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• 151.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
The distribution of free path lengths in a onedimensional quasicrystal2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 152.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
The Orbit Method and Geometric Quantisation2018Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 153.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Optimal Liquidation of Stock2016Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 154.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Optimal stopping and incomplete information in finance2011Licentiate thesis, comprehensive summary (Other academic)
1. Recovering a piecewise constant volatility from perpetual put option prices
Open this publication in new window or tab >>Recovering a piecewise constant volatility from perpetual put option prices
2010 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 47, no 3, p. 680-692Article in journal (Refereed) Published
##### Abstract [en]

In this paper we present a method to recover a time-homogeneous piecewise constant volatility from a finite set of perpetual put option prices. The whole calculation process of the volatility is decomposed into easy computations in many fixed disjoint intervals. In each interval, the volatility is obtained by solving a system of nonlinear equations.

##### Keywords
Perpetual put option, calibration of models, piecewise constant volatility
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-134147 (URN)10.1239/jap/1285335403 (DOI)000282856000005 ()
Available from: 2010-11-24 Created: 2010-11-22 Last updated: 2017-12-12Bibliographically approved
2. Optimal selling of an asset under incomplete information
Open this publication in new window or tab >>Optimal selling of an asset under incomplete information
2011 (English)In: International Journal of Stochastic Analysis, ISSN 2090-3332, E-ISSN 2090-3340, Vol. 2011, p. 543590-Article in journal (Refereed) Published
##### Abstract [en]

We consider an agent who wants to liquidate an asset with unknown drift. The agent believes that the drift takes one of two given values and has initially an estimate for the probability of either of them. As time goes by, the agent observes the asset price and can thereforeupdate his beliefs about the probabilities for the drift distribution. We formulate an optimal stopping problem that describes the liquidation problem, and we demonstrate that the optimal strategy is to liquidate the first time the asset price falls below a certain time-dependent boundary. Moreover, this boundary is shown to be monotonically increasing, continuous and to satisfy a nonlinear integral equation.

Mathematics
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-141329 (URN)10.1155/2011/543590 (DOI)
Available from: 2011-01-11 Created: 2011-01-11 Last updated: 2017-12-11Bibliographically approved
3. Optimal selling of an asset with jumps under incomplete information
Open this publication in new window or tab >>Optimal selling of an asset with jumps under incomplete information
2013 (English)In: Applied Mathematical Finance, ISSN 1350-486X, E-ISSN 1433-4313, Vol. 20, no 6, p. 599-610Article in journal (Refereed) Published
##### Abstract [en]

We study the optimal liquidation strategy of an asset with price process satisfying a jump diffusion model with unknown jump intensity. It is assumed that the intensity takes one of two given values, and we have an initial estimate for the probability of both of them. As time goes by, by observing the price fluctuations, we can thus update our beliefs about the probabilities for the intensity distribution. We formulate an optimal stopping problem describing the optimal liquidation problem. It is shown that the optimal strategy is to liquidate the first time the point process falls below (goes above) a certain time-dependent boundary.

Mathematics
##### Identifiers
urn:nbn:se:uu:diva-164690 (URN)10.1080/1350486X.2013.810462 (DOI)
Available from: 2011-12-22 Created: 2011-12-22 Last updated: 2017-12-08Bibliographically approved
• 155.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Kolmogorov Equations2013Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 156.
Umea Univ, Dept Math & Math Stat, SE-90187 Umea, Sweden.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. KTH Royal Inst Technol, Dept Math, SE-10044 Stockholm, Sweden.
Existence, uniqueness and regularity of solutions to systems of nonlocal obstacle problems related to optimal switching2019In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 475, no 1, p. 13-31Article in journal (Refereed)

We study viscosity solutions to a system of nonlinear degenerate parabolic partial integro-differential equations with interconnected obstacles. This type of problem occurs in the context of optimal switching problems when the dynamics of the underlying state variable is described by an n-dimensional Levy process. We first establish a continuous dependence estimate for viscosity sub- and supersolutions to the system under mild regularity, growth and structural assumptions on the partial integro-differential operator and on the obstacles and terminal conditions. Using the continuous dependence estimate, we obtain the comparison principle and uniqueness of viscosity solutions as well as Lipschitz regularity in the spatial variables. Our main contribution is construction of suitable families of viscosity sub- and supersolutions which we use as "barrier functions" to prove Holder continuity in the time variable, and, through Perron's method, existence of a unique viscosity solution. This paper generalizes parts of the results of Biswas, Jakobsen and Karlsen (2010) [5] and of Lundstrom, Nystrom and Olofsson (2014) [21,22] to hold for more general systems of equations.

• 157.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type2014In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 193, no 4, p. 1213-1247Article in journal (Refereed)
• 158.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Multi-scale Inference of Interaction Rules in Animal Groups Using Bayesian Model Selection2013In: PloS Computational Biology, ISSN 1553-734X, E-ISSN 1553-7358, Vol. 9, no 3, article id e1002961Article in journal (Refereed)

Inference of interaction rules of animals moving in groups usually relies on an analysis of large scale system behaviour. Models are tuned through repeated simulation until they match the observed behaviour. More recent work has used the fine scale motions of animals to validate and fit the rules of interaction of animals in groups. Here, we use a Bayesian methodology to compare a variety of models to the collective motion of glass prawns (Paratya australiensis). We show that these exhibit a stereotypical 'phase transition', whereby an increase in density leads to the onset of collective motion in one direction. We fit models to this data, which range from: a mean-field model where all prawns interact globally; to a spatial Markovian model where prawns are self-propelled particles influenced only by the current positions and directions of their neighbours; up to non-Markovian models where prawns have 'memory' of previous interactions, integrating their experiences over time when deciding to change behaviour. We show that the mean-field model fits the large scale behaviour of the system, but does not capture the observed locality of interactions. Traditional self-propelled particle models fail to capture the fine scale dynamics of the system. The most sophisticated model, the non-Markovian model, provides a good match to the data at both the fine scale and in terms of reproducing global dynamics, while maintaining a biologically plausible perceptual range. We conclude that prawns' movements are influenced by not just the current direction of nearby conspecifics, but also those encountered in the recent past. Given the simplicity of prawns as a study system our research suggests that self-propelled particle models of collective motion should, if they are to be realistic at multiple biological scales, include memory of previous interactions and other non-Markovian effects.

