Blar i utgivelsesdato Mastergradsoppgaver i matematikk
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Kompleksiteten til noen kryptologisk viktige algoritmer
(Master thesis; Mastergradsoppgave, 19900404)Denne hovedfagsoppgaven har som mål å sammenligne teoretisk og praktisk kompleksitet til algoritmer som har stor kryptologisk betydning. Et av målene er å avsløre den skjulte konstanten bak Onotasjonen, slik at algoritmene kan sammenlignes på et reelt grunnlag. I tillegg er det sett på sammenheng mellom sikkerhet, størrelsen på tall, asymptotisk og praktisk kompleksitet. Spesielt algoritmer som ... 
The use of elliptic curves in cryptography
(Master thesis; Mastergradsoppgave, 200706)The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor Miller in 1985. Being a relatively new field, there is still a lot of ongoing research on the subject, but elliptic curve cryptography, or ECC for short, has already been implemented in reallife applications. Its strength was proved in 2003 when the U.S. National Security Agency adopted ECC for protecting ... 
Hopf algebras and monoidal categories
(Master thesis; Mastergradsoppgave, 20070614)In this thesis we study the correspondence between categorical notions and bialgebra notions, and make a kind of dictionary and grammar book for translation between these notions. We will show how to obtain an antipode, and how to define braidings and quantizations. The construction is done in two ways. First we use the properties of a bialgebra to define a monoidal structure on (co)modules over ... 
Differential invariants of the 2D conformal Lie algebra action
(Master thesis; Mastergradsoppgave, 20080215)In this thises we consider the Lie algebra that corresponds to the Lie pseudogroup of all conformal transformations on the plane. This conformal Lie algebra is canonically represented as a Lie algebra of vector fields on R^2. We will find all possible representations of vector fields in R^3=J^0R^2 which projects to the canonical representation and find the algebra of scalar differential invariants ... 
Light induced forces on dielectric nanospheres
(Master thesis; Mastergradsoppgave, 20080215)Waves that are reflected and refracted by material bodies also transfer momentum to these bodies. This means that the wave field induces a force on the bodies, and multiple reflections between bodies induce forces between them. Light is an electromagnetic wave phenomenon, and the waves carry energy and momentum. Hence, any object that is scattering and refracting light is also acted upon by a ... 
Distributing a private key generator in Ad hoc Networks
(Master thesis; Mastergradsoppgave, 20090515)A Mobile Ad hoc Network (MANET) is a wireless network that does not rely on a fixed infrastructure. These characteristics make algorithms that route network traffic particularly vulnerable to attack. Mechanisms used to protect against such attacks often depend on cryptographic keys. Since the nodes in a MANET have limited resources, designing methods for cryptographic key management is ... 
Extensions of groups and modules
(Master thesis; Mastergradsoppgave, 2010)The main goal of this thesis is to build up detailed constructions and give complete proofs for the extension functors of modules and groups, which we define using cohomology functors. Further, we look at the relations that appear between these and short exact sequences of modules, respectively groups. We calculate also several concrete cohomology groups, and build extensions that are described by ... 
Local classification of 2dimensional solvable Lie algebra actions on the plane.
(Master thesis; Mastergradsoppgave, 201005)In the thesis the local classification of 2dimensional solvable Lie algebra action on the plane is given. Normal forms of such actions are found. The classification applied to classifcation of 2nd order differential equations that are solvable in quadratures. 
Matroids, demimatroids and chains of linear codes
(Master thesis; Mastergradsoppgave, 20101209)The central theme of this thesis is the study of matroids and related concepts such as linear codes and graphs. Demimatroids, structures which arise from a relaxation of the definition of a matroid are explored along with related themes. Finally we examine the fact that some results in coding theory are essentially consequences of results for demimatroids. 
Almost complex homogeneous spaces with semisimple isotropy
(Master thesis; Mastergradsoppgave, 201205)We classify the almost complex structures on homogeneous spaces M = G/H of real dimension less than or equal to 6 with semisimple isotropy group H. 
