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Lund University
Faculty of Egineering, LTH
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#### Education - Master Theses/Examensarbeten

If you are interested in doing your Master Thesis at the division of Mathematical Physics, some inspiration can be found in the list below. For more information and to discuss suggestions of topics, you are welcome to contact Sven Åberg or anyone that you would be interested in working with.

Earlier Master Theses at Mathematical Physics

Tunneling through nanostructures in the presence of Phonons

Examiner: Carl Erik Magnusson
Supervisor: Andreas Wacker
Co-supervisor Olov Karlström
Year: 2011

In this thesis work electron transport through a double quantum dot in a nanowire in the presence of phonon scattering was investigated. By using the first order von Neumann approach, the equations of motion for the density matrix were calculated including electron phonon scattering. By considering deformation potential scattering mechanism, the matrix element for the electron phonon coupling for the electrons and the first acoustic phonon mode in the nanowire was calculated. An increase in transport in the presence of phonon scattering was found as the phonons soften the resonance conditions.

Simulation of Acoustic Phonons in Nanowire and Nanowire Hetrostructure

Student: Hussein Ayedh
Examiner: Carl Erik Magnusson
Supervisor: Andreas Wacker
Year: 2011

This thesis presents a master project which involves simulations of acoustic phonons in finite semiconductor nanostructures by using COMSOL Multiphysics environments according to the elastic continuum model. In this work the phonon dispersion relations have been explored and discussed in several nanostructures under different conditions, and the phonon velocities in these structures are calculated. We have proven that the dispersion relations of phonon modes can be obtained in a ﬁnite nanostructure. The numerical simulations of acoustic phonons have been done in two types of nanostructures: The first structure is nanowire made of pure III-V semiconductors such as GaAs, InAs and GaN. The second is Nanowire-heterostructure made of two materials like GaAs/AlAs, InAs/InP and GaN/AlN. The phonon modes with small wavevectors in these two structures are compared to each other, where they manifest similar behavior in both of these two struc-tures. Strain energy and strain components of phonon modes are discussed along line at the middle of different nanostructures. They are compared to each other to prove the likeness of acoustic phonons behavior in pure nanowire and nano- heterostructure for small wavevectors.

Thermal Conduction in Heterostructure Nanowires

Student: Martin Lindskog
Examiner: Carl Erik Magnusson
Supervisor: Andreas Wacker
Year: 2010
Populärvetenskaplig sammanfattning på svenska finns här

Controlling the heat conduction of nanowires is important for operating transistors and thermoelectric elements. The purpose of the thesis is to investigate the effect on the thermal conduction by heterostructures in the nanowire, and to what extent it could be manipulated by the choice of materials and structure of the nanowire. To this end, an estimation for the behavior of the heat conduction for a simple nanowire was computed, and the dispersion relations for real and imaginary wave vectors was determined. By the method of matching modes, the calculation of the transmission and reflection amplitudes of the elastic continuum vibrational modes was attempted, but this method failed in that the equations of the boundary conditions at the interface could not be solved. It is suggested that the real and imaginary modes split up into complex modes at the extrema of the dispersion relations.

Student: Johan Silfverberg
Examiner: Sven Åberg
Year: 2010
Populärvetenskaplig sammanfattning på svenska finns här

In this thesis nuclei with high spin have been studied, or more specifically their deformation at high spin. The total energy which is a renormalized sum of single- particle energies, is minimized in a set of multiple parameters. The most simple deformation corresponds to elliptic shape described by two so called quadrupole parameters epsilon2 and gamma. The aim of this thesis is to explore the next lowest order in deformation, so called hexadecapole deformation. There are three hexadecapole deformation parameters but they have generally been combined to one parameter. In this thesis, however, the total energy is minimized in all three hexadecapole parameters.

European Barrier Options: Analytical and Numerical Valuation

Student: Nils Rosendahl
Examiner: Ragnar Bengtsson
Supervisor: Sven Åberg
Year: 2010

European barrier options offer investors opportunities to invest at a lower cost compared to the corresponding ordinary European options. European barrier options can either be of out type, meaning that the option becomes worthless if the barrier is crossed, or of in type meaning that a barrier has to be crossed, otherwise the payoff is zero. This paper derives the well known closed form solution for European barrier options, where the underlying is assumed to follow a geometric Brownian motion. Two numerical methods are applied to approximate the price of a barrier option , the Monte Carlo method and the finite difference method. The Monte Carlo method is first implemented for a barrier option written on an asset that is assumed to follow the so called Cox Ingersoll Ross dynamics as well as a Poisson driven jump process. The finite difference method is used to valuate a barrier option on an underlying asset that follows a geometric Brownian motion. Two implementations of the explicit finite difference method are made and compared, where the second is an improvement of the first in the sense that it on forehand can determine the stability of the implementation.

