In this thesis, the sales of bread and coffee bread at Coop Värmland AB have been studied. The aim was to find what factors that are important for the sales and then make predictions of how the sales will look like in the future to reduce waste and increase profits. Big data analysis and data exploration was used to get to know the data and find the factors that affect the sales the most. Time series forecasting and supervised machine learning models were used to predict future sales. The main focus was five different models that were compared and analysed, they were; Decision tree regression, Random forest regression, Artificial neural networks, Recurrent neural networks and a time series model called Prophet. Comparing the observed values to the predictions made by the models indicated that using a model based on the time series is to be preferred, that is, Prophet and Recurrent neural network. These two models gave the lowest errors and by that, the most accurate results. Prophet yielded mean absolute percentage errors of 8.295% for bread and 9.156% for coffee bread. The Recurrent neural network gave mean absolute percentage errors of 7.938% for bread and 13.12% for coffee bread. That is about twice as good as the models they are using today at Coop which are based on the mean value of the previous sales. / I denna avhandling har försäljningen av matbröd och fikabröd på Coop Värmland AB studerats. Målet var att hitta vilka faktorer som är viktiga för försäljningen och sedan förutsäga hur försäljningen kommer att se ut i framtiden för att minska svinn och öka vin- ster. Big data- analys och explorativ dataanalys har använts för att lära känna datat och hitta de faktorer som påverkar försäljningen mest. Tidsserieprediktion och olika mask- ininlärningsmodeller användes för att förutspå den framtida försäljningen. Huvudfokus var fem olika modeller som jämfördes och analyserades. De var Decision tree regression, Random forest regression, Artificial neural networks, Recurrent neural networks och en tidsseriemodell som kallas Prophet. Jämförelse mellan de observerade värdena och de värden som predicerats med modellerna indikerade att de modeller som är baserade på tidsserierna är att föredra, det vill säga Prophet och Recurrent neural networks. Dessa två modeller gav de lägsta felen och därmed de mest exakta resultaten. Prophet gav genomsnittliga absoluta procentuella fel på 8.295% för matbröd och 9.156% för fikabröd. Recurrent neural network gav genomsnittliga absoluta procentuella fel på 7.938% för matbröd och 13.12% för fikabröd. Det är ungefär dubbelt så korrekt som de modeller de använder idag på Coop som baseras på medelvärdet av tidigare försäljning.

This essay proves the Universal Approximation Theorem for discriminatory activation functions, in particular continuous sigmoidal functions, over compact spaces. In other words, a neural network with a discriminatory activation function can approximate any continuous function over a compact space. The theorem guarantees the effectivity of neural networks. / Denna uppsats bevisar Universala Approximations Satsen för diskriminerande funktioner, särskillt kontinuerliga sigmoidala funktioner, över kompakta rum. Med andra ord, ett neuralt nätverk med en discriminerande aktiveringsfunktion kan approximera alla kontinguerliga functioner över kompakta rum. Satsen garanterar effektivitet av neurala nätverk.

We study the application of Lie symmetry methods to solve some cases of the Boltzmann equation, a cornerstone of kinetic theory. The study explores hidden invariances and symmetry-based solutions that help to clarify the complexities inherent in the structure of the equation. Moreover, the study demonstrates a novel approach to solving the equation by rewriting it using the Fourier transform in the velocity variable, which resulted in a non-trivial solution to the Boltzmann equation. The findings not only clarify the mathematical underpinnings of the Boltzmann equation but also enhance our understanding of particle interactions in gases. Overall, this thesis not only enriches the theoretical understanding of integro-differential equations through its rigorous approach but also highlights the efficacy of Lie symmetry methods in unraveling the complexities of fundamental equations in physical sciences.