• 159.
Uppsala University, Disciplinary Domain of Science and Technology, Biology, Department of Cell and Molecular Biology, Computational Biology and Bioinformatics. Fjordforsk A/S.
Derivation and Numerical analysis of an Attenuation Operator for non-relativistic waves2018In: Scientific Reports, ISSN 2045-2322, E-ISSN 2045-2322, Vol. 8, article id 16572Article in journal (Refereed)

Quantum mechanical models for particles are strictly dependent on the Schrödinger equation, where the solutions and the Hermitian polynomials form a mathematical foundation to derive expectation values for observables. As for all quantum systems, the solutions are derived in discrete energy levels, and yield probability density, the kinetic energy and average momentum. In this study however, an attenuation Hamiltonian is derived by the algebraic relation of the momentum and position operators, and the derived equation, where the attenuation of kinetic energy is the eigenvalue, is studied numerically. The numerical solutions suggest that the change in kinetic energy from one transition to the next proceed in an undular fashion, and not in a definite manner. This suggests that any sub-atomic particle which experiences a transition from one level to the next, does so by both gaining and losing energy in an undular manner before reaching an equilibrium with a new and stabilized kinetic energy. The results show also that the phase of the change in kinetic energy between transitions differs between high and low momenta and that higher levels of momentum attenuate more smoothly than transitions between lower energy levels. The investigated attenuation operator may be important for future pinning and quasipinning approaches and play a role in future quantum information processing. Future research is required on the spectrum of the operator and on its potential analytical solutions.

• 160.
Uppsala University, Science for Life Laboratory, SciLifeLab. Uppsala University, Disciplinary Domain of Science and Technology, Biology, Department of Cell and Molecular Biology, Computational Biology and Bioinformatics. Fjordforsk A/S, Midtun, Vangsnes, Norway.
Mathematical Modeling of Rogue Waves: A Survey of Recent and Emerging Mathematical Methods and Solutions2018In: Axioms, E-ISSN 2075-1680, Vol. 7, no 2, article id 42Article, review/survey (Refereed)

Anomalous waves and rogue events are closely associated with irregularities and unexpected events occurring at various levels of physics, such as in optics, in oceans and in the atmosphere. Mathematical modeling of rogue waves is a highly active field of research, which has evolved over the last few decades into a specialized part of mathematical physics. The applications of the mathematical models for rogue events is directly relevant to technology development for the prediction of rogue ocean waves and for signal processing in quantum units. In this survey, a comprehensive perspective of the most recent developments of methods for representing rogue waves is given, along with discussion of the devised forms and solutions. The standard nonlinear Schrödinger equation, the Hirota equation, the MMT equation and other models are discussed and their properties highlighted. This survey shows that the most recent advancement in modeling rogue waves give models that can be used to establish methods for the prediction of rogue waves in open seas, which is important for the safety and activity of marine vessels and installations. The study further puts emphasis on the difference between the methods and how the resulting models form the basis for representing rogue waves in various forms, solitary or with a wave background. This review has also a pedagogic component directed towards students and interested non-experts and forms a complete survey of the most conventional and emerging methods published until recently.

• 161.
Uppsala University, Disciplinary Domain of Science and Technology, Biology, Department of Cell and Molecular Biology, Computational Biology and Bioinformatics. Fjordforsk AS.
A & E Trounev IT Consulting, Toronto, Canada.
Derivation of a Hamiltonian for formation of particles in a rotating system subjected to a homogeneous magnetic field2019Report (Other academic)

Bose-einstein condensates have received wide attention in the last decades given their unique properties. In this study, we apply supersymmetry rules on an existing Hamiltonian and derive a new model which represents an inverse scattering problem with similarities to the circuit and the harmonic oscillator equations. The new Hamiltonian is analysed and its numerical solutions are presented. The results suggest that the SUSY Hamiltonian describes the formation of vortices in a homogeneous magnetic field in a rotating system.

• 162.
Uppsala University, Disciplinary Domain of Science and Technology, Biology, Department of Cell and Molecular Biology, Computational Biology and Bioinformatics. Fjordforsk A/S, Vangsnes, Norway.
A&E Trounev Consulting, Toronto, Canada; Fjordforsk A/S, Vangsnes, Norway.
Formation of quantum vortices in a rotating sphere in an electromagnetic field2019Report (Other academic)

Vortexes in three dimensional systems are an emerging topic of study in the realm of quantum physics, particle physics and magnetism. In this study we describe a new Hamiltonian, and solve it numerically for the description of vortices in a spherical model subjected to an electromagnetic field. The numerical analysis shows also that supersymmetric Hamiltonian describes quantized orbitals for atomic systems with vorticity included, which is to our knowledge novel and also describes electricity as composed of quasi-symmetric vortex bundles. Further work on analytical solutions is under development. The 3D solutions of the Hamiltonian are relevant for nuclear physics, in the physics of elementary particles, in geophysics, and in the physics of stellar bodies.

• 163.
Uppsala University, Disciplinary Domain of Science and Technology, Biology, Department of Cell and Molecular Biology, Computational Biology and Bioinformatics. Fjordforsk A/S, Vangsnes, Norway.
A&E Trounev Consulting, Toronto, Canada; Fjordforsk ASFjordforsk A/S, Vangsnes, Norway.
Modelling the formation of atmospheric vortices on planet earth using a supersymmetric operator2019Report (Other academic)

Macrophysical phenomena, such as turbulence, vorticity are normally modelled using fluid-mechanical differential equations and geodesics systems. Other methods may also be of importance to the meteorology community, such as quantum physical operators. In this study we use a novel Hamiltonian and study the vortex formation in the atmosphere of planet Earth under the effects of the gravity density stratification and the electromagnetic field of the planet. The results propose that vorticity in the atmosphere (high and low pressure systems) is driven in major part by the interplay between the earths magnetic field and gravity density. The results show that the quantized behaviour of atmospheric vortices lies in their dominant occurrence on the northern and southern hemispheres. The use of quantum mechanical operators in modelling planetary vorticity reveals also that these vortices arise from the core of the planet and manifest in a most pronounced manner on the surface of the earth where gravity density is experiencing an abrupt phase change. Further research is made on combining this model with earths atmospheric parameters, such as ocean temperatures and circulation, terrestrial oscillation and the sun's magnetic field. The results are important for future developments of climate and weather prediction models.

• 164.
Uppsala University, Disciplinary Domain of Science and Technology, Biology, Department of Cell and Molecular Biology, Computational Biology and Bioinformatics. Fjordforsk A/S, Vangsnes, Norway.
A&E Trounev Consulting, Toronto, Canada; Fjordforsk A/S, Vangsnes, Norway.
Quantum vorticity in a rotating magnetic field2019Report (Other academic)

Vortexes in superfluids are a critical part of quantum optics, quantum dynamics studies and quantum physics in general. The behavior of vortexes is modeled by models such as the Ginzburg-Landau formula, and other systems, where symmetry-breaking is an inevitable event upon vortex formation. In this paper, we present and study a supersymmetric Hamiltonian which allows formation of vortexes in a quantum hall system to occur from the boundaries, as in natural phenomena, without breaking the symmetry. We study the numerical solutions of the Hamiltonian under rotating magnetic field.

