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Sobolev functions generalize the concept of differentiability for functions beyond classical settings. The spaces of Sobolev functions are fundamental in mathematics and physics, particularly in the study of partial differential equations and functional analysis. This thesis provides an overview of construction of an extension operator on the space of Sobolev functions on a locally uniform domain. The primary reference is Luke Rogers' work "A Degree-Independent Sobolev Extension Operator". Locally uniform domains satisfy certain geometric properties, for example there are not too thin cusps. However locally uniform domains can possess highly non-rectifiable boundaries. For instance, the interior of the Koch snowflake represents a locally uniform domain with a non-rectifiable boundary. First we will divide the interior points of the complement of our locally uniform domain into dyadic cubes and use a collection of the cubes having certain geometric properties. The collection is called Whitney decomposition of the locally uniform domain. To extend a Sobolev function to a small cube in the Whitney decomposition one approach is to use polynomial approximations to the function on an nearby piece of the domain. We will use a polynomial reproducing kernel in order to obtain a degree independent extension operator. This involves defining the polynomial reproducing kernel in sets of the domain that we call here twisting cones. These sets are not exactly cones, but have some similarity to cones. Although a significant part of Rogers' work deals extensively with proving the existence of the kernel with the desired properties, our focus will remain in the construction of the extension operator so we will discuss the polynomial reproducing kernel only briefly. The extension operator for small Whitney cubes will be defined as convolution of the function with the kernel. For large Whitney cubes it is enough to set the extension to be 0. Finally the extension operator will be the smooth sum of the operators defined for each cube. Ultimately, since the domain is locally uniform the boundary is of measure zero and no special definition for the extension is required there. However it is necessary to verify that the extension "matches" the function correctly at the boundary, essentially that their k-1-th derivatives are Lipschitz there. This concludes the construction of a degree independent extension operator for Sobolev functions on a locally uniform domain.

As part of the SAFER2028, ABCRad (Alternative Buffer/Backfill Characterization and Radionuclide Interactions) research project, two alternative bentonite materials supplied by Posiva Oy were investigated in this thesis. The aim of the thesis was to investigate and determine the sorption behavior of these two buffer material candidates, with a deliberate reference to a well- known Na-Wyoming type bentonite serving as a benchmark. In order to closely imitate conditions relevant to repository settings, a synthetic reference water was prepared, and the experiments were conducted within a glove box in N2 atmosphere excluding CO2 and O2. This thesis provides valuable perspectives on the behavior and attributes of the alternative bentonite materials, which is crucial for guiding decisions in the design of repositories for radioactive waste and strategies for managing spent nuclear fuel. More specifically, this thesis provides thermodynamic sorption models (TMS) for two risk-driving radionuclides, uranium (U) and cesium (Cs). Batch sorption isotherms were made using a 1:20 solid-to-liquid ratio including 0.5 g of bentonite in 10 cm3 of reference water. Gamma spectroscopy and Liquid Scintillation Counting (LSC) were employed for the analysis of reaction supernatants. Complementary to these techniques, additional bentonite properties, including Cation Exchange Capacity (CEC) and Exchangeable Cations (EC), were determined. Pre-characterization was done for the bentonites using Fourier Transform Infrared Spectroscopy (FTIR) and the Specific Surface Areas (SSA) were determined. These analyses provide a comprehensive characterization of the alternative backfill materials under investigation. This thesis focuses on combining quantitative sorption data (e.g., distribution coefficient, Kd) with mechanistic understanding (e.g., FTIR spectroscopy). This contributes to an improved understanding of radionuclide sorption mechanisms, thereby bolstering safety considerations. The CEC determined for the bentonites, Laviosa, LMS, and Na-Wyoming were 87 (±0,048) meq/100 g, 95 (±0,34) meq/100 g, and 91 (± 1,23) meq/100 g, respectively. The distribution coefficient (Kd) values of uranium ranged from 130–135 cm3/g with Laviosa and 78–110 cm3/g with LMS, while those of cesium ranged from 130–280 cm3/g with Laviosa and from 150–425 cm3/g with LMS. Cesium demonstrated sorption of 95% within the 10-10 to 10-6 M range, decreasing slightly to 85–95% at concentrations up to 10-2 M. Uranium showed sorption in the range 80–100% across both clays, peaking at lower concentrations and declining at higher concentrations. These data align with those of the reference buffer material, indicating that these bentonites could potentially serve as feasible alternatives if they exhibit additional favorable sorption capacity with other risk-driving radionuclides (e.g., Eu, Ni, Th).

