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In this master's thesis we explore the mathematical model of classical Lagrangian mechanics with constraints. The main focus is on the nonholonomic case which is obtained by letting the constraint distribution to be nonintegrable. Motivation for the study arises from various physical examples, such as a rolling rigid body or a snakeboard. In Chapter 2, we introduce the model and derive the associated equations of motion in several different forms while using the Lagrangian variational principle as a basis for the kinematics. We also show how nonintegrability of the constraint distribution is linked to some external forces via the Frobenius theorem. Symmetric mechanical systems are discussed in Chapter 3. We define the concept for a Lagrangian system with constraints and show how any free and proper Lie group action induces an intrinsic vertical structure to the tangent bundle of the configuration manifold. The associated bundle is used to define the nonholonomic momentum which is a constrained version of the form that appears in the modern formulation of the classical Noether's theorem. One applies the classical Noether's theorem to a symmetric system with integrable constraints by restricting observation to an integral submanifold. This procedure, however, is not always possible. In nonholonomic mechanics, a Lie group symmetry implies only an additional equation of motion rather than actual conservation law. In Chapter 4, we introduce a coordinate free technique to split the Lagrangian variational principle in two equations, based on the Lie group invariance. The equations are intrinsic, that is to say, independent of the choice of connections, related parallel transports and covariant differentiation. The vertical projection, associated to the symmetry, may be varied to alter the representation and shift balance between the two equations. In Chapter 5, the results are applied to the rattleback which is a Lagrangian model for a rigid, convex object that rolls without sliding on a plane. We calculate the nonholonomic momentum and state the equations of motion for a pair of simple connections. One of the equation is also solved with respect to a given solution for the other one. The thesis is mainly based on the articles 'Nonholonomic Mechanical Systems with Symmetry' (A.M. Bloch, P.S. Krishnaprasad, J.E. Marsden, and R M. Murray, 1996), 'Lagrangian reduction by stages' (H. Cendra, J.E. Marsden, and T.S. Ratiu, 2001), 'Geometric Mechanics, Lagrangian Reduction and Nonholonomic Systems' (H. Cendra, J.E. Marsden, and T.S. Ratiu, 2001) and the book 'Nonholonomic mechanics and control' (A.M. Bloch, 2003).

The purpose of this study is to develop a method for optimizing the data assimilation system of the HIROMB-BOOS -model at the Finnish Meteorological Institute by finding an optimal time interval and an optimal grid for the data assimilation. This is needed to balance the extra time the data assimilation adds to the runtime of the model and the improved accuracy it provides. Data assimilation is the process of combining observations with a numerical model to improve the accuracy of the model. There are different ways of doing this, some of which are covered in this work. The HIROMB-BOOS -circulation model is a 3D-forecast model for the Baltic Sea. The variables forecast are temperature, salinity, sea surface height, currents, ice thickness and ice coverage. Some of the most important model equations are explained here. The HIROMB-BOOS -model at the Finnish Meteorological Institute has a preoperational data assimilation system that is based on the optimal interpolation method. In this study the model was run for a 2-month test period with different time intervals of data assimilation and different assimilation grids. The results were compared to data from five buoys in the Baltic Sea. The model gives more accurate results when the time interval of the data assimilation is small. The thicker the data assimilation grid is, the better the results. An optimal time interval was determined taking into account the time the assimilation takes. An optimal grid was visually determined based on an optimal grid thickness, for which the added time had to be considered as well. The optimized data assimilation scheme was tested by performing a 12-month test run and comparing the results to buoy data. The optimized data assimilation has a positive effect on the model results.