• 165.
Uppsala University, Disciplinary Domain of Science and Technology, Technology, Department of Engineering Sciences, Electricity.
Magnetic Leakage Fields and End Region Eddy Current Power Losses in Synchronous Generators2017Doctoral thesis, comprehensive summary (Other academic)

The conversion of mechanical energy to electrical energy is done mainly with synchronous generators. They are used in hydropower generators and nuclear plants that presently account for about 80% of the electric energy production in Sweden. Because of the dominating role of the synchronous generators, it is important to minimize the power losses for efficient use of natural resources and for the economies of the electric power companies and their customers. For a synchronous machine, power loss means undesired heat production. In electric machines, there are power losses due to windage, friction in bearings, resistance in windings, remagnetization of ferromagnetic materials, and induced voltages in windings, shields and parts that are conductive but ideally should be non-conductive.

The subject of this thesis is prediction of end region magnetic leakage fields in synchronous generators and the eddy current power losses they cause. The leakage fields also increase the hysteresis losses in the end regions. Magnetic flux that takes paths such that eddy current power losses increase in end regions of synchronous generators is considered to be leakage flux. Although only a small fraction of the total magnetic flux is end region leakage flux, it can cause hot spots, discoloration and reduce the service life of the insulation on the core laminations. If unattended, damaged insulation could lead to electric contact and eddy currents induced by the main flux between the outermost laminations. That gives further heating and deterioration of the insulation of laminations deeper into the core. In a severe case, the core can melt locally, cause a cavity, buckling and a short circuit of the main conductors. The whole stator may have to be replaced. However, the end region leakage flux primarily causes heating close to the main stator conductors which makes the damage possible to discover by visual inspection before it has become irrepairable.

1. Axial Magnetic Fields at the Ends of a Synchronous Generator at Different Points of Operation
Open this publication in new window or tab >>Axial Magnetic Fields at the Ends of a Synchronous Generator at Different Points of Operation
2015 (English)In: IEEE transactions on magnetics, ISSN 0018-9464, E-ISSN 1941-0069, Vol. 51, no 2, article id 8100208Article in journal (Refereed) Published
##### Abstract [en]

Axial magnetic fields leaking out at the ends of a conventional rotating synchronous machine cause losses. Therefore, it is important to be able to predict the axial magnetic fields. A linear steady-state model for the axial magnetic flux density phasor in the end regions of non-salient synchronous generators has previously been verified experimentally. This paper describes an extension of the model to salient pole synchronous generators and a method for calculating the coefficients. Experiments and 3-D finite element simulations justify a distinction between axial flux density contributions from the d and q components of the stator current. How the coefficients and the axial magnetic fields in the ends of a small synchronous generator change with steady-state operation conditions is here shown with measurements and to some extent with 3-D finite element simulations.

##### Keywords
Axial magnetic flux, hydropower generator, operation conditions
##### National Category
Physical Sciences Electrical Engineering, Electronic Engineering, Information Engineering
##### Identifiers
urn:nbn:se:uu:diva-255312 (URN)10.1109/TMAG.2014.2347269 (DOI)000353595800015 ()
Available from: 2015-06-16 Created: 2015-06-15 Last updated: 2017-12-04Bibliographically approved
2. Axial Magnetic Fields, Axial Force, and Losses in the Stator Core and Clamping Structure of a Synchronous Generator with Axially Displaced Stator
Open this publication in new window or tab >>Axial Magnetic Fields, Axial Force, and Losses in the Stator Core and Clamping Structure of a Synchronous Generator with Axially Displaced Stator
2017 (English)In: Electric power components and systems, ISSN 1532-5008, E-ISSN 1532-5016, Vol. 45, no 4, p. 410-419Article in journal (Refereed) Published
##### Abstract [en]

Axial displacement of the stator in a synchronous machine gives rise to axial magnetic field both at the ends and deep inside the stator. The axial magnetic field causes losses. This article contains results from two studies with an axially displaced stator. In the first study, axial magnetic leakage fields in the ends of a small synchronous generator at load and no load were measured and simulated. In the second study, axial force and iron losses at no load were calculated with non-linear materials and a three-dimensional, time-stepped finite element method. For some machines with vertical shafts, the sum of iron losses and thrust bearing losses can be reduced if the rotor is lowered or the stator raised, whichever is best.

##### Place, publisher, year, edition, pages
TAYLOR & FRANCIS INC, 2017
##### Keywords
axial magnetic field, axial force, losses, synchronous generator, finite element analysis
##### National Category
Electrical Engineering, Electronic Engineering, Information Engineering
##### Identifiers
urn:nbn:se:uu:diva-319683 (URN)10.1080/15325008.2016.1266062 (DOI)000397048400005 ()
Available from: 2017-04-07 Created: 2017-04-07 Last updated: 2017-11-29Bibliographically approved
3. Harmonically Time Varying, Traveling Electromagnetic Fields along a Plate and a Laminate with a Rectangular Cross Section, Isotropic Materials and Infinite Length
Open this publication in new window or tab >>Harmonically Time Varying, Traveling Electromagnetic Fields along a Plate and a Laminate with a Rectangular Cross Section, Isotropic Materials and Infinite Length
2017 (English)In: Progress in Electromagnetics Research B, ISSN 1937-6472, E-ISSN 1937-6472, Vol. 77, p. 117-136Article in journal (Refereed) Published
##### Abstract [en]

This article contains derivation of propagation factors and Fourier series for harmonically time varying, traveling electromagnetic fields in a plate and a laminate with rectangular cross sections, isotropic materials and infinite length. Different and quite general fields are taken into account on all boundaries. Choices of boundary conditions and continuity conditions are discussed. Certain combinations of types of boundary conditions make the derivation possible for a laminate. Comparisons are made between results of Fourier series and finite element calculations.

