| 1. |
- Kurujyibwami, Celestin
(författare)
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Admissible transformations and the group classification of Schrödinger equations
- 2017
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We study admissible transformations and solve group classification problems for various classes of linear and nonlinear Schrödinger equations with an arbitrary number n of space variables.The aim of the thesis is twofold. The first is the construction of the new theory of uniform seminormalized classes of differential equations and its application to solving group classification problems for these classes. Point transformations connecting two equations (source and target) from the class under study may have special properties of semi-normalization. This makes the group classification of that class using the algebraic method more involved. To extend this method we introduce the new notion of uniformly semi-normalized classes. Various types of uniform semi-normalization are studied: with respect to the corresponding equivalence group, with respect to a proper subgroup of the equivalence group as well as the corresponding types of weak uniform semi-normalization. An important kind of uniform semi-normalization is given by classes of homogeneous linear differential equations, which we call uniform semi-normalization with respect to linear superposition of solutions.The class of linear Schrödinger equations with complex potentials is of this type and its group classification can be effectively carried out within the framework of the uniform semi-normalization. Computing the equivalence groupoid and the equivalence group of this class, we show that it is uniformly seminormalized with respect to linear superposition of solutions. This allow us to apply the version of the algebraic method for uniformly semi-normalized classes and to reduce the group classification of this class to the classification of appropriate subalgebras of its equivalence algebra. To single out the classification cases, integers that are invariant under equivalence transformations are introduced. The complete group classification of linear Schrödinger equations is carried out for the cases n = 1 and n = 2.The second aim is to study group classification problem for classes of generalized nonlinear Schrödinger equations which are not uniformly semi-normalized. We find their equivalence groupoids and their equivalence groups and then conclude whether these classes are normalized or not. The most appealing classes are the class of nonlinear Schrödinger equations with potentials and modular nonlinearities and the class of generalized Schrödinger equations with complex-valued and, in general, coefficients of Laplacian term. Both these classes are not normalized. The first is partitioned into an infinite number of disjoint normalized subclasses of three kinds: logarithmic nonlinearity, power nonlinearity and general modular nonlinearity. The properties of the Lie invariance algebras of equations from each subclass are studied for arbitrary space dimension n, and the complete group classification is carried out for each subclass in dimension (1+2). The second class is successively reduced into subclasses until we reach the subclass of (1+1)-dimensional linear Schrödinger equations with variable mass, which also turns out to be non-normalized. We prove that this class is mapped by a family of point transformations to the class of (1+1)-dimensional linear Schrödinger equations with unique constant mass.
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| 2. |
- Malik, Adam, 1991
(författare)
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Mathematical Modelling of Cell Migration and Polarization
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Cell migration plays a fundamental role in both development and disease. It is a complex process during which cells interact with one another and with their local environment. Mathematical modelling offers tools to investigate such processes and can give insights into the underlying biological details, and can also guide new experiments. The first two papers of this thesis are concerned with modelling durotaxis, which is the phenomena where cells migrate preferentially up a stiffness gradient. Two distinct mechanisms which potentially drive durotaxis are investigated. One is based on the hypothesis that adhesion sites of migrating cells become reinforced and have a longer lifespan on stiffer substrates. The second mechanism is based on cells being able to generate traction forces, the magnitude of which depend on the stiffness of the substrate. We find that both mechanisms can indeed give rise to biased migration up a stiffness gradient. Our results encourages new experiments which could determine the importance of the two mechanisms in durotaxis. The third paper is devoted to a population-level model of cancer cells in the brain of mice. The model incorporates diffusion tensor imaging data, which is used to guide the migration of the cells. Model simulations are compared to experimental data, and highlights the model’s difficulty in producing irregular growth patterns observed in the experiments. As a consequence, the findings encourage further model development. The fourth paper is concerned with modelling cell polarization, in the absence of environmental cues, referred to as spontaneous symmetry breaking. Polarization is an important part of cell migration, but also plays a role during division and differentiation. The model takes the form of a reaction diffusion system in 3D and describes the spatio-temporal evolution of three forms of Cdc42 in the cell. The model is able to produce biologically relevant patterns, and numerical simulations show how model parameters influence key features such as pattern formation and time to polarization.
