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(2013)Gabrielin torvi on kolmiulotteinen matemaattinen kappale, jonka pinta-ala on ääretön, mutta jonka tilavuus on korkeintaan piin suuruinen. Kappaleen olemassaolon huomasi italialainen fyysikko Evangelista Torricelli. Gabrielin torven olemassaolo hämmensi 1600-luvun matemaatikkoja, koska matemaattiset metodit ja tieto eivät olleet vielä kehittyneet niin paljon, että Gabrielin torven kaltaisille kappaleille olisi pystytty antamaan ymmärrettäviä todisteita. Kappaleen nimi viittaa kristillisen mytologian tuomiopäivään, jolloin arkkienkeli Gabriel puhaltaa torveensa ja tunnettu, äärellinen maailmamme loppuu yhdistyen päättymättömään jumalaiseen todellisuuteen. Nimi kuvaa kappaletta hyvin, sillä Gabrielin torvessa yhdistyy äärellinen ja ääretön mielenkiintoisella tavalla. Kochin lumihiutale on Gabrielin torven analogia tasossa. Se on tasokuvio, jonka pinta-ala on äärellinen, mutta sen piirin pituus on ääretön. Tämän kaiken ja muut lumihiutaleen mielenkiintoiset ominaisuudet löysi ruotsalainen matemaatikko Helge von Koch vuonna 1904. Kochin löydökset olivat merkittävä etappi fraktaalitutkimuksen saralla. Gabrielin torvea ja Kochin lumihiutaletta voisi ensisilmäyksellä luulla paradokseiksi, mutta sitä ne eivät kuitenkaan ole. Sekä torvea, että lumihiutaletta voi käyttää peruskoulun ja lukion matematiikan opetuksessa elävöittämään opiskelua ja tuomaan lisää syvyyttä käsitteisiin.
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(2018)Työn päätavoitteena on osoittaa viidennen asteen polynomiyhtälön ratkaisukaavan mahdottomuus. Ratkaisukaava on mahdollista muodostaa vain polynomeille, jotka ovat juurtamalla ratkeavia. Juurtamalla ratkeavan polynomin kukin juuri voidaan ilmaista kerroinkunnan alkioiden muodostamana päättyvänä lausekkeena, joka käyttää vain kunnan laskutoimituksia ja juurenottoa. Työn lähtökohdaksi otetaan kuntien laajennukset ja ennen kaikkea polynomin kerroinkunnan laajennukset polynomin juurilla. Kun kerroinkuntaa laajennetaan juuri kerrallaan, syntyy useiden sisäkkäisten kuntalaajennusten torni, jonka huipulla on polynomin kaikki juuret sisältävä polynomin juurikunta. Galois'n teorian keskeisimpiä työvälineitä ovat automorfismit eli kunnan isomorfismit itselleen. Sellaiset laajennuskunnan automorfismit, jotka kiinnittävät laajennuksen lähtökunnan, muodostavat laajennuksen Galois'n ryhmän. Myös polynomille on mahdollista määritellä Galois'n ryhmä: polynomin Galois'n ryhmä on sen juurikunnan Galois'n ryhmä polynomin kerroinkunnan suhteen. Osoittautuu, että kukin Galois'n ryhmän alkio on samaistettavissa jonkin polynomin juurten permutaation kanssa, joten Galois'n ryhmä on siis aina symmetrisen ryhmän aliryhmä. Työn loppupuolella keskiöön nousevat juurilaajennukset eli kunnan laajennukset kunnan alkioiden juurroksilla. Kun sopivaan juurilaajennukseen sovelletaan kuudennessa luvussa todistettavaa Galois'n teorian peruslausetta, osoittautuu, että juurtamalla ratkeavan polynomin Galois'n ryhmästä löytyy aina tietty sisäinen rakenne, jota kutsutaan ratkeavuudeksi. Viimeisessä luvussa osoitetaan, että polynomin Galois'n ryhmän ratkeavuus on välttämätön ja riittävä ehto polynomin juurtamalla ratkeavuudelle. Viiden ja sitä useamman alkion symmetrinen ryhmä ei kuitenkaan ole ratkeava, mutta on olemassa polynomeja, joiden Galois'n ryhmä se on. Näin ollen polynomeille, joiden aste on viisi tai sitä korkeampi, ei ole mahdollista muodostaa yleistä ratkaisukaavaa. Työn päättää esimerkki viidennen asteen polynomista, joka ei ole juurtamalla ratkeava.