Työssä konstruoidaan euklidisen kaksiulotteisen pallonkuoren kanssa melkein varmasti homeomorfinen satunnainen metrinen avaruus, Brownin graafi, ja tarjotaan mahdollinen diskretisaatio pallonkuorelle käyttäen neliötahkoisia tasoverkkoja. Aluksi konstruoidaan Gromovin-Hausdorffin metriikka kompaktien metristen avaruuksien joukkoon. Tämän jälkeen konstruoidaan Corin-Vauquelin-Schaefferin bijektio olennaisesti tason puiden ja neliötahkoisten tasoverkkojen joukkojen välille, missä puiden kaarien ja tasoverkkojen tahkojen lukumäärä on sama kiinnitetty luonnollinen luku ja puiden solmuihin on lisäksi liitetty kokonaisluku. Tämän bijektiivisen vastaavuuden perusteella n-tahkoisten neliötasoverkkojen lukumäärä on helppo laskea kaikille luonnollisille luvuille n. Huomataan, että tasaisesti jakautunut n-tahkoinen neliötasoverkko on satunnaismuuttuja kompaktien metristen avaruuksien avaruudessa ja todetaan, että on mielekästä tutkia näiden satunnaismuuttujien suppenemista jakauman mielessä. Sen jälkeen kun Brownin graafi on konstruoitu esitetään Jean-François Le Gall'n ja Grégory Miermont'n todistama tuore tulos, jonka mukaan Brownin graafi on sopiva skaalaraja diskreeteistä tasoverkoista jakaumien suppenemisen mielessä. Tutkielman lopuksi arvioidaan lyhyesti, kuinka hyvin Brownin graafi kuvaa tasaisesti jakautunutta satunnaista metriikkaa pallon kuorella sekä esitellään aiheeseen liittyviä avoimia ongelmia. Työn motivaationa ovat osaltaan sovellukset kvanttigravitaatioteoriaan.

Self-Sovereign Identity is a new concept of managaging digital identities in the digital services. The purpose of the Self-Sovereign Identity is to place the user in the center and move towards decentralized model of identity management. Verifiable Credentials, Verifiable Presentations, Identity Wallets and Decentralized Identifiers are part of the Self-Sovereign Identity model. They have also been recently included in the OpenID Connect specifications to be used with the widely used authentication layer built on OAuth 2.0. The OpenID Connect authentication can now be leveraged with the Decetralized Identifiers (DIDs) and the public keys contained in DID Documents. This work assessed the feasibility of integrating the Verifiable Credentials, Verifiable Presentations and Decentralized Identifiers with OpenID Connect in the context of two use cases. The first use case is to integrate the Verifiable Credentials and Presentations into an OpenID Connect server and utilise Single Sign-On in federated environment. The second use case is to ignore the OpenID Provider and enable the Relying Party to authenticate directly with the Identity Wallet. Custom software components, the Relying Party, the Identity Wallet and the Verifiable Credential Issuer were built to support the assessments. Two new authorization flows were designed for the two use cases. The Federated Verifiable Presentation Flow describes the protocol of Relying Party authenticating with OpenID Provider which receives the user information from the Wallet. The flow does not require any changes for any Relying Party using the same OpenID Provider to authenticate and utilise Single Sign-On. The Verifiable Presentation Flow enables the Relying Party to authenticate directly with the Wallet. However, this flow requires multiple changes to Relying Party and benefits of federated environment are not available, e.g., the Single Sign-On. Both of the flows are useful for their own specific use cases. The new flows are utilising the new segments of the Self-Sovereign Identity and are promising steps towards self-sovereignty.

Context: Factors that affect software team performance are a highly studied subject. One of the reasons for this is the subject’s meaningfulness to companies and software teams since anyone interested in improving team performance wants to know which factors affect positively on the team performance. What motivated us to do this thesis on this subject was our interest in both software teams and social sciences. Objective: This thesis’s aim was to better understand how the factors selected in our unofficial interviews will affect the software team performance and how big this affect is. These selected factors are psychological safety, team leader’s behaviour and team’s gender diversity. Method: We conducted a literature review with a keyword search. When we needed to specify the search by a factor we used factor-related words and if needed limit the subject area to computer science. All in all 23 reference papers were selected in the search. Results: Our analysis shows that all of our factors have a positive impact on the performance of the team, though how big this impact is depended on the factor. Psychological safety seems to have the biggest impact while the behaviour of team leader has a decent impact, not huge but not minuscule, lastly the gender diversity of the team has only a very small impact. Conclusions: Ultimately we have concluded that all three chosen factors have a positive effect on software team performance. Though from these three factors, psychological safety and team leader’s behaviour have the most significant impact on software team performance. So for software team leaders, it’s important to pay attention to these two factors, especially since they are even linked to each other.