Statistical measures of simulated financial correlation matrices

Student: Martin Ravn
Examiner: Ragnar Bengtsson
Supervisor: Sven Åberg
Year: 2009

Stock market time series are noisy and unpredictable because the market dynamics is very complex. Still there is interesting information. Some of this information comes from correlations between branches and from the market. There are also for small time lags, interesting time correlations. Because of ergodicity, there will be noise in the correlation matrix calculated from a portfolio with uncorrelated stocks. The noise can be simulated with Random Matrix Theory, or more precisely with Wishart matrices. By simulating the time evolution of stock prices and add financial correlations between the stocks, we can generate financial correlation matrices. How will the simulated financial correlation matrices be different from the Wishart matrices? I will diagonalize the correlation matrix and analyze how the eigenvalue spectra for the correlation matrices will change when I add financial correlations. The eigenvalues will be tested using extreme value statistics. How will the extreme value distributions depend on for example the number of companies? Analyzing the eigenvalue time dependence, several other measures can be done. For example curvature distributions and autocorrelation functions.

Spin-orbit and L2 coupling strengths in the Nilsson mean field model

Student: Jerker Tapper
Supervisor: Ragnar Bengtsson, Sven Åberg
Year: 2008

The Nilsson mean field model in nuclear physics uses a spin-orbit term and an L2 term without having standardised coupling strengths as function of the neutron and proton numbers. Analytic expression were produced for these coupling strengths so as to optimize the agreement between the Nilsson model and the Folded Yukawa model. The implications of the new isospin-dependent single-particle coupling coefficients are explored by comparing the root mean square deviances of spectra produced by other existing varietes of spin-orbit and L2 coefficients. It is concluded that improvements on the Nilsson model can be made in many regions.

Phenomenological realtions between 2+-energies and transition rates in nuclei

Examiner: Carl Erik Magnusson, Ingemar Ragnarsson
Supervisor: Sven Åberg
Year: 2008

The 2+-state is the first excited state of deformed even-even nuclei, and the 0+ state is the groundstate of all even-even nuclei. This makes them ideal for studying relations between the energy of an excited state and the transition rate to or from that state. The relations of this kind that work and are presently used are empirical fits to data of formulas that are base on on inaccurate nuclear structure assumptions, such as the irrational flow model. Relations based on this model are called Grodzins relations and they relate the energy of a state to the transition rate of that state through the deformation parameter \beta. The eperimental value of one observable gives an estimate of the other.

The purpose of this thesis is to investigate if a non-empirical nuclear model that takes pairing effects into account can produce better results tha the Grodzins relation and modifications fo it. This model uses both the experimental mass and the energy to produce an estimate of the transition rate.

Capacitance-Voltage characteristics, a way to determine doping profiles in wrap-gated nanowire transistors.

Student: Olov Karlström
Examiner: Ragnar Bengtsson
Supervisor: Andreas Wacker
Year: 2008
Populärvetenskaplig sammanfattning på svenska finns här

A wrap-gated doped nanowire structure is studied by implementing a Poisson-Schrödinger solver. Doping is assumed to be only radially dependant. Special interest is taken to the relation between doping profile and capacitance-voltage characteristics. Good agreement with experimental data is achieved.

Quantum Information in Nanowires

Student: Katja Szybek
Examiner: Sven Åberg
Supervisors: Peter Samuelsson, Claudio Verdozzi
Year: 2007

This thesis deals with some key aspects of quantum information and quantum computation. We have developed a theoretical model of the basic element of the quantum computer - the quantum bit (qubit). The qubit was defined as the two bottom levels of an electronically controlled double qantum dot system created in a homogeneous nanowire and charged with a single electron. We discussed the system's ability to work in a sequence of three operations: initialization, manipulation and measurement. We also analyzed the decoherence in the double quantum dot due to electron-phonon interaction under conditions of free electron evolution and when an external field was applied.

