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Tutkielmassa perehdytään Fermat’n pieneen lauseeseen ja sen todistuksiin. Fermat’n pientä lausetta tarkastellaan myös alkulukutestauksen näkökulmasta. Loppupuolella määritellään Eulerin φ-funktio ja esitetään Eulerin lause. Eulerin lauseen käytännöllisyyttä tarkastellaan jakojäännösten selvittämisessä. Tutkielman johdanto on pieni katsaus Pierre de Fermat’n ja Leonhard Eulerin elämään. Johdannossa käsitellään myös Fermat’n pienen lauseen sekä Eulerin lauseen historiaa. Tutkielmassa on käytetty useita lukuteoreettisia käsitteitä, jotka määritellään heti johdannon jälkeen luvussa Tutkielmassa käytettyjä määritelmiä. Tutkielma esittää Fermat’n pienen lauseen kolmessa eri muodossa, jotka kaikki ovat keskenään ekvivalentteja. Lauseen käyttöä havainnollistetaan myös muutamalla esimerkillä. Luvussa Fermat’n pienen lauseen todistuksia kyseinen lause todistetaan ensin suoraviivaisesti ja sen jälkeen induktiolla. Lopuksi lausetta havainnollistetaan kuvitellun helminauhan avulla. Tutkielma osoittaa, että Fermat’n pieni lause toteutuu millä tahansa alkuluvulla p. Fermat’n pienen lauseen toteutuminen jollain luvulla ei kuitenkaan yksin riitä osoittamaan lukua alkuluvuksi. Otsikon Pseudoalkuluvut alla käsitellään lukuja, jotka eivät ole alkulukuja, mutta joilla Fermat’n pieni lause toteutuu. Jotta voitaisiin varmistua, että luku on alkuluku, E. Lucas kehitti 1800-luvun loppupuolella alkulukutestin, joka hyödyntää Fermat’n pientä lausetta. Testi on esitetty, todistettu ja sitä on havainnollistettu esimerkein kohdassa Lucas-Lehmer alkulukutesti. Testin todistukseen vaaditaan muutamia aputuloksia, jotka on esitetty ennen varsinaista testiä. Tutkielma määrittelee Eulerin φ-funktion ja havainnollistaa sen käyttöä esimerkillä. Tämän jälkeen tutkielmassa johdetaan kaava, jonka avulla φ-funktion arvon voi kätevästi laskea. Kaavan johtamista varten on todistettu muutama aputulos. Kaavan käytöstä on esimerkki. Tutkielmassa käsitellään Eulerin lause. Heti määritelmän jälkeen Eulerin lauseella ratkaistaan jakojäännöksiä. Sitten Eulerin lause todistetaan ensin induktion ja binomikaavan avulla ja sitten redusoidun jäännösluokkasysteemin avulla. Ennen kumpaakin todistusta esitellään ja todistetaan todistuksissa käytettäviä aputuloksia. Lopuksi tutkielma käsittelee suurten potenssien jakojäännösten ratkaisemista Eulerin lauseen ja binäärijärjestelmän avulla.

The plastic pollution has become a massive problem in the Arctic, affecting aquatic, and terrestrial ecosystems, the cryosphere, and the atmosphere. One of the solutions proposed by the Arctic Council is to improve waste management by using renewable and sustainable materials. This is where bioplastics reveal their importance. They can be bio-produced by microorganisms from organic waste, they are biodegradable and can be reused. Their production relies on a circular economy system making it sustainable. Here lies the relevance of developing the bioplastic bioproduction and technology. The present research focused on the development of a specific production of polyhydroxyalkanoates (PHAs) from organic waste, in collaboration with the start-up Dionymer (Bordeaux, France). First, the purpose of the study was to up scale the process from the fermentation of chemical volatile fatty acids in flasks (400 mL culture medium) to 2 L bioreactors (BR) by characterizing the main differences in the two processes. Secondly, the research consisted in implementing and testing different set-up for the BR to enhance and improve bioplastic and biomass yields, including aeration and agitation. The characterization of the culture parameters differences between BR and flask pointed out; a higher viscosity of the medium at the end of the process, a darker PHA product and a lower final optical density (OD) (8 versus 12) respectively. Secondly, the focus was on the increase of the OD in BR and finding the origin of the stress, to do so, the following parameters were tested; - three aerations strategies; pO2<10%, <20% and >20%; - two agitations blades; marine and Rushton with baffles; - two aerations spargers; circular and micro. The results revealed that; the pO2 needs to be higher to 20% and it may be linked with the reduction of stress induced to the cells; the marine blades increased the OD and reduced the medium viscosity; the impact of the micro sparger seemed to improve aeration and tent to be very sensitive to antifoam agent that reduced the aeration of the medium. So far, the optimum BR set-up seemed to include the use of marine blades and a pO2 above 20%. More experiments of optimization still need to be performed to unsure a stable and higher production performance.

