Mikä yliopisto-opiskelijaa kiihdyttää? : Opiskeluympäristöjen tutkimusta CASS-kokemusotantamenetelmällä flow-teoriaa ja ydintunneteorioita soveltaen

Flow experience is often defined either as an experience of high concentration and enjoyment or as a situation, where high challenges are matched with high skills. According to core-emotion theories, the experience of any emotion contains two core emotions: valence and arousal. Through an accurate mathematical model, the present study investigated, whether the experience of concentration and enjoyment is related to situations where both challenge and skills are high and in balance. Further, it was investigated what sort of core emotions are related to differing relationships between challenge and skills. Finally, university students’ experiences of their natural study environments were described in terms of core emotions and in terms of relationships between challenge and skills. Participants were 55 university students who participated two weeks research period. Altogether 3367 questionnaire answers were collected with the CASS experience-sampling method, operating in 3G-mobile phones. The relationship between challenge and skills (competence) was defined in an exact way in polar coordinates. An enjoyable and concentrated flow experience was defined as a sum variable of absorption, interest and enthusiasm. Core emotions were calculated with factor analysis from nine emotion variables. As expected, an experience of concentration and enjoyment was, on average, related to the situations where both challenge and skills were high and in balance. This was not, however, the case in every situation. Thus, it should be taken into consideration how flow experience is operationalised in experience sampling studies. When flow experience was defined as a situation of high challenge and high skills, it was often related to high valence and arousal emotions such as excitement or enthusiasm. A happier or a more tranquil enjoyment was related to situations of moderate challenge and high skills. Experiences differed clearly between various natural study environments. At lectures students were often bored or mentally absent, and did not experience challenges. In a small group students were often excited or enthusiastic, and showed optimal balance between challenge and skills. At library students felt satisfied and were engaged in highly challenging work.

Activator Protein-1 and Epidermal Growth Factor Receptor Interplay : In Vivo & In Vitro Studies

Critical cellular decisions such as should the cell proliferate, migrate or differentiate, are regulated by stimulatory signals from the extracellular environment, like growth factors. These signals are transformed to cellular responses through their binding to specific receptors present at the surface of the recipient cell. The epidermal growth factor receptor (EGF-R/ErbB) pathway plays key roles in governing these signals to intracellular events and cell-to-cell communication. The EGF-R forms a signaling network that participates in the specification of cell fate and coordinates cell proliferation. Ligand binding triggers receptor dimerization leading to the recruitment of kinases and adaptor proteins. This step simultaneously initiates multiple signal transduction pathways, which result in activation of transcription factors and other target proteins, leading to cellular alterations. It is known that mutations of EGF-R or in the components of these pathways, such as Ras and Raf, are commonly involved in human cancer. The four best characterized signaling pathways induced by EGF-R are the mitogen-activated protein kinase cascades (MAPKs), the lipid kinase phosphatidylinositol 3 kinase (PI3K), a group of transcription factors called Signal Transducers and Activator of Transcription (STAT), and the phospholipase Cγ; (PLCγ) pathways. The activation of each cascade culminates in kinase translocation to the nucleus to stimulate various transcription factors including activator protein 1 (AP-1). AP-1 family proteins are basic leucine zipper (bZIP) transcription factors that are implicated in the regulation of a variety of cellular processes (proliferation and survival, growth, differentiation, apoptosis, cell migration, transformation). Therefore, the regulation of AP-1 activity is critical for the decision of cell fate and their deregulated expression is widely associated with many types of cancers, such as breast and prostate cancers. The aims of this study were to characterize the roles of EGF-R signaling during normal development and malignant growth in vitro and in vivo using different cell lines and tissue samples. We show here that EGF-R regulates cell proliferation but is also required for regulation of AP-1 target gene expression in fibroblasts in a MAP-kinase mediated manner. Furthermore, EGF-R signaling is essential for enterocyte proliferation and migration during intestinal maturation. EGF-R signaling network, especially PI3-K-Akt pathway mediated AP-1 activity is involved in cellular survival in response to ionizing radiation. Taken together, these results elucidate the connection of EGF-R and AP-1 in various cellular contexts and show their importance in the regulation of cellular behaviour presenting new treatment cues for intestinal perforations and cancer therapy.

