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  • 1.
    Adamowicz, Tomasz
    et al.
    Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland.
    Lundström, Niklas L.P.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The boundary Harnack inequality for variable exponent p-Laplacian, Carleson estimates, barrier functions and p(⋅)-harmonic measures2016In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 195, no 2, p. 623-658Article in journal (Refereed)
    Abstract [en]

    We investigate various boundary decay estimates for p(⋅)-harmonic functions. For domains in Rn,n≥2satisfying the ball condition (C1,1-domains), we show the boundary Harnack inequality for p(⋅)-harmonic functions under the assumption that the variable exponent p is a bounded Lipschitz function. The proof involves barrier functions and chaining arguments. Moreover, we prove a Carleson-type estimate for p(⋅)-harmonic functions in NTA domains in Rn and provide lower and upper growth estimates and a doubling property for a p(⋅)-harmonic measure.

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  • 2.
    Amanuel, Meron
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    ON GENERATING THE PROBABILITY MASS FUNCTION USING FIBONACCI POWER SERIES2022Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    This thesis will focus on generating the probability mass function using Fibonacci sequenceas the coefficient of the power series.

    The discrete probability, named Fibonacci distribution,was formed by taking into consideration the recursive property of the Fibonacci sequence,the radius of convergence of the power series, and additive property of mutually exclusiveevents. This distribution satisfies the requisites of a legitimate probability mass function.

    It's cumulative distribution function and the moment generating function are then derived and the latter are used to generate moments of the distribution, specifically, the mean and the variance.

    The characteristics of some convergent sequences generated from the Fibonacci sequenceare found useful in showing that the limiting form of the Fibonacci distribution is a geometricdistribution. Lastly, the paper showcases applications and simulations of the Fibonacci distribution using MATLAB.

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  • 3.
    Andersson, Henrik
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Classification of spectral signatures in biological aerosols2013Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    In this thesis multivariate methods were used to evaluate pretreatment methods, such as normalization, as well as classification possibilities of data collected with Laser Induced Breakdown Spectroscopy (LIBS). The LIBS system that FOI is currently developing for the purpose of classifying biological airborne threats was used to collect data from ten different samples in a laboratory environment. Principal component analysis (PCA) shows that it is possible to observe differences between samples using the two types of data acquired from the LIBS system, i.e., 2D CCD camera images and 1D spectra extracted from the image. Further results using partial least squares discriminant analysis (PLS-DA) show that normalization of the data only has visual effects in the PCA score-plots and do not affect the models predictability. Results also show that cropping and binning the pixels in the image is possible to some extent without losing significant predictability.

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    Classification of spectral signatures in biological aerosols
  • 4.
    Bengzon, Fredrik
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larson, Mats G
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Adaptive finite element approximation of multiphysics problems: a fluid structure interaction model problem2010In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 84, no 12, p. 1451-1465Article in journal (Refereed)
    Abstract [en]

    We consider computation of the displacement of an elastic object immersed into a viscous incompressible flow. For simplicity, we assume that the mechanics of the solid is governed by linear elasticity and the motion of the fluid by the Stokes equation. We derive an a posteriori error estimate for this one way coupled problem using duality techniques. Based on these estimates we develop an adaptive algorithm that automatically constructs a suitable adapted mesh for the fluid and solid domains given goal quantities specified on the solid problem.

  • 5.
    Berglund, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Evaluating ∫0f(x)dx and ∫ab f(x)dx using residue calculus2014Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    In this essay we use complex analysis, in particular modern residue calculus, to compute certain Riemann integrals.

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    Andre_Berglund_Bachelors_Umea
  • 6.
    Berglund, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Numerical Simulations of Linear Stochastic Oscillators: driven by Wiener and Poisson processes2017Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    The main component of this essay is the numerical analysis of stochastic differential equations driven by Wiener and Poisson processes. In order to do this, we focus on two model problems, the geometric Brownian motion and the linear stochastic oscillator, studied in the literature for stochastic differential equations only driven by a Wiener process. This essay covers theoretical as well as numerical investigations of jump - or more specifically, Poisson - processes and how they influence the above model problems.

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  • 7.
    Bergqvist, Ewa
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Positive currents related to polynomial convexity1999Licentiate thesis, comprehensive summary (Other academic)
  • 8.
    Bergström, Martin
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Feature extraction and cluster analysis of oil slicks using optical satellite data2021Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
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  • 9.
    Bjurulf, Harald
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The Riemann Mapping Theorem2024Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    The Riemann Mapping Theorem is presented and proven in this essay. The theorem, first published 1851, is essential for the study of holomorphic functions on simply connected, proper subsets of C.