Ilmatieteen laitoksen ylläpitämä Suomen virallisten sademittarien havaintoverkko on harva ja sen laajentaminen vaatisi enemmän resursseja. Kansalaisten sadehavaintojen hyödyntäminen olisi yksi keino laajentaa havaintoverkkoa ja siten muun muassa parantaa sääennusteiden laatua. Tässä tutkielmassa selvitettiin kansalaisten Netatmo-kesäsadehavaintojen käyttökelpoisuutta Suomessa. Vuosien 2019–2022 kesä-, heinä- ja elokuun Netatmo-sadehavaintojen laatua tutkittiin vertailemalla niitä Ilmatieteen laitoksen automaattisadeasemien havaintoihin. Vertailua tehtiin tilastollisten suureiden, keskiarvon, korrelaation ja absoluuttisen keskivirheen, avulla. Ennen varsinaista analyysia pyrittiin rajauksilla selkeyttämään aineistoa sekä poistamaan siitä selvästi virheellisiä Netatmo-sadeasemia ja -havaintoja, kuten yli 150 mm:n tunti- ja yli 200 mm:n vuorokausisademäärät. Pääsääntöisesti Netatmo-sadehavainnot näyttävät tilastollisten suureiden valossa hyviltä, sillä esimerkiksi 75 % Netatmo- ja lähimpien Ilmatieteen laitoksen asemien välisistä vuorokausisateiden korrelaatioista oli vähintään 0.6. Netatmo-havaintojen välinen vaihtelu oli kuitenkin suurempi kuin Ilmatieteen laitoksen asemien havaintojen, mikä kertoo osan Netatmo-havainnoista olevan virheellisiä. Virheitä löytyi useita erilaisia. Yleisesti monien Netatmo-asemien havaittiin aliarvioivan sademäärää, koska keskimäärin Netatmo-asemat olivat mitanneet sateita vajaat 10 % vähemmän kuin niiden vertailuasemat. Lisäksi Netatmo-asemien havainnoissa oli huomattavasti enemmän pieniä 0.1 mm mittauksia kuin Ilmatieteen laitoksen vertailuasemilla eikä osa asemista ollut mitannut mitään 0.1 mm virhemittauksia lukuunottamatta. Jotkut Netatmo-asemat puolestaan mittasivat yksittäin tai jopa jatkuvasti virheellisiä hyvin suuria sademääriä. Osa asemista myös yliarvioi sademäärää, sillä asemien sateet korreloivat hyvin vertailuasemien sateiden kanssa ollen vain paljon suurempia. Toisaalta joidenkin Netatmo-asemien realistiset sadehavainnot oli mitattu eri aikoihin kuin vertailuasemien sateet, joten asemien koordinaatit voivat olla väärät. Välillä taas Netatmo-havaintojen laatu muuttui ajan myötä, sillä kyse on kansalaisten havainnoista. Asemat saattoivat ensin tuottaa hyviä havaintoja ja sitten huonoja tai päinvastoin. Kaikkiaan Netatmo-kesäsadehavainnot vaikuttavat käyttökelpoisilta, koska suurin osa havainnoista on hyviä. Netatmo-asemat myös saavat suuria sateita kiinni hyvin. Lisäksi huonoja havaintoja korrelaation perusteella tuottavat Netatmo-asemat ovat hajallaan eri puolilla Suomea ja hyviä asemia on kaikkialla enemmän. Koska virheellisiä Netatmo-asemia ja -havaintoja on silti varsin paljon, Netatmo-sadehavainnot tarvitsevat kattavaa laadun varmistusta ennen havaintojen hyödyntämistä. Laadun varmistusta voisi tehdä tämän tutkimuksen tavoin vertailemalla havaintoja tilastollisesti Ilmatieteen laitoksen asemien havaintoihin. Lisäksi Netatmo-havaintoja voisi verrata keskenään.