##### National Category
Mathematics Engineering and Technology
##### Identifiers
urn:nbn:se:uu:diva-331171 (URN)10.2528/PIERB17061909 (DOI)
Available from: 2017-10-11 Created: 2017-10-11 Last updated: 2017-11-29Bibliographically approved
4. Harmonically Time Varying, Traveling Electromagnetic Fields along a Laminate Approximated by a Homogeneous, Anisotropic Block with Infinite Length
Open this publication in new window or tab >>Harmonically Time Varying, Traveling Electromagnetic Fields along a Laminate Approximated by a Homogeneous, Anisotropic Block with Infinite Length
2017 (English)In: Progress in Electromagnetics Research B, ISSN 1937-6472, E-ISSN 1937-6472Article in journal (Refereed) Submitted
##### Abstract [en]

Analytical expressions that include arbitrarily directed fields on all laminate boundaries can be used for calculation of the fields inside the laminate when the boundary fields are known from, e.g., measurements. A linear laminate block could be used in non-destructive testing for comparisons between different laminates. This article contains derivation of Fourier series of harmonically time varying, traveling electromagnetic fields in homogeneous, anisotropic approximations of laminates. The component of the magnetic field strength in the stacking direction is used as a source term in two-dimensional Poisson equations for the magnetic field strength in other directions. This approximation is here used in three dimensions under the precondition that the conductivity is much smaller in the laminate stacking direction than in the other directions. Sine interpolation and different choices of types of boundary conditions are discussed. Different alternative subdivisions of the Poisson boundary value problems are treated. Shorted derivations of simple analytical expressions are given for both traveling and standing waves in two dimensions. Results from Fourier series in the three-dimensional case are compared with results from finite element calculations.

##### National Category
Mathematical Analysis Engineering and Technology
##### Identifiers
urn:nbn:se:uu:diva-331175 (URN)
Available from: 2017-10-11 Created: 2017-10-11 Last updated: 2017-11-29
5. A Loss Model and Finite Element Analyses of the Influence of Load Angle Oscillation on Stator Eddy Current Losses in a Synchronous Generator
Open this publication in new window or tab >>A Loss Model and Finite Element Analyses of the Influence of Load Angle Oscillation on Stator Eddy Current Losses in a Synchronous Generator
2017 (English)In: IEEE transactions on magnetics, ISSN 0018-9464, E-ISSN 1941-0069Article in journal (Refereed) Submitted
##### Abstract [en]

The load angle of a synchronous generator connected to a power grid has an eigenfrequency that depends on the operating conditions. The existence of an eigenfrequency can make the generator sensitive to electrical and mechanical disturbances and motivates the use of damper windings and power stabilizing systems. The eddy current losses in the stator core and clamping structure increase as a consequence of the load angle oscillations. This is shown both with transient finite element analyses and analytically via a loss model derived from a steady state phasor model of the eddy current loss density. The model is also applicable to the quasi-steady states occurring during load angle oscillations.

##### National Category
Electrical Engineering, Electronic Engineering, Information Engineering
##### Identifiers
urn:nbn:se:uu:diva-331176 (URN)
Available from: 2017-10-11 Created: 2017-10-11 Last updated: 2017-10-17
• 166.
Uppsala University, Disciplinary Domain of Science and Technology, Technology, Department of Engineering Sciences, Electricity.
Uppsala University, Disciplinary Domain of Science and Technology, Technology, Department of Engineering Sciences, Electricity.
Harmonically Time Varying, Traveling Electromagnetic Fields along a Laminate Approximated by a Homogeneous, Anisotropic Block with Infinite Length2017In: Progress in Electromagnetics Research B, ISSN 1937-6472, E-ISSN 1937-6472Article in journal (Refereed)

Analytical expressions that include arbitrarily directed fields on all laminate boundaries can be used for calculation of the fields inside the laminate when the boundary fields are known from, e.g., measurements. A linear laminate block could be used in non-destructive testing for comparisons between different laminates. This article contains derivation of Fourier series of harmonically time varying, traveling electromagnetic fields in homogeneous, anisotropic approximations of laminates. The component of the magnetic field strength in the stacking direction is used as a source term in two-dimensional Poisson equations for the magnetic field strength in other directions. This approximation is here used in three dimensions under the precondition that the conductivity is much smaller in the laminate stacking direction than in the other directions. Sine interpolation and different choices of types of boundary conditions are discussed. Different alternative subdivisions of the Poisson boundary value problems are treated. Shorted derivations of simple analytical expressions are given for both traveling and standing waves in two dimensions. Results from Fourier series in the three-dimensional case are compared with results from finite element calculations.

• 167.
Univ Bristol, Bristol, England.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
The Three Gap Theorem and the Space of Lattices2017In: The American mathematical monthly, ISSN 0002-9890, E-ISSN 1930-0972, Vol. 124, no 8, p. 741-745Article in journal (Refereed)

The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence alpha, 2 alpha,..., N alpha, for any integer N and real number alpha. This statement was proved in the 1950s independently by various authors. Here we present a different approach using the space of two-dimensional Euclidean lattices.

• 168.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Volatility Derivatives – Variance and Volatility Swaps2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 169.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Analysis of the IRB asset correlation coefficient with an application to a credit portfolio2013Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 170.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Abstract Harmonic Analysis on Locally Compact Abelian Groups2018Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 171.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
A note on mass-minimising extensions2015In: General Relativity and Gravitation, ISSN 0001-7701, E-ISSN 1572-9532Article in journal (Refereed)
• 172.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
First Law of Black Hole Mechanics as a Condition for Stationarity2014In: Physical Review D. Particles and fields, ISSN 0556-2821, E-ISSN 1089-4918Article in journal (Refereed)
• 173.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
The phase space for the Einstein-Yang-Mills equations and the first law of black hole thermodynamics2014In: Advances in Theoretical and Mathematical Physics, ISSN 1095-0761, E-ISSN 1095-0753Article in journal (Refereed)
• 174.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
CONFINEMENT EFFECTS ON SCATTERING FOR A NANOPARTICLE2007In: Proceedings of the International Conference "APLIMAT 2007", February 6-9, 2007, Bratislava, Slovak Republic., Bratislava: APLIMAT 2007 , 2007, p. 225-233Conference paper (Other academic)

The motion of a nanoparticle in a narrow, bend channel is used to illustrate features of scattering in systems with semi-open geometries. Under certain general constraints on the geometry, results on the scattering process axe established.

• 175.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
On the maximal ionization for the atomic Pauli operator2005In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 461, no 2063, p. 3355-3364Article in journal (Refereed)
• 176. Mikulevicius, Remigijus
Convergence of Weak Euler Scheme for Nondegenerate Levy-Driven Stochastic Differential EquationsManuscript (preprint) (Other academic)
• 177. Mikulevicius, Remigijus
On the Rate of Convergence of Weak Euler Approximation for Nondegenerate SDEs Driven by Levy Processes2011In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 121, no 8, p. 1720-1748Article in journal (Refereed)
• 178. Mikulevicius, Remigijus
Rate of Convergence of Weak Euler Approximation for Nondegenerate Stochastic Differential EquationsManuscript (preprint) (Other academic)
• 179. Mikulevicius, Remigijus
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Weak Euler Approximation for Ito Diffusion and Jump Processes2015In: Stochastic Analysis and Applications, ISSN 0736-2994, E-ISSN 1532-9356, Vol. 33, no 3, p. 549-571Article in journal (Refereed)

This article studies the rate of convergence of the weak Euler approximation for Itô diffusion and jump processes with Hölder-continuous generators. It covers a number of stochastic processes including the nondegenerate diffusion processes and a class of stochastic differential equations driven by stable processes. To estimate the rate of convergence, the existence of a unique solution to the corresponding backward Kolmogorov equation in Hölder space is first proved. It then shows that the Euler scheme yields positive weak order of convergence.