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  • 10.
    Björnfot, Fredrik
    Umeå University, Faculty of Science and Technology, Department of Physics.
    GDP Growth Rate Nowcasting and Forecasting2017Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    The main purpose of this project was to help Swedbank get a better understandingof how gross domestic product growth rate develops in the future froma data set of macroeconomic variables. Since GDP values are released long aftera quarter has ended Swedbank would like to have a model that could predictupcoming GDP from these data sets. This was solved by a combination ofgrowth rate predictions from a dynamic factor model, a vector autoregressivemodel and two machine learning models. The predictions were combined usinga weighting method called system averaging model where the model predictionwith least historical error receives the largest weight in the nal future prediction.In previous work a simple moving average model has been implementedto achieve this eect however there are several aws in a simple moving averagemodel. Most of these defects could in theory be avoided by using an exponentialweighting scheme instead. This resulted in the use of an exponentialweighting method that is used to calculate weights for future predictions. Themain conclusions from this project were that some predictions could get betterwhen removing bad performing models which had too large of a weight. Puttingtoo high weight on a single well performing model is also not optimal since thepredictions could get very unstable because of varying model performance. Theexponential weighting scheme worked well for some predictions however whenthe parameter , that controls how the weight is distributed between recent andhistorical errors, got too small a problem arose. Too few values were used toform the nal weights for the prediction and the estimate got unsteady results.

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  • 11.
    Bjørnstad Pedersen, Lars
    Umeå University, Faculty of Science and Technology, Department of Physics.
    An analysis of a shared mating in V2.2014Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    In this master thesis we investigate, from a topological point of view and without applying Thurston´s Theorem, why the mating of the so called basilica polynomial  and the dendrite  is shared with the mating of  and the dendrite . Both these matings equal the rational map .

    Defined in the thesis are for both matings homeomorphic changes of coordinates from the set  to the Fatou and Julia set of . Here  is the filled Julia set of  and is the -fixed point of .

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    An analysis of a shared mating in V2 - 2014 - Lars Pedersen
  • 12.
    Borgqvist, Johannes G.
    et al.
    Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, United Kingdom.
    Ohlsson, Fredrik
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Baker, Ruth E.
    Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, United Kingdom.
    Energy translation symmetries and dynamics of separable autonomous two-dimensional ODEs2023In: Physica D: Non-linear phenomena, ISSN 0167-2789, E-ISSN 1872-8022, Vol. 454, article id 133876Article in journal (Refereed)
    Abstract [en]

    We study symmetries in the phase plane for separable, autonomous two-state systems of ordinary differential equations (ODEs). We prove two main theoretical results concerning the existence and non-triviality of two orthogonal symmetries for such systems. In particular, we show that these symmetries correspond to translations in the internal energy of the system, and describe their action on solution trajectories in the phase plane. In addition, we apply recent results establishing how phase plane symmetries can be extended to incorporate temporal dynamics to these energy translation symmetries. Subsequently, we apply our theoretical results to the analysis of three models from the field of mathematical biology: a canonical biological oscillator model, the Lotka–Volterra (LV) model describing predator–prey dynamics, and the SIR model describing the spread of a disease in a population. We describe the energy translation symmetries in detail, including their action on biological observables of the models, derive analytic expressions for the extensions to the time domain, and discuss their action on solution trajectories.

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  • 13. Brännström, Niklas
    et al.
    Persson, Leif A.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A measure theoretic approach to linear inverse atmospheric dispersion problems2015In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 31, no 2, article id 025009Article in journal (Refereed)
    Abstract [en]

    Using measure theoretic arguments, we provide a general framework for describing and studying the general linear inverse dispersion problem where no a priori assumptions on the source function has been made (other than assuming that it is indeed a source, i.e. not a sink). We investigate the source-sensor relationship and rigorously state solvability conditions for when the inverse problem can be solved using a least-squares optimization method. That is, we derive conditions for when the least-squares problem is well-defined.

  • 14. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Augmented Lagrangian finite element methods for contact problems2019In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 53, no 1, p. 173-195Article in journal (Refereed)
    Abstract [en]

    We propose two different Lagrange multiplier methods for contact problems derived from the augmented Lagrangian variational formulation. Both the obstacle problem, where a constraint on the solution is imposed in the bulk domain and the Signorini problem, where a lateral contact condition is imposed are considered. We consider both continuous and discontinuous approximation spaces for the Lagrange multiplier. In the latter case the method is unstable and a penalty on the jump of the multiplier must be applied for stability. We prove the existence and uniqueness of discrete solutions, best approximation estimates and convergence estimates that are optimal compared to the regularity of the solution.