Digitoitujen väri-infrailmakuvien numeerinen monikuvatulkinta metsäninventoinnissa

The use of remote sensing imagery as auxiliary data in forest inventory is based on the correlation between features extracted from the images and the ground truth. The bidirectional reflectance and radial displacement cause variation in image features located in different segments of the image but forest characteristics remaining the same. The variation has so far been diminished by different radiometric corrections. In this study the use of sun azimuth based converted image co-ordinates was examined to supplement auxiliary data extracted from digitised aerial photographs. The method was considered as an alternative for radiometric corrections. Additionally, the usefulness of multi-image interpretation of digitised aerial photographs in regression estimation of forest characteristics was studied. The state owned study area located in Leivonmäki, Central Finland and the study material consisted of five digitised and ortho-rectified colour-infrared (CIR) aerial photographs and field measurements of 388 plots, out of which 194 were relascope (Bitterlich) plots and 194 were concentric circular plots. Both the image data and the field measurements were from the year 1999. When examining the effect of the location of the image point on pixel values and texture features of Finnish forest plots in digitised CIR photographs the clearest differences were found between front-and back-lighted image halves. Inside the image half the differences between different blocks were clearly bigger on the front-lighted half than on the back-lighted half. The strength of the phenomenon varied by forest category. The differences between pixel values extracted from different image blocks were greatest in developed and mature stands and smallest in young stands. The differences between texture features were greatest in developing stands and smallest in young and mature stands. The logarithm of timber volume per hectare and the angular transformation of the proportion of broadleaved trees of the total volume were used as dependent variables in regression models. Five different converted image co-ordinates based trend surfaces were used in models in order to diminish the effect of the bidirectional reflectance. The reference model of total volume, in which the location of the image point had been ignored, resulted in RMSE of 1,268 calculated from test material. The best of the trend surfaces was the complete third order surface, which resulted in RMSE of 1,107. The reference model of the proportion of broadleaved trees resulted in RMSE of 0,4292 and the second order trend surface was the best, resulting in RMSE of 0,4270. The trend surface method is applicable, but it has to be applied by forest category and by variable. The usefulness of multi-image interpretation of digitised aerial photographs was studied by building comparable regression models using either the front-lighted image features, back-lighted image features or both. The two-image model turned out to be slightly better than the one-image models in total volume estimation. The best one-image model resulted in RMSE of 1,098 and the two-image model resulted in RMSE of 1,090. The homologous features did not improve the models of the proportion of broadleaved trees. The overall result gives motivation for further research of multi-image interpretation. The focus may be improving regression estimation and feature selection or examination of stratification used in two-phase sampling inventory techniques. Keywords: forest inventory, digitised aerial photograph, bidirectional reflectance, converted image co­ordinates, regression estimation, multi-image interpretation, pixel value, texture, trend surface

