Masteruppsatserhttps://hdl.handle.net/2077/288872024-11-07T19:40:57Z2024-11-07T19:40:57ZSimilarity Problems: Which Groups Are Unitarizable?Westlund, Timhttps://hdl.handle.net/2077/830592024-08-20T20:08:13Z2024-08-20T00:00:00ZSimilarity Problems: Which Groups Are Unitarizable?
Westlund, Tim
This thesis covers some theory on similarity of group representations to unitary representations.
We discuss the notion of amenability and give some classes of groups that are amenable. We then
prove the Dixmier-Day theorem, that states that a locally compact group G is unitarizable if it is
amenable. We also investigate the converse of this statement, which is still an open problem. We
will give some statements where we make some assumptions on the similarity that are equivalent
to amenability. We will also investigate when bounded algebra homomorphism A → B(H), where
A is a C∗-algebra, are similar to a *-homomorphism. We will present connections between the
unitarizability of groups and unitarizability of group C∗-algebras, and this will be useful for some
results about the converse of the Dixmier-Day theorem. We will also investigate the notions of
completely positive and completely bounded maps and prove Stinespring’s theorem for completely
positive maps followed by Wittstock’s theorem for completely bounded maps. We then prove
Haagerup’s theorem that states that unitarizability of homomorphisms is equivalent to the property
of being completely bounded.
2024-08-20T00:00:00ZEfficient Implementation of the 3D Helmholtz equation in C++/PETScKöhle, Renéhttps://hdl.handle.net/2077/828612024-08-12T20:10:21Z2024-08-12T00:00:00ZEfficient Implementation of the 3D Helmholtz equation in C++/PETSc
Köhle, René
The paper describes the comparison of different preconditioners for the solution of
the Helmholtz equation with Krylov subspace methods in three dimensions. The
solution of this equation has applications in microwave imaging and microwave hyperthermia
for cancer detection and treatment. Due to the challenging nature of
the Helmholtz equation, we employ a frequency and convergence analysis in two
and three dimensions. We examine the sensitivity of the equation to various parameters
and determine the effectiveness of various preconditioners. The use of finite
difference approximation and preconditioned Krylov subspace methods allows for
a convergence order of 2 to be achieved. The numerical results provide support
for the aforementioned statement. Provided that there are no issues with resonant
frequencies the desired convergence is achieved. This is applicable to the results
obtained for different parameter functions and frequencies, as well as for two- and
three-dimensional problems.
2024-08-12T00:00:00ZPrime number racesElofsson, Carlhttps://hdl.handle.net/2077/828602024-08-12T20:01:29Z2024-08-12T00:00:00ZPrime number races
Elofsson, Carl
In this thesis we investigate the behaviour of primes in arithmetic progressions, with
a focus on the phenomenon known as Chebyshev’s bias. Under the assumption of
the Generalized Riemann Hypothesis and the Linear Independence Hypothesis, we
prove that there is a bias towards quadratic non-residues. Additionally we extend
the investigation to the setting of function fields. In the function field setting, we
investigate the behaviour of prime polynomials in residue classes modulo a fixed
monic polynomial. Moreover, we prove that for an irreducible polynomial m there
is a bias towards quadratic non-residues modulo m.
2024-08-12T00:00:00ZPoint Process Learning estimation of bandwidth selection for the pair correlation function.Östling, Jenshttps://hdl.handle.net/2077/828592024-08-12T20:11:01Z2024-08-12T00:00:00ZPoint Process Learning estimation of bandwidth selection for the pair correlation function.
Östling, Jens
In this master thesis I use the newly developed method of Point Process Learning
(PPL) to select the bandwidth used when estimating the Pair correlation function
with kernel estimation. The selection of bandwidth when using kernel estimation is
non trivial as different bandwidths give different results. Cronie et. al showed in
their paper that the PPL method out performed the current state of the art when
selecting the bandwidth for the intensity function. In this thesis I try the method
when selecting the bandwidth for the pair correlation function. The results are
promising but the scale of this pilot study is not large enough to give any conclusive
results.
2024-08-12T00:00:00Z