Ylioppilaskirjoitukset ovat suomalaisen koulutuksen kentällä merkittävä ja laajasti vaikuttava instituutio. 2018 tapahtuneen sähköistymisen myötä ylioppilaskirjoituksissa tapahtui paljon muutoksia, joiden vaikutuksia ei ole vielä tutkittu kovin laajalti. Tässä tutkielmassa jatketaan aiemman opettaja työnsä tutkijana -tutkielman aihetta, ja haetaan ymmärrystä sähköistymisen vaikutuksista fysiikan ylioppilaskirjoituksiin. Fysiikan ylioppilaskirjoituksia tarkastellaan tässä tutkielmassa \emph{uudistetun Bloomin taksonomian} kautta. Taksonomia esitellään ja sen aiempaa käyttöä kasvatustieteessä, sekä erityisesti päättöarvioinnin tutkimuksessa tarkastellaan. Kun taksonomia ymmärretään, siirrytään hyödyntämään sitä fysiikan ylioppilaskirjoitusten tarkastelussa. Uudistettua Bloomin taksonomiaa hyödyntäen analysoidaan yhteensä kaksitoista fysiikan ylioppilaskoetta vuosilta 2015 - 2018 ja 2021 - 2023. Kuusi kokeista edustavat paperikokeiden ajan loppua, ja kuusi uusimpia sähköisiä ylioppilaskokeita. Kokeiden tehtävät analysoidaan ja luokitellaan uudistetun Bloomin taksonomian mukaiseen taksonomiatauluun erityisesti huomioiden hyvän vastauksen piirteiden perusteella tunnettua kunkin tehtävän pistejakaumaa. Näin saadaan tietoa erikseen paperisten ja sähköisten fysiikan ylioppilaskokeiden tehtävien mittaamasta kognitiivisesta osaamisesta, sekä niiden vaatimista tiedon tyypeistä. Analyysin perusteella havaitaan, että fysiikan ylioppilaskokeet mittaavat laajasti erilaisia kognitiivisen osaamisen ja tiedon tasoja, kuitenkin selvästi painottuen ymmärtämistä ja soveltamista mittaaviin tehtäviin. Lisäksi havaitaan, että erityisesti painoarvoa on käsitteellisellä tiedolla ja menetelmätiedolla. Vertaamalla paperikokeiden ja sähköisten kokeiden tuloksia havaitaan, että sähköistymisen yhteydessä ymmärtämistä vaativa osuus tehtäväpisteistä on pienentynyt 11,5 %-yksikköä. Myös muita pienempiä muutoksia havaitaan, mutta ne eivät ole tilastollisesti merkitseviä. Lopuksi tutkielmassa arvioidaan muutosten syitä, erityisesti sähköistymisen tarjoaman laajemman työkaluvalikoiman vaikutusta tehtävänlaadintaan. Toisaalta taksonomiasta luonteesta ja tutkielman koejärjestelystä tunnistetaan puutteita, joiden seurauksena tulosten luotettavuus nousee kyseenalaiseksi.

Statistician C. R. Rao made many contributions to multivariate analysis over the span of his career. Some of his earliest contributions continue to be used and built upon almost eighty years later, while his more recent contributions spur new avenues of research. This thesis discusses these contributions, how they helped shape multivariate analysis as we see it today, and what we may learn from reviewing his works. Topics include his extension of linear discriminant analysis, Rao’s perimeter test, Rao’s U statistic, his asymptotic expansion of Wilks’ Λ statistic, canonical factor analysis, functional principal component analysis, redundancy analysis, canonical coordinates, and correspondence analysis. The examination of his works shows that interdisciplinary collaboration and the utilization of real datasets were crucial in almost all of Rao’s impactful contributions.

Buildings consume approximately 40% of global energy, hence, understanding and analyzing energy consumption patterns of buildings is essential in bringing desirable insights to building management stakeholders for better decision-making and energy efficiency. Based on a specific use case of a Finnish building management company, this thesis presents the challenge of optimizing energy consumption forecasting and building management by addressing the shortcomings of current individual building-level forecasting approaches and the dynamic nature of building energy use. The research investigates the plausibility of a system of building clusters by studying the representative cluster profiles and dynamic cluster changes. We focus on a dataset comprising hourly energy consumption time series from a variety of Finnish university buildings, employing these as subjects to implement a novel stream clustering approach called ClipStream. ClipStream is an attibute-based stream clustering algorithm to perform continuous online clustering of time series data batches that involves iterative data abstraction, clustering, and change detection phases. This thesis shows that it was plausible to build clusters of buildings based on energy consumption time series. 23 buildings were successfully clustered into 3-5 clusters during each two-week window of the period of investigation. The study’s findings revealed distinct and evolving energy consumption clusters of buildings and characterized 7 predominant cluster profiles, which reflected significant seasonal variations and operational changes over time. Qualitative analyses of the clusters primarily confirmed the noticeable shifts in energy consumption patterns from 2019 to 2022, underscoring the potential of our approach to enhance forecasting efficiency and management effectiveness. These findings could be further extended to establish energy policy, building management practices, and broader sustainability efforts. This suggests that improved energy efficiency can be achieved through the application of machine learning techniques such as cluster analysis.