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(2018)Tutkielmassa esitellään Eulerin gammafunktio ja siihen liittyviä keskeisiä tuloksia. Gammafunktio on kertomafunktion yleistys reaaliluvuille lukuun ottamatta ei-positiivisia kokonaislukuja. Tutkielma liittyy matemaattisen analyysin alaan, joka käsittelee reaaliarvoisia funktioita. Tutkielmassa käytetään lauseita, jotka on todistettu matematiikan perusopinnoissa, joten ne oletetaan tunnetuiksi. Kertomafunktion yleistäminen oli 1600-luvulla merkittävä interpolaatio-ongelma, jota pohtivat monet suuret matemaatikot. Vuonna 1729 Euler ratkaisi ongelman esittämällä gammafunktion äärettömänä tulona ja seuraavana vuonna esitti sen integraalimuodon. Tämä integraalimuoto esitellään nykyisin yleensä ensimmäisenä, kun puhutaan gammafunktiosta. Tutkielman alussa perustellaan, miksi gammafunktio on sellainen kuin se on. Gammafunktion eri esitysmuotoja esitellään kronologisessa järjestyksessä tukeutuen oivaltaviin näkökulmiin, minkä jälkeen gammafunktio määritellään tarkasti. Gammafunktioon liittyvät keskeiset lauseet todistetaan. Tärkeimpänä lauseena Bohrin-Mollerupin lause, jonka mukaan kaikista kertomafunktion yleistyksistä vain gammafunktio on logaritmisesti konveksi. Viidennessä luvussa todistetaan gammafunktiolle Weierstrassin tuloesitys, johon liittyy oleellisesti myös Eulerin-Mascheronin vakio. Weierstrassin tuloesitystä käytetään tutkielmassa muissa todistuksissa. Tämän jälkeen esitellään joitakin esimerkkejä ja sovelluksia. Gammafunktiota sovelletaan erittäin laajasti monilla aloilla. Se on keskeinen työkalu toki analyysissä, mutta myös tilastotieteessä, todennäköisyyslaskennassa ja lukuteoriassa. Tutkielmassa esitellään vain osa näistä sovelluksista. Gammafunktion avulla saadaan laskettua myös n-ulotteisen pallon tilavuus. Tutkielman lopuksi esitellään kompleksiarvoinen gammafunktio. Luvussa esitellään myös gammafunktion yhteys Riemannin zetafunktioon. Tämä analyyttisen lukuteorian sovellus on gammafunktion yksi tärkeimmistä sovelluksista.
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(2015)Pollution is of concern for human health and the environment. Authorities have set concentration frames for many pollutants. Therefore, monitoring is required in industrial and farming activities, for example. Methods based on gas chromatography - tandem mass spectrometry are mainly utilized in the study of trace pollutants in the environmental samples. The main reason for that being versatility and robustness of the technique, which is capable of analysis of numerous compound groups covering all the main pollutants. This thesis focuses on recent gas chromatography - tandem mass spectrometry method development in environmental analysis. Samples of air particulate matter and from aqueous environment are covered. Optimization of tandem mass spectrometry parameters, such as excitation energies, the choice of precursor ions and product ions are shown. Tandem mass spectrometry techniques increase sensitivity and selectivity of the analysis by reducing baseline due to specific fragmentation. For the reason of comparison, conventional one dimensional mass spectrometric applications are shown and advantages and disadvantages of both approaches are discussed. Research papers published during past decade were reviewed. Studied material included research results of gas chromatography- tandem mass spectrometry applications for determination of organic compounds, such as endocrine disruptive compounds, organometals, illicit drugs, pharmaceuticals, pesticides, brominated and chlorinated organic compounds, some volatile organic compounds, organonitrogen compounds and polycyclic aromatic hydrocarbons, in environmental samples. Sample collection and preparation techniques utilized in applications are reviewed as well. In the experimental part of this study, gas chromatography - mass spectrometry method for the determination of carboxylic acids, phenols and sugars in aerosol particles was developed. The experimental part covers also the study of fatty acids, which are analyzed utilizing the same method. Fatty acids are qualitative trace markers for agriculture based pollution in air. Result of the practical application of fatty acids as trace marker is presented as well. Developed method for determining fatty acids was utilized in field campaign in Bologna, Italy. C18/C16 –ratio is trace marker of agricultural based emission. Campaign results of C18/C16 –ratio and is presented in this thesis. Day-of-week trend of fatty acid emission is covered as well. The utilization of tandem mass spectrometry will reduce common matrix effects of gas chromatography - mass spectrometry and in this way will make analysis more sensitive. Tandem mass spectrometry methods will complete to the requirements of environment monitoring and thus are highly recommended in environmental analysis.
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(2022)Ultrasonic guided lamb waves can be used to monitor structural conditions of pipes and other equipment in industry. An example is to detect accumulated precipitation on the surface of pipes in a non-destructive and non-invasive way. The propagation of Lamb waves in a pipe is influenced by the fouling on its surface, which makes the fouling detection possible. In addition, multiple helical propagation paths around pipe structure provides rich information that allows the spatial localization of the fouled area. Gaussian Processes (GP) are widely used tools for estimating unknown functions. In this thesis, we propose machine learning models for fouling detection and spatial localization of potential fouled pipes based on GPs. The research aims to develop a systematic machine learning approach for ultrasonic detection, interpret fouling observations from wave signals, as well as reconstruct fouling distribution maps from the observations. The lamb wave signals are generated in physics experiments. We developed a Gaussian Process Regression model as a detector, to determine whether each propagation path is going across the fouling or not, based on comparison with clean pipe. This binary classification can be regarded as one case of the different fouling observations. Latent variable Gaussian Process models are deployed to model the observations over the unknown fouling map. Then Hamiltonian Monte Carlo sampling is utilized to perform full Bayesian inference for the GP hyper-parameters. Thus, the fouling map can be reconstructed based on the estimated parameters. We investigate different latent variable GP models for different fouling observation cases. In this thesis, we present the first unsupervised machine learning methods for fouling detection and localization on the surface of pipe based on guided lamb waves. In these thesis we evaluate the performance of our methods with a collection of synthetic data. We also study the effect of noise on the localization accuracy.