  • 15.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, Gower Street, London, United Kingdom.
    Hansbo, Peter
    Department of Mechanical Engineering, Jönköping University, 551 11, Jönköping, Sweden.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    CutFEM based on extended finite element spaces2022In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 152, no 2, p. 331-369Article in journal (Refereed)
    Abstract [en]

    We develop a general framework for construction and analysis of discrete extension operators with application to unfitted finite element approximation of partial differential equations. In unfitted methods so called cut elements intersected by the boundary occur and these elements must in general by stabilized in some way. Discrete extension operators provides such a stabilization by modification of the finite element space close to the boundary. More, precisely the finite element space is extended from the stable interior elements over the boundary in a stable way which also guarantees optimal approximation properties. Our framework is applicable to all standard nodal based finite elements of various order and regularity. We develop an abstract theory for elliptic problems and associated parabolic time dependent partial differential equations and derive a priori error estimates. We finally apply this to some examples of partial differential equations of different order including the interface problems, the biharmonic operator and the sixth order triharmonic operator.

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  • 16.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Department of Materials and Manufacturing, Jönköping University, Jönköping, Sweden.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions2024In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 93, no 345, p. 35-54Article in journal (Refereed)
    Abstract [en]

    We show error estimates for a cut finite element approximation of a second order elliptic problem with mixed boundary conditions. The error estimates are of low regularity type where we consider the case when the exact solution u ∈ Hs with s ∈ (1, 3/2]. For Nitsche type methods this case requires special handling of the terms involving the normal flux of the exact solution at the the boundary. For Dirichlet boundary conditions the estimates are optimal, whereas in the case of mixed Dirichlet-Neumann boundary conditions they are suboptimal by a logarithmic factor.

  • 17.
    Carlsson, Linus
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    An equivalence to the Gleason problem2010In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 370, no 2, p. 373-378Article in journal (Refereed)
    Abstract [en]

    In this article we study the Gleason problem locally. A new method for solving the Gleason A problem is presented. This is done by showing an equivalent statement to the Gleason A problem. In order to prove this statement, necessary and a sufficient conditions for a bounded domain to have the Gleason A property are found. Also an example of a bounded but not smoothly-bounded domain in C(n) is given, which satisfies the sufficient condition at the origin, and hence has the Gleason A property there.

  • 18.
    Carlsson, Linus
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Analytic properties in the spectrum of certain Banach algebras2009In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 261, no 1, p. 189-200Article in journal (Refereed)
  • 19.
    Carlsson, Linus
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nebenhülle and the Gleason problem2010In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 138, no 1, p. 267-273Article in journal (Refereed)
  • 20.
    Carlsson, Linus
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Spectrum of certain Banach algebras and $\overline\partial$-problems2007In: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 90, no 1, p. 51-58Article in journal (Refereed)
  • 21.
    Carlsson, Linus
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Trivial generators for nontrivial fibres2008In: MATHEMATICA BOHEMICA, ISSN 0862-7959, Vol. 133, no 2, p. 121-131Article in journal (Refereed)
    Abstract [en]

    Pseudoconvex domains are exhausted in such a way that we keep a part of the boundary fixed in all the domains of the exhaustion. This is used to solve a problem concerning whether the generators for the ideal of either the holomorphic functions continuous up to the boundary or the bounded holomorphic functions, vanishing at a point in $\mathbb{C}^n$ where the fibre is nontrivial, has to exceed $n$. This is shown not to be the case.

  • 22.
    Carlsson, Linus
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Fällström, Anders
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A note on B-envelope of holomorphy and B-extendable domains2008In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 53, no 4, p. 307-309Article in journal (Refereed)
    Abstract [en]

    Let B subset of H(infinity)(X) be a Banach Algebra on a Riemann domain X over C(n). We show that under certain conditions on B and X, all functions in B can be extended to functions in B(E(B, X)) where E(B, X) is the B-envelope of holomorphy.

  • 23.
    Cegrell, Urban
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Measures of finite pluricomplex energy2019In: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 123, no 1, p. 203-213Article in journal (Refereed)
    Abstract [en]

    We study a complex Monge-Ampere type equation of the form (dd(c)u)(n) = ke(-u)dV/integral e(-u)dV.

  • 24.
    Chronéer, Zackarias
    Umeå University, Faculty of Science and Technology, Department of Physics. Umeå Universitet.
    Generating Artificial Portfolios: Exploring the possibility of using GANs to recreate realistic portfolios2024Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    In this thesis a method for generating option portfolios using machine learning, more specifically WGAN-GP (Wasserstein Generative Adversarial Networks with Gradient Penalty), is presented. To reduce the complexity however, the model does not immediately generate portfolios with option series, but instead option classes, which includes the underlying asset, option type and direction of position. The generated portfolios are then transformed such that they include option series. A comparison between the real and generated portfolios was conducted, using a range of different metrics, such as number of positions, total market value and margin. Which concluded in that the model, presented in this thesis, effectively functions as a portfolio generator.