Noncommutative Gravitation as a Gauge Theory of Twisted Poincaré Symmetry

Gravitaation kvanttiteorian muotoilu on ollut teoreettisten fyysikkojen tavoitteena kvanttimekaniikan synnystä lähtien. Kvanttimekaniikan soveltaminen korkean energian ilmiöihin yleisen suhteellisuusteorian viitekehyksessä johtaa aika-avaruuden koordinaattien operatiiviseen ei-kommutoivuuteen. Ei-kommutoivia aika-avaruuden geometrioita tavataan myös avointen säikeiden säieteorioiden tietyillä matalan energian rajoilla. Ei-kommutoivan aika-avaruuden gravitaatioteoria voisi olla yhteensopiva kvanttimekaniikan kanssa ja se voisi mahdollistaa erittäin lyhyiden etäisyyksien ja korkeiden energioiden prosessien ei-lokaaliksi uskotun fysiikan kuvauksen, sekä tuottaa yleisen suhteellisuusteorian kanssa yhtenevän teorian pitkillä etäisyyksillä. Tässä työssä tarkastelen gravitaatiota Poincarén symmetrian mittakenttäteoriana ja pyrin yleistämään tämän näkemyksen ei-kommutoiviin aika-avaruuksiin. Ensin esittelen Poincarén symmetrian keskeisen roolin relativistisessa fysiikassa ja sen kuinka klassinen gravitaatioteoria johdetaan Poincarén symmetrian mittakenttäteoriana kommutoivassa aika-avaruudessa. Jatkan esittelemällä ei-kommutoivan aika-avaruuden ja kvanttikenttäteorian muotoilun ei-kommutoivassa aika-avaruudessa. Mittasymmetrioiden lokaalin luonteen vuoksi tarkastelen huolellisesti mittakenttäteorioiden muotoilua ei-kommutoivassa aika-avaruudessa. Erityistä huomiota kiinnitetään näiden teorioiden vääristyneeseen Poincarén symmetriaan, joka on ei-kommutoivan aika-avaruuden omaama uudentyyppinen kvanttisymmetria. Seuraavaksi tarkastelen ei-kommutoivan gravitaatioteorian muotoilun ongelmia ja niihin kirjallisuudessa esitettyjä ratkaisuehdotuksia. Selitän kuinka kaikissa tähänastisissa lähestymistavoissa epäonnistutaan muotoilla kovarianssi yleisten koordinaattimunnosten suhteen, joka on yleisen suhteellisuusteorian kulmakivi. Lopuksi tutkin mahdollisuutta yleistää vääristynyt Poincarén symmetria lokaaliksi mittasymmetriaksi --- gravitaation ei-kommutoivan mittakenttäteorian saavuttamisen toivossa. Osoitan, että tällaista yleistystä ei voida saavuttaa vääristämällä Poincarén symmetriaa kovariantilla twist-elementillä. Näin ollen ei-kommutoivan gravitaation ja vääristyneen Poincarén symmetrian tutkimuksessa tulee jatkossa keskittyä muihin lähestymistapoihin.

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.

A computational study of Compton profiles of water - methanol solutions

The molecular level structure of mixtures of water and alcohols is very complicated and has been under intense research in the recent past. Both experimental and computational methods have been used in the studies. One method for studying the intra- and intermolecular bindings in the mixtures is the use of the so called difference Compton profiles, which are a way to obtain information about changes in the electron wave functions. In the process of Compton scattering a photon scatters inelastically from an electron. The Compton profile that is obtained from the electron wave functions is directly proportional to the probability of photon scattering at a given energy to a given solid angle. In this work we develop a method to compute Compton profiles numerically for mixtures of liquids. In order to obtain the electronic wave functions necessary to calculate the Compton profiles we need some statistical information about atomic coordinates. Acquiring this using ab-initio molecular dynamics is beyond our computational capabilities and therefore we use classical molecular dynamics to model the movement of atoms in the mixture. We discuss the validity of the chosen method in view of the results obtained from the simulations. There are some difficulties in using classical molecular dynamics for the quantum mechanical calculations, but these can possibly be overcome by parameter tuning. According to the calculations clear differences can be seen in the Compton profiles of different mixtures. This prediction needs to be tested in experiments in order to find out whether the approximations made are valid.