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(2022)Ultrasonic guided lamb waves can be used to monitor structural conditions of pipes and other equipment in industry. An example is to detect accumulated precipitation on the surface of pipes in a non-destructive and non-invasive way. The propagation of Lamb waves in a pipe is influenced by the fouling on its surface, which makes the fouling detection possible. In addition, multiple helical propagation paths around pipe structure provides rich information that allows the spatial localization of the fouled area. Gaussian Processes (GP) are widely used tools for estimating unknown functions. In this thesis, we propose machine learning models for fouling detection and spatial localization of potential fouled pipes based on GPs. The research aims to develop a systematic machine learning approach for ultrasonic detection, interpret fouling observations from wave signals, as well as reconstruct fouling distribution maps from the observations. The lamb wave signals are generated in physics experiments. We developed a Gaussian Process Regression model as a detector, to determine whether each propagation path is going across the fouling or not, based on comparison with clean pipe. This binary classification can be regarded as one case of the different fouling observations. Latent variable Gaussian Process models are deployed to model the observations over the unknown fouling map. Then Hamiltonian Monte Carlo sampling is utilized to perform full Bayesian inference for the GP hyper-parameters. Thus, the fouling map can be reconstructed based on the estimated parameters. We investigate different latent variable GP models for different fouling observation cases. In this thesis, we present the first unsupervised machine learning methods for fouling detection and localization on the surface of pipe based on guided lamb waves. In these thesis we evaluate the performance of our methods with a collection of synthetic data. We also study the effect of noise on the localization accuracy.
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(2015)Gaussin-Bonnet'n lause on differentiaaligeometriassa keskeinen tulos, joka on nimetty Carl Friedrich Gaussin ja Pierre Ossian Bonnet'n mukaan. Lause liittää avaruuden geometrian ja topologian toisiinsa, hyödyntäen geometrisia ja topologisia invariantteja, eli sopivissa muunnoksissa muuttumattomia suureita. Ensimmäisessä luvussa esitellään sileät monistot ja ensimmäisiä perustavanlaatuisia askelia differentiaaligeometrian suuntaan. Aiheen geometrinen luonne ei pelkkien sileiden monistojen tapauksessa nouse vielä ilmiselvästi esiin, vaikka tangenttivektorit ja -avaruudet pystytäänkin jo määrittelemään. Siinä missä topologinen monisto yleistää euklidisen avaruuden ''hyvät'' topologiset ominaisuudet, saadaan sileisiin monistoihin siirryttäessä käyttöön myös keinoja käsitellä suuntia sekä tehdä differentiaalilaskentaa. Perusmääritelmien lisäksi käsitellään muita tärkeitä tuloksia, kuten differentiaalimuotoja ja pintapuolisesti monistoilla integroimisen teoriaa. Toisessa luvussa tartutaan varsinaiseen Riemannin geometriaan ja edetään sileistä monistoista Riemannin monistoihin. Vihdoin selkeämpi geometrisyys tulee esille, kun monistoille määritelty Riemannin metriikka mahdollistaa etäisyyksistä ja kaarevuudesta puhumisen. Konnektioiden avulla mahdollistetaan sileä siirtymä tangenttiavaruudesta toiseen ja päästään käsiksi suunnistetun derivaatan yleistykseen kovarianttiin derivaattaan sekä euklidisen avaruuden suoran viivan yleistyksiin geodeeseihin. Tämän koneiston avulla pystytään määrittelemään monta eri tilanteisiin sopivaa kaarevuuden käsitettä, joiden avulla avaruuden muodosta saadaan tietoa. Viimeisessä, kolmannessa, luvussa määritellään muutamia aiempiin aiheisiin kuulumattomia Gaussin-Bonnet'n lauseen kannalta tarpeellisia käsitteitä. Tilan säästämiseksi algebrallista topologiaa vaativat todistukset sivuutetaan. Lisäksi ''Umlaufsatzin'' eli kiertokulmalauseen todistus sivuutetaan. Lopuksi todistetaan Gaussin-Bonnet'n kaava ja sen avulla itse Gaussin-Bonnet'n lause. Gaussin-Bonnet'n lause on erittäin merkittävä tulos mm. siksi, että se yhdistää niin erilaiset suureet toisiinsa: lokaalista geometriasta kumpuavan Gaussin kaarevuden ja Eulerin karakteristikan, joka on globaali topologinen invariantti. Gaussin-Bonnet'n lause toimii vain 2-ulotteisten monistojen tapauksessa, mutta sille on olemassa monia korkeampiulotteisia yleistyksiä. Näiden yleistysten avulla joitain lauseen geometrisistä ja topologisista seurauksista saadaan hyödynnettyä muissakin monistoissa. Näitä edistyneempiä tuloksia ja muuta, esoteerisempaa, Riemannin geometriaa ei tässä tutkielmassa käsitellä.