Numerical calculation of Casimir forces
(Master thesis; Mastergradsoppgave, 20120619)In this thesis a set of regularized boundary integral equation are introduced that can be used to calculate the Casimir force induced by a two dimensional scalar field. The boundary integral method is compared to the functional integral method and mode summation where possible. Comparisons are done for the case of two parallel plates, two concentric circles and two adjacent circles. The results ... 
Deforming the vacuum. On the physical origin and numerical calculation of the Casimir effect.
(Master thesis; Mastergradsoppgave, 20140514)A new method for calculating the Casimir force between compact objects was introduced in May 2012 by Per Jakobsen and Isak Kilen. In this method a regularization procedure is used to reduce the pressure to the solution of an integral equation defined on the boundaries of the objects. In this thesis the method is further developed by extending from a 2D to a 3D massless scalar field, subject to ... 
Symmetry transformation groups and differential invariants
(Master thesis; Mastergradsoppgave, 20141115)There exists a local classification of finitedimensional Lie algebras of vector fields in two complex dimensions. We lift the Lie algebras from this classification to three complex dimensions. 
Towers of Betti Numbers of Matroids and Weight Distribution of Linear Codes and their Duals
(Master thesis; Mastergradsoppgave, 20150515)The main notion behind the study of matroids is linear dependence. In this thesis, we give a survey of the concepts and properties of linear errorcorrecting codes over finite fields being dependent only on the matroids derived from these codes. In particular, the weight distributions of linear codes, and their extensions, over bigger fields are only dependent on the Ngraded Betti numbers of these ... 
Simplicial complexes, Demimatroids, Flag of linear codes and pair of matroids.
(Master thesis; Mastergradsoppgave, 20150515)We first describe linear errorcorrecting codes, and show how many of their most important properties are determined by their associated matroids . We also introduce the simplicial complex of the independent sets of a matroid. We then proceed to study flags of linear codes, and recall the definition of demimatroids, and how such demimatroids associated to flags can describe important properties ... 
Homological methods applied to theory of codes and matroids
(Master thesis; Mastergradsoppgave, 20150515)In this thesis we first give a survey of linear errorcorrecting codes, and how many of their most important properties only depend on the matroids derived from their parity check matrices. We also introduce the StanleyReisner ring associated to the simplicial complex of the independent sets of a matroid. We then recall in particular how some important properties of linear codes, including their ... 
Tipping points and crises in financial markets
(Master thesis; Mastergradsoppgave, 20150515)Electricity spot markets and other financial markets are complex systems, and it is difficult to forecast their behaviour, especially uncontrolled and unmanageable situations, such as power crises and deflation of financial bubbles. An energy crisis is any price rise in the supply of energy resources to an economy. It has undesirable consequences, occasionally irreversible. The most known of these ... 
Arctic tipping points
(Master thesis; Mastergradsoppgave, 20150515)The Arctic is warming much faster than the entire planet, and this causes severe melting of sea ice. However, the climate of different regions of the Earth is interconnected, and changes in the amount of ice in the Arctic can dramatically affect the climate across the whole planet. Some scientists claim that a possible tipping point is the event of the icefree Arctic Ocean in summer. Certain ... 
The Unidirectional Pulse Propagation Equation for Cylindrical Vector modes
(Master thesis; Mastergradsoppgave, 20150727)A new model for the unidirectional pulse propagation equations (UPPE) was developed by Per Jacobsen[1], this model is based on the assumption of cylindrical vector (CV) modes. The model will be strong for CV type electrical eld representations where only a few modes will be excited. In this thesis we will investigate the model further. The model will be implemented as a pseudo spectral method where ... 
Modelling the evolution of ideal, infinite domain patterns, on a finite domain using a Perfectly Matched Layer
(Master thesis; Mastergradsoppgave, 20160128)The SwiftHohenberg equation is an evolution equation which can produce a Pattern, or a patternlike picture, to be more precise. For example, it could be used to model some simple natural patterns, like stripes and rolls that one may observe in a RayleighBenard convection experiment. But for any pattern formation obtained by an evolution equation to look ideal, we have to consider this equation ...