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    Generating Artificial Portfolios
  • 25.
    Chuaqui, Martin
    et al.
    Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile.
    Hernández, Rodrigo
    Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, Viña del Mar, Chile.
    Llinares, Adrián
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Madrid, Spain.
    Mas, Alejandro
    Departamento de Análisis Matemático, Universidad de Valencia, Burjassot, Spain.
    On the convolution of convex 2-gons2024In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 538, no 2, article id 128387Article in journal (Refereed)
    Abstract [en]

    We study the convolution of functions of the form [Formula presented] which map the open unit disk of the complex plane onto polygons of 2 edges when α∈(0,1). Inspired by a work of Cima, we study the limits of convolutions of finitely many fα and the convolution of arbitrary unbounded convex mappings. The analysis for the latter is based on the notion of angle at infinity, which provides an estimate for the growth at infinity and determines whether the convolution is bounded or not. A generalization to an arbitrary number of factors shows that the convolution of n randomly chosen unbounded convex mappings has a probability of 1/n! of remaining unbounded. We provide the precise asymptotic behavior of the coefficients of the functions fα.

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  • 26. Cinti, Chiara
    et al.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Polidoro, Sergio
    A Note on Harnack Inequalities and Propagation Sets for a Class of Hypoelliptic Operators2010In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 33, no 4, p. 341-354Article in journal (Refereed)
    Abstract [en]

    In this paper we are concerned with Harnack inequalities for non-negative solutions u:Ω→ℝ to a class of second order hypoelliptic ultraparabolic partial differential equations in the form Lu:=∑j=1mX2ju+X0u−∂tu=0 where Ω is any open subset of ℝN + 1, and the vector fields X1, ..., Xm and X0t are invariant with respect to a suitable homogeneous Lie group. Our main goal is the following result: for any fixed (x0, t0) ∈ Ω we give a geometric sufficient condition on the compact sets K⊆Ω for which the Harnack inequality supK u≤CK u(x0, t0) holds for all non-negative solutions u to the equation Lu=0. We also compare our result with an abstract Harnack inequality from potential theory.

  • 27.
    Daghighi, Abtin
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Gauthier, Paul M.
    Université de Montréal, Montréal, Canada.
    An algebra of polyanalytic functions2021In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 165, no 2, p. 225-240Article in journal (Refereed)
    Abstract [en]

    The most important uniform algebra is the family of continuous functions on a compact subset K of the complex plane C which are analytic on the interior int(K). For polyanalytic functions and compact sets K which are regular (i.e. K = int(K)), we introduce analogous spaces, which are Banach spaces with respect to the sup-norm, but are not closed with respect to the usual pointwise multiplication. We introduce a multiplication on these spaces and investigate the resulting algebras.

  • 28.
    Dmytryshyn, Andrii
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Miniversal deformations of pairs of skew-symmetric matrices under congruence2016In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 506, p. 506-534Article in journal (Refereed)
    Abstract [en]

    Miniversal deformations for pairs of skew-symmetric matrices under congruence are constructed. To be precise, for each such a pair (A, B) we provide a normal form with a minimal number of independent parameters to which all pairs of skew-symmetric matrices ((A) over tilde (,) (B) over tilde), close to (A, B) can be reduced by congruence transformation which smoothly depends on the entries of the matrices in the pair ((A) over tilde (,) (B) over tilde). An upper bound on the distance from such a miniversal deformation to (A, B) is derived too. We also present an example of using miniversal deformations for analyzing changes in the canonical structure information (i.e. eigenvalues and minimal indices) of skew-symmetric matrix pairs under perturbations.

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  • 29.
    Dmytryshyn, Andrii
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Miniversal deformations of pairs of symmetric matrices under congruence2019In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 568, p. 84-105Article in journal (Refereed)
    Abstract [en]

    For each pair of complex symmetric matrices (A, B) we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices ((A) over tilde (B) over tilde), close to (A, B) can be reduced by congruence transformation that smoothly depends on the entries of (A ) over tilde and (B) over tilde. Such a normal form is called a miniversal deformation of (A, B) under congruence. A number of independent parameters in the miniversal deformation of a symmetric matrix pencil is equal to the codimension of the congruence orbit of this symmetric matrix pencil and is computed too. We also provide an upper bound on the distance from (A, B) to its miniversal deformation.