Syftet med den här studien är att undersöka hur utomhusundervisning kan ge eleverna möjligheter att föra och följa resonemang i matematik. Studien har två frågeställningar, den ena är vilka typer av matematiska resonemang ges eleverna möjlighet att föra och följa vid utomhusmatematik? Den andra frågeställningen är hur tänker lärarna när det gäller möjligheter för eleverna att föra och följa matematiska resonemang? För att uppnå syftet och besvara frågeställningarna genomförs två observationer av utomhusundervisning samt intervju med lärarna som bedrev undervisningen.   För att kategorisera vilka typer av resonemang eleverna använder utformas ett observationsschema som tar stöd av Lithners ramverk, som även är studiens teori. Detta ramverk förklarar att det finns två huvudtyper, imitativa och kreativa resonemang, där de dessutom finns olika undergrupper. Resultatet visar att eleverna får möjlighet att föra och följa båda kreativa (CMR) och imitativa resonemang (IR) i utomhusundervisning. På grund av uppgifternas utformning förekommer det mest IR i observationerna. Lärarna är medvetna om vad resonemangsförmågan innebär och denna insikt kan hjälpa eleverna att utveckla sin resonemangsförmåga. Utomhusmiljön blir dessutom ett gott inslag i matematikundervisning. Genom att arbeta ute får eleverna använda fler sinnen och arbeta mer kreativt. Detta kan i sin tur skapa motivation samtidigt som eleverna får gripa för att begripa.

In the past decade, cooperative communications has been emerging as a pertinent technology for the current and upcoming generations of mobile communication infrastructure. The indispensable benefits of this technology have motivated numerous studies from both academia and industry on this area. In particular, cooperative communications has been developed as a means of alleviating the effect of fading and hence improve the reliability of wireless communications. The key idea behind this technique is that communication between the source and destination can be assisted by several intermediate nodes, so-called relay nodes. As a result, cooperative communication networks can enhance the reliability of wireless communications where the transmitted signals are severely impaired because of fading. In addition, through relaying transmission, communication range can be extended and transmit power of each radio terminal can be reduced as well. The objective of this thesis is to analyze the system performance of cooperative relay networks integrating advanced radio transmission techniques and using the two major relaying protocols, i.e., decode-and-forward (DF) and amplify-and-forward (AF). In particular, the radio transmission techniques that are considered in this thesis include multiple-input multiple-output (MIMO) systems and orthogonal space-time block coding (OSTBC) transmission, adaptive transmission, beamforming transmission, coded cooperation, and cognitive radio transmission. The thesis is divided into an introduction section and six parts based on peer-reviewed journal articles and conference papers. The introduction provides the readers with some fundamental background on cooperative communications along with several key concepts of cognitive radio systems. In the first part, performance analysis of cooperative single and multiple relay networks using MIMO and OSTBC transmission is presented wherein the diversity gain, coding gain, outage probability, symbol error rate, and channel capacity are assessed. It is shown that integrating MIMO and OSTBC transmission into cooperative relay networks provides full diversity gain. In the second part, the performance benefits of MIMO relay networks with OSTBC and adaptive transmission strategies are investigated. In the third part, the performance improvement with respect to outage probability of coded cooperation applied to opportunistic DF relay networks over conventional cooperative networks is shown. In the fourth part, the effects of delay of channel state information feedback from the destination to the source and co-channel interference on system performance is analyzed for beamforming AF relay networks. In the fifth part, cooperative diversity is investigated in the context of an underlay cognitive AF relay network with beamforming. In the sixth part, finally, the impact of the interference power constraint on the system performance of multi-hop cognitive AF relay networks is investigated.