The role of artificial viscosity in Smoothed Particle Hydrodynamics (SPH)

Student: Joakim Darelid
Examiner: Ragnar Bengtsson
Supervisors: Enrique Garcia-Berro, Domingo Garcia-Senz (Universitat Politecnica de Catalunya, Barcelona)
Year: 2007
Populärvetenskaplig sammanfattning på svenska finns här

The simulation of extremely violent events, such as novae and supernovae explosions or the rapid merging of two compact objects (like double white dwarfs, double neutron stars and black holes) is an exciting subject in modern astrophysics.
With the advent of massive parallel computational facilities and the associated increase of the computational power available today, many of the above mentioned problems can be handled and simulated in a quite realistic way. One such numerical schemes is a widely spread numerical technique called Smoothed Particle Hydrodynamics (SPH), which allows to approximately simulate astrophysical phenomena like those previously mentioned. Here, an SPH code is tested and its performance is evaluated. In particular, several recipes for the artificial viscosity are adapted and checked. The tests are done using as benchmark a problem which has an analytic solution, the Sedov problem, which consists in studying the propagation of a shock wave in a polytropic gas of constant initial density. The tests indicate that, given an appropriate choice of the artificial viscosity, the SPH code under study yields satisfactory results that are in very good agreement with the analytical solution.

Creating an agent-based artificial market

Student: Per Berseus
Examiner: Gunnar Ohlén
Supervisors: Thomas Guhr, Rudi Schäfer
Year: 2007
Full version is available

This thesis is a study within econophysics, a research field where financial problems are investigated using methods from statistical physics. An artificial stock market was implemented in Java and used to assess some assertions made in empirical studies by J.-P. Bouchaud and collegues. The model was able to reproduce several characteristics of high-fequency data from financial markets, e.g. a leptocurtic return distribution and a nearly constant return function. Much of the emphasis of this report is made on the methodology itself, since many modelling issues remain to be solved in the agent-based computational economics field. However, the market was also used to simlulate trading scenarios, where the impact of different trading strategies was analysed. More specifically, it was shown in simulation that an overall diffusive price development can result from a competition between two populations of traders: one of liquidity providers, who create persistence in the prices, and another of liquidity takers, who create anti-persistence. It was also shown that a slowly decaying trade sign autocorrelation function could result from similar trading criteria, but not in any obvious way from liquidity provider's division of large market orders into several smaller ones.

Loschmidt echo for one- and many-body systems

Student: Anders Nilsson
Examiner: Thomas Guhr
Supervisor: Sven Åberg
Year: 2007

In this thesis, the Loschmidt echo for one- and many-body systems are studied. Random matrix theory is used to simulate Hamiltonian systems with different statistical properties depending on if the systems are quantum regular or quantum chaotic. The systems are then subjected to a time independent perturbation. The Loschmidt echoes are then studied for these systems. The results are studied for different time regimes and for different strengths of the perturbation. Comparison is made to find out if the Loschmidt echo has different features depending on if the unperturbed Hamiltonian system is quantum chaotic or quantum regular.

Student: Emanuel Ydrefors
Supervisor: Sven Åberg
Year: 2006

In this thesis the charge radius and the neutron radius for the ground state and the well-known 16+ isometric state of 178Hf are investigated. In an experiment published 12 years ago the charge radius of this isometric state was measured to be slightly smaller than for the ground state. Different effects which influence the radii are investigated in this thesis. The nilsson model is used to calculate single-particle energies and radii. The BCS-theory is then used to treat the pairing interactions in the ground state. The deformation of the nucleus in the two states is also discussed. Another effect on the radius comes from the collective motion of the nucleus around its equilibrium deformation according to Heisenberg's uncertainty principle is discussed. The charge radius for the excited state is found to have a slightly smaller charge radius than the ground. Compared to the experimental value the result has the correct sign but the absolute value is smaller.

Optical Response of Quantum Dots embedded in a Quantum wire

Student: Greta Lindwall
Examiner: Ragnar Bengtsson
Supervisor: Andreas Wacker
Year: 2006
Populärvetenskaplig sammanfattning på svenska finns här

Linear and non-linear properties of semiconductor quantum dots embedded in a quantum wire are studied. The pure dephasing of the optical polarization due to the interaction of the exciton to acoustic phonons is calculated. Signatures of the lowest phonon modes in the absorption spectrum is observed and, in contrast to bulk phonons, a broadened zero-phonon line. Furthermore, the dephasing of Rabi oscillations is discussed.