• 180.
Bates Coll, Lewiston, USA. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Invertibility Properties of Singular Integral Operators Associated with the Lam, and Stokes Systems on Infinite Sectors in Two Dimensions2017In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 89, no 2, p. 151-207Article in journal (Refereed)

In this paper we establish sharp invertibility results for the elastostatics and hydrostatics single and double layer potential type operators acting on , , whenever is an infinite sector in . This analysis is relevant to the layer potential treatment of a variety of boundary value problems for the Lam, system of elastostatics and the Stokes system of hydrostatics in the class of curvilinear polygons in two dimensions, such as the Dirichlet, the Neumann, and the Regularity problems. Mellin transform techniques are used to identify the critical integrability indices for which invertibility of these layer potentials fails. Computer-aided proofs are produced to further study the monotonicity properties of these indices relative to parameters determined by the aperture of the sector and the differential operator in question.

• 181.
Chalmers University of Technology and the University of Gothenburg.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Convergence of Finite Volume Scheme for a Three-Dimensional Poisson Equation2014In: Journal of Mathematical Sciences, Vol. 202, no 2, p. 130-153Article in journal (Refereed)

We construct and analyze a finite volume scheme for numerical solution of a three-dimensional Poisson equation. We derive optimal convergence rates in the discrete H1 norm and sub-optimal convergence in the maximum norm, where we use the maximal available regularity of the exact solution and minimal smoothness requirement on the source term. The theoretical results are justified through implementing some canonical examples in 3D.

• 182.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Three Systems of Orthogonal Polynomials and Associated Operators2012Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 183.
Univ Milano Bicocca, Dipartimento Fis, Piazza Sci 3, I-20126 Milan, Italy.;INFN, Sez Milano Bicocca, I-20126 Milan, Italy..
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics. Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
q-Virasoro Modular Double and 3d Partition Functions2017In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 353, no 3, p. 1059-1102Article in journal (Refereed)

We study partition functions of 3d gauge theories on compact manifolds which are S (1) fibrations over S-2. We show that the partition functions are free field correlators of vertex operators and screening charges of the q-Virasoro modular double, which we define. The inclusion of supersymmetric Wilson loops in arbitrary representations allows us to show that the generating functions of Wilson loop vacuum expectation values satisfy two-related commuting sets of q-Virasoro constraints. We generalize our construction to 3d unitary quiver gauge theories and as an example we give the free boson realization of the ABJ(M) model.

• 184.
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics. Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
q-Virasoro Modular Triple2019In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 366, no 1, p. 397-422Article in journal (Refereed)

Inspired by 5d supersymmetric Yang-Mills theories placed on the compact space S5, we propose an intriguing algebraic construction for the q-Virasoro algebra. We show that, when multiple q-Virasoro chiral sectors have to be fused together, a natural SL(3,Z) structure arises. This construction, which we call the modular triple, is consistent with the observed triple factorization properties of supersymmetric partition functions derived from localization arguments. We also give a 2d CFT-like construction of the modular triple, and conjecture for the first time a (non-local) Lagrangian formulation for a q-Virasoro model, resembling ordinary Liouville theory.

• 185.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
The Vladimirov Heat Kernel in the Program of Jorgenson-Lang2018Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 186.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Uppsala Univ, Dept Math, Uppsala, Sweden..
The A∞-Property of the Kolmogorov Measure2017In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 10, no 7, p. 1709-1756Article in journal (Refereed)

We consider the Kolmogorov-Fokker-Planck operator K := Sigma(m)(i=1) partial derivative(xixi) + Sigma(m)(i=1) x(i)partial derivative(yi) - partial derivative(t) in unbounded domains of the form Omega={(x,x(m),y,y(m),t) RN+1 vertical bar x(m)>psi(x,y,t)}. Concerning and psi, we assume that Omega is what we call an (unbounded) admissible Lip(K)-domain: psi satisfies a uniform Lipschitz condition, adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator K, as well as an additional regularity condition formulated in terms of a Carleson measure. We prove that in admissible Lip(K)-domains the associated parabolic measure is absolutely continuous with respect to a surface measure and that the associated Radon-Nikodym derivative defines an A(infinity) weight with respect to this surface measure. Our result is sharp.

• 187.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Reflected BSDE of Wiener-Poisson type in time-dependent domains2016In: Stochastic Models, ISSN 1532-6349, E-ISSN 1532-4214, Vol. 32, no 2, p. 275-300Article in journal (Refereed)

In the paper we study multi-dimensional reflected backward stochasticdifferential equations driven by Wiener-Poisson type processes. We prove existence and uniqueness of solutions, with reflection in the inward spatial normal direction, in the setting of certain time-dependent domains.

• 188.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
On the parabolic Lipschitz approximation of parabolic uniform rectifiable sets2017In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 33, no 4, p. 1397-1422Article in journal (Refereed)

We prove the existence of big pieces of regular parabolic Lip-schitz graphs for a class of parabolic uniform rectifiable sets satisfying what we call a synchronized two cube condition. An application to the fine properties of parabolic measure is given.

• 189.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
On the properties of nonlinear nonlocal operators arising in neural field models2013In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 398, no 1, p. 335-351Article in journal (Refereed)

We study the existence and continuous dependence of stationary solutions of the one-population Wilson-Cowan model on the steepness of the firing rate functions. We investigate the properties of the nonlinear nonlocal operators which arise when formulating the stationary one-population Wilson-Cowan model as a fixed point problem. The theory is used to study the existence and continuous dependence of localized stationary solutions of this model on the steepness of the firing rate functions. The present work generalizes and complements previously obtained results as we relax on the assumptions that the firing rate functions are given by smoothed Heaviside functions.