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| 3. |
- Vasilis, Jonatan, 1981
(författare)
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Harmonic measures
- 2010
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis uses both analytic and probabilistic methods to study continuous and discrete problems. The main areas of study are the asymptotic properties of p-harmonic measure, and various aspects of the square root of the Poisson kernel. Fix a domain and a boundary point, subject to certain regularity conditions. Consider the part of the boundary that lies within a disc, centered at the fixed boundary point. It is shown that as the radius of the disc tends to zero, the p-harmonic measure of the boundary set decays as an explicitly given power of the radius. The square root of the Poisson kernel is studied in both continuous and discrete settings. In the continuous case the domain is the unit disc, and a Hardy space related to the square root of the Poisson kernel is defined. The main result is that, as opposed to the classical Hardy space, the positive functions do not admit a characterization in terms of an Orlicz space. Similar results are given also in the discrete case, where the domain is instead a regular tree. Further results in the discrete setting include the construction of a nearest neighbor random walk on the tree with exit distribution determined by powers of the Poisson kernel. The minimally thin sets of these random walks are characterized. Finally, we suggest a generalization of a two-dimensional geometric result – the ring lemma – to three dimensions.
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| 4. |
- Hultgren, Jakob, 1986
(författare)
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Real and complex Monge-Ampère equations, statistical mechanics and canonical metrics
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Recent decades has seen a strong trend in complex geometry to study canonical metrics and the way they relate to geometric analysis, algebraic geometry and probability theory. This thesis consists of four papers each contributing to this field. The first paper sets up a probabilistic framework for real Monge-Ampère equations on tori. We show that solutions to a large class of real Monge-Ampère equations arise as the many particle limit of certain permanental point processes. The framework can be seen as a real, compact analog of the probabilistic framework for Kähler-Einstein metrics on Kähler manifolds. The second paper introduces a variational approach in terms of optimal transport to real Monge-Ampère equations on compact Hessian manifolds. This is applied to prove existence and uniqueness results for various types of canonical Hessian metrics. The results can, on one hand, be seen as a first step towards a probabilistic approach to canonical metrics on Hessian manifolds and, on the other hand, as a remark on the Gross-Wilson and Kontsevich-Soibelmann conjectures in Mirror symmetry. The third paper introduces a new type of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, that generalises Kähler-Einstein metrics. Existence and uniqueness theorems are given as well as a proof of one direction of a generalised Yau-Tian-Donaldson conjecture, establishing a connection between this new notion of canonical metrics and stability in algebraic geometry. The fourth paper gives a necessary and sufficient condition for existence of coupled Kähler-Einstein metrics on toric manifolds in terms of a collection of associated polytopes, proving this generalised Yau-Tian-Donaldson conjecture in the toric setting.
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| 5. |
- La Cognata, Cristina, 1985-
(författare)
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High order summation-by-parts based approximations for discontinuous and nonlinear problems
- 2017
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Numerical approximations using high order finite differences on summation-byparts (SBP) form are investigated for discontinuous and fully nonlinear systems of partial differential equations. Stability and conservation properties of the approximations are obtained through a weak imposition of interface and boundary conditions with the simultaneous-approximation-term (SAT) technique. The SBP-SAT approximations replicate the continuous integration by parts rule. From this property, well-posedness and integral properties of the continuous problem are mimicked, and energy estimates leading to stability are obtained.The first part of the thesis focuses on the simulations of discontinuous linear advection problems. An artificial interface is introduced, separating parts of the spatial domain characterized by different wave speeds. A set of flexible stability conditions at the interface are derived, which can be adapted to yield conservative or non-conservative approximations. This model can be interpreted as a simplified version of nonlinear problems involving jumps at shocks, or as a prototypical of wave propagation through different materials.In the second part of the thesis, the vorticity/stream function formulation of the nonlinear momentum equation for an incompressible inviscid fluid is considered. SBP operators are used to derive a new Arakawa-like Jacobian with mimetic properties by combining different consistent approximations of the convection terms. Energy and enstrophy conservation is obtained for periodic problems using schemes with arbitrarily high order of accuracy. These properties are crucial for long-term numerical calculations in climate and weather forecasts or ocean circulation predictions.The third and final contribution of the thesis is dedicated to the incompressible Navier-Stokes problem. First, different completely general formulations of energy bounding boundary conditions are derived for the nonlinear equations. The boundary conditions can be used at both far field and solid wall boundaries. The discretisation in time and space with weakly imposed initial and boundary conditions using the SBP-SAT framework is proved to be stable and the divergence free condition is approximated with the design order of the scheme. Next, the same formulations are considered in a linearised setting, whereupon the spectra associated with the initial boundary value problem and its SBP-SAT discretisation are derived using the Laplace-Fourier technique. The influence of different boundary conditions on the spectrum and in particular the convergence to steady state is studied.