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(2021)The introductory section of the thesis discusses on the European General Data Protection Regulation, abbreviated GDPR, background information and historical facts. The second section covers basic concepts of personal data and GDPR enforcement. The third section gives detailed analysis on data subject rights as well as best practices for GDPR compliance to avoid penalties. The fourth section concentrates on the technical aspects of the right to be forgotten, solely concentrating on the technical aspects of permanent erasure/deletion of personal or corporate data in compliance with the customer’s desire. Permanent deletion or erasure of data, technically addressing the issue of the right to be forgotten and block chain network technology are the main focus areas of the thesis. The fifth section of the thesis profoundly elaborates block chain and the relation with GDPR compliance in particular. Then the thesis resumes explaining about security aspects and encryption, confidentiality, integrity and availability of data as well as authentication, authorization and auditing mechanisms in relation to the GDPR. The last section of the thesis is the conclusion and recommendation section which briefly summarizes the entire discussion and tries to suggest further improvements
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(2015)Involuntary displacements are more common than ever, and the reasons vary from natural disasters, wars and conflicts to environmental degradation and development-induced displacement. Typically, the victims of these phenomena inhabit the Global South, and are further impoverished due to the lack of having a say where and how to live. The lack of social justice and recognition of social development is typical in large-scale involuntary displacements, and also affects to the abilities of people to reconstruct and recover after resettlement. The linkage between forced migration and risk to impoverishment has been widely acknowledged especially in cases that lack participatory measures and proper compensation for the victims. When the impacts are recognized, it is understood that forced resettlement has effects in the economic, social, and physical spheres of life, and can be a major burden for the urban poor. However, involuntary displacement also disrupts the attachments and constructions of sense of place, which have further impacts on social cohesion. The question of community reorganization is crucial in the context where resettlement is simultaneous and combines various heterogeneous groups. The aim of this research is to examine the extent and quality of rehabilitation after involuntary displacement in an urban environment in Sri Lanka. The focus in this research is given to women, as gender is a significant factor in resettlement outcomes, yet it is often ignored in planning and implementation. Gender roles and norms in Sri Lanka are still fixed and conventional, making it an interesting approach to study resettlement and rehabilitation and the daily interactions and perceptions on them. A case study for this research took place in four resettlement sites in the outskirts of Colombo Metro Region, Sri Lanka, which is in the midst of significant urban development plans that aim to relocate tens of thousands of slum dwellers in the need to release prime lands for investments, simultaneously beautifying the city and fighting frequent flooding. The methodological approach applied in this study addresses feminist geography while it looks at rehabilitation measures in the everyday life point of view. It utilized semi-structured interviews of resettled women as a key research method. The case study took place in upgraded underserved settlements that inhabit tsunami-displaced people and development-induced displaced people. Also local experts were interviewed in order to gain a broader understanding of the dynamics and urban policies in the Colombo Metro Region. The case evidences that rehabilitation is a slow process of adaptation, and that physical assets alone cannot improve the livelihoods of slum dwellers. More emphasis should be put into the social relations and community dynamics if participatory measures and development-from-within are to fully deliver. Also full tenure is needed in order to the people to adapt to the mainstream society. For women the question of belonging and reconstructing the sense of place is essential as they spend a lot of their time at home and the immediate environment, and are traditionally in charge of homemaking. Therefore the sense of place and community are essential in the creation of social cohesion and management of the new neighborhood, and along with gender sensitive approaches should not be overlooked if sustainable resettlement is to be pursued as a consequence of ever more crowded cities of the South.
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(2020)Tutkielman tavoitteena on esittää, että lukion todennäköisyyslaskennan käsitteet ja teoriat voidaan opettaa ja opiskella oppikirjojen perinteisistä esimerkeistä poikkeavassa kontekstissa. Kontekstiksi valitaan todennäköisyyslaskennan mallien todellinen sovellusalue. Samalla luodaan tehtäväpaketin muodossa ainerajat ylittävä opetusprojekti lukioon. Tutkielmassa perustellaan, miksi evoluutio ja populaatiogenetiikka ovat hyvä asiayhteys lukion todennäköisyyslaskennen opetuksessa ja toisin päin. Opetusprojektin toteutus noudattaa soveltaen tutkivan oppimisen menetelmää. Oppiainekohtaisten tavoitteiden saavuttamisen lisäksi yksi opetusprojektin päätavoitteista on oppilaiden ajattelutaitojen kehittäminen, sillä opiskeltavien asioiden syvällinen ymmärtäminen tapahtuu ajattelun kautta. Tutkielmassa on rakennettu tehtäväpaketti, joka koostuu 15 tehtävästä. Tehtävät liittyvät todennäköisyyslaskentaan, evoluutioon ja populaatiogenetiikkaa. Tehtävät alkavat yksinkertaisista käsitteiden määrittelytehtävistä ja peruslaskutehtävistä muuttuen vähitellen tiedon soveltamista vaativiksi ongelmanratkaisutehtäviksi. Opiskelijalla on aktiivinen rooli tiedonhakijana, ja opiskelijoiden keskinäiset keskusteltu ja yhteistyö ovat keskeisessä roolissa prosessissa, jonka tavoite on uuden tiedon rakentaminen.
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(2014)The study of obesity has drawn wide-spread interest because of its far-reaching consequences. Obesity is on the rise and has been linked to several clinical complications such as type 2 diabetes and hypertension. Obesity is defined as a condition in which body mass index (BMI) is greater than 3m2/kg. BMI itself is highly heritable with the rate of heritability in twin and adoption studies ranging from 45%–85%. The study was designed to analyze the differences in gene expression in MZ twin pairs discordant for BMI. A total of 26 twin pairs were selected based on a within-pair BMI difference of more than 3 kg/m2 when compared to his or her twin. The twins for this study belonged to either FinnTwin16 (birth cohort 1975-1979) or FinnTwin12 (birth cohort 1983-1987). Samples were extracted from the participants and used in microarray experiments. The resulting data was processed using various packages of the Bioconductor software. The quality control process identified one sample as faulty and as a result the sample as well as the sample of the twin were discarded. This resulted in a sample size of 50 twins. Differential analysis carried out using the limma package of Bioconductor revealed 980 genes that were differentially expressed. These genes were then processed further in the Ingenuity Pathway Analysis (IPA) tool as well as the BiNGO tool. This revealed the molecular networks, gene pathways and gene annotations that were pertinent to the genes uploaded. IPA also provided a list of functions and diseases these genes were involved in. As the final step, the prevalent themes across these results were summarized. This analysis provided many leads that should be investigated further in future studies. It is suggested that future studies start with precise, concrete biological questions that narrow down the scope of biological analysis. This is in view of the extensive amount of data available via microarray studies and the myriad of hypothesizes that can be investigated.