År 2045 ska Sverige uppnå territoriella nettonollutsläpp och till år 2030 ska utsläppen från transportsektorn ha minskat med 70 % jämfört med år 2010. Sveriges vägtrafik står för en tredjedel av de totala växthusgasutsläppen. För att uppnå klimatmålen bör de mest lämpade styrmedlen och åtgärderna prioriteras. En systematisk undersökning av de faktorer som påverkat utsläppsutvecklingen kan vägleda beslutsfattare att fördela resurserna där de gör mest nytta. Dekomponeringsanalys är en potentiell metod för detta syfte då flera olika faktorers effekter kan särskiljs och mätas. Fem additiva LMDI-I dekomponeringsanalyser genomfördes på utsläppsutvecklingen av fossilt CO2 inom personbilstrafiken mellan åren 1990–2015. De faktorer som undersöktes var befolkning, bil per capita, bränsleteknologier, motorstorlekar, trafikarbete per bil, emissioner och biobränsle. Data från emissionsmodellen HBEFA, Trafikverket och SCB användes i analyserna. Under hela perioden 1990–2015 minskade CO2-utsläppen och dekomponeringsanalyserna visade att alla de ingående faktorerna påverkat utvecklingen. Sett över hela tidsperioden 1990–2015 hade faktorerna påverkat utvecklingen mest i storleksordningen trafikarbete per bil (35 %), bränsleteknologier (15 %), befolkning (15 %), bil per capita (13 %), emissioner (11 %), biobränsle (7 %) samt motorstorlekar (5 %). Procenten anger andelen som faktorn utgjorde av effekternas absoluta summa. Trafikarbete per bil, emissioner, biobränsle och motorstorlekar minskade utsläppen. Bränsleteknologier, befolkning och bil per capita ökade utsläppen. Resultaten kan användas som en indikation för vilka faktorer som kan påverka den framtida utsläppsutvecklingen mest och för vilka åtgärder bör vidtas. Åtgärderförslag är incitament för att välja mer hållbara transportsätt, öka andelen av bilar med lägre utsläpp i fordonsflottan och använda mer biobränsle. / By year 2045 Sweden shall reach zero territorial net emissions and by year 2030 the emissions from the transport sector shall be reduced by 70% compared to year 2010. In Sweden the road traffic stands for a third of the total greenhouse gas emissions. In order to achieve the climate targets, the most suited policies and actions should be prioritized. A systematic investigation into the factors that affect the change in emissions can guide decision makers to distribute resources where they contribute the most. A decomposition analysis is a potential method for this purpose since the effect of different factors can be separated and measured. Five additive LMDI-I decomposition analyses were made on the change in fossil CO2 emission from passenger cars in Sweden between year 1990–2015. The factors that were investigated were: population, vehicle per capita, fuel technologies, engine sizes, distance travelled per vehicle, emissions and biofuel share. Data from the emissions model HBEFA, the Swedish Transport Administration and Statistics Sweden were used in the analyses. During the period of year 1990–2015 the CO2 emissions were reduced, and the decomposition analyses showed that all ingoing factors affected the change. Throughout the period the factors that contributed the most were in order of size: distance travelled per vehicle (35%), fuel technologies (15%), population (15%), car per capita (13%), emissions (11%), biofuel (7%) and engine size (5%). The percentage is the share of the factor’s effect of the absolute sum of all the different effects. Distance travelled per vehicle, emissions, bio fuels and engine size reduced the emissions. Fuel technologies, population and car per capita increased the emissions. The suggestions of actions are incentive for people to use more sustainable means for transportation, increase the share of cars with lower emissions in the fleet and use more biofuel.

<p>For many questions of complex analysis of several variables classical potential theory does not provide suitable tools and is replaced by pluripotential theory. The latter got many important applications within complex analysis and related fields. Pluripolar sets play a special role in pluripotential theory. These are the exceptional sets this theory. Complete pluripolar sets are especially important. In the thesis we study complete pluripolar sets and pluripolar hulls. We show that in some sense there are many complete pluripolar sets. We show that on each closed subset of the complex plane there is continuous function whose graph is complete pluripolar. On the other hand we study the propagation of pluripolar sets, equivalently we study pluripolar hulls. We relate the pluripolar hull of a graph to fine analytic continuation of the function. Fine analytic continuation of an analytic function over the unit disk is related to the fine topology introduced by Cartan and to the previously known notion of finely analytic functions. We show that fine analytic continuation implies non-triviality of the pluripolar hull. Concerning the inverse direction, we show that the projection of the pluripolar hull is finely open. The difficulty to judge from non-triviality of the pluripolar hull about fine analytic continuation lies in possible multi-sheetedness. If however the pluripolar hull contains the graph of a smooth extension of the function over a fine neighborhood of a boundary point we indeed obtain fine analytic continuation.</p>