Two-component Bose gas under rotation

Student: Jonas Christensson
Examiner: Sven Åberg
Supervisor: Stephanie Reimann
Year: 2006

A rotating two-component Bose-Einstein condensate has been investigated theoretically, using exact diagonalization. Equal repulsive interactions were assumed between all particles. With different number of particles in the two components, they were found to separate in space with one component orbiting the other. With equal numbers, no such separation was found.

A study of five high-z Type Ia supernovae

Student: Jakob Nordin
Examiner: Ragnar Bengtsson
Supervisor: Ariel Goobar
Year: 2006

Observations of type Ia supernovae (SNE), as a type of standard candle, give constraints on the behaviour of the scale factor of the universe. These observations now seem to indicate an accelerated expansion, caused by some still unknown Dark Energy. In this work a study of Hubble Space Tele scope images of five distant SNE is presented, together with a description of some of the theories, techniques and uncertainties involved.

Three-Dimensional Spiral Waves in Cardiac Tissue

Student: Johathan Johnsson
Supervisor: Gunnar Ohlén
Year: 2006

The properties of some simple types of 3-D spiral waves, scroll waves, in a model of cardiac tissue are investigated, using the simplified Wohlfart-Arlock model to describe the excitability of the cells, and their interconnections. Algorithms are developed to find and follow the tip of two-dimensional spirals, which leads to the discovery of the flower like quasi-periodical trajectory of the spiral tip. The tissue size needed for a spiral to survive for a long time is determined to be slightly larger than the area that the trajectory of a single spiral tip covers, when a spiral propagate in a large piece of tissue. When comparing the propagation of the 2-D double spiral and the 3-D scroll ring, the extra curvature dependency added when going from two to three dimensions is found to be negligible.

Regular and chaotic contributions to the shell structure of a finite Fermi system

Student: Marc Puig von Friesen
Supervisor: Sven Åberg
Year: 2006

The purpose of this paper is to study the shell effects in a dilute interacting fermionic gas trapped in a 3-dimensional harmonic oscillator. The expressions of the variance of the oscillating part of the energy will be studied within the \tau_{min} approximation, a semiclassical model based upon periodic orbit theory, where one separates the total expression into a regular and a chaotic contribution. These results will be compared with the outcome from the Hartree-Fock procedure and the explicit expressions for two periodic orbits.

Exact Diagonalization of a Rotating Bose-Einstein Condensate Confined in an Anharmonic Potential

Student: Sara Bargi
Supervisors: Stephanie Reimann & Georgios Kavoulakis
Year: 2005

The purpose of this Master Thesis is to perform an exact diagonalization for a system of cold bosons confined in an anharmonic potential subject to rotation. We examine the lowest state of the system as a function of the rotational frequency and the interaction strength for both repulsive and attractive interactions, and find the resulting phases of vortex structures and center-of-mass rotation to be consistent with previous mean-field calculations.

Investigation of the 'magnetic bands' in the Pb-region in a 'tilted' cranked Nilsson-Strutinsky formalism

Student: Claes-Göran Zander
Year: 2005

In this thesis the Cranking Nilsson-Strutinsky (CNS) model is extended by adding the spin component, perpendicular to the rotation axis, for specific orbitals. This is done by first transforming the total angular momentum operator into the stretched representation, which leads to supplementary operators $f_i$ in addition to the stretched angular momentum operators $(j_i)_t$. Single and double commutators are used to describe the first and second order terms of the $f_i$-operators and the functions inside the commutators are represented with spherical harmonics. Matrix elements of the spherical harmonics, the stretched representation of the orbital operators and the spin operators are calculated in the |N,l,j,omega>-basis. The expectation values of the j-operator along the perpendicular axis are then calculated and added to the total spin in a semiclassical approach using Pythagoras' theorem. The modified CNS model is used for different fixed configurations in 199Pb and the results are compared with previous calculations, using standard CNS, and with experimental results. From this comparison the expected result is obtained, that is the added component is mainly contributing to the total spin for low and intermediate spin values. This results in a lower total energy in this spin region compared with the standard CNS model. In addition, a better correspondence to the experimental values for the transition energies between different spin-states is achieved.