Extremely water-repellent, i.e. superhydrophobic, surfaces display self-cleaning, anti-icing and anti-fogging characteristics, and can be used to reduce viscous drag in liquid flow. Despite the auspicious prospects of superhydrophobicity, methods for measuring wettability and energy dissipation in droplets moving on these surfaces have been insufficient. In this thesis I present the theoretical basis of non-wetting phenomena and review contemporary wetting characterization methods. I also discuss a measuring technique we developed for superhydrophobic surfaces based on magnetically induced oscillations of water-like ferrofluid droplets, which was recently published in Nature Communications. This method can simultaneously measure forces caused by two independent dissipative processes at a precision of ca. 10 nN. One is related to the contact angle hysteresis at the three-phase contact line and the other to viscous dissipation near the droplet-solid contact area. The ability to provide quantitative information about droplet dynamics, inhomogeneities and defects on extremely non-wetting surfaces can make this a useful research and quality control technique for academic and commercial use.

Tutkielmassa tutustutaan Fibonaccin lukuihin ja Lucasin lukuihin. Tavoitteena on ensinnäkin tarkastella Fibonaccin lukuja ja näiden yhteyttä Lucasin lukuihin ja kultaiseen leikkaukseen. Toiseksi tavoitteena on tarkastella erityisesti Fibonaccin lukuja lukuteorian näkökulmasta tutkimalla alkulukuihin ja jaollisuuteen liittyviä ominaisuuksia. Lisäksi todistetaan Zeckendorfin lause. Tutkielmassa on neljä lukua. Ensimmäisessä luvussa on johdanto ja viimeisessä luvussa on loppusanat. Johdannossa on lyhyt historiallinen katsaus aiheeseen. Toisessa luvussa on ominaisuuksia ja määritelmiä. Luvussa 2.1 määritellään Fibonaccin luvut. Ensimmäiset Fibonaccin luvut ovat nolla ja yksi. Seuraava Fibonaccin luku saadaan aina määrittämällä kahden edellisen Fibonaccin luvun summa. Luvussa 2.2 tarkastellaan lyhyesti Fibonaccien lukujen yhteyttä matriiseihin ja todistetaan tämän avulla kaksi tulosta Fibonaccin luvuille. Luvussa 2.3 esitetyt Lucasin luvut määritellään alkuarvoja lukuun ottamatta samoin kuin Fibonaccin luvut. Ensimmäiset Lucasin luvut ovat kaksi ja yksi. Luvussa 2.4 tutkitaan kultaista leikkausta. Kultainen leikkaus saadaan, kun jaetaan jana kahteen osaan. Jaon ehtona on, että pidemmän ja lyhyemmän osan pituuksien suhde on yhtä suuri kuin koko janan ja pidemmän osan pituuksien suhde. Luvussa 2.5 todistetaan Binet'n kaava Fibonaccin luvuille ja Lucasin luvuille. Binet'n kaavan avulla voidaan määrittää valittu Fibonaccin tai Lucasin luku tuntematta muita Fibonaccin tai Lucasin lukuja. Binet'n kaavan avulla osoitetaan, että peräkkäisten Fibonaccin lukujen suhde lähestyy kultaista leikkausta indeksin kasvaessa rajatta. Lucasin luvuille saadaan samanlainen tulos. Kolmannessa luvussa tarkastellaan lukuteorian ominaisuuksia. Luvussa 3.1 Määritellään Fibonaccin ja Lucasin alkuluvut. Nämä ovat alkulukuja, jotka ovat myös Fibonaccin tai Lucasin lukuja. Osoitetaan, että peräkkäiset Fibonaccin luvut ovat suhteellisia alkulukuja. Luvussa 3.2 on jaollisuusominaisuuksia. Osoitetaan Fibonaccin lukujen olevan jaollisia toisillaan, jos niiden indeksit ovat jaollisia toisillaan. Tulos pätee myös toiseen suuntaan, jos jakajan indeksi ei ole kaksi. Todistetaan lisäksi, että Fibonaccin lukujen suurin yhteinen tekijä on aina Fibonaccin luku. Jaollisuusominaisuuksien perusteella osoitetaan, että alkulukuja on ääretön määrä. Luvussa 3.3 todistetaan Zeckendorfin lause. Sen perusteella jokainen positiivinen kokonaisluku voidaan esittää yksikäsitteisesti Fibonaccin lukujen summana, jos Fibonaccin luvut eivät ole peräkkäisiä ja jos niiden indeksi on suurempi kuin yksi.