In this paper we define and study the Julia set and the Fatou set of an arbitrary polynomial f, which is defined on the closed complex plane and whose degree is at least two. We are especially interested in the structure of these sets and in approximating the size of the Julia set. First, we define the Julia and Fatou sets by using the concepts of normal families and equicontinuity. Then we move on to proving many of the essential facts concerning these sets, laying foundations for the main theorems of this paper presented in the fifth chapter. By the end of this chapter we achieve quite a good understanding of the basic structure of the Julia set and the Fatou set of an arbitrary polynomial f. In the fourth chapter we introduce the Hausdorff measure and dimension along with some theorems regarding them. In this chapter we also say more about fractals and self-similar sets, for example the Cantor set and the Koch curve. The main goal of this chapter is to prove a well-known result which allows to easily determine the Hausdorff dimension of any self-similar set that fulfils certain conditions. We end this chapter by calculating the Hausdorff dimension of the one-third Cantor set and the Koch-curve by using the result described earlier and notice, that their Hausdorff dimension is not integer-valued. In the fifth chapter we study the structure of the Julia set further, concentrating on its connectedness, and introduce the Mandelbrot set. In this chapter we also prove the three main theorems of this paper. First we show a sufficient condition for the Julia set of a polynomial to be totally disconnected. This result, with some theorems proven in the third chapter, shows that in this case the Julia set is a Cantor-like set. The second result shows when the Julia set of a quadratic polynomial of the form f(z) = z^2 + c is a Jordan curve. The third and final result shows that given an arbitrary polynomial f, there exists a lower bound for the Hausdorff dimension of the Julia set of the polynomial f, which depends on the polynomial f. This is the most important result of this paper.

Crowdsourcing has been used in computer science education to alleviate the teachers’ workload in creating course content, and as a learning and revision method for students through its use in educational systems. Tools that utilize crowdsourcing can act as a great way for students to further familiarize themselves with the course concepts, all while creating new content for their peers and future course iterations. In this study, student-created programming assignments from the second week of an introductory Java programming course are examined alongside the peer reviews these assignments received. The quality of the assignments and the peer reviews is inspected, for example, through comparing the peer reviews with expert reviews using inter-rater reliability. The purpose of this study is to inspect what kinds of programming assignments novice students create, and whether the same novice students can act as reliable reviewers. While it is not possible to draw definite conclusions from the results of this study due to limitations concerning the usability of the tool, the results seem to indicate that novice students are able to recognise differences in programming assignment quality, especially with sufficient guidance and well thought-out instructions.

Sums of log-normally distributed random variables arise in numerous settings in the fields of finance and insurance mathematics, typically to model the value of a portfolio of assets over time. In particular, the use of the log-normal distribution in the popular Black-Scholes model allows future asset prices to exhibit heavy tails whilst still possessing finite moments, making the log-normal distribution an attractive assumption. Despite this, the distribution function of the sum of log-normal random variables cannot be expressed analytically, and has therefore been studied extensively through Monte Carlo methods and asymptotic techniques. The asymptotic behavior of log-normal sums is of especial interest to risk managers who wish to assess how a particular asset or portfolio behaves under market stress. This motivates the study of the asymptotic behavior of the left tail of a log-normal sum, particularly when the components are dependent. In this thesis, we characterize the asymptotic behavior of the left and right tail of a sum of dependent log-normal random variables under the assumption of a Gaussian copula. In the left tail, we derive exact asymptotic expressions for both the distribution function and the density of a log-normal sum. The asymptotic behavior turns out to be closely related to Markowitz mean-variance portfolio theory, which is used to derive the subset of components that contribute to the tail asymptotics of the sum. The asymptotic formulas are then used to derive expressions for expectations conditioned on log-normal sums. These formulas have direct applications in insurance and finance, particularly for the purposes of stress testing. However, we call into question the practical validity of the assumptions required for our asymptotic results, which limits their real-world applicability.