• 190.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
A Brownian optimal switching problem under incomplete information2018In: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 23, article id 67Article in journal (Refereed)

In this paper we study an incomplete information optimal switching problem in which the manager only has access to noisy observations of the underlying Brownian motion {W-t}(t)>= 0. The manager can, at a fixed cost, switch between having the production facility open or closed and must find the optimal management strategy using only the noisy observations. Using the theory of linear stochastic filtering, we reduce the incomplete information problem to a full information problem, show that the value function is non-decreasing with the amount of information available, and that the value function of the incomplete information problem converges to the value function of the corresponding full information problem as the noise in the observed process tends to 0. Our approach is deterministic and relies on the PDE-representation of the value function.

• 191.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Optimal Switching Problems and Related Equations2015Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of five scientific papers dealing with equations related to the optimal switching problem, mainly backward stochastic differential equations and variational inequalities. Besides the scientific papers, the thesis contains an introduction to the optimal switching problem and a brief outline of possible topics for future research.

Paper I concerns systems of variational inequalities with operators of Kolmogorov type. We prove a comparison principle for sub- and supersolutions and prove the existence of a solution as the limit of solutions to iteratively defined interconnected obstacle problems. Furthermore, we use regularity results for a related obstacle problem to prove Hölder continuity of this solution.

Paper II deals with systems of variational inequalities in which the operator is of non-local type. By using a maximum principle adapted to this non-local setting we prove a comparison principle for sub- and supersolutions. Existence of a solution is proved using this comparison principle and Perron's method.

In Paper III we study backward stochastic differential equations in which the solutions are reflected to stay inside a time-dependent domain. The driving process is of Wiener-Poisson type, allowing for jumps. By a penalization technique we prove existence of a solution when the bounding domain has convex and non-increasing time slices. Uniqueness is proved by an argument based on Ito's formula.

Paper IV and Paper V concern optimal switching problems under incomplete information. In Paper IV, we construct an entirely simulation based numerical scheme to calculate the value function of such problems. We prove the convergence of this scheme when the underlying processes fit into the framework of Kalman-Bucy filtering. Paper V contains a deterministic approach to incomplete information optimal switching problems. We study a simplistic setting and show that the problem can be reduced to a full information optimal switching problem. Furthermore, we prove that the value of information is positive and that the value function under incomplete information converges to that under full information when the noise in the observation vanishes.

1. Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type
Open this publication in new window or tab >>Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type
2014 (English)In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 193, no 4, p. 1213-1247Article in journal (Refereed) Published
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-192711 (URN)10.1007/s10231-013-0325-y (DOI)000339962000017 ()
Available from: 2013-01-25 Created: 2013-01-24 Last updated: 2017-12-06Bibliographically approved
2. Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness
Open this publication in new window or tab >>Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness
2014 (English)In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 145, no 3-4, p. 407-432Article in journal (Refereed) Published
##### Abstract [en]

In this paper we study viscosity solutions to the system  \begin{eqnarray*}&&\min\biggl\{-\mathcal{H}u_i(x,t)-\psi_i(x,t),u_i(x,t)-\max_{j\neq i}(-c_{i,j}(x,t)+u_j(x,t))\biggr\}=0,\notag\\&&u_i(x,T)=g_i(x),\ i\in\{1,\dots,d\},\end{eqnarray*}where $(x,t)\in\mathbb R^{N}\times [0,T]$. Concerning $\mathcal{H}$ we assume that $\mathcal{H}=\mathcal{L}+\mathcal{I}$ where$\mathcal{L}$ is a linear, possibly degenerate, parabolic operator of second order and $\mathcal{I}$ is a non-local integro-partial differential operator. A special case of this type of system of variational inequalities with terminal data occurs in the context of optimal switching problems when thedynamics of the underlying state variables is described by $N$-dimensional Levy processes.  We establish a general comparison principle for viscosity sub- and supersolutions to the system under mild regularity, growth andstructural assumptions on the data, i.e., on the operator $\mathcal{H}$ and on continuous functions $\psi_i$, $c_{i,j}$, and$g_i$.   Using the comparison principle we establish the existence of a unique viscosity solution $(u_1,\dots,u_d)$  to the system by using Perron's method. Our contribution, compared to the existing literature, is that we establish existence and uniqueness of viscosity solutions in the setting of Levy processes and non-local operators with no sign assumption on the switching costs $\{c_{i,j}\}$ and allowing $c_{i,j}$  to depend on $x$ as well as $t$.

Mathematics
##### Identifiers
urn:nbn:se:uu:diva-204876 (URN)10.1007/s00229-014-0683-9 (DOI)000343881600008 ()
Available from: 2013-08-12 Created: 2013-08-12 Last updated: 2017-12-06Bibliographically approved
3. Reflected BSDE of Wiener-Poisson type in time-dependent domains
Open this publication in new window or tab >>Reflected BSDE of Wiener-Poisson type in time-dependent domains
2016 (English)In: Stochastic Models, ISSN 1532-6349, E-ISSN 1532-4214, Vol. 32, no 2, p. 275-300Article in journal (Refereed) Published
##### Abstract [en]

In the paper we study multi-dimensional reflected backward stochasticdifferential equations driven by Wiener-Poisson type processes. We prove existence and uniqueness of solutions, with reflection in the inward spatial normal direction, in the setting of certain time-dependent domains.

##### Keywords
Backward stochastic differential equation, convex domain, reflected backward stochastic differential equation, time-dependent domain, 60H10, 60H20
##### National Category
Mathematical Analysis Probability Theory and Statistics
##### Identifiers
urn:nbn:se:uu:diva-224258 (URN)10.1080/15326349.2015.1116011 (DOI)000377141900005 ()
Available from: 2014-05-07 Created: 2014-05-07 Last updated: 2017-12-05Bibliographically approved
4. Optimal switching problems under partial information
Open this publication in new window or tab >>Optimal switching problems under partial information
2015 (English)In: Monte Carlo Methods and Applications, ISSN 1569-3961, Vol. 21, no 2, p. 91-120Article in journal (Refereed) Published
##### Abstract [en]

In this paper we formulate and study an optimal switching problem under partial information. In our model the agent/manager/investor attempts to maximize the expected reward by switching between different states/investments. However, he is not fully aware of his environment and only an observation process, which contains partial information about the environment/underlying, is accessible. It is based on the partial information carried by this observation process that all decisions must be made. We propose a probabilistic numerical algorithm based on dynamic programming, regression Monte Carlo methods, and stochastic filtering theory to compute the value function. In this paper, the approximation of the value function and the corresponding convergence result are obtained when the underlying and observation processes satisfy the linear Kalman-Bucy setting. A numerical example is included to show some specifc features of partial information.