Tämän Pro gradun aiheena on Fibonaccin luvut ja työn tavoitteena oli antaa yleisluontoinen kuva niistä; siitä, mitä ne ovat, minkälaisia ominaisuuksia niillä on ja miten ne käyttäytyvät sekä koota lukuihin liittyviä olennaisia teemoja ja tuloksia johdonmukaiseksi kokonaisuudeksi. Työn johdantoluvussa perustellaan aihevalinta ja annetaan yleiskuvaus työstä. Luvussa 2 käsitellään Fibonaccin lukujonon historiaa lähinnä sen henkilön kautta, jonka nimeä se kantaa. Luvussa 3 annetaan Fibonaccin luvuille rekursiivinen ja analyyttinen määritelmä sekä esitellään samalla tavoin rekursiivisesti määriteltävät Lucasin luvut, joita tarvitaan työssä myöhemmin esitettävissä todistuksissa. Tästä siirrytään osoittamaan, miten kultainen leikkaus konstruoidaan Fibonaccin lukujonon avulla, esitetään muutamia yksinkertaisia Fibonaccin lukujonolle ja sen jäsenille päteviä kaavoja ja tuloksia, tuotetaan Fibonaccin lukuja generoiva funktio ja osoitetaan, miten Fibonaccin lukujen avulla voidaan tuottaa Pythagoraan kolmioita. Luvun loppupuolella käsitellään Fibonaccin lukujen jaollisuusominaisuuksia ja todistetaan niihin liittyviä tuloksia ja päädytään lopulta antamaan ratkaisu johdannossa esitettävään kaniongelmaan osoittamalla Fibonaccin lukujen ja Pascalin kolmion välinen yhteys. Työn viimeisessä luvussa analysoidaan työn kirjoitusprosessia suhteessa asetettuihin tavoitteisiin ja suhteutetaan teemaa yleisempään matemaattiseen kontekstiin sekä pohditaan lukujen mahdollisia opetuksellisia sovelluksia.

Eddy covariance (EC)-flux measurement technique is based on measurement of turbulent motions of air with accurate and fast measurement devices. For instance, in order to measure methane flux a fast methane gas analyser is needed which measures methane concentration at least ten times in a second in addition to a sonic anemometer, which measures the three wind components with the same sampling interval. Previously measurement of methane flux was almost impossible to carry out with EC-technique due to lack of fast enough gas analysers. However during the last decade new instruments have been developed and thus methane EC-flux measurements have become more common. Performance of four methane gas analysers suitable for eddy covariance measurements are assessed in this thesis. The assessment and comparison was performed by analysing EC-data obtained during summer 2010 (1.4.-26.10.) at Siikaneva fen. The four participating methane gas analysers are TGA-100A (Campbell Scientific Inc., USA), RMT-200 (Los Gatos Research, USA), G1301-f (Picarro Inc., USA) and Prototype-7700 (LI-COR Biosciences, USA). RMT-200 functioned most reliably throughout the measurement campaign and the corresponding methane flux data had the smallest random error. In addition, methane fluxes calculated from data obtained from G1301-f and RMT-200 agree remarkably well throughout the measurement campaign. The calculated cospectra and power spectra agree well with corresponding temperature spectra. Prototype-7700 functioned only slightly over one month in the beginning of the measurement campaign and thus its accuracy and long-term performance is difficult to assess.