Spatial variation of climate and the impact of disturbances on local climate and forest recovery in northern Finland

The structure and function of northern ecosystems are strongly influenced by climate change and variability and by human-induced disturbances. The projected global change is likely to have a pronounced effect on the distribution and productivity of different species, generating large changes in the equilibrium at the tree-line. In turn, movement of the tree-line and the redistribution of species produce feedback to both the local and the regional climate. This research was initiated with the objective of examining the influence of natural conditions on the small-scale spatial variation of climate in Finnish Lapland, and to study the interaction and feedback mechanisms in the climate-disturbances-vegetation system near the climatological border of boreal forest. The high (1 km) resolution spatial variation of climate parameters over northern Finland was determined by applying the Kriging interpolation method that takes into account the effect of external forcing variables, i.e., geographical coordinates, elevation, sea and lake coverage. Of all the natural factors shaping the climate, the geographical position, local topography and altitude proved to be the determining ones. Spatial analyses of temperature- and precipitation-derived parameters based on a 30-year dataset (1971-2000) provide a detailed description of the local climate. Maps of the mean, maximum and minimum temperatures, the frost-free period and the growing season indicate that the most favourable thermal conditions exist in the south-western part of Lapland, around large water bodies and in the Kemijoki basin, while the coldest regions are in highland and fell Lapland. The distribution of precipitation is predominantly longitudinally dependent but with the definite influence of local features. The impact of human-induced disturbances, i.e., forest fires, on local climate and its implication for forest recovery near the northern timberline was evaluated in the Tuntsa area of eastern Lapland, damaged by a widespread forest fire in 1960 and suffering repeatedly-failed vegetation recovery since that. Direct measurements of the local climate and simulated heat and water fluxes indicated the development of a more severe climate and physical conditions on the fire-disturbed site. Removal of the original, predominantly Norway spruce and downy birch vegetation and its substitution by tundra vegetation has generated increased wind velocity and reduced snow accumulation, associated with a large variation in soil temperature and moisture and deep soil frost. The changed structural parameters of the canopy have determined changes in energy fluxes by reducing the latter over the tundra vegetation. The altered surface and soil conditions, as well as the evolved severe local climate, have negatively affected seedling growth and survival, leading to more unfavourable conditions for the reproduction of boreal vegetation and thereby causing deviations in the regional position of the timberline. However it should be noted that other factors, such as an inadequate seed source or seedbed, the poor quality of the soil and the intensive logging of damaged trees could also exacerbate the poor tree regeneration. In spite of the failed forest recovery at Tunsta, the position and composition of the timberline and tree-line in Finnish Lapland may also benefit from present and future changes in climate. The already-observed and the projected increase in temperature, the prolonged growing season, as well as changes in the precipitation regime foster tree growth and new regeneration, resulting in an advance of the timberline and tree-line northward and upward. This shift in the distribution of vegetation might be decelerated or even halted by local topoclimatic conditions and by the expected increase in the frequency of disturbances.

Topological stability through tame retractions

A smooth map is said to be stable if small perturbations of the map only differ from the original one by a smooth change of coordinates. Smoothly stable maps are generic among the proper maps between given source and target manifolds when the source and target dimensions belong to the so-called nice dimensions, but outside this range of dimensions, smooth maps cannot generally be approximated by stable maps. This leads to the definition of topologically stable maps, where the smooth coordinate changes are replaced with homeomorphisms. The topologically stable maps are generic among proper maps for any dimensions of source and target. The purpose of this thesis is to investigate methods for proving topological stability by constructing extremely tame (E-tame) retractions onto the map in question from one of its smoothly stable unfoldings. In particular, we investigate how to use E-tame retractions from stable unfoldings to find topologically ministable unfoldings for certain weighted homogeneous maps or germs. Our first results are concerned with the construction of E-tame retractions and their relation to topological stability. We study how to construct the E-tame retractions from partial or local information, and these results form our toolbox for the main constructions. In the next chapter we study the group of right-left equivalences leaving a given multigerm f invariant, and show that when the multigerm is finitely determined, the group has a maximal compact subgroup and that the corresponding quotient is contractible. This means, essentially, that the group can be replaced with a compact Lie group of symmetries without much loss of information. We also show how to split the group into a product whose components only depend on the monogerm components of f. In the final chapter we investigate representatives of the E- and Z-series of singularities, discuss their instability and use our tools to construct E-tame retractions for some of them. The construction is based on describing the geometry of the set of points where the map is not smoothly stable, discovering that by using induction and our constructional tools, we already know how to construct local E-tame retractions along the set. The local solutions can then be glued together using our knowledge about the symmetry group of the local germs. We also discuss how to generalize our method to the whole E- and Z- series.