This thesis focuses on statistical topics that proved important during a research project involving quality control in chemical forensics. This includes general observations about the goals and challenges a statistician may face when working together with a researcher. The research project involved analyzing a dataset with high dimensionality compared to the sample size in order to figure out if parts of the dataset can be considered distinct from the rest. Principal component analysis and Hotelling's T^2 statistic were used to answer this research question. Because of this the thesis introduces the ideas behind both procedures as well as the general idea behind multivariate analysis of variance. Principal component analysis is a procedure that is used to reduce the dimension of a sample. On the other hand, the Hotelling's T^2 statistic is a method for conducting multivariate hypothesis testing for a dataset consisting of one or two samples. One way of detecting outliers in a sample transformed with principal component analysis involves the use of the Hotelling's T^2 statistic. However, using both procedures together breaks the theory behind the Hotelling's T^2 statistic. Due to this the resulting information is considered more of a guideline than a hard rule for the purposes of outlier detection. To figure out how the different attributes of the transformed sample influence the number of outliers detected according to the Hotelling's T^2 statistic, the thesis includes a simulation experiment. The simulation experiment involves generating a large number of datasets. Each observation in a dataset contains the number of outliers according to the Hotelling's T^2 statistic in a sample that is generated from a specific multivariate normal distribution and transformed with principal component analysis. The attributes that are used to create the transformed samples vary between the datasets, and in some datasets the samples are instead generated from two different multivariate normal distributions. The datasets are observed and compared against each other to find out how the specific attributes affect the frequencies of different numbers of outliers in a dataset, and to see how much the datasets differ when a part of the sample is generated from a different multivariate normal distribution. The results of the experiment indicate that the only attributes that directly influence the number of outliers are the sample size and the number of principal components used in the principal component analysis. The mean number of outliers divided by the sample size is smaller than the significance level used for the outlier detection and approaches the significance level when the sample size increases, implying that the procedure is consistent and conservative. In addition, when some part of the sample is generated from a different multivariate normal distribution than the rest, the frequency of outliers can potentially increase significantly. This indicates that the number of outliers according to Hotelling's T^2 statistic in a sample transformed with principal component analysis can potentially be used to confirm that some part of the sample is distinct from the rest.

Perinteinen tekstihaku vertaa toisiinsa tekstistä löytyviä merkkijonoja, jolloin esimerkiksi hakusanalla 'Nokia' voidaan tulokseksi saada dokumentteja matkapuhelinvalmistajasta, Nokian kaupungista tai F.E Sillanpään Ihmiset suviyössä teoksen päähenkilöstä. Tässä tutkielmassa esitetään informaation haussa (engl. Information Retrieval, IR) käytettävä menetelmä, jolla on mahdollista hakea tekstidokumentteja tarkasti määritellyllä käsitteellä. Tarkasti määritellyllä käsitteellä tarkoitetaan ontologiassa, koneymmärrettävässä sanastossa, määriteltyä käsitettä. Tässä tutkielmassa keskitytään erityisesti historiaontologiassa määriteltyihin tapahtumiin. Tutkielmassa esitetty menetelmä pyrkii tunnistamaan dokumentissa esiintyvät käsitteet sanoja ympäröivän semantiikan perusteella. Täsmällisesti sanaa ympäröivä semantiikka saadaan niin kutsutusta semanttisesta avaruudesta, joka muodostetaan piilevän semantiikan analyysiksi (engl. Latent Semantic Analysis, LSA) kutsutulla matemaattisella menetelmällä, ja ympäröivää semantiikkaa sovelletaan ontologiseen kyselyn laajentamiseen. Mallin toimivuutta pyrittiin arvioimaan koejärjestelyllä, jossa aineistona käytetään Suomalaista historiaontologiaa ja suomenkielisen Wikipedia-tietosanakirjan artikkeleita. Koejärjestelyssä ilmenneiden vaikeuksien vuoksi toimivuuden arviointi jäi puutteelliseksi. Tutkielman lopussa on pohdittu menetelmän merkitystä informaation haussa yleisesti, sillä tutkielmassa kuvattu menetelmä ontologiassa määriteltyjen käsitteiden kuvaamisesta tekstidokumenttien määräämään semanttiseen avaruuteen on uusi, eikä aiempaa tutkimusta menetelmän toiminnasta tai kehittämisestä ole tehty.