The question of reunification, or ‘the Taiwan issue’, stands as one of the paramount geopolitical conundrums of the 21st century. China asserts that Taiwan is an inalienable part its historical geo-body and socio-cultural chronicles under the unifying idea of ‘Chineseness’. Nevertheless, since Taiwan’s democratization process began to thrive in the 1990s, perceptions of national identity have diverged drastically from those in mainland China. Corollary, the appeal for reunification in Taiwan is almost non-existent, and hence achieving peaceful unification under the ‘one country, two systems’ proposal seems highly unlikely. Furthermore, the United States assumes a pivotal role in cross-strait geopolitics, intricately tangling the question of Taiwan into the broader scheme of great power politics. This thesis examines the intricate dynamics of the Taiwan issue by analyzing the practical geopolitical reasoning of the PRC intellectuals of statecraft over Taiwan. The theoretical and methodological foundations of this study draw from critical geopolitics and critical discourse analysis. The primary empirical research materials comprise the three Taiwan white papers published by the PRC. In addition, the analysis is supplemented by other official documents as well as vast array of research literature published on cross-strait geopolitics. Building upon Ó Tuathail’s theorization of practical geopolitical reasoning, the paper presents the ‘grammar of geopolitics’ of the Taiwan issue from the perspective of the PRC. Within this analytical framework, three guiding geopolitical storylines were identified: 1) Historical Sovereignty, 2) National Unity under ‘Chineseness’, and 3) Separatism and External Powers as Antagonist Forces. The results reveals that the CCP has constructed the imperative of reunification as an historically and geographically bound inevitability. Nevertheless, China's increasing geopolitical anxiety over achieving the objective of reunification with Taiwan is evidential in its discourses. This increasing geopolitical anxiety is likely to compel the CCP to adopt more coercive actions in the near and mid-term future if it deems it necessary. Given the developments in Taiwan, Sino-U.S. relations and domestically in China, it seems probable that pressure on Taiwan will continue to mount throughout the 2020s. Much of the strategic calculations and geopolitical discourses constructed regarding the Taiwan issue can be attributed to the CCP's concerns about its own legitimacy to rule. Within its geopolitical discourses, the issue of reunification is rendered to an existential question for China and arguably it constitutes a significant part of the modern CCP’s raison d'être. China’s increasing self-confidence as a superpower is continually trembling the dynamics of international affairs and the geopolitical landscape, particularly within the Indo-Pacific region. Consequently, the project of Chinese geopolitics remains an unfinished business, and warrants further contributions from researchers in the field of critical geopolitics.

Despite the continuous efforts to unveil the true nature of Dark Matter (DM), it still remains as a mystery. In this thesis we propose one model that can produce the correct relic abundance of DM in the current Universe, while fitting into the existing experimentally obtained constraints. In this model we add a singlet fermion, which is a not completely sterile right-handed neutrino, and a heavy real scalar singlet into the Standard Model of Particle Physics (SM) and carry out the relic density calculations. The DM candidate here is the singlet fermion, which acts as a thermally decoupled Weakly Interacting Massive Particle. Theoretical framework is laid out in detail. Special attention is given to obtaining the definition for relic abundance from Lee-Weinberg equation in terms of yield, and the decoupling temperature. It is found, that the usual way of handling the thermally averaged cross section appearing in these definitions is not suitable in this case. In fact, the usual approximations can only be done when thermally averaged cross section is almost linear in $s$, and this is a demand that very few models can satisfy. The correct way to treat the cross section by taking the expansion in terms of the relative velocity is presented with careful attention to detail. This methodology is then applied to the extension of the SM we introduced. Only tree-level processes are being considered. Cross sections are calculated for each possible process to obtain the total cross section needed for the DM relic density calculations. We present how the different free parameters in the theory affect the relic abundance and what masses are allowed for the right-handed neutrino to obtain. It is found out that the parameters in this model are heavily constrained. Yet the model is able to fit into the constraints obtained from branching ratio and direct detection (DD) experiments, while producing the correct relic density. This is true when the mixing angle $\theta$ is of the order $1 \times 10^{-4}$, and right-handed neutrino has the mass of exactly half of the mass of the heavy scalar or higher than the mass of the heavy scalar. It is proposed that allowing lepton mixing and adding a separate mass term for fermion in the model could make the model less restricted. Investigating this would be interesting thing to do in the future. However, the proposed DM candidate remains viable and the upcoming DD experiments will relatively soon reveal if the singlet fermion is the DM particle we are seeking.