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| 6. |
- Raufi, Hossein, 1983
(författare)
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Positive vector bundles in complex and convex geometry
- 2014
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis concerns various aspects of the geometry of holomorphic vector bundles and their analytical theory which all, vaguely speaking, are related to the notion of positive curvature in general, and L^2-methods for the dbar-equation in particular. The thesis contains four papers.In Paper I we introduce and study the notion of singular hermitian metrics on holomorphic vector bundles. We define what it means for such metrics to be positively curved in the sense of Griffiths, and investigate the assumptions needed in order to define the curvature tensor of such metrics as currents with measure coefficients. We also investigate the regularisation of such metrics.In Paper II we prove the Nakano vanishing theorem with Hörmander L^2-estimates on a compact Kähler manifold using Siu's d-dbar-Bochner-Kodaira method. We then introduce the singular hermitian metrics and regularisation results of Paper I, and use these to prove a Demailly-Nadel type of vanishing theorem for vector bundles over Riemann surfaces.A fundamental tool in complex geometry closely related to the notion of positivity is the Ohsawa-Takegoshi extension theorem. In Paper III the d-dbar-Bochner-Kodaira method is applied to extend this theorem from line bundles to vector bundles over compact Kähler manifolds. Another way of obtaining a vector bundle version of this theorem is to reduce it to the line bundle setting through the useful algebraic geometric procedure of studying the projective bundle associated with the vector bundle. In Paper III we also investigate the relationship between these two different approaches.On a trivial line bundle, a positively curved metric is the complex-analytic counterpart of a log concave function in the real-variable setting. In Paper IV we extend this link between complex and convex geometry to trivial vector bundles. We define two new notions of log concavity for real, matrix-valued functions, corresponding to Griffiths and Nakano positivity, and we prove a matrix-valued Prekopa theorem.
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| 7. |
- Schoug, Lukas
(författare)
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On two-dimensional conformal geometry related to the Schramm-Loewner evolution
- 2019
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis contains three papers, one introductory chapter and one chapter with overviews of the papers and some additional results. The topic of this thesis is the geometry of models related to the Schramm-Loewner evolution.In Paper I, we derive a multifractal boundary spectrum for SLEκ(ρ) processes with κ<4 and ρ chosen so that the curves hit the boundary. That is, we study the sets of points where the curves hit the boundary with a prescribed ``angle'', and compute the Hausdorff dimension of those sets. We study the moments of the spatial derivatives of the conformal maps gt, use Girsanov's theorem to change to an appropriate measure, and use the imaginary geometry coupling to derive a correlation estimate.In Paper II, we study the two-valued sets of the Gaussian free field, that is, the local sets such the associated harmonic function only takes two values. It turns out that the real part of the imaginary chaos is large close to these sets. We use this to derive a correlation estimate which lets us compute the Hausdorff dimensions of the two-valued sets.Paper III is dedicated to studying quasislits, that is, images of the segment [0,i] under quasiconformal maps of the upper half-plane into itself, fixing ∞, generated by driving the Loewner equation with a Lip-1/2 function. We improve estimates on the cones containing the curves, and hence on the Hölder regularity of the curves, in terms of the Lip-1/2 seminorm of the driving function.
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| 8. |
- Kirchner, Kristin, 1987
(författare)
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Numerical Approximation of Solutions to Stochastic Partial Differential Equations and Their Moments
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The first part of this thesis focusses on the numerical approximation of the first two moments of solutions to parabolic stochastic partial differential equations (SPDEs) with additive or multiplicative noise. More precisely, in Paper I an earlier result (A. Lang, S. Larsson, and Ch. Schwab, Covariance structure of parabolic stochastic partial differential equations, Stoch. PDE: Anal. Comp., 1(2013), pp. 351–364), which shows that the second moment of the solution to a parabolic SPDE driven by additive Wiener noise solves a well-posed deterministic space-time variational problem, is extended to the class of SPDEs with multiplicative Lévy noise. In contrast to the additive case, this variational formulation is not posed on Hilbert tensor product spaces as trial–test spaces, but on projective–injective tensor product spaces, i.e., on non-reflexive Banach spaces. Well-posedness of this variational problem is derived for the case when the multiplicative noise term is sufficiently small. This result is improved in Paper II by disposing of the smallness assumption. Furthermore, the deterministic equations in variational form are used to derive numerical methods for approximating the first and the second moment of solutions to stochastic ordinary and partial differential equations without Monte Carlo sampling. Petrov–Galerkin discretizations are proposed and their stability and convergence are analyzed. In the second part the numerical solution of fractional order elliptic SPDEs with spatial white noise is considered. Such equations are particularly interesting for applications in statistics, as they can be used to approximate Gaussian Matérn fields. Specifically, in Paper III a numerical scheme is proposed, which is based on a finite element discretization in space and a quadrature for an integral representation of the fractional inverse involving only non-fractional inverses. For the resulting approximation, an explicit rate of convergence to the true solution in the strong mean-square sense is derived. Subsequently, in Paper IV weak convergence of this approximation is established. Finally, in Paper V a similar method, which exploits a rational approximation of the fractional power operator instead of the quadrature, is introduced and its performance with respect to accuracy and computing time is compared to the quadrature approach from Paper III and to existing methods for inference in spatial statistics.