Mechanisms for alphavirus nonstructural polyprotein processing

Replication and transcription of the RNA genome of alphaviruses relies on a set of virus-encoded nonstructural proteins. They are synthesized as a long polyprotein precursor, P1234, which is cleaved at three processing sites to yield nonstructural proteins nsP1, nsP2, nsP3 and nsP4. All the four proteins function as constitutive components of the membrane-associated viral replicase. Proteolytic processing of P1234 polyprotein is precisely orchestrated and coordinates the replicase assembly and maturation. The specificity of the replicase is also controlled by proteolytic cleavages. The early replicase is composed of P123 polyprotein intermediate and nsP4. It copies the positive sense RNA genome to complementary minus-strand. Production of new plus-strands requires complete processing of the replicase. The papain-like protease residing in nsP2 is responsible for all three cleavages in P1234. This study addressed the mechanisms of proteolytic processing of the replicase polyprotein in two alphaviruses Semliki Forest virus (SFV) and Sindbis virus (SIN) representing different branches of the genus. The survey highlighted the functional relation of the alphavirus nsP2 protease to the papain-like enzymes. A new structural motif the Cys-His catalytic dyad accompanied with an aromatic residue following the catalytic His was described for nsP2 and a subset of other thiol proteases. Such an architecture of the catalytic center was named the glycine specificity motif since it was implicated in recognition of a specific Gly residue in the substrate. In particular, the presence of the motif in nsP2 makes the appearance of this amino acid at the second position upstream of the scissile bond a necessary condition for the cleavage. On top of that, there were four distinct mechanisms identified, which provide affinity for the protease and specifically direct the enzyme to different sites in the P1234 polyprotein. Three factors RNA, the central domain of nsP3 and the N-terminus of nsP2 were demonstrated to be external modulators of the nsP2 protease. Here I suggest that the basal nsP2 protease specificity is inherited from the ancestral papain-like enzyme and employs the recognition of the upstream amino acid signature in the immediate vicinity of the scissile bond. This mechanism is responsible for the efficient processing of the SFV nsP3/nsP4 junction. I propose that the same mechanism is involved in the cleavage of the nsP1/nsP2 junction of both viruses as well. However, in this case it rather serves to position the substrate, whereas the efficiency of the processing is ensured by the capability of nsP2 to cut its own N-terminus in cis. Both types of cleavages are demonstrated here to be inhibited by RNA, which is interpreted as impairing the basal papain-like recognition of the substrate. In contrast, processing of the SIN nsP3/nsP4 junction was found to be activated by RNA and additionally potentiated by the presence of the central region of nsP3 in the protease. The processing of the nsP2/nsP3 junction in both viruses occurred via another mechanism, requiring the exactly processed N-terminus of nsP2 in the protease and insensitive to RNA addition. Therefore, the three processing events in the replicase polyprotein maturation are performed via three distinct mechanisms in each of two studied alphaviruses. Distinct sets of conditions required for each cleavage ensure sequential maturation of P1234 polyprotein: nsP4 is released first, then the nsP1/nsP2 site is cut in cis, and liberation of the nsP2 N-terminus activates the cleavage of the nsP2/nsP3 junction at last. The first processing event occurs differently in SFV and SIN, whereas the subsequent cleavages are found to be similar in the two viruses and therefore, their mechanisms are suggested to be conserved in the genus. The RNA modulation of the alphavirus nonstructural protease activity, discovered here, implies bidirectional functional interplay between the alphavirus RNA metabolism and protease regulation. The nsP2 protease emerges as a signal transmitting moiety, which senses the replication stage and responds with proteolytic cleavages. A detailed hypothetical model of the alphavirus replicase core was inferred from the data obtained in the study. Similar principles in replicase organization and protease functioning are expected to be employed by other RNA viruses.

