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Abelian categories provide an abstract generalization of the category of modules over a unitary ring. An embedding theorem by Mitchell shows that one can, whenever an abelian category is sufficiently small, find a unitary ring such that the given category may be embedded in the category of left modules over this ring. An interesting consequence of this theorem is that one can use it to generalize all diagrammatic lemmas (where the conditions and claims can be formulated by exactness and commutativity) true for all module categories to all abelian categories.\\ The goal of this paper is to prove the embedding theorem, and then derive some of its corollaries. We start from the very basics by defining categories and their properties, and then we start constructing the theory of abelian categories. After that, we prove several results concerning functors, "homomorphisms" of categories, such as the Yoneda lemma. Finally, we introduce the concept of a Grothendieck category, the properties of which will be used to prove the main theorem. The final chapter contains the tools in generalizing diagrammatic results, a weaker but more general version of the embedding theorem, and a way to assign topological spaces to abelian categories. The reader is assumed to know nothing more than what abelian groups and unitary rings are, except for the final theorem in the proof of which basic homotopy theory is applied.

The aim of this masters thesis was to examine subjective wellbeing and personal happiness. Empirical study of happiness is part of broader wellbeing research and is based on an idea that the best experts of personal wellbeing are the individuals themselves. In addition to perceptions of personal happiness, the aim was also to acquire knowledge about personal values and components personal happiness is based on. In this study, moving into certain community and the characteristics of neigbourhood contributing happiness, were defined to represent these values. The object was, through comparative case-study, to obtain knowledge about subjective wellbeing of the individuals in two different residential areas inside metropolitan area of Helsinki. In comparative case study the intention usually is that the examined units represent spesific 'cases' from something broader and therefore the results can be somehow generalized. Consequently the chosen cases in this study were selected due to their image of 'urban village' and thus the juxtapositioning was constructed between secluded post-suburban village and more heterogeneous urban village better attached to existing urban structure. The research questions were formed as follows: Are there any differences between the areas regarding the components personal happiness is based on? Are there any differences between the areas regarding the level of residents subjective wellbeing? Based on the residents assessments, what are the most important characteristics of neighbourhood contributing personal happiness? The data used in order to gain answers to these questions was obtained from internet-based survey questionnaire. Based on the data residents of post-suburban village Sundsberg seem to share highly family oriented set of values and actualizing these values is ensured with high income, wealth and secure work situation. Instead in Kumpula the components of happiness seem place more towards learning and personal development, interesting leisure and hobbies and specially having an influence regarding communal decisions. Concerning subjective wellbeing of residents there can be seen some differences as well. Personal life is experienced a bit more happier in Sundsberg than in Kumpula. People are more satisfied with their personal health and job satisfaction in Sundsberg and additionally feelings of loneliness, inadequancy and frustration are bit more common in Kumpula. Regarding the characteristics of neighbourhood contributing happiness data suggests that key characteristics of area are peacefulness and safely, good location and connections and proximity of parks and recreational areas. These characteristics were considered highly significant in both areas but they were experienced to actualize better in Kumpula. In addition to these components the residents in Kumpula were overall more satisfied with various characteristics contributing happiness in their residential area. Besides these attributes mentioned above residents in Kumpula emphasize also some 'softer' elements connecting into social, functional and communal side of area. From Sundsberg point of view residential area best contributing happiness is child friendly and safe community based on likeminded people who share the same socio-economical situation. The results of this study can be linked back into the society and metropolitan area, which they were chosen from as a cases to be studied. The results can thereby be seen as an example of differentation of conditions of personal happiness between certain population segments. It is possible to detect an spatial dimension to this process as well and thereby the results suggests that regional segmentation affects between high-ranking residential areas as well. Thereby the results of this research contributes to the debate on innovative, diverse and dynamic urban area and as well cohesion of metropolitan area and the society in whole.