High-Speed Astrophysics: Chasing Neutron-Star Oscillations

Student: Ricky Nilsson
Examiner: Ragnar Bengtsson
Supervisor: Dainis Dravins
Year: 2005

The purpose of this thesis is to investigate the potential of high time-resolution optical observations to explore high-speed astrophysicsl phenomena, focusing on neutron star oscillations, and also which methods and instruments that should be required for detection of such. In addition, the thesis contains an account of the initial testing of a high time-resolution photon-correlator, intended to be part of a future mobile observation instrument, QVANTOS (Quantum-Optical Spectrometer) Mark II, and the programming of its control- and data acquisition-program. We had the opportunity to test the correlator both in actual astronomical observations at Skinakas Observatory in Crete together with the OPTIMA group from Max Planck Institute for Extraterrestrial Physics (MPE) in Garching near Munich and their OPTIMA-detector instrument, and prior to that in the optics laboratory at lund Observatory. Subsequent preliminary analyses of data from the observations at Mt. Sinakas are also presented.
To comprehend the implications on neutron star oscillation detection on the research of neutron star structure and nuclear physics, some introductory theory on these subjects will be given, concentrating on p-modes in the oscillation part. My hope is that this thesis will serveas a presentation of one possible application offered by the promisisng future of high time-resolution astrophysics.

Spontaneous magnetisation in quantum dots

Student: Keith Edmonds
Supervisor: Stephanie Reimann
Year: 2004

This thesis analuses magnetism in quantum dots for both isolated dots and lattices of dots. The formalism used is spin Density Functional Theory in local denstiy approximation with the correlation functional from the paper by Attaccalite et al. (2002). A six electron parabolic quantum dot is found to have a transition between the zero spin ground state and the spin one ground state relative to coupling , $r_s$, and ellpsoidal deformation $\delta$. A lattice of similar dots display anti-ferromagnetic behaviour for all lattice constants, a, investigated. 7 electron isotropic dots are modelled collectively in a lattice. A transition between metallic, ferromagnetic and anti-feromagnetic is found relative to the lattice constant, 2, and coupling $r_s$. A crude estimate of hte magnetic moment and field for basic lattices is given.

Modeling of Static and Dynamic properties for 1-D Bose-gases at T=0

Student: Magnus Ögren
Supervisor: Sven Åberg
Year: 2004

Starting from Lieb's and Liniger's work on Bose gases with a homogeneous particle density, we derive three different examples of non-linear Schrödinger-like equations for modeling static and dynamical properties of (effectively) one dimensional Bose gases at zero temperature (1-D-BEC) with different atom-atom-interactions.

The equations are studied numerically using an imaginary time method for obtaining the many-particle ground state density, after an excitaion of the lowest breathing mode (to a harmonic oscillator potential). The oscillator frequency ($\omega_B$) is easured by averaging over a few periods in the real time development. The breathing mode ratio ($\omega^2_B/\omega^2$) is studied in the intermediate regime, i.e. in the transition from the Gross-Pitaevski i regime to the Tonk-Girardeau regime where the bosons obey a fermionic behaviour.

The major result is that we can achieve confidence breathing mode ratios for the entire intermediate regime with only a simple correction term to the 1-D-Gross-Pitaevskii equation. This gives us confidence that also other properties can be modeled this way.

From one of the equations we also develop a parameter dependent generalization from of the Thomas-Fermi approximation useful as a 'first estimation' of properties in the intermediate regime when the number of particles is large.

Anharmonically-trapped Bose-Einstein condensates under rotation

Student: Johan Bergelin
Supervisor: Georgios Kavoulakis
Year: 2004

We investigate the lowest state of a trapped Bose-Einstein condensate under rotatio, for various forms of confinement, namely a harmonic, a weakly anharmonic, and a purely quartic trapping potential. In the harmonic confinement there are only two singly-quantized vortex states. In the other two cases there are phases involving both singly as well as multiply-quantised vortices depending on the rotational frequencuof the trap andof the coupling constant. We derive the corresponding phase diagram variationally.

The Nilsson model for Two-Dimensional Quantum Dots

Student: Johnny Kvistholm
Supervisor: Ingemar Ragnarsson and Stephanie Reimann
Year: 2004
Full version is available

An anisotropic two-dimensional model for a quantum dot is constructed. An analogous treatment as for the construction of the Nilsson model is done in two dimensions, with an $l_z^2$ term and a Rashba spin-orbit term. The spectrum of the two-dimensional Nilsson model is investigated for modulations of the Rashba spin orbit coupling, the Nilsson interpolation variable and the elongation parameter. A previously unknown distinct shell pattern for the Rashba spin-orbit coupling is discovered. A study of the magic numbers shows a characteristic level drop between the shells and the 'bent' shape of the pattern account for a beating mode. An investigation of the hard wall limit, energy spectra, addition energies, energy shifts and fluctuations, shell corrections, total ground state energies and magic numbers is performed. A method of determining the interpolation variable for the two-dimensional Nilsson model is presented. The two-dimensional Nilsson model is compared with spin-density functional theory with encouraging results.