  • 30.
    Dmytryshyn, Andrii
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Futorny, Vyacheslav
    Klymchuk, Tetiana
    Sergeichuk, Vladimir V.
    Generalization of Roth's solvability criteria to systems of matrix equations2017In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 527, p. 294-302Article in journal (Refereed)
    Abstract [en]

    W.E. Roth (1952) proved that the matrix equation AX - XB = C has a solution if and only if the matrices [Graphics] and [Graphics] are similar. A. Dmytryshyn and B. Kagstrom (2015) extended Roth's criterion to systems of matrix equations A(i)X(i')M(i) - (NiXi"Bi)-B-sigma i = Ci (i = 1,..., s) with unknown matrices X1,, X-t, in which every X-sigma is X, X-T, or X*. We extend their criterion to systems of complex matrix equations that include the complex conjugation of unknown matrices. We also prove an analogous criterion for systems of quaternion matrix equations. (C) 2017 Elsevier Inc. All rights reserved.

  • 31.
    Dmytryshyn, Andrii
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå Univ, HPC2N, SE-90187 Umeå, Sweden.
    Kågstrom, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå Univ, HPC2N, SE-90187 Umeå, Sweden.
    Coupled Sylvester-type Matrix Equations and Block Diagonalization2015In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 36, no 2, p. 580-593Article in journal (Refereed)
    Abstract [en]

    We prove Roth-type theorems for systems of matrix equations including an arbitrary mix of Sylvester and $\star$-Sylvester equations, in which the transpose or conjugate transpose of the unknown matrices also appear. In full generality, we derive consistency conditions by proving that such a system has a solution if and only if the associated set of $2 \times 2$ block matrix representations of the equations are block diagonalizable by (linked) equivalence transformations. Various applications leading to several particular cases have already been investigated in the literature, some recently and some long ago. Solvability of these cases follow immediately from our general consistency theory. We also show how to apply our main result to systems of Stein-type matrix equations.

  • 32.
    Efraimidis, Iason
    et al.
    Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain.
    Llinares, Adrián
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Vukotić, Dragan
    Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain.
    Korenblum’s principle for bergman spaces with radial weights2024In: Computational methods in Function Theory, ISSN 1617-9447, E-ISSN 2195-3724Article in journal (Refereed)
    Abstract [en]

    We show that the Korenblum maximum (domination) principle is valid for weighted Bergman spaces Apw with arbitrary (non-negative and integrable) radial weights w in the case 1≤p<∞. We also notice that in every weighted Bergman space the supremum of all radii for which the principle holds is strictly smaller than one. Under the mild additional assumption lim infr→0+w(r)>0, we show that the principle fails whenever 0<p<1.

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  • 33.
    Elfverson, Daniel
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Målqvist, Axel
    Multiscale methods for problems with complex geometry2017In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 321, p. 103-123Article in journal (Refereed)
    Abstract [en]

    We propose a multiscale method for elliptic problems on complex domains, e.g. domains with cracks or complicated boundary. For local singularities this paper also offers a discrete alternative to enrichment techniques such as XFEM. We construct corrected coarse test and trail spaces which takes the fine scale features of the computational domain into account. The corrections only need to be computed in regions surrounding fine scale geometric features. We achieve linear convergence rate in the energy norm for the multiscale solution. Moreover, the conditioning of the resulting matrices is not affected by the way the domain boundary cuts the coarse elements in the background mesh. The analytical findings are verified in a series of numerical experiments.

  • 34.
    Enflo, Karin
    Umeå University, Faculty of Arts, Department of historical, philosophical and religious studies.
    Measuring one-dimensional diversity2022In: Inquiry, ISSN 0020-174X, E-ISSN 1502-3923Article in journal (Refereed)
    Abstract [en]

    A large number of diversity measures have been proposed in the literature, almost all of which are designed for sets with elements that differ in multiple aspects. However, these types of measures are not appropriate for sets with elements that differ in a single aspect only, such as duration, value or probability. In this essay I present a new measure of diversity, designed specifically for single aspect diversity. The measure captures the intuitive idea that single aspect diversity is affected by the maximal dissimilarity between the elements, the number of different elements, and the even spacing of the elements’ aspect values, when represented as points in one dimension. I show that the measure satisfies six plausible conditions for a single aspect diversity measure, which concern scale independence, minimal diversity, maximal dissimilarity, restricted function growth, strict monotonicity, and even spacing. The measure is also characterised.

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  • 35.
    Engström, Christian
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Torshage, Axel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Accumulation of Complex Eigenvalues of a Class of Analytic