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(2024)Essential thrombocythemia (ET) is a clonal hematopoietic disease characterized by an abnormal increase of platelets in the circulation, with increased risk of thrombosis and hemorrhage. Despite megakaryocytes having a central role in the disease, few studies have investigated their gene expression in ET. The aim of this study is to characterize the gene expression profiles of megakaryocytes from ET patients harboring different driver mutations, and increase the knowledge of the molecular mechanisms underlying the pathophysiology of the disease. In this study, samples were obtained from healthy donors and ET patients with JAK2 V617F, CALR Type I, CALR Type II driver mutations and triple-negative patients. Following megakaryocyte culture from peripheral blood and RNA sequencing, the data was pre-processed and analyzed using differential gene expression analysis. The downstream analysis was conducted using pathway enrichment analysis tools. The analysis revealed that all mutants shared common deregulated genes related to processes involving platelets and coagulation. However, it was shown that CALR and JAK2 V617F mutants also have distinct patterns of gene expression. CALR Type I mutants had a unique gene expression signature consisting of genes related to immune response, as well as metabolic, regulatory, proliferative, and inflammatory pathways, while CALR Type II mutants had unique genes related to ribosomes. The CALR mutants also shared a common anti-inflammatory response signature which set them apart from JAK2 V617F mutants. In conclusion, this study shows that the gene expression profiles of ET mutants are heterogeneous. Moreover, the results provide new insights into the gene expression profiles of CALR mutants that distinguish them from the other mutants. Further experiments using single-cell RNA sequencing methods could build upon these findings and uncover the observed gene expression discrepancies between CALR and JAK2 mutants with increased accuracy.
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(2020)Predicting patient deterioration in an Intensive Care Unit (ICU) effectively is a critical health care task serving patient health and resource allocation. At times, the task may be highly complex for a physician, yet high-stakes and time-critical decisions need to be made based on it. In this work, we investigate the ability of a set of machine learning models to algorithimically predict future occurrence of in hospital death based on Electronic Health Record (EHR) data of ICU-patients. For one, we will assess the generalizability of the models. We do this by evaluating the models on hospitals the data of which has not been considered when training the models. For another, we consider the case in which we have access to some EHR data for the patients treated at a hospital of interest. In this setting, we assess how EHR data from other hospitals can be used in the optimal way to improve the prediction accuracy. This study is important for the deployment and integration of such predictive models in practice, e.g., for real-time algorithmic deterioration prediction for clinical decision support. In order to address these questions, we use the eICU collaborative research database, which is a database containing EHRs of patients treated at a heterogeneous collection of hospitals in the United States. In this work, we use the patient demographics, vital signs and Glasgow coma score as the predictors. We devise and describe three computational experiments to test the generalization in different ways. The used models are the random forest, gradient boosted trees and long short-term memory network. In our first experiment concerning the generalization, we show that, with the chosen limited set of predictors, the models generalize reasonably across hospitals but that only a small data mismatch is observed. Moreover, with this setting, our second experiment shows that the model performance does not significantly improve when increasing the heterogeneity of the training set. Given these observations, our third experiment shows that
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(2022)The focus of this work is to efficiently sample from a given target distribution using Monte Carlo Makov Chain (MCMC). This work presents No-U-Turn Sampler Lagrangian Monte Carlo with the Monge metric. It is an efficient MCMC sampler, with adaptive metric, fast computations and with no need to hand-tune the hyperparameters of the algorithm, since the parameters are automatically adapted by extending the No-U-Turn Sampler (NUTS) to Lagrangian Monte Carlo (LMC). This work begins by giving an introduction of differential geometry concepts. The Monge metric is then constructed step by step, carefully derived from the theory of differential geometry giving a formulation that is not restricted to LMC, instead, it is applicable to any problem where a Riemannian metric of the target function comes into play. The main idea of the metric is that it naturally encodes the geometric properties given by the manifold constructed from the graph of the function when embedded in higher dimensional Euclidean space. Hamiltonian Monte Carlo (HMC) and LMC are MCMC samplers that work on differential geometry manifolds. We introduce the LMC sampler as an alternative to Hamiltonian Monte Carlo (HMC). HMC assumes that the metric structure of the manifold encoded in the Riemannian metric to stay constant, whereas LMC allows the metric to vary dependent on position, thus, being able to sample from regions of the target distribution which are problematic to HMC. The choice of metric affects the running time of LMC, by including the Monge metric into LMC the algorithm becomes computationally faster. By generalizing the No-U-Turn Sampler to LMC, we build the NUTS-LMC algorithm. The resulting algorithm is able to estimate the hyperparameters automatically. The NUTS algorithm is constructed with a distance based stopping criterion, which can be replaced by another stopping criteria. Additionally, we run LMC-Monge and NUTS-LMC for a series of traditionally challenging target distributions comparing the results with HMC and NUTS-HMC. The main contribution of this work is the extension of NUTS to generalized NUTS, which is applicable to LMC. It is found that LMC with Monge explores regions of target distribution which HMC is unable to. Furthermore, generalized NUTS eliminates the need to choose the hyperparameters. NUTS-LMC makes the sampler ready to use for scientific applications since the only need is to specify a twice differentiable target function, thus, making it user friendly for someone who does not wish to know the theoretical and technical details beneath the sampler.