Student: Nils Lavesson
Examiner: Ragnar Bengtsson
Supervisor: Leif Lönnblad, theoretical high energy physics
Year: 2004

There are two main ways of describing multi particle production: fixed order matrix elements and parton showers. They provide a good description in complementary regions of phase space. There is an algorithm available for correcting the dipole cascade model with fixed order matrix elements in e+e- collisions. This algorithm is modified to be used for hadron-hadron collision in general and W+ production in proton antiproton collisions at the Tevatron in particular.

Determinations of Oscillator Strengths in Zr II using Fourier Transform Spectrometry

Student: Gunilla Ljung
Examiner: Ragnar Bengtsson
Supervisors: Sveneric Johansson & Hampus Nilsson, Atomic Astrophysics, Lund Observatory
Year: 2004

Using the Fourier Transform Spectrometer at Lund Observatory, spectra from zirconium have been recorded and analyzed. From these spectra measured relative intensities have been used to derive branching fractions for singly ionized zirconium, Zr II. Oscillator strengths for 263 spectral lines in Zr II have been derived by combining the branching fractions with previously measured radiative lifetimes.

Stabilization and synchronization behavior of a large lattice of coupled limit cycle oscillators

Student: Jan Eric Sträng
Supervisor: Per Östborn and Gunnar Ohlén
Year: 2003

The stabilization and synchronization of a large lattice of coupled oscillators is studied. This is done by numerical integration of a system resembling the sino atrial node of the heart. Under the integration process, iterated for different inter oscillator couplings, composite values of the state of the system are measured. From quantities describing phases and periods of oscillators, a stabilization pattern is recognized. A stabilization time can therefore be introduced. This stabilization time seems to diverge at two inter-oscillator couplings consistent with prior investigations where only stabilized periods and period distributions were taken into account. The possibly of a third new critical inter-oscillator coupling is suggested. At this critical coupling, the phase landscapes changes drastically.

An unexpected connection in Random Matrix Theory

Student: Johan Grönqvist
Supervisor: Thomas Guhr
Year: 2003

Random Matrix Theory was proposed for the study of excitations in nuclei, and is now used to study systems in many different areas, including e.g. quantum chaos, mesoscopic physics and signal transmission. In order to compare theory and data, one often wants to compare correlations between eigenvalue distributions. To this end, k-Point correlation functions are introduced. A method of calculation of these is by introducing so called generating functions, and making use of supersymmetry. Here, a previously unknown connection is proven, which makes it possible to obtain the k-point correlations from the simplest cases of the generating functions.

Quantum Chaos in Nuclear Masses

Student: Henrik Olofsson
Supervisor: Sven Åberg
Year: 2003
Full version is available

A random matrix model is set up to incorporate possible chaotic motion in the nucleus in the shell correction energy. This gives a contribution to the nucleus total energy i.e. its mass. The discrepancy between calculated and measured masses can be reproduced in a statistical sense.

Rotational band structures in 86Nb, 87Nb and 86Zr

Student: Gillis Carlsson
Year: 2003
Full version is available.

The rotational band structures in 86Nb, 87Nb and 86Zr were analysed and most of the bands could be explained using the cranked Nilsson-Strutinsky model. Investiations were also made to improve the $\kappa$ and $\mu$ values in the A=90 region. The investigation revealed difficulties in producing a consistent set of $\kappa$ and $\mu$ values valid in all three nuclei.

Credit risk modeling - A structural approach

Students: Andreas Sundin & Michal Wolanski
Supervisor: Thomas Guhr
Year: 2003

Accurate and reliable credit risk models are of great importance for creditors like banks and other financial institutions. In the credit risk management business there is a wide range of vendor credit risk models at hand, both reduced form models and structural models. A structural model is based on an underlying microeconomic process to which it is possible to apply the mathematical tools of statistical physics. This thesis presents a structural credit risk model based on a jump-diffusion process. The most important input parameters of the model are asset volatility, asset drift, asset correlations, jump intensity, leverage and portfolio size. The primary model outputs are the default probability and the distribution of credit losses. The model is analyzed by means of both analytical and numerical calculations. The analytical analysis shows that it is not possible to get a closed form solution of the loss distribution unless a number of crucial simplifications are done. The numerical analysis of the model is more extensive and includes a comprehensive investigation of the parameter space of the model. The most important conclusion of the thesis is that a structural credit risk model is a powerful instrument to acquire valuable knowledge of how different economical parameters affect the default probability and the distribution of credit losses.