Filamentary structures appear to be very common in molecular clouds which are known to be the birth sites of stars. These interstellar filaments of gas and dust can have a dominant role in star formation. According to the current understanding of star formation, large-scale supersonic MHD turbulence present in the clouds tends to form these narrow elongated structures which are then broken into dense clumps through gravitational instability. The aim of my Master's thesis is to study the formation of stars by first reviewing the classical stability analysis of the most common self-gravitating structures subject to harmonic disturbances, having an emphasis on cylindrical symmetry which can be used to describe the behaviour of filamentary structures. Especially the density distribution, the critical line mass and the stability criterion (Jeans length) for self-gravitating isothermal cylinder are presented. Then the theory is applied to an individual target, a filament known as Taurus Molecular Cloud–1 (TMC–1). The goal is to investigate if the observed structures are in agreement with theoretical models and if stars can be formed in the filament through gravitational instability. By examining the properties of the cloud it is possible to find out when gravity (instead of external pressure) begins to dominate the evolution of the cloud. Important supportive forces, thermal pressure and turbulence, can be studied by observing standard molecular line tracers as ammonia. Ammonia NH3(1,1) and NH3(2,2) inversion lines are measured with the Effelsberg 100-m telescope in order to derive e.g. gas kinetic temperatures and non-thermal velocity dispersions along the TMC–1 filament. Furthermore, supplementing the molecular line data with the data from SCUBA 850 micron dust emission map of the cloud (Nutter et al., 2008) and assuming that the gas kinetic temperature equals to the dust temperature (Tkin = Tdust), hydrogen column densities can be calculated. Moreover, the cross-sectional column density profiles in several places along the filament can be fitted with the classical model of a hydrostatic isothermal cylinder by Stodolkiewicz (1963). The model fits roughly to the cross-sectional column density profiles. The fragment lengths or the projected clump separations along the ‘backbone’ of the filament (longitudinal profile) are derived from the intensity maxima seen in the SCUBA data. The observed fragment lengths along the filament are compared with the Jeans lengths and the fastest growing modes of disturbance predicted from Stodolkiewicz's model in order to investigate the star-forming potential of TMC-1. The observed clump separations are in a rough agreement with the local theoretical Jeans lengths. Moreover, the longitudinal SCUBA intensity profile is expanded into Fourier series, and the wavelengths corresponding to the dominating amplitudes are calculated. These wavelengths are compared with theoretical Jeans lengths in order to investigate if the perturbations travelling in the filament can cause collapse or if they are only ‘sound waves’. Two wavelengths, lambda = 0.23 pc and lambda = 0.30 pc, exceed the local theoretical Jeans length and hence may cause collapse within the filament and trigger star formation in the fragments.

The task of organizing a given graph into a structure called a tree decomposition is relevant in multiple areas of computer science. In particular, many NP-hard problems can be solved in polynomial time if a suitable tree decomposition of a graph describing the problem instance is given as a part of the input. This motivates the task of finding as good tree decompositions as possible, or ideally, optimal tree decompositions. This thesis is about finding optimal tree decompositions of graphs with respect to several notions of optimality. Each of the considered notions measures the quality of a tree decomposition in the context of an application. In particular, we consider a total of seven problems that are formulated as finding optimal tree decompositions: treewidth, minimum fill-in, generalized and fractional hypertreewidth, total table size, phylogenetic character compatibility, and treelength. For each of these problems we consider the BT algorithm of Bouchitté and Todinca as the method of finding optimal tree decompositions. The BT algorithm is well-known on the theoretical side, but to our knowledge the first time it was implemented was only recently for the 2nd Parameterized Algorithms and Computational Experiments Challenge (PACE 2017). The author’s implementation of the BT algorithm took the second place in the minimum fill-in track of PACE 2017. In this thesis we review and extend the BT algorithm and our implementation. In particular, we improve the eciency of the algorithm in terms of both theory and practice. We also implement the algorithm for each of the seven problems considered, introducing a novel adaptation of the algorithm for the maximum compatibility problem of phylogenetic characters. Our implementation outperforms alternative state-of-the-art approaches in terms of numbers of test instances solved on well-known benchmarks on minimum fill-in, generalized hypertreewidth, fractional hypertreewidth, total table size, and the maximum compatibility problem of phylogenetic characters. Furthermore, to our understanding the implementation is the first exact approach for the treelength problem.