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(2018)Mid-Proterozoic mafic dyke swarms in southern Finland are associated with rapakivi magmatism. The dyke swarms are commonly referred to as “Subjotnian” (1.64–1.54 Ga), being older than the rift-filling Jotnian sandstones. Mafic rocks from five dyke swarms located in Åland, Satakunta, Häme, Suomenniemi and Sipoo were studied in this thesis. An X-ray fluorescence (XRF) analysis was made of 110 rock samples from 101 mafic dykes and one mafic intrusion. The analyses were made of the same rock samples as previous paleomagnetic studies. Overall, the Subjotnian mafic dykes in southern Finland are hyperstene-normative tholeiitic basalts or basaltic andesites with varying MgO contents (3–15 wt%). Some dykes show alkaline features with higher total alkali and/or Nb/Y values. They vary from quartz- to olivine-normative types. The dykes of the Åland swarm form two geohemical groups. The division is accompanied with a switch in magnetic polarity and distinct virtual geomagnetic pole positions. These observations imply that two separate magmatic events/pulses that have an age difference have taken place in Åland. The Satakunta dykes form two geochemical groups of which the other includes presumably Svecofennian dykes that show high Nb/Y values at given Zr/Y ratios. The dykes of the Häme swarm form three geochemical groups. Although some Suomenniemi dykes show geochemical and paleomagnetic affinities to Häme dykes, they probably represent a distinct igneous event of the event that formed the nearby Häme swarm. The Sipoo dykes are very homogeneous in their geochemistry and can be distinguished from the emplacement events that formed the other Subjotnian swarms. The Subjotnian dyke swarms in southern Finland that are believed to have emplacement ages of >1.63 Ga (Häme, Suomenniemi and Sipoo swarms in S-SE Finland) generally have higher Nb/Y (and Zr/Y) values than the dyke swarms that are believed to record younger magmatic events at <1.58 Ga (Åland and Satakunta swarms in SW Finland). Some Satakunta dykes, however, have geochemical and/or paleomagnetic implications that suggest they have an older Subjotnian age than the dated 1.57 Ga dyke in Satakunta. Further chronological work on the Satakunta dyke swarm is needed to verify the age of the dykes. Many of the Subjotnian dykes show a secondary magnetization component, called the “B-component”, whose direction is always close to, but distinct of, the Present Earth Field (PEF) at the sampling location. There was no correlation between the B-component and the magma types of the dykes. The B-component occurs mostly in dykes that are very altered. Thus, the results support previous suggestions that the B-component formed due to hydrothermal alteration of the rocks and the subsequent formation of new magnetic minerals.
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(2020)The oldest rocks in Finland are the Archaean grey gneisses of eastern and northern Finland. The Archaean of the Karelian craton spans about 1000 Ma of crustal growth and evolution and forms the core of the Fennoscandian shield. The Karelian province is a complex patchwork of different rock types. The individual formations are of small territorial extent in accordance with often postulated small Archaean plates. Overall, the Karelian craton is a granitoid-greenstone terrain with prevailing TTGs and younger granites, which show increasing level of potassium. The craton also includes a distinct sodic variety of granites that combines features of classical Archaean TTGs and late Archaean high-K granites. A minor number of Mg-rich lithologic units, including adakites and sanukitoids, are reported as well. A small number of A-type granites, syenites and S-type granites are widely distributed and of local nature only. Peculiarly, a large number of TTGs is peraluminous. The formation of Karelian craton may be explained by accretion of small plates, perhaps during the late Archaean supercraton event in a process that at least in later stages included active plate marginal processes.
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(2015)Lake Pyhäjärvi, the largest lake in southwestern Finland, has been under considerable external nutrient loading for the past decades. Rivers Pyhäjoki and Yläneenjoki are the only major input rivers of the lake and are the source of most of the external nutrient loading to the lake. The hydrogeochemistry of the Lake Pyhäjärvi catchment, as well as the catchments of the two input rivers, and groundwater-surface water interaction were evaluated using a wide array of geochemical tracers (major ions, dissolved silica, stable isotopes of oxygen and hydrogen, electrical conductivity and radon-222). Additionally, the feasibility of using mass balance based methods to separate river waters of the two input rivers to their respective source components was evaluated independently with each tracer analysed. In the context of this study, stable isotopes alone were deemed plausible and were only usable in the River Pyhäjoki catchment as there was not enough difference between river water and precipitation (new water) stable isotope proportions in River Yläneenjoki. Employing the stable isotopes of oxygen, mass balance based modelling was attempted to separate the hydrograph of River Pyhäjoki to its end-members (new water and old water). Based on the obtained data, the hydrogeochemical content of the surface waters of each subcatchment (Pyhäjärvi, Pyhäjoki and Yläneenjoki) differed significantly. Additionally, the groundwaters were clearly distinguished from surface waters. Differences were most apparent in stable isotope proportions as well as dissolved silica, and in the case of groundwaters in radon-222 concentrations. This categorization was further supported by a hierarchical cluster analysis. Surface waters showed varying signs of evaporation, whereas groundwaters retained the stable isotope characteristics of mean annual precipitation. Dissolved silica concentrations appeared to be mostly affected by the amount of easily soluble silica in the sediment, water residence time, as well as biological uptake in the surface waters. Lithology seemed to be the controlling factor in radon concentrations, with areas of granitic bedrock having the highest concentrations. The hydrograph separation of River Pyhäjoki gave results between 66–88% of old water in the river at the time of sampling, consisting likely mostly of groundwater. Though this result corresponds well with recent similar studies in the area, there were considerable sources of uncertainty, therefore making the result best thought of as indicative. However, there was clear evidence of groundwater-surface water interaction in all of the subcatchments of Lake Pyhäjärvi, with clear evidence of groundwater discharging into the lake, lake water infiltrating into an aquifer near the shore of the lake, as well as signs of groundwater discharging into the two input rivers.