Mathematics
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-219911 (URN)10.1515/mcma-2014-0013 (DOI)
Available from: 2014-03-06 Created: 2014-03-06 Last updated: 2016-04-13Bibliographically approved
5. A Brownian optimal switching problem under incomplete information
Open this publication in new window or tab >>A Brownian optimal switching problem under incomplete information
2018 (English)In: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 23, article id 67Article in journal (Refereed) Published
##### Abstract [en]

In this paper we study an incomplete information optimal switching problem in which the manager only has access to noisy observations of the underlying Brownian motion {W-t}(t)>= 0. The manager can, at a fixed cost, switch between having the production facility open or closed and must find the optimal management strategy using only the noisy observations. Using the theory of linear stochastic filtering, we reduce the incomplete information problem to a full information problem, show that the value function is non-decreasing with the amount of information available, and that the value function of the incomplete information problem converges to the value function of the corresponding full information problem as the noise in the observed process tends to 0. Our approach is deterministic and relies on the PDE-representation of the value function.

##### Keywords
optimal switching problem, stochastic filtering, incomplete information
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-242556 (URN)10.1214/18-ECP146 (DOI)000445501300003 ()
Available from: 2015-01-27 Created: 2015-01-27 Last updated: 2018-11-22Bibliographically approved
• 192.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Cherry flows with non-trivial attractors2019In: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 244, no 3, p. 243-253Article in journal (Refereed)

We provide an example of a Cherry flow (i.e. a C-infinity flow on the 2-dimensional torus with a sink and a saddle) having a quasi-minimal set which is an attractor. The first return map for such a flow, also constructed in the paper, is a C-infinity circle map having a flat interval and a non-trivial wandering interval.

• 193.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Studies of the Boundary Behaviour of Functions Related to Partial Differential Equations and Several Complex Variables2015Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of a comprehensive summary and six scientific papers dealing with the boundary behaviour of functions related to parabolic partial differential equations and several complex variables.

Paper I concerns solutions to non-linear parabolic equations of linear growth. The main results include a backward Harnack inequality, and the Hölder continuity up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary of an NTA cylinder. It is also shown that the Riesz measure associated with such solutions has the doubling property.

Paper II is concerned with solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a weight in the Muckenhoupt class 1+2/n. Two main results are that non-negative solutions which vanish continuously on the lateral boundary of an NTA cylinder satisfy a backward Harnack inequality and that the quotient of two such functions is Hölder continuous up to the boundary. Another result is that the parabolic measure associated to such equations has the doubling property.

In Paper III, it is shown that a bounded pseudoconvex domain whose boundary is α-Hölder for each 0<α<1, is hyperconvex. Global estimates of the exhaustion function are given.

In Paper IV, it is shown that on the closure of a domain whose boundary locally is the graph of a continuous function, all plurisubharmonic functions with continuous boundary values can be uniformly approximated by smooth plurisubharmonic functions defined in neighbourhoods of the closure of the domain.

Paper V studies  Poletsky’s notion of plurisubharmonicity on compact sets. It is shown that a function is plurisubharmonic on a given compact set if, and only if, it can be pointwise approximated by a decreasing sequence of smooth plurisubharmonic functions defined in neighbourhoods of the set.

Paper VI introduces the notion of a P-hyperconvex domain. It is shown that in such a domain, both the Dirichlet problem with respect to functions plurisubharmonic on the closure of the domain, and the problem of approximation by smooth plurisubharmoinc functions in neighbourhoods of the closure of the domain have satisfactory answers in terms of plurisubharmonicity on the boundary.

1. Boundary estimates for non-negative solutions to non-linear parabolic equations
Open this publication in new window or tab >>Boundary estimates for non-negative solutions to non-linear parabolic equations
2015 (English)In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 54, no 1, p. 847-879Article in journal (Refereed) Published
##### Abstract [en]

This paper is mainly devoted to  the boundary behavior of non-negative solutions to the equation$\H u =\partial_tu-\nabla\cdot \operatorname{A}(x,t,\nabla u) = 0$in domains of the form $\Omega_T=\Omega\times (0,T)$ where $\Omega\subset\mathbb R^n$ is a bounded non-tangentially accessible (NTA) domain and $T>0$. The assumptions we impose on$A$ imply that $H$ is a non-linear parabolic operator with linear growth. Our main results include a backward Harnackinequality, and the H\"older continuity  up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary. Furthermore, to each such solution one can associate a natural Riesz measure supported on the lateral boundary and one of our main result is a proof of the doubling property for this measure. Our results generalize,  to the setting of non-linear equations with linear growth, previous results concerningthe boundary behaviour, in Lipschitz cylinders and time-independent NTA-cylinders, established for  non-negative solutions to equations of the type $\partial_tu-\nabla\cdot (\operatorname{A}(x,t)\nabla u)=0$, where $A$ is a measurable, bounded and uniformly positive definite matrix-valued function. In the latter case the measure referred to above is essentially the caloric or parabolic measure associated to  the operator and related to Green's function. At the end of the paper we also remark that our arguments are general enough to allow us to generalize parts of our results to general fully non-linear parabolic partial differential equations of second order.

Mathematics
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-204871 (URN)10.1007/s00526-014-0808-8 (DOI)000359941200033 ()
Available from: 2013-08-12 Created: 2013-08-12 Last updated: 2017-12-06Bibliographically approved
2. Boundary estimates for solutions to linear degenerate parabolic equations
Open this publication in new window or tab >>Boundary estimates for solutions to linear degenerate parabolic equations
2015 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 259, no 8, p. 3577-3614Article in journal (Refereed) Published
##### Abstract [en]

Let $\Omega\subset\mathbb R^n$ be a bounded NTA-domain and let $\Omega_T=\Omega\times (0,T)$ for some $T>0$.  We study the boundary behaviour of non-negativesolutions to the equation$Hu =\partial_tu-\partial_{x_i}(a_{ij}(x,t)\partial_{x_j}u) = 0, \ (x,t)\in \Omega_T.$We assume that $A(x,t)=\{a_{ij}(x,t)\}$ is measurable, real, symmetric and that\begin{equation*}\beta^{-1}\lambda(x)|\xi|^2\leq \sum_{i,j=1}^na_{ij}(x,t)\xi_i\xi_j\leq\beta\lambda(x)|\xi|^2\mbox{ for all }(x,t)\in\mathbb R^{n+1},\ \xi\in\mathbb R^{n},\end{equation*}for some constant $\beta\geq 1$ and for some non-negative and real-valued function $\lambda=\lambda(x)$belonging to the Muckenhoupt class $A_{1+2/n}(\mathbb R^n)$.Our main results includethe doubling property of the associated parabolic measure andthe H\"older continuity  up to the boundary of quotients of non-negative solutionswhich vanish continuously on a portion of the boundary. Our resultsgeneralize previous results of Fabes, Kenig, Jerison, Serapioni, see \cite{FKS}, \cite{FJK}, \cite{FJK1}, to a parabolic setting.

