Hole deviation is one of the most significant problems in drilling applications. Deviated holes result in inefficient blasting and have severe economic impacts due to increased equipment consumption. Conversely, straighter holes help in increasing production scales and reduce operational costs. The reason for hole deviations could range from anisotropic behavior of the rocks being drilled to the behavior of the drill string under the action of imposed forces. Deviations caused due to rock anisotropy are complex in nature and non-controllable to some extent but deviations due to mechanics of drill strings can be controlled as they lie in the operator’s area of influence. In this thesis, a 2D mathematical model is constructed which predicts the bending behavior of the drill rod and the model is further extended to predict the buckled profile of the entire drill string. Two bit state parameters are defined which help in better characterization of hole deviation and understanding of deflected state of the drill bit. Epiroc’s drilling data is inserted in the model developed and the buckled profile of the drill string is studied. The developed model is used to further understand the effect of various drilling parameters like bit thrust, angle of inclination, hole length, etc. on hole deviation. Through the results it was concluded that the bending stiffness of the drill rods is the most critical parameter influencing hole deviation for Epiroc’s drilling data and drill string setup. Finally, potential improvements and techniques to validate results obtained from the mathematical model are proposed. / Hålavvikelse är ett av de mest avgörande problemen i borrapplikationer. Hålavvikelse resulterar i ineffektiv sprängning och får allvarliga ekonomiska konsekvenser på grund av ökad utrustningskonsumtion. Omvänt hjälper rakare hål att öka produktionsskalorna och minska driftskostnaderna. Anledningen till hålavvikelser variera från anisotropi hos bergformationerna som borras till borrsträngens beteende under påverkan av pålagda krafter. Avvikelser orsakade på grund av berganisotropi är komplexa till sin natur och kan inte kontrolleras till viss del, men avvikelser på grund av mekanik i borrsträngar kan kontrolleras eftersom de ligger i operatörens inflytandeområde. I denna avhandling konstrueras en matematisk 2D-modell som förutsäger borrstångens böjningsbeteende och modellen utökas ytterligare för att förutsäga den böjda profilen för hela borrsträngen. Två olika borrkroneparametrar definieras för att bättre karakterisera hålavvikelsen och förstå borrkronans böjning. Epirocs borrsträngsdata används i den utvecklade modellen och borrsträngens böjda profil studeras. Den utvecklade modellen används för att ytterligare förstå effekten av hålavvikelse för olika borrparametrar som kraften på borrkronan, lutningsvinkel, hållängd etc. Utifrån resultaten drogs slutsatsen att borrsträngens böjstyvhet är den mest kritiska parametern som påverkar hålavvikelsen för Epirocs borrdata och borrsträngsinstallation. Slutligen föreslås potentiella förbättringar och tekniker för att validera resultat som erhållits från den matematiska modellen.

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.

Studien undersöker förskollärares syn på matematikundervisningen samt hur de arbetar för att främja barns matematiska kunskaper i förskolans verksamheter. Studien var en kvalitativ intervju där det intervjuades åtta förskollärare från olika förskolor. Studiens resultat visade många aspekter som är grundläggande för matematikundervisning och matematikinlärning i förskolan, där förskolläraren har en central roll i att främja barns matematiska kunskaper. Resultaten analyserades med hjälp av utvecklingspedagogiskt perspektiv och dess centrala begrepp. Studiens resultat kopplades och jämfördes med olika tidigare forskningar som visade en tydlig koppling mellan resultaten och den tidigare forskningen. Studiens slutsats byggdes på förskollärares resonemang och deras arbetssätt mot matematikundervisning. I slutsatsen framkom det även att matematik är ett ämne som finns överallt i förskolan, och att matematikundervisningen främst sker spontant och är sällan planerad.

This thesis consists of five papers related to the theory of unequal probability sampling from a finite population. Generally, it is assumed that we wish to make modelassisted inference, i.e. the inclusion probability for each unit in the population is prescribed before the sample is selected. The sample is then selected using some random mechanism, the sampling design. Mostly, the thesis is focused on three particular unequal probability sampling designs, the conditional Poisson (CP-) design, the Sampford design, and the Pareto design. They have different advantages and drawbacks: The CP design is a maximum entropy design but it is difficult to determine sampling parameters which yield prescribed inclusion probabilities, the Sampford design yields prescribed inclusion probabilities but may be hard to sample from, and the Pareto design makes sample selection very easy but it is very difficult to determine sampling parameters which yield prescribed inclusion probabilities. These three designs are compared probabilistically, and found to be close to each other under certain conditions. In particular the Sampford and Pareto designs are probabilistically close to each other. Some effort is devoted to analytically adjusting the CP and Pareto designs so that they yield inclusion probabilities close to the prescribed ones. The result of the adjustments are in general very good. Some iterative procedures are suggested to improve the results even further. Further, balanced unequal probability sampling is considered. In this kind of sampling, samples are given a positive probability of selection only if they satisfy some balancing conditions. The balancing conditions are given by information from auxiliary variables. Most of the attention is devoted to a slightly less general but practically important case. Also in this case the inclusion probabilities are prescribed in advance, making the choice of sampling parameters important. A complication which arises in the context of choosing sampling parameters is that certain probability distributions need to be calculated, and exact calculation turns out to be practically impossible, except for very small cases. It is proposed that Markov Chain Monte Carlo (MCMC) methods are used for obtaining approximations to the relevant probability distributions, and also for sample selection. In general, MCMC methods for sample selection does not occur very frequently in the sampling literature today, making it a fairly novel idea.