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(2018)As a part of Kumpula Campus Drill Hole Project, a 370 m deep drill hole was drilled on the University of Helsinki, Kumpula campus area in December 2015. Drilling took place on an amphibolitic outcrop, which is the main rock type of the area and part of the 1.9 Ga old Svecofennian orogenic belt. In this work, the geochemistry of the campus bedrock is analyzed, focusing on the amphibolite. Granite, actinolite rock and diopside-actinolite skarn are additional rock types described from the core in this work. The geochemical methods utilized are a portable X-ray fluorescence (P-XRF) spectrometer Niton XL3t GOLDD+ by Thermo Scientific and a laboratory wavelength dispersive X-ray fluorescence (WD-XRF) spectrometer PANAlytical Axios mAX 4kW. WD-XRF device is utilized in quantitative analysis and semi-quantitative Omnian scans. In addition to geochemical interpretation of the bedrock, feasibility of the P-XRF device in outcrop and drill core related studies is evaluated by comparing the methods. The surface of the drill core was analyzed with the P-XRF device. Representative samples of each rock type were sawed of the core and analyzed with both P-XRF and WD-XRF Omnian scans. In addition to surface analyses, a fused bead was prepared from one representative amphibolite sample and analyzed with WD-XRF quantitative method. Outcrop studies focused on the feasibility of the P-XRF device in in situ analyses. Compared to nearest temporally related amphibolite units, the amphibolite of the campus bedrock seems to be more felsic on average. All described rock types are connected to former petrogenetic interpretations of the local bedrock. However, further geochemical analyses are required to verify the interpretations. WD-XRF quantitative method and Omnian scans suggest almost similar results for fused bead of the amphibolite. Changing the sample type to solid rock surface introduces heterogeneity related problems to the quantitative determination of Omnian scans and quality of the results decreases almost to the level of P-XRF. Yet, the advantage of the Omnian scans method in rock surface analyses compared to P-XRF is better detection of light elements. For example, P-XRF device detects Mg, Al and K poorly and Na is not detected at all. On the other hand, SiO2 is on average detected quite accurately from rock surface with P-XRF when compared to WD-XRF quantitative method for fused bead. WD-XRF Omnian scans and quantitative application results of fused bead do not seem to differ remarkably. Broad rock type classification can be made with P-XRF device for drill core, but results cannot be considered quantitative. It should also be noticed, that the major element oxide sum values of P-XRF drill core surface analyses are quite low on average (84.00 wt.%). In outcrop analyses, different features lower the quality of the rock surfaces, resulting in even lower major element oxide sum values in analysis. Although major oxide sum values are very low on outcrops, relatively high amounts of for example Cl, S and P are detected for unknown reasons. Major advantages of the P-XRF device are the ease of use, light weight and rather good detection of for example SiO2. Developing the quantitativeness of the device would make it more comparable to laboratory XRF devices but it already has multiple features that are highly beneficial in a wide range of scientific fields.
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(2019)The Vihanti-Lampinsaari group in the Raahe-Ladoga belt, central Finland, hosts massive sulphide deposits. Host rocks are highly deformed and metamorphosed (amphibolite to granulite facies) felsic to intermediate volcanic rocks with minor mafic metavolcanics. Due to intense metamorphism, it is difficult to figure out the nature of the original protolith of hosts rocks and there is no systematic examination of the trace elements behaviour found in the literature for the rocks in the Lampinsaari region. Two main topics were studied: (1) The trace elements behaviour of the rocks in the Vihanti (Lampinsaari) group, and (2) origin of the volcanic rocks and volcanic architecture of the Vihanti group. Better knowledge on the geochemistry of the immobile trace elements in these highly metamorphosed rocks led to the more accurate and precise determination of the origin of volcanic rocks and their volcanic architecture. Igneous (Felsic-intermediate-mafic) metavolcanic rocks of the Vihanti-Lampinsaari region have all calc-alkaline magmatic affinity according to (Ross and Bedard; 2009) Zr vs Y and major elements classification by Miyashiro (1974). The chondrite normalized patterns of the trace elements having enriched concentrations of LREE relative to low HREE indicates island arc settings. There is no significant hydrothermal material present in metasedimentary rocks of the Vihanti Lampinsaari area. Meta-sedimentary rocks of the Vihanti group are largely silica-rich with the presence of detrital components. Two calc-silicate bearing paragneisses have higher concentrations of Fe which are due to the pyrite while three distinct samples with higher P₂O₅ having higher concentrations of Mn represents phosphate minerals. Negative Eu anomaly due to Ca-replacement and enriched LREE with respect to HREE is an indication of evolved source areas. Sedimentary carbonates with one rock type of serpentinite-dolomite are mineralized ones have a higher concentration of Mn and are enriched in FeO which is due to the hydrothermal alteration in the sediments. Volcanic sedimentary rocks are highly evolved in composition but show depletion in Ba, Ti and Sr. Investigated sulfide ore deposits of the Vihanti-Lampinsaari region have diverse REE behaviour in chondrite normalized patterns due to high fluid/rock ratios. Due to high fluid/rock ratios, it is more likely to lose parent texture of the rocks with dissimilar trends in primitive and chondritic normalized patterns. Positive anomalies of Eu represent ore mineralization and considered for better chances of ore-bearing rocks. While negative Eu anomalies indicate late-stage mineralization in these mineralized rocks of the Vihanti-Lampinsaari region.