Improving GPS time series for geodynamic studies

Accurate and stable time series of geodetic parameters can be used to help in understanding the dynamic Earth and its response to global change. The Global Positioning System, GPS, has proven to be invaluable in modern geodynamic studies. In Fennoscandia the first GPS networks were set up in 1993. These networks form the basis of the national reference frames in the area, but they also provide long and important time series for crustal deformation studies. These time series can be used, for example, to better constrain the ice history of the last ice age and the Earth s structure, via existing glacial isostatic adjustment models. To improve the accuracy and stability of the GPS time series, the possible nuisance parameters and error sources need to be minimized. We have analysed GPS time series to study two phenomena. First, we study the refraction in the neutral atmosphere of the GPS signal, and, second, we study the surface loading of the crust by environmental factors, namely the non-tidal Baltic Sea, atmospheric load and varying continental water reservoirs. We studied the atmospheric effects on the GPS time series by comparing the standard method to slant delays derived from a regional numerical weather model. We have presented a method for correcting the atmospheric delays at the observational level. The results show that both standard atmosphere modelling and the atmospheric delays derived from a numerical weather model by ray-tracing provide a stable solution. The advantage of the latter is that the number of unknowns used in the computation decreases and thus, the computation may become faster and more robust. The computation can also be done with any processing software that allows the atmospheric correction to be turned off. The crustal deformation due to loading was computed by convolving Green s functions with surface load data, that is to say, global hydrology models, global numerical weather models and a local model for the Baltic Sea. The result was that the loading factors can be seen in the GPS coordinate time series. Reducing the computed deformation from the vertical time series of GPS coordinates reduces the scatter of the time series; however, the long term trends are not influenced. We show that global hydrology models and the local sea surface can explain up to 30% of the GPS time series variation. On the other hand atmospheric loading admittance in the GPS time series is low, and different hydrological surface load models could not be validated in the present study. In order to be used for GPS corrections in the future, both atmospheric loading and hydrological models need further analysis and improvements.

Quantum Space-Time and Noncommutative Gauge Field Theories

Arguments arising from quantum mechanics and gravitation theory as well as from string theory, indicate that the description of space-time as a continuous manifold is not adequate at very short distances. An important candidate for the description of space-time at such scales is provided by noncommutative space-time where the coordinates are promoted to noncommuting operators. Thus, the study of quantum field theory in noncommutative space-time provides an interesting interface where ordinary field theoretic tools can be used to study the properties of quantum spacetime. The three original publications in this thesis encompass various aspects in the still developing area of noncommutative quantum field theory, ranging from fundamental concepts to model building. One of the key features of noncommutative space-time is the apparent loss of Lorentz invariance that has been addressed in different ways in the literature. One recently developed approach is to eliminate the Lorentz violating effects by integrating over the parameter of noncommutativity. Fundamental properties of such theories are investigated in this thesis. Another issue addressed is model building, which is difficult in the noncommutative setting due to severe restrictions on the possible gauge symmetries imposed by the noncommutativity of the space-time. Possible ways to relieve these restrictions are investigated and applied and a noncommutative version of the Minimal Supersymmetric Standard Model is presented. While putting the results obtained in the three original publications into their proper context, the introductory part of this thesis aims to provide an overview of the present situation in the field.