<p>Brownian motion has met growing interest in mathematics, physics and particularly in finance since it was introduced in the beginning of the twentieth century. Stochastic processes generalizing Brownian motion have influenced many research fields theoretically and practically. Moreover, along with more refined techniques in measure theory and functional analysis more stochastic processes were constructed and studied. Lévy processes, with Brownian motionas a special case, have been of major interest in the recent decades. In addition, Lévy processes include a number of other important processes as special cases like Poisson processes and subordinators. They are also related to stable processes.</p><p>In this thesis we generalize a result by S. Chandrasekhar [2] and Edward Nelson who gave a detailed proof of this result in his book in 1967 [12]. In Nelson’s first result standard Ornstein-Uhlenbeck processes are studied. Physically this describes free particles performing a random and irregular movement in water caused by collisions with the water molecules. In a further step he introduces a nonlinear drift in the position variable, i.e. he studies the case when these particles are exposed to an external field of force in physical terms.</p><p>In this report, we aim to generalize the result of Edward Nelson to the case of α-stable Lévy processes. In other words we replace the driving noise of a standard Ornstein-Uhlenbeck process by an α-stable Lévy noise and introduce a scaling parameter uniformly in front of all vector fields in the cotangent space, even in front of the noise. This corresponds to time being sent to infinity. With Chandrasekhar’s and Nelson’s choice of the diffusion constant the stationary state of the velocity process (which is approached as time tends to infinity) is the Boltzmann distribution of statistical mechanics.The scaling limits we obtain in the absence and presence of a nonlinear drift term by using the scaling property of the characteristic functions and time change, can be extended to other types of processes rather than α-stable Lévy processes.</p><p>In future, we will consider to generalize this one dimensional result to Euclidean space of arbitrary finite dimension. A challenging task is to consider the geodesic flow on the cotangent bundle of a Riemannian manifold with scaled drift and scaled Lévy noise. Geometrically the Ornstein-Uhlenbeck process is defined on the tangent bundle of the real line and the driving Lévy noise is defined on the cotangent space.</p>

<p>Undersökningen beskriver den matematiska kommunikationen mellan speciallärare och matematiksvaga elever. Den empiriska datan består av två matematiklektioner från år 9 i grundskolan. Datan analyserades för att finna olika typer av kommunikation.</p><p>Lärarna var i många delar medvetna i sitt sätt att kommunicera med eleverna och undervisningsmiljön kändes trygg. Under lektionerna observerades två parallella språk, ett vardagsanknutet och ett matematiskt språk. Det framkom att ett tydligt förhållningssätt beträffande matematiskt språk kan underlätta för elevens inlärning samt medvetandegöra läraren om den kommunikativa betydelsen i undervisningen med styrdokumentens mål i fokus.</p><p> </p> / <p>The present investigation describes the mathematical communication between the special needs teacher and pupils with mathematical disabilities. The empirical data consists of two mathematics lessons from grade 9 in the compulsory school. The data was analyzed in order to find different kinds of communication. The teachers were in many ways aware of their way to communicate with pupils and the teaching environment felt confident. During the lessons two parallel languages were observed, one everyday related and one mathematical language. It was found that keeping a distinct approach to the mathematical language can facilitate student learning and make the teacher aware about teaching communicatively with steering documents goals in focus.</p><p> </p>

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.

Matematisk modellering är en av de förmågor eleverna ska ges möjlighet att utveckla på gymnasiet. Det är dock inte helt självklart bland lärare vad modellering innebär och frågan är om de undervisningsaktiviteter som används i relation till undervisning av modellering idag verkligen är ändamålsenliga. Denna studie undersöker elevernas potentiella utveckling i att modellera inom en lektionsserie, en så kallad modelleringsmodul, riktad mot kursen Matematik 5 på gymnasiet. Elevernas potentiella utveckling studeras utifrån deras involvering i de olika delprocesser som anses utgöra modelleringsprocessen i sin helhet. Resultatet indikerar att modelleringsmodulen tycks involvera eleverna i samtliga dessa delprocesser. Samtidigt visar resultatet att detta inte är en självklarhet och att vissa delprocesser tycks vara svårare att involvera eleverna i än andra. / Mathematical modelling is one of the competences students will have the possibility to develop in upper secondary school. However, it is not entirely obvious among teachers what modelling really means and it is questionable if the learning activities that are being used in relation to modelling education today really are effective. This study focuses on the students’ potential development within a series of lessons, a so-called modelling module, adapted for the course Mathematics 5 in upper secondary school. The students’ potential development is studied based on their involvement in the various sub-processes that are considered to be the modelling process as a hole. The result indicates that the modelling module involves students in all of these sub-processes. At the same time, the results show this is not self-evident and some of the subprocesses seem to be harder to involve the students in than others.