Complex Cardiac Excitation in Inhomogeneous Tissue

Student: Liselotte Rydqvist
Supervisor: Gunnar Ohlén and Per Östborn
Year: 2002

This study deals with the propagation of electric waves in cardiac tissue. The object of this project was to investigate whether chaos is possible for the simplified Wohlfart-Arlock model if the tissue is inhomogeneous. Specific questions investigated in this project were:

• Can a region of idle cells cause chaos?
• Can inhomogeneous coupling between adjacent cells give rise to an aperiodic behaviour
• If there is one single spiral wave rotating in the tissue, can this spiral wave split into several spiral waves? If that is the case, what is the underlying reason?

The investigation showed that changed coupling between adjacent cells was the only possibility to cause chaos. For idel cells there was no such behaviour found.
For a single spiral wave rotating in the tissue, we found that the wave could multiply if a region, containing idel cells or containing cells with changed coupling, was placed in a special way.

Synchronisation of pulse-coupled non-linear oscillators

Student: Martin Folkesson
Supervisor: Gunnar Ohlén and Per Östborn
Year: 2001

In this thesis a system similar to the sino-atrial node, or sinus node, is studied. The sinus node is the natural pacemaker of the heart. Experiments show that sinus nodal cells fire an action potential periodically in time. Such firings, pulses, affect the firings of neighbouring cells. The effect of these pulses is dependent on the state of the cell subject to the pulse. This is an important feature of the system. In this study a large system of such pulse-coupled, non-linear oscillators was simulated. The model of the network was applied on a two-dimensional lattice. Each oscillator was connected to it four nearest neighbours.
Synchronisation is in our model studied outgoing from the oscillators' mean periods of firing. Nodes that become stable with the same mean period, ideally measured over an infinite amout of time, are said to be synchronised. Interacting synchronised nodes are assigned to one and the same cluster.
Defining clusters as above, a sinus-nodal system seems to have two
phase transitions. The first takes places as the largest cluster of a network grows dramatically in size. The simulations performed show that this growth follows a power law with respect to the coupling strength. The cluster sizes are at this point critically distributed, thus the behaviour seems analogous to a second order phase transition. The second assumed phase transition takes place when the system becomes entirely synchronised, which is seen for a finite coupling strength. This is in contrast to perfect spin alignment in the Ising model, which appear only for infinite coupling strength or at absolute zero temperature.

Spin ~ 60 hbar i 156Dy
Rotationsband och partikel-hål excitationer vid höga spinn

Student: Fikret Saric
Year: 2000

En beräkningsmetod för kärnans högspinn-tillstånd med cranking Nilsson-Strutinsky modellen har studerats. Med denna metod har flera olika egenskaper studerats. Bland annat: kärnans formändring, band korsningar och band terminering. 156Dy kärnan har studerats i ett område av deformation från epsilon = 0 till epsilon = 0.30 och vissa ändringar har gjorts i de numeriska beräkningarna som möjliggör att man kan särskilja rotationsband med en partikel i (d 3/2,s 1/2) från rotationsband utan någon sådan partikel. En jämförelse har gjorts med experimentella data och med de tidigare teoretiska värdena för proton-neutron konfigurationer för 156Dy kärnan.

Modelling of Cardiac Excitation

Student: Åsa Bergkvist
Supervisor: Gunnar Ohlén and Per Östborn
Year: 2000

Two existing models for describing the cardiac action potential, the Wohlfart- Arlock model and the FitzHugh-Nagumo equations, are investigated and compared. A new simplified version of the Wohlfart-Arlock model is derived and compared to the ordinary one. Spiral waves are initiated and the effect of reentry is studied.