African climate science is still underdeveloped; the related infrastructure is limited and climate observation network is sparse. The lack of observations is usually compensated with coarse macro-climate models (1 km2), of which only few concentrates on tropical Africa and on the spatial variability of rains. The need for fine-resolution data is growing, as more accurate information (5-100m) is utilized in different sciences. There has been a surge in micro-climate studies utilizing small dataloggers to observe climate variability, as loggers can be placed to different environments regardless of the complexity of a terrain. Accurate, fine-scale climate models are needed in ecological research, where species distributions are related to environmental gradients. Most of the studies describing species occurrences rely on macro-scale climate models that have been averaged over years, although the factors restricting species occurrences in an environment are developed even under one growth season, and are dependent on climate extremes. Accurate information of species and its occurrence environment is even more needed as climate is changing; species habitats are threatened through the warming of climate and its resulted effects. Fine-resolution data of micro-climate is needed especially in the studying of epiphytes living in tropical montane cloud forests, as epiphytes are dependent on the surrounding environments' specific atmospheric composition, namely relative humidity. The knowledge of epiphytic species distributions is poorly studied, although epiphytes are known to react fast to quick changes in environmental conditions. In this study small dataloggers were placed around the Taita Hills in a mountainous region of Kenya to record spatial and temporal micro-climate variability. Annual, monthly and extreme temperature and humidity parameters were studied through statistical spatial modelling, with generalized additive models (GAMs). The significance of each thriving factor of climate parameters were studied separately with bootstrapping. Of thousands of models created, temperature was best explained by elevation, and humidities by land cover (e.g. distance to forests), a proxy of canopy cover. Modelling extremes requires more research, as extreme models proved to be most difficult to model. More weight should be put on accurate environment gradients (e.g. canopy cover) when modelling spatial climate variability in a fine-scale. Created fine-resolution grids were tested in modelling the distributions of epiphytic species through Boosted Regression Trees (BRT). Models created gave an insight to the drivers of epiphytic species distributions and to the limitations of their occurrences in a limited environment. Different explanatory variable groups were tested (topographical, in situ, climate groups), of which climate explained most of the variations in epiphytic species occurrences when considered with AUC-values. Some epiphytes seemed to be specialized to certain habitat trees, highlighting the value of not only indigenous trees but exotic ones. Epiphytes were restricted to occur under certain temperatures and humidities ranges that is an indication of the vulnerability of epiphytes to changing climate.

Nowadays, billions of people access the Internet on mobile phones and a significant portion of the traffic comes from browsers. Mobile browsers could be used as a gateway to access the underlying resources of mobile devices for fingerprinting purposes. Browsers include APIs to access the underlying hardware and software resources, such as sensors, audio and media devices, battery, and so on. The growing number of APIs have created new opportunities for browser fingerprinting mechanisms. However, the widely used browser fingerprint systems are designed for the desktop environment and the identifying information gathered using these systems do not include the unique features of mobile phones such as device sensors. The goal of this thesis is to explore additional fingerprintable metrics in the mobile context and analyze their contribution in fingerprinting browsers. In this thesis, we investigated time evolution of browser's features fingerprints and fingerprinting in the wild in the context of mobile devices. In time evolution of feature's fingerprinting, we have examined the change in permission requirements of browsers over time and evolution of browser's features fingerprints for both Google Chrome and Firefox. In our experiment, we have seen that permission requirements have increased over time, e.g. Firefox 4.0 requires only four permissions, while Firefox 55.0 requires 24 permissions. In evolution of browser's features, we have seen fingerprints that are related to media, audio, WebGL, and canvas elements of the browser show a frequent change across versions. In addition, we have seen, for both Chrome and Firefox, the user agent string is unique for each version and media devices for Chrome is unique for each version as well in our dataset. In fingerprinting in the wild, we have collected fingerprints from 134 browsing sessions of which 96 were unique. From the gathered dataset, we have calculated the identifying information, entropy, contribution of each browser's feature in our test. The result shows that IP address, user agent, and media devices are the highest entropy contributors. In addition, we have observed that the maximum possible entropy gain in our dataset, 6.58 bits, can be obtained by joining only media devices and user agent strings. To sum up, in our experiment, we have acquired additional fingerprintable metrics form modern APIs, such as sensors, audio and media devices, and battery. In time evolution of browser feature's fingerprint experiments, we have seen that modern API feature's fingerprints show frequent change across versions. Similarly, in fingerprinting in the wild experiments, these APIs are among the highest entropy contributors.