<p>Vi innfører klassen av kvasikonforme avbildninger i det komplekse planet, og viser en rekke fundamentale egenskaper ved disse. Herunder den målbare avbildningssatsen. Heretter anvendes disse resultatene til å vise klassifikasjonssetningen for fikserte fatoukomponenter i sin optimale form. Det vil si, vi viser riktigheten av Fatous formodning om vandrende komponenter, og eksistens av siegeldisker og hermanringer. Fokus i bevisførelsen er på anvendelser av kvasikonforme avbildninger.</p>

<p>Oppgaven handler om fouriertransformasjon av generaliserte funksjoner, med spesiell vekt på fouriertransformasjon av hyperfunksjoner. Transformasjonen på hyperfunksjoner er deretter sammenlignet med Carlemans fouriertransform, som er en av de tidlige generaliseringene av den klassiske fouriertransformen. Det er vist at begge transformene er symmetriske, det vil si at invers fouriertransform er også definert på samme rom. Videre vises det at begge transformene generaliserer både den klassiske fouriertransformen og Schwartz' fouriertransform av distribusjoner med kompakt support.</p>

<p>Today, UltraSound (US) images are often used in medical examination and surgery. An improvement of the quality of these US-images will lead to many advantages, which is a big motivation for research on this field. One obstacle in improving the quality of the images is the presence of noise and texture. In order to distinguish this unwanted information from the interesting objects, different techniques can be used. Characteristic features, such as the ability to find vague contours, small objects or edges of small strength, decides if the technique is suitable for analysing noisy signals. This thesis presents different techniques for finding objects in US-images by using the continuous wavelet-transform. One observation from the analysis is that for edge-detectors using the wavelet-transform at a single scale, there is a compromise between accuracy and reliability. One has to choose between detecting small objects or vague contours. At fine scales one is able to detect small objects, but not objects with a vague contour without including redundant information. At coarse scales one is able to detect vague contours without including redundant information, but one will not detect small objects. The Lipschitz-regularity and the length of a maxima-line in the time-scale plane works well to find the points where the signal changes with a long duration, but is less suitable to find small objects and to remove unwanted information. By using the value of the wavelet-transform at several scales, it is possible to find vague contours in images, small objects, and edges of small strength compared to the strength of the noise. Another important observation from the analysis is that use of the circumference of objects is appropriate in order to find the most important objects in an image. Using this information has been very useful with respect to the analysis of US-images. Medical ultra-sound images are in general of varying quality. In addition the quality of a US-image will typically change within the signal, and changes with respect to the quality of the contour of objects and the influence of noise. The technique which in general is most reliable and produces the best representations of the US-images analysed in this thesis, uses information about the amplitude of the wavelet-transform both within and across scales, in addition to information about the circumference of the objects. This combined edge-detector is reliable with respect to represent the important objects in the image, and this representation is often easily obtained by the edge-detector.</p>

<p>We consider systems of complex exponential functions in spaces of square integrable functions. Some classical one-dimensional theory is reviewed, in particular, we emphasize the duality between the Riesz bases of complex exponential functions in $L^2$-spaces and complete interpolating sequences in $PW^2$-spaces of entire functions of exponential type. Basis properties for $L^2$-spaces over planar convex domains are then studied in detail. The convex domain in question is shown to be crucial for what basis properties the corresponding $L^2$-space possesses. We explain some results related to Fuglede's conjecture about existence of orthonormal bases and then a result by Lyubarskii and Rashkovskii regarding Riesz bases for $L^2$-spaces over convex polygons, symmetric with respect to the origin. Finally, we make a modest attempt to apply the techniques by Lyubarskii and Rashkovskii combined with approximation of plurisubharmonic functions using logarithms of moduli of entire functions, to construct a complete system of exponential functions in the space of square integrable functions over a disk. This work is not completed yet.</p>

<p>e will evaluate forward contracts in the electricity market. A thorough presentation of stochastic analysis for processes with discontinuous paths are provided, and some results concerning these from mathematical finance are stated. Using a Feynman-Kac-type theorem by Pham we derive a partial integro-differential equation giving the forward price from the spot dynamics taken from Geman and Roncoroni. This spot model is regime switching, so we get two equations. These equations are then attempted solved numerically. We suggest the following approach: When implementing boundary-conditions numerically we use values obtained from a Monte Carlo simulation of the spot dynamics to calibrate the boundary.</p>

