedness in a pure stellar disc. Both approaches are then compared and interpreted in terms of long-lived global m = 1 instabilities, with almost zero pattern speed.

Multi-Tone Testing of Linear and Nonlinear Analog Circuits using Polynomial Coefficients

A method of testing for parametric faults of analog circuits based on a polynomial representation of fault-free function of the circuit is presented. The response of the circuit under test (CUT) is estimated as a polynomial in the applied input voltage at relevant frequencies in addition to DC. Classification or Cur is based on a comparison of the estimated polynomial coefficients with those of the fault free circuit. This testing method requires no design for test hardware as might be added to the circuit fly some other methods. The proposed method is illustrated for a benchmark elliptic filter. It is shown to uncover several parametric faults causing deviations as small as 5% from the nominal values.

Acceleration of the Cosmological Expansion as an Effect of Inhomogeneities

The cosmological observations of light from type Ia supernovae, the cosmic microwave background and the galaxy distribution seem to indicate that the expansion of the universe has accelerated during the latter half of its age. Within standard cosmology, this is ascribed to dark energy, a uniform fluid with large negative pressure that gives rise to repulsive gravity but also entails serious theoretical problems. Understanding the physical origin of the perceived accelerated expansion has been described as one of the greatest challenges in theoretical physics today. In this thesis, we discuss the possibility that, instead of dark energy, the acceleration would be caused by an effect of the nonlinear structure formation on light, ignored in the standard cosmology. A physical interpretation of the effect goes as follows: due to the clustering of the initially smooth matter with time as filaments of opaque galaxies, the regions where the detectable light travels get emptier and emptier relative to the average. As the developing voids begin to expand the faster the lower their matter density becomes, the expansion can then accelerate along our line of sight without local acceleration, potentially obviating the need for the mysterious dark energy. In addition to offering a natural physical interpretation to the acceleration, we have further shown that an inhomogeneous model is able to match the main cosmological observations without dark energy, resulting in a concordant picture of the universe with 90% dark matter, 10% baryonic matter and 15 billion years as the age of the universe. The model also provides a smart solution to the coincidence problem: if induced by the voids, the onset of the perceived acceleration naturally coincides with the formation of the voids. Additional future tests include quantitative predictions for angular deviations and a theoretical derivation of the model to reduce the required phenomenology. A spin-off of the research is a physical classification of the cosmic inhomogeneities according to how they could induce accelerated expansion along our line of sight. We have identified three physically distinct mechanisms: global acceleration due to spatial variations in the expansion rate, faster local expansion rate due to a large local void and biased light propagation through voids that expand faster than the average. A general conclusion is that the physical properties crucial to account for the perceived acceleration are the growth of the inhomogeneities and the inhomogeneities in the expansion rate. The existence of these properties in the real universe is supported by both observational data and theoretical calculations. However, better data and more sophisticated theoretical models are required to vindicate or disprove the conjecture that the inhomogeneities are responsible for the acceleration.

Non-Gaussianities and Preheating from N-flation and Magnetogenesis via rotating cosmic string loops

In this thesis we examine multi-field inflationary models of the early Universe. Since non-Gaussianities may allow for the possibility to discriminate between models of inflation, we compute deviations from a Gaussian spectrum of primordial perturbations by extending the delta-N formalism. We use N-flation as a concrete model; our findings show that these models are generically indistinguishable as long as the slow roll approximation is still valid. Besides computing non-Guassinities, we also investigate Preheating after multi-field inflation. Within the framework of N-flation, we find that preheating via parametric resonance is suppressed, an indication that it is the old theory of preheating that is applicable. In addition to studying non-Gaussianities and preheatng in multi-field inflationary models, we study magnetogenesis in the early universe. To this aim, we propose a mechanism to generate primordial magnetic fields via rotating cosmic string loops. Magnetic fields in the micro-Gauss range have been observed in galaxies and clusters, but their origin has remained elusive. We consider a network of strings and find that rotating cosmic string loops, which are continuously produced in such networks, are viable candidates for magnetogenesis with relevant strength and length scales, provided we use a high string tension and an efficient dynamo.

Studies of the star-forming structures in the dense interstellar medium: a view by dust extinction

New stars in galaxies form in dense, molecular clouds of the interstellar medium. Measuring how the mass is distributed in these clouds is of crucial importance for the current theories of star formation. This is because several open issues in them, such as the strength of different mechanism regulating star formation and the origin of stellar masses, can be addressed using detailed information on the cloud structure. Unfortunately, quantifying the mass distribution in molecular clouds accurately over a wide spatial and dynamical range is a fundamental problem in the modern astrophysics. This thesis presents studies examining the structure of dense molecular clouds and the distribution of mass in them, with the emphasis on nearby clouds that are sites of low-mass star formation. In particular, this thesis concentrates on investigating the mass distributions using the near infrared dust extinction mapping technique. In this technique, the gas column densities towards molecular clouds are determined by examining radiation from the stars that shine through the clouds. In addition, the thesis examines the feasibility of using a similar technique to derive the masses of molecular clouds in nearby external galaxies. The papers presented in this thesis demonstrate how the near infrared dust extinction mapping technique can be used to extract detailed information on the mass distribution in nearby molecular clouds. Furthermore, such information is used to examine characteristics crucial for the star formation in the clouds. Regarding the use of extinction mapping technique in nearby galaxies, the papers of this thesis show that deriving the masses of molecular clouds using the technique suffers from strong biases. However, it is shown that some structural properties can still be examined with the technique.

Formation of structure in dark energy cosmologies

Acceleration of the universe has been established but not explained. During the past few years precise cosmological experiments have confirmed the standard big bang scenario of a flat universe undergoing an inflationary expansion in its earliest stages, where the perturbations are generated that eventually form into galaxies and other structure in matter, most of which is non-baryonic dark matter. Curiously, the universe has presently entered into another period of acceleration. Such a result is inferred from observations of extra-galactic supernovae and is independently supported by the cosmic microwave background radiation and large scale structure data. It seems there is a positive cosmological constant speeding up the universal expansion of space. Then the vacuum energy density the constant describes should be about a dozen times the present energy density in visible matter, but particle physics scales are enormously larger than that. This is the cosmological constant problem, perhaps the greatest mystery of contemporary cosmology. In this thesis we will explore alternative agents of the acceleration. Generically, such are called dark energy. If some symmetry turns off vacuum energy, its value is not a problem but one needs some dark energy. Such could be a scalar field dynamically evolving in its potential, or some other exotic constituent exhibiting negative pressure. Another option is to assume that gravity at cosmological scales is not well described by general relativity. In a modified theory of gravity one might find the expansion rate increasing in a universe filled by just dark matter and baryons. Such possibilities are taken here under investigation. The main goal is to uncover observational consequences of different models of dark energy, the emphasis being on their implications for the formation of large-scale structure of the universe. Possible properties of dark energy are investigated using phenomenological paramaterizations, but several specific models are also considered in detail. Difficulties in unifying dark matter and dark energy into a single concept are pointed out. Considerable attention is on modifications of gravity resulting in second order field equations. It is shown that in a general class of such models the viable ones represent effectively the cosmological constant, while from another class one might find interesting modifications of the standard cosmological scenario yet allowed by observations. The thesis consists of seven research papers preceded by an introductory discussion.