In this Thesis I present the general theory of semigroups of linear operators. From the philosophical point of view I begin by connecting deterministic evolution in time to dynamic laws that are stated in terms of a differential equation. This leads us to associate semigroups with the models for autonomic deterministic motion. From the historical point of view I reflect upon the history of the exponential function and its generalizations. I emphasize their role as solutions to certain linear differential equations that characterize both exponential functions and semigroups. This connection then invites us to consider semigroups as generalizations of the exponential function. I believe this angle of approach provides us with motivation as well as useful ideas. From the mathematical point of view I construct the basic elements of the theory. First I consider briefly uniformly and strongly continuous semigroups. After that I move on to the more general σ(X, F)-continuous case. Here F is a so called norming subspace of the dual X^*. I prove the existence of both the infinitesimal generator S of the semigroup and the resolvent (λ - S)^(-1) as well as some of their basic properties. Then I turn to the other direction and show how to create a semigroup starting from its generator. That is the content of the famous Hille—Yosida Theorem. From the practical point of view I give some useful characterizations of the generator in terms of dissipativity and accretivity. These techniques also lead us to an effortless proof of Stone's Theorem on unitary groups. Finally, from an illustrational point of view I give two applications. The first is about multiplicative semigroups on L^p spaces, where the setting is simple enough to allow intuition to accompany us. The second takes on a problem of generating a particular stochastic weak*-continuous semigroup. It serves to illustrate some of our results.

A model in mathematic logic is called pseudo-finite, in case it satisfies only such sentences of first-order predicate logic that have a finite model. Its main part modelled based on Jouko Väänänen's article 'Pseudo- finite model theory', this text studies classic model theory restricted to pseudo-finite models. We provide a range of classic results expressed in pseudo-finite terms, while showing that a set of other well-known theorems fail when restricted to the pseudo-finite, unless modified substantially. The main finding remains that a major portion of the classic theory, including Compactness Theorem, Craig Interpolation Theorem and Lidström Theorem, holds in an analogical form in the pseudo-finite theory. The thesis begins by introducing the basic first-order model theory with the restriction to relational formulas. This purely technically motivated limitation doesn't exclude any substantial results or methods of the first-order theory, but it simplifies many of the proofs. The introduction behind, the text moves on to present all the classic results that will later on be studied in terms of the pseudo-finite. To enable and ease this, we also provide some powerful tools, such as Ehrenfeucht-Fraïssé games. In the main part of the thesis we define pseudo-finiteness accurately and build a pseudo-finite model theory. We begin from easily adaptable results such as Compactness and Löwenheim-Skolem Theorems and move on to trickier ones, examplified by Craig Interpolation and Beth Definability. The section culminates to a Lidström Theorem, which is easy to formulate but hard to prove in pseudo-finite terms. The final chapter has two independent sections. The first one studies the requirements of a sentence for having a finite model, illustrates a construction of a finite model for a sentence that has one, and culminates into an exact finite model existence theorem. In the second one we define a class of models with a certain, island-like structure. We prove that the elements of this class are always pseudo-finite, and at the very end the text, we present a few examples of this class.

After 2013, the environmental protection department in China has significantly reduced on-road emission through the upgrade of emission standards, the improvement of fuel quality and economic tools. However, the specific effect of the control policies on emission and air quality is still difficult to quantify. This is mainly due to the data shortage on vehicle emission factors and vehicle activities. In this research, we developed the 2008-2018 on-road emissions inventory based on Emission Inventory Preparation Guide (GEI) and existing vehicle activity database. Our estimates suggest that CO and PM2.5 showed a relatively significant decrease, by 66.2% and 58.8%. However, the trend of NOx (5.8%) and NMVOC (-4.8%) was relatively stable. The Beijing-Tianjin-Hebei (BTH), Yangtze River Delta (YRD), Pearl River Delta (PRD) and Sichuan Basin (SCB) regions all showed a uniform trend especially in NOx. For Beijing-Tianjin-Hebei, the significant decline in NOx might be caused by earlier implementations in emission standard and fuel quality. In addition to this, we designed additional evaporation emission scenarios to verify the application of GEI in quantify emission impact on secondary pollutant (PM2.5 and O3). The results indicate that evaporation emission contributed to Maximum Daily Average 8-hour (MDA8) O3 concentration by about 3.5%, for Beijing, Shanghai and Nanjing. This value can reach up to 5.9%, 5.3% and 7.3%, but the impact on PM2.5is extremely limited. Our results indicate the feasibility of GEI in improving and lowering the technical barrier of on-road emission inventory establishment at the same time and its further application in quantifying on-road emission contribution to air quality. Besides that, it shows a strong potential in on-road policy environmental assessment and short-term air quality assessment.