Mathematics
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-204869 (URN)10.1016/j.jde.2015.04.028 (DOI)000363434300004 ()
Available from: 2013-08-12 Created: 2013-08-12 Last updated: 2017-12-06Bibliographically approved
3. A note on the hyperconvexity of pseudoconvex domains beyond Lipschitz regularity
Open this publication in new window or tab >>A note on the hyperconvexity of pseudoconvex domains beyond Lipschitz regularity
2015 (English)In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, no 3, p. 531-545Article in journal (Refereed) Published
##### Abstract [en]

We show that bounded pseudoconvex domains that are Hölder continuous for all α < 1 are hyperconvex, extending the well-known result by Demailly (Math. Z. 184 1987) beyond Lipschitz regularity.

##### Keywords
plurisubharmonic functions, continuous boundary, hyperconvexity, bounded exhaustion function, Hölder for all exponents, log-lipschitz, boundary regularity, Reinhardt domains.
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-251330 (URN)10.1007/s11118-015-9486-1 (DOI)000365769100010 ()
Available from: 2015-04-15 Created: 2015-04-15 Last updated: 2017-12-04Bibliographically approved
4. Approximation of plurisubharmonic functions
Open this publication in new window or tab >>Approximation of plurisubharmonic functions
2016 (English)In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 61, no 1, p. 23-28Article in journal (Refereed) Published
##### Abstract [en]

We extend a result by Fornaaess and Wiegerinck [Ark. Mat. 1989;27:257-272] on plurisubharmonic Mergelyan type approximation to domains with boundaries locally given by graphs of continuous functions.

##### Keywords
plurisubharmonic functions, approximation, continuous boundary, boundary regularity, Mergelyan type approximation
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-251324 (URN)10.1080/17476933.2015.1053473 (DOI)000365643500003 ()
Available from: 2015-04-15 Created: 2015-04-15 Last updated: 2017-12-04Bibliographically approved
5. Plurisubharmonic functions on compact sets
Open this publication in new window or tab >>Plurisubharmonic functions on compact sets
2012 (English)In: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 106, p. 133-144Article in journal (Refereed) Published
##### Abstract [en]

Poletsky has introduced a notion of plurisubharmonicity for functions defined on compact sets in C-n. We show that these functions can be completely characterized in terms of monotone convergence of plurisubharmonic functions defined on neighborhoods of the compact.

##### Keywords
plurisubharmonic functions on compacts, Jensen measures, monotone convergence
##### National Category
Natural Sciences
##### Identifiers
urn:nbn:se:uu:diva-189931 (URN)10.4064/ap106-0-11 (DOI)000311525700011 ()
Available from: 2013-01-04 Created: 2013-01-04 Last updated: 2017-12-06Bibliographically approved
6. Plurisubharmonic approximation and boundary values of plurisubharmonic functions
Open this publication in new window or tab >>Plurisubharmonic approximation and boundary values of plurisubharmonic functions
2014 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 413, no 2, p. 700-714Article in journal (Refereed) Published
##### Abstract [en]

We study the problem of approximating plurisubharmonic functions on a bounded domain Omega by continuous plurisubharmonic functions defined on neighborhoods of (Omega) over bar. It turns out that this problem can be linked to the problem of solving a Dirichlet type problem for functions plurisubharmonic on the compact set (Omega) over bar in the sense of Poletsky. A stronger notion of hyperconvexity is introduced to fully utilize this connection, and we show that for this class of domains the duality between the two problems is perfect. In this setting, we give a characterization of plurisubharmonic boundary values, and prove some theorems regarding the approximation of plurisubharmonic functions.

##### Keywords
Plurisubharmonic functions on compacts, Jensen measures, Approximation, Plurisubharmonic extension, Plurisubharmonic boundary values
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-220970 (URN)10.1016/j.jmaa.2013.12.041 (DOI)000331344600014 ()
Available from: 2014-03-26 Created: 2014-03-24 Last updated: 2017-12-05Bibliographically approved
• 194.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Lebesgue Theory: A Brief Overview2016Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 195.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Option Pricing with Extreme Events2011Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 196.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Morse homology of RPn2013Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 197.
Aalto Univ, Dept Math & Syst Anal, POB 11100, FI-00076 Aalto, Finland.
Univ Manitoba, Dept Math, Winnipeg, MB R3T 2N2, Canada. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
A Model of the Teichmüller space of genus-zero bordered surfaces by period maps2019In: Conformal Geometry and Dynamics, ISSN 1088-4173, E-ISSN 1088-4173, Vol. 23, p. 32-51Article in journal (Refereed)

We consider Riemann surfaces Σ withnborders homeomorphic to S1 and no handles. Using generalized Grunsky operators, we define a period mapping from the infinite-dimensional Teichmüller space of surfaces of this type into the unit ball in the linear space of operators on an n-fold direct sum of Bergman spaces of the disk. We show that this period mapping isholomorphic and injective.

• 198.
Aalto Univ, Dept Math & Syst Anal, POB 11100, FI-00076 Aalto, Finland.
Univ Manitoba, Dept Math, Winnipeg, MB R3T 2N2, Canada. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Dirichlet's problem and Sokhotski-Plemelj's jump formula on Weil-Petersson-Class quasidisks2016In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 41, no 1, p. 119-127Article in journal (Refereed)

We show the solvability of the Dirichlet problem on Weil–Petersson class quasidisks and establish a Sokhotski–Plemelj jump formula for Weil–Petersson class quasicircles. Furthermore we show that the resulting Cauchy projections are bounded. In both cases the boundary data belongs to a certain conformally invariant Besov space. Moreover we show that the WP-class quasicircles are chord-arc curves.

Quasiconformal Teichmüller theory as an analytical foundation for two dimensional conformal field theoryIn: Contemporary Mathematics, ISSN 0271-4132, E-ISSN 1098-3627Article in journal (Refereed)

The functorial mathematical definition of conformal field theory was first formulated approximately 30 years ago. The underlying geometric category is based on the moduli space of Riemann surfaces with parametrized boundary components and the sewing operation. We survey the recent and careful study of these objects, which has led to significant connections with quasiconformal Teichmüller theory and geometric function theory. In particular we propose that the natural analytic setting for conformal field theory is the moduli space of Riemann surfaces with so-called Weil-Petersson class parametrizations. A collection of rigorous analytic results is advanced here as evidence. This class of parametrizations has the required regularity for CFT on one hand, and on the other hand are natural and of interest in their own right in geometric function theory.

• 200.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Analytical Valuation of American-Style Asian Options under Jump-Diffusion Processes2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
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