<p>This thesis consists of four papers and a summary. The common topic of the included papers are the pricing equations of path-dependent options. Various properties of barrier options and American options are studied, such as convexity of option prices, the size of the continuation region in American option pricing and pricing formulas for turbo warrants. In Paper I we study the effect of model misspecification on barrier option pricing. It turns out that, as in the case of ordinary European and American options, this is closely related to convexity properties of the option prices. We show that barrier option prices are convex under certain conditions on the contract function and on the relation between the risk-free rate of return and the dividend rate. In Paper II a new condition is given to ensure that the early exercise feature in American option pricing has a positive value. We give necessary and sufficient conditions for the American option price to coincide with the corresponding European option price in at least one diffusion model. In Paper III we study parabolic obstacle problems related to American option pricing and in particular the size of the non-coincidence set. The main result is that if the boundary of the set of points where the obstacle is a strict subsolution to the differential equation is C¹-Dini in space and Lipschitz in time, there is a positive distance, which is uniform in space, between the boundary of this set and the boundary of the non-coincidence set. In Paper IV we derive explicit pricing formulas for turbo warrants under the classical Black-Scholes assumptions.</p>

<p>In 1989 Jun Kigami made an analytic construction of a Laplacian on the Sierpiński gasket, a construction that he extended to post critically finite fractals. Since then, this field has evolved into a proper theory of analysis on fractals. The new results obtained in this thesis are all in the setting of Kigami's theory. They are presented in three papers.</p><p>Strichartz recently showed that there are first order linear differential equations, based on the Laplacian, that are not solvable on the Sierpiński gasket. In the first paper we give a characterization on the polynomial p so that the differential equation p(Δ)u=f is solvable on any open subset of the Sierpiński gasket for any f continuous on that subset. For general p we find the open subsets on which p(Δ)u=f is solvable for any continuous f.</p><p>In the second paper we describe the infinitesimal geometric behavior of a large class of smooth functions on the Sierpiński gasket in terms of the limit distribution of their local eccentricity, a generalized direction of gradient. The distribution of eccentricities is codified as an infinite dimensional perturbation problem for a suitable iterated function system, which has the limit distribution as an invariant measure. We extend results for harmonic functions found by Öberg, Strichartz and Yingst to larger classes of functions.</p><p>In the third paper we define and study intrinsic first order derivatives on post critically finite fractals and prove differentiability almost everywhere for certain classes of fractals and functions. We apply our results to extend the geography is destiny principle, and also obtain results on the pointwise behavior of local eccentricities. Our main tool is the Furstenberg-Kesten theory of products of random matrices.</p>

For many questions of complex analysis of several variables classical potential theory does not provide suitable tools and is replaced by pluripotential theory. The latter got many important applications within complex analysis and related fields. Pluripolar sets play a special role in pluripotential theory. These are the exceptional sets this theory. Complete pluripolar sets are especially important. In the thesis we study complete pluripolar sets and pluripolar hulls. We show that in some sense there are many complete pluripolar sets. We show that on each closed subset of the complex plane there is continuous function whose graph is complete pluripolar. On the other hand we study the propagation of pluripolar sets, equivalently we study pluripolar hulls. We relate the pluripolar hull of a graph to fine analytic continuation of the function. Fine analytic continuation of an analytic function over the unit disk is related to the fine topology introduced by Cartan and to the previously known notion of finely analytic functions. We show that fine analytic continuation implies non-triviality of the pluripolar hull. Concerning the inverse direction, we show that the projection of the pluripolar hull is finely open. The difficulty to judge from non-triviality of the pluripolar hull about fine analytic continuation lies in possible multi-sheetedness. If however the pluripolar hull contains the graph of a smooth extension of the function over a fine neighborhood of a boundary point we indeed obtain fine analytic continuation.