At the end of the inflationary epoch, about 10^(−12) seconds after the Big Bang singularity, the universe was filled with plasma consisting of quarks and gluons. At some stage the cooling of the universe could have led to the occurrence of first-order cosmological phase transitions that proceed by nucleation and expansion of bubbles all over the primordial plasma. Cosmological turbulence is generated as a consequence of bubble collisions and acts as a source of primordial gravitational waves. The purpose of this thesis is to provide an overview of cosmological turbulence as well as the corresponding gravitational wave production, and compile some of the results obtained to this day. We also touch on the onset of cosmological turbulence by analysing shock formation. In the one-dimensional case considering only right-moving waves, the result is Burgers’ equation. The development of a power spectrum with random initial conditions under Burgers’ equation is calculated numerically using the Euler method with sufficiently low step sizes. Both in the viscid and inviscid cases, the result is the presence of a −8/3 power law in the inertial range at the time of shock formation.

This thesis will present the concept of arbitrage and some applications of arbitrage pricing. Arbitrage opportunity means that there is a possibility to make money without any initial investment and without a risk of losing money. To start, some definitions are introduced in the fields of measure theory, probability theory and mathematical finance. Then the guidelines of market models considered throughout the thesis will be defined. The mathematical definition of arbitrage and arbitrage pricing are introduced first in simple setting of one period market model and then in multi-period market model. As a main result of this thesis are introduced and proven The fundamental theorems of arbitrage pricing. The first fundamental theorem of arbitrage pricing shows that a market is arbitrage free if and only if there exists at least one risk neutral probability measure equivalent to original probability measure such that the discounted prices are martingales with respect to this risk neutral measure. This will be proven for multi-period market model. The second fundamental theorem of arbitrage pricing shows that the completeness of a market model is equivalent to existence of unique risk-neutral probability measure. This will be proven for one period market model. Finally, I look into some investing and hedging strategies replicating payoffs and portfolio insurance. Some examples of commonly used options strategies will be introduced such as butterfly spread and iron condor.

The purpose of this thesis is to act as a guide for the 2017 article A study guide for the l^2 decoupling theorem by J. Bourgain and C. Demeter. However, this thesis is self-sufficient. The aim has been to give a detailed presentation and handle the weight exponent E especially carefully in the arguments. We begin by presenting the decoupling inequality of the l^2 decoupling theorem and the associated Fourier transform -like operator. The theorem concerns finding a satisfactory upper bound for the decoupling constant related to the inequality. We also list some general results that a graduate student might not be very familiar with; among them are a few consequences of Hölder's inequality. We move on to study the properties of the weight functions that we use in the L^p-norms in the decoupling. We present two operator lemmas to which we can reduce many of our arguments. The other lemma gives us the opportunity to use certain Schwartz functions in our proofs. We then move on to prove the l^2 decoupling theorem in the lower range 2<= p <= (2n)/(n-1). This includes the definition of multilinear decoupling constants and an iterative process.

Freshwater ecosystems are an important part of the carbon cycle. Boreal lakes are mostly supersaturated with CO2 and act as sources for atmospheric CO2. Dissolved CO2 exhibits considerable temporal variation in boreal lakes. Estimates for CO2 emissions from lakes are often based on surface water pCO2 and modelled gas transfer velocities (k). The aim of this study was to evaluate the use of a water column stratification parameter as proxy for surface water pCO2 in lake Kuivajärvi. Brunt-Väisälä frequency (N) was chosen as the measure of water column stratification due to simple calculation process and encouraging earlier results. The relationship between N and pCO2 was evaluated during 8 consecutive May–October periods between 2013 and 2020. Optimal depth interval for N calculation was obtained by analysing temperature data from 16 different measurement depths. The relationship between N and surface pCO2 was studied by regression analysis and effects of other environmental conditions were also considered. Best results for the full study period were obtained via linear fit and N calculation depth interval spanning from 0.5 m to 12 m. However, considering only June–October periods resulted in improved correlation and the relationship between the variables more closely resembling exponential decay. There was also strong inter-annual variation in the relationship. The proxy often underestimated pCO2 values during the spring peak, but provided better estimates in summer and autumn. Boundary layer method (BLM) was used with the proxy to estimate CO2 flux, and the result was compared to fluxes from both BLM with measured pCO2 and eddy covariance (EC) technique. Both BLM fluxes compared poorly with the EC flux, which was attributed to the parametrisation of k.