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| 9. |
- Muhumuza, Asaph Keikara, 1975-
(författare)
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Extreme points of the Vandermonde determinant in numerical approximation, random matrix theory and financial mathematics
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis discusses the extreme points of the Vandermonde determinant on various surfaces, their applications in numerical approximation, random matrix theory and financial mathematics. Some mathematical models that employ these extreme points such as curve fitting, data smoothing, experimental design, electrostatics, risk control in finance and method for finding the extreme points on certain surfaces are demonstrated.The first chapter introduces the theoretical background necessary for later chapters. We review the historical background of the Vandermonde matrix and its determinant, some of its properties that make it more applicable to symmetric polynomials, classical orthogonal polynomials and random matrices.The second chapter discusses the construction of the generalized Vandermonde interpolation polynomial based on divided differences. We explore further, the concept of weighted Fekete points and their connection to zeros of the classical orthogonal polynomials as stable interpolation points.The third chapter discusses some extended results on optimizing the Vandermonde determinant on a few different surfaces defined by univariate polynomials. The coordinates of the extreme points are shown to be given as roots of univariate polynomials.The fourth chapter describes the symmetric group properties of the extreme points of Vandermonde and Schur polynomials as well as application of these extreme points in curve fitting.The fifth chapter discusses the extreme points of Vandermonde determinant to number of mathematical models in random matrix theory where the joint eigenvalue probability density distribution of a Wishart matrix when optimized over surfaces implicitly defined by univariate polynomials.The sixth chapter examines some properties of the extreme points of the joint eigenvalue probability density distribution of the Wishart matrix and application of such in computation of the condition numbers of the Vandermonde and Wishart matrices. The seventh chapter establishes a connection between the extreme points of Vandermonde determinants and minimizing risk measures in financial mathematics. We illustrate this with an application to optimal portfolio selection.The eighth chapter discusses the extension of the Wishart probability distributions in higher dimension based on the symmetric cones in Jordan algebras. The symmetric cones form a basis for the construction of the degenerate and non-degenerate Wishart distributions.The ninth chapter demonstrates the connection between the extreme points of the Vandermonde determinant and Wishart joint eigenvalue probability distributions in higher dimension based on the boundary points of the symmetric cones in Jordan algebras that occur in both the discrete and continuous part of the Gindikin set.
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| 10. |
- Petersson, Andreas, 1990
(författare)
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Approximating Stochastic Partial Differential Equations with Finite Elements: Computation and Analysis
- 2019
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Stochastic partial differential equations (SPDE) must be approximated in space and time to allow for the simulation of their solutions. In this thesis fully discrete approximations of such equations are considered, with an emphasis on finite element methods combined with rational semigroup approximations. A quantity of interest for SPDE simulations often takes the form of an expected value of a functional applied to the solution. This is the major theme of this thesis, which divides into two minor themes. The first is how to analyze the error resulting from the fully discrete approximation of an SPDE with respect to a given functional, which is referred to as the weak error of the approximation. The second is how to efficiently compute the quantity of interest as well as the weak error itself. The Monte Carlo (MC) and multilevel Monte Carlo (MLMC) methods are common approaches for this. The thesis consists of five papers. In the first paper the additional error caused by MC and MLMC methods in simulations of the weak error is analyzed. Upper and lower bounds are derived for the different methods and simulations illustrate the results. The second paper sets up a framework for the analysis of the asymptotic mean square stability, the stability as measured in a quadratic functional, of a general stochastic recursion scheme, which is applied to several discretizations of an SPDE. In the third paper, a novel technique for efficiently generating samples of SPDE approximations is introduced, based on the computation of discrete covariance operators. The computational complexities of the resulting MC and MLMC methods are analyzed. The fourth paper considers the analysis of the weak error for the approximation of the semilinear stochastic wave equation. In the fifth paper, a Lyapunov equation is derived, which allows for the deterministic approximation of the expected value of a quadratic functional applied to the solution of an SPDE. The paper also includes an error analysis of an approximation of this equation and an analysis of the weak error, with respect to the quadratic functional, of an approximation of the considered SPDE.
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