Equilibrium and stability of Riemann ellipsoids

Student: Peter Olivius
Supervisor: Ragnar Bengtsson
Year: 1998

The Riemann ellipsoid is a homogeneous, incompressible ellipsoidal fluid, without viscosity or temperature gradients. The equation of motion and constants of the motion are determined. Equilibria are found when the rotational axis is along a principal axis, and when the rotational axis lies in a principal plane. In the principal plane case, the equilibria are always shape potential unstable, but in the principal axis case, there are both stable and unstable equilibria. The regions of stable and unstable equilibria in both principal axis and principal plane cases are graphically illustrated. It is proved that when the rotational axis is not lying in a principal plane and not along a principal axis, there cannot be equilibrium. A paper of Bernard Riemann concerning these ellipsoids was studied in detail, and some minor mistakes were found.

Simulation of sinus nodal behaviour

Student: Per Östborn
Supervisor: Gunnar Ohlén, Björn Wohlfart
Year: 1998

We compare mathematical models of sinus nodal (SN) cell activity. An elaborate model is found to be more sensitive to external pertubations than simpler models. We conclude that phase response plots and circle mapls can be successfully used to predict dynamical behaviour of a coupled pair of SN cells. We find that small, weakly coupled cells in the SN interior and a high resistance in the immediately surrouding tissue help meet the atial current load. The interwined SN cell arrangement make possible an effective protection against spread of disturbances and helps reducing phase difference between cells at different locations in the SN. Simulations suggest that a heavy atrial current drain and/or internal SN conduction problems can give rise to slow rhythms, sometimes with pauses which are multiples of normal interbeat intervals, quasiperiodic rhythms, chaotic rhythms and frequency locked states. Frequency ratios of the latter form devil's staircases in bifurcation trees. Quasiperiodic motion appears when intercellular coupling is very weak. We suggest how time series of interbeat intervals can be used to identify two- and three-periodic rhythms. Some information about the base frequencies ca be gained. Two-periodic rhythms may arise when there are two weakly interacting areas of reasonably well-coupled SN cells. A ring of model SN cells carry a reentrant of reasonably well-coupled SN cells. A ring of model SN cells can carry a reentrant rhythm with up to twice the normal beating frequency. In contrast to the case of ordinary cardiac cells, no beating irregularities can be obtained in the high frequency limit.

A Microscopic Model of Friction in a Complex Quantum Mechanical System

Student: Daniel Almehed
Supervisor: Sven Åberg
Year: 1997

A time dependent random matrix model is used to study chaos in complex quantum mechanical systems. An energy level density from a deformed nuclear model is simulated. The heating effect given by the gradient of the level density can give a microscopic explanation to a friction term in a macroscopic model of a chaotic quantum mechanical system. An expression of how the friction-energy depends on system parameters is presented.

Chaos assisted tunnelling
A simple model to describe the deexcitation of a superdeformed nucleus

Student: Jan Johansson
Supervisor: Sven Åberg
Year: 1997

A simple model, based on random matrices, is proposed to model the deexcitation of a superdeformed nucleus. The class of random matrices which is used is the Gaussian Orthogonal Ensembles. The system can be transferred from an ordered to a chaotic system. The transformation depends implicit on the excitation energy.

Creation and Propagation of Spiral Waves in Myocardial Tissue

Supervisor: Gunnar Ohlén, Björn Wohlfart
Year: 1997

The behavior of spiral-waves in a model of myocardial tissue is studied numerically. Questions as how spirals are initiated, how they interact, how they move, and how they can be destroyed are discussed.

Neutrino Oscillations and The Solar Neutrino Problem

Supervisor: Cecilia Jarlskog
Year: 1996

(No Abstract)

Aspects of Spiral Wave Dynamics in Myocardial Tissue

Student: Mattias Christensson
Supervisor: Gunnar Ohlén, Björn Wohlfart
Year: 1996

(No Abstract)

Controling Chaos and Hyperchaos in Dynamical Systems

Student: Mattias Christensson
Supervisor: Gunnar Ohlén
Year: 1996

In this study we shall deal with dynamical systems which show chaotic behaviour. These systems are inherently unpredictable in their long term evolution and therefore it may be desirable to develop mehtods that can be used to control them. One such method (originally developed by Ott and coworkers for time discrete systems, see Ott et al. 1990 and Romeires et al. 1992) which uses the sensitive dynamics of a chaotic system in order to control it is described here and we will subsequently develop a way to apply this method to both autonomous an nonautonomous time continuous systems. We will also control a system displaying hyperchaotic dynamics, which has not been done before.

A Critical Analysis of Modifications of Gravity as an Alternative to Dark Matter

Student: Erik Lindahl
Supervisor: Ingemar Bengtsson, Cecilia Jarlskog
Year: 1995

(No Abstract)