Semi-supervised learning of WLAN radio maps

In this thesis a manifold learning method is applied to the problem of WLAN positioning and automatic radio map creation. Due to the nature of WLAN signal strength measurements, a signal map created from raw measurements results in non-linear distance relations between measurement points. These signal strength vectors reside in a high-dimensioned coordinate system. With the help of the so called Isomap-algorithm the dimensionality of this map can be reduced, and thus more easily processed. By embedding position-labeled strategic key points, we can automatically adjust the mapping to match the surveyed environment. The environment is thus learned in a semi-supervised way; gathering training points and embedding them in a two-dimensional manifold gives us a rough mapping of the measured environment. After a calibration phase, where the labeled key points in the training data are used to associate coordinates in the manifold representation with geographical locations, we can perform positioning using the adjusted map. This can be achieved through a traditional supervised learning process, which in our case is a simple nearest neighbors matching of a sampled signal strength vector. We deployed this system in two locations in the Kumpula campus in Helsinki, Finland. Results indicate that positioning based on the learned radio map can achieve good accuracy, especially in hallways or other areas in the environment where the WLAN signal is constrained by obstacles such as walls.

Embracing Integrability in Stellar and Planetary Dynamics

Hamiltonian systems in stellar and planetary dynamics are typically near integrable. For example, Solar System planets are almost in two-body orbits, and in simulations of the Galaxy, the orbits of stars seem regular. For such systems, sophisticated numerical methods can be developed through integrable approximations. Following this theme, we discuss three distinct problems. We start by considering numerical integration techniques for planetary systems. Perturbation methods (that utilize the integrability of the two-body motion) are preferred over conventional "blind" integration schemes. We introduce perturbation methods formulated with Cartesian variables. In our numerical comparisons, these are superior to their conventional counterparts, but, by definition, lack the energy-preserving properties of symplectic integrators. However, they are exceptionally well suited for relatively short-term integrations in which moderately high positional accuracy is required. The next exercise falls into the category of stability questions in solar systems. Traditionally, the interest has been on the orbital stability of planets, which have been quantified, e.g., by Liapunov exponents. We offer a complementary aspect by considering the protective effect that massive gas giants, like Jupiter, can offer to Earth-like planets inside the habitable zone of a planetary system. Our method produces a single quantity, called the escape rate, which characterizes the system of giant planets. We obtain some interesting results by computing escape rates for the Solar System. Galaxy modelling is our third and final topic. Because of the sheer number of stars (about 10^11 in Milky Way) galaxies are often modelled as smooth potentials hosting distributions of stars. Unfortunately, only a handful of suitable potentials are integrable (harmonic oscillator, isochrone and Stäckel potential). This severely limits the possibilities of finding an integrable approximation for an observed galaxy. A solution to this problem is torus construction; a method for numerically creating a foliation of invariant phase-space tori corresponding to a given target Hamiltonian. Canonically, the invariant tori are constructed by deforming the tori of some existing integrable toy Hamiltonian. Our contribution is to demonstrate how this can be accomplished by using a Stäckel toy Hamiltonian in ellipsoidal coordinates.

Analysing local algorithms in location-aware quasi-unit-disk graphs

A local algorithm with local horizon r is a distributed algorithm that runs in r synchronous communication rounds; here r is a constant that does not depend on the size of the network. As a consequence, the output of a node in a local algorithm only depends on the input within r hops from the node. We give tight bounds on the local horizon for a class of local algorithms for combinatorial problems on unit-disk graphs (UDGs). Most of our bounds are due to a refined analysis of existing approaches, while others are obtained by suggesting new algorithms. The algorithms we consider are based on network decompositions guided by a rectangular tiling of the plane. The algorithms are applied to matching, independent set, graph colouring, vertex cover, and dominating set. We also study local algorithms on quasi-UDGs, which are a popular generalisation of UDGs, aimed at more realistic modelling of communication between the network nodes. Analysing the local algorithms on quasi-UDGs allows one to assume that the nodes know their coordinates only approximately, up to an additive error. Despite the localisation error, the quality of the solution to problems on quasi-UDGs remains the same as for the case of UDGs with perfect location awareness. We analyse the increase in the local horizon that comes along with moving from UDGs to quasi-UDGs.