Kaupunkien kasvu, energian kulutuksen lisääntyminen sekä uusiutuvan energian murros ovat aikaansaaneet tuulivoimahankkeiden nopean lisääntymisen Suomessa. Tämän seurauksena tuulivoimahankkeet sijoittuvat entistä syrjäisemmille alueille, jolloin ne ovat merkittävä osa planetaarista urbanisaatiota sekä sen yhtä vaihetta, laajentunutta urbanisaatiota. Planetaarinen urbanisaatio tarkoittaa urbaanin verkoston levittäytymistä kaikkialle yli hallinnollisten tai maantieteellisten rajojen. Laajentunut urbanisaatio taas ilmenee kaupunkikeskusten ulkopuolisille alueille kohdistuvina suurina infrastruktuuri-, teollisuus- ja kehityshankkeina, jotka tukevat keskusten toimintoja ja kasvua. Näin ollen laajentunut urbanisaatio on osa planetaarisen urbanisaation kaikkialle ulottuvaa verkostoa: energia virtaa kaupunkikeskuksista poispäin investointien ja työvoiman muodossa ja jälleen takaisin keskuksiin muun muassa sähkönä ja taloudellisena tuottona. Tutkielmassa perehdytään laajentuneen urbanisaation ilmenemiseen Vaara-Kainuussa tuulivoimaa vastustavien kommenttien kautta. Tutkielman tavoitteena on tunnistaa tuulivoimaa vastustavista kommenteista yhteys laajentuneen urbanisaation teoriaan sisällönanalyysin teemoittelun avulla. Lisäksi tavoitteena on ymmärtää millä tavoin laajentunut urbanisaatio ilmenee tuulivoiman yhteydessä tutkimusalueella. Analyysissä tunnistetaan Vaara-Kainuuseen suunnitellun tuulivoiman rooli laajentuneen urbanisaation prosessissa. Lisäksi tutkielman keskeisenä tuloksena havaitaan laajentuneen urbanisaation ajallinen monikerroksellisuus sekä urbanisaation jatkuva muutos alueella. Tuulivoiman saapuminen Vaara-Kainuuseen muuttaa ja vahvistaa alueella jo ennestään muun muassa luontomatkailun muodossa lymyillyttä laajentuneen urbanisaation verkostoa. Näin ollen tuulivoima, yhtenä laajentuneen urbanisaation kerroksena, vahvistaa energian virtaamista planetaarisen urbanisaation verkostossa ja yhdistää alueen entistä tiiviimmin tukemaan kaupunkikeskusten kasvua.

In this thesis, we aim to introduce the reader to profinite groups. Profinite groups are defined by two characteristics: firstly, they have a topology defined on them (notably, they are compact). Secondly, they are constructed from some collection of finite groups, each equipped with a discrete topology and forming what is known as an inverse system. The profinite group emerges as an inverse limit of its constituent groups. This definition is, at this point, necessarily quite abstract. Thus, before we can really understand profinite groups we must examine two areas: first, we will study topological groups. This will give us the means to deal with groups as topological spaces. Topological groups have some characteristics that differentiate them from general topological spaces: in particular, a topological group is always a homogeneous space. Secondly, we will explore inverse systems and inverse limits, which will take us into category theory. While we could explain these concepts without categories, this thesis takes the view that category theory gives us a useful “50000-feet view” by giving these ideas a wider mathematical context. In the second chapter, we will go through preliminary information concerning group theory, general topology and category theory that will be needed later. We will begin with some basic concepts from group theory and point-set topology. These sections will mostly contain information that is familiar from the introductory university courses. The chapter will then continue by introducing some basic concepts of category theory, including inverse systems and inverse limits. For these, we will give an application by showing how the Cantor set is homeomorphic to an inverse limit of a collection of finite sets. In the third chapter, we will examine topological groups and prove some of their properties. In the fourth chapter, we will introduce an example of profinite groups: Zp, the additive group of p-adic integers. This will be expanded into a ring and then into the field Qp. We will discuss the uses of Zp and Qp and show how to derive them as an inverse limit of finite, compact groups.