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(2019)The Central Finland Granitoid Complex (CFGC) is a large (44,000 km2) plutonic core of a Svecofennian (Paleoproterozoic, 1.91–1.82 Ga) arc complex, formed from collisions of several volcanic arcs and their accretion over the Karelian craton. The CFGC consists mostly of granitic to granodioritic rock types. Mafic-ultramafic plutonic rock types are not common, and they consist of mostly small gabbro-diorite intrusions, which may have ultramafic parts. There are two distinct belts around the CFGC, where Ni-Cu potential mafic-ultramafic intrusions are situated – Vammala and Kotalahti. The intrusions within these belts were formed during the height of magmatism within the CFGC (1.89–1.87 Ga). They host Ni-Cu mineralizations, some of which have been economically exploited. The mineralizations are hosted by olivine-rich ultramafic cumulates. The intrusions formed from hydrous tholeiitic basalts (10–12 wt-% MgO) with arc-type trace element chemistry. The difference between Vammala and Kotalahti type intrusions (clinopyroxene and orthopyroxene-dominated, respectively) are attributed to the rock type of the assimilated country rock. In this thesis, three previously unknown or poorly studied mafic-ultramafic intrusions (Matokulma, Palojärvi, and Hongonniittu) within the CFGC are studied in detail. The petrology, similarity to Vammala-Kotalahti type intrusions, parental magma compositions, ore potential, and petrogenesis of the intrusions are described. Rock samples and field observations were gathered during the summer of 2017. Whole-rock geochemistry, mineral geochemistry, isotope geochemistry, and geophysics are used to describe the petrology of the intrusions. Matokulma and Palojärvi intrusions are studied in detail, compared to Hongonniittu intrusion, which was not studied as intricately. The Matokulma intrusion is the least evolved (whole-rock median Mg#=72) of the studied intrusions and consists of tholeiitic melagabbros where clinopyroxene±orthopyroxene and plagioclase are the main cumulus phases within interstitial, magmatic amphibole (magnesiohastingsite to pargasite in composition). Orthopyroxene and plagioclase are intercumulus phases in some samples. There are also mafic dikes that intrude the tonalitic country rock that surrounds the gabbro. The dikes are similar to the gabbros in geochemistry although they are generally more evolved. Trace element geochemistry suggests that the gabbros and dikes are genetically connected, and the dikes possibly represent the residual magmas of the gabbros. The Palojärvi intrusion is noticeably more evolved than the Matokulma intrusion (median Mg#=49), which is apparent in the iron and titanium rich mineral and whole-rock geochemistry. The strongly tholeiitic melagabbros are composed of both orthopyroxene and clinopyroxene as cumulus phases with plagioclase and common Fe-Ti oxide, often within interstitial magmatic amphibole (magnesio-hastingsite to magnesioferri-hornblende in composition). The Fe-Ti oxides are mostly ilmenomagnetite but both magnetite and ilmenite grains are present in same samples. Based on a few mineral analyzes, the ilmenomagnetite contains up to 1.4 wt-% V2O3. U-Pb age determination samples from a leucogabbro dike within the intrusion and granite that crosscuts the intrusion yielded weighted average 206Pb/207Pb ages of 1883.4±4.8 Ma and 1893.8±7.1 Ma, respectively. The age results are in contrast to the intrusive relationship observed in the field. However, considering the margin of error of the results, the granite can be younger than the gabbro, 1887 Ma and 1888 Ma, respectively. The age of ca. 1.89 Ga is at the early stage of the most voluminous mafic-ultramafic magmatism in the Svecofennian terrane. The parental magmas of the Matokulma and Palojärvi intrusions were evolved and contained approximately 5 wt-% and 2 wt-% MgO, respectively. The presence of magmatic amphiboles in most samples indicate that the parental magmas were hydrous. Samples from all intrusions plot similarly in primitive mantle normalized Rare Earth Element (REE) and Normal-Mid-Ocean Ridge Basalt (NMORB) normalized spider diagrams. Similar patterns indicate a similar source for the parental magmas. The trace element geochemistry has signatures of subduction related fluid metasomatism. The rocks are enriched in large ion lithophile elements (LILE) and depleted in High Field Strength Elements (HFSE). These geochemical characteristics indicate that the studied intrusions crystallized from a hydrous, NMORB-like evolved basaltic magma, which has experienced fluid metasomatism. The studied intrusions differ from olivine-rich ultramafic cumulates of Vammala and Kotalahti type intrusions based on their more evolved, gabbroic composition and because of this, they are not Ni-Cu ore potential. Palojärvi may host a Fe-Ti-V mineralization, if there are magnetite rich layers within the intrusion.
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