<p>The main result in this thesis is a new generalization of Selberg's inequality in Hilbert spaces with a proof. In Chapter 1 we define Hilbert spaces and give a proof of the Cauchy-Schwarz inequality and the Bessel inequality. As an example of application of the Cauchy-Schwarz inequality and the Bessel inequality, we give an estimate for the dimension of an eigenspace of an integral operator. Next we give a proof of Selberg's inequality including the equality conditions following [Furuta]. In Chapter 2 we give selected facts on positive semidefinite matrices with proofs or references. Then we use this theory for positive semidefinite matrices to study inequalities. First we give a proof of a generalized Bessel inequality following [Akhiezer,Glazman], then we use the same technique to give a new proof of Selberg's inequality. We conclude with a new generalization of Selberg's inequality with a proof. In the last section of Chapter 2 we show how the matrix approach developed in Chapter 2.1 and Chapter 2.2 can be used to obtain optimal frame bounds. We introduce a new notation for frame bounds.</p>

<p>We give a brief account of the general analytic theory of continued fractions and state and prove the Lorentzen bestness theorem. We investigate the possibility of a new proof of the Lorentzen bestness theorem and gives a related convergence theorem together with a conjecture. We explore some connections between the limit-periodic continued fractions and other parts of mathematics and we give a few suggestions of topics suitable for further research.</p>

<p>This master's thesis explores the area of combinatorics concerned with counting mathematical objects with regards to symmetry. Two main theorems in this field are Burnside's Lemma and P'{o}lya's Enumeration Theoremfootnote{P'{o}lya's Enumeration Theorem is also known as Redfield--P'{o}lya's Theorem.}. Both theorems yield a formula that will count mathematical objects with regard to a group of symmetries. Burnside's Lemma utilizes the concept of orbits to count mathematical objects with regard to symmetry. As a result of the Burnside Lemma's reliance on orbits, implementation of the lemma can be computationally heavy. In comparison, P'{o}lya's Enumeration Theorem's use of the cycle index of a group eases the computational burden. In addition, P'{o}lya's Enumeration Theorem allows for the introduction of weights allowing the reader to tackle more complicated problems. Building from basic definitions taken from abstract algebra a presentation of the theory leading up to P'{o}lya's Enumeration Theorem is given, complete with proofs. Examples are given throughout to illustrate these concepts. Applications of this theory are present in the enumeration of graphs and chemical compounds.</p>

Today, UltraSound (US) images are often used in medical examination and surgery. An improvement of the quality of these US-images will lead to many advantages, which is a big motivation for research on this field. One obstacle in improving the quality of the images is the presence of noise and texture. In order to distinguish this unwanted information from the interesting objects, different techniques can be used. Characteristic features, such as the ability to find vague contours, small objects or edges of small strength, decides if the technique is suitable for analysing noisy signals. This thesis presents different techniques for finding objects in US-images by using the continuous wavelet-transform. One observation from the analysis is that for edge-detectors using the wavelet-transform at a single scale, there is a compromise between accuracy and reliability. One has to choose between detecting small objects or vague contours. At fine scales one is able to detect small objects, but not objects with a vague contour without including redundant information. At coarse scales one is able to detect vague contours without including redundant information, but one will not detect small objects. The Lipschitz-regularity and the length of a maxima-line in the time-scale plane works well to find the points where the signal changes with a long duration, but is less suitable to find small objects and to remove unwanted information. By using the value of the wavelet-transform at several scales, it is possible to find vague contours in images, small objects, and edges of small strength compared to the strength of the noise. Another important observation from the analysis is that use of the circumference of objects is appropriate in order to find the most important objects in an image. Using this information has been very useful with respect to the analysis of US-images. Medical ultra-sound images are in general of varying quality. In addition the quality of a US-image will typically change within the signal, and changes with respect to the quality of the contour of objects and the influence of noise. The technique which in general is most reliable and produces the best representations of the US-images analysed in this thesis, uses information about the amplitude of the wavelet-transform both within and across scales, in addition to information about the circumference of the objects. This combined edge-detector is reliable with respect to represent the important objects in the image, and this representation is often easily obtained by the edge-detector.

We consider systems of complex exponential functions in spaces of square integrable functions. Some classical one-dimensional theory is reviewed, in particular, we emphasize the duality between the Riesz bases of complex exponential functions in $L^2$-spaces and complete interpolating sequences in $PW^2$-spaces of entire functions of exponential type. Basis properties for $L^2$-spaces over planar convex domains are then studied in detail. The convex domain in question is shown to be crucial for what basis properties the corresponding $L^2$-space possesses. We explain some results related to Fuglede's conjecture about existence of orthonormal bases and then a result by Lyubarskii and Rashkovskii regarding Riesz bases for $L^2$-spaces over convex polygons, symmetric with respect to the origin. Finally, we make a modest attempt to apply the techniques by Lyubarskii and Rashkovskii combined with approximation of plurisubharmonic functions using logarithms of moduli of entire functions, to construct a complete system of exponential functions in the space of square integrable functions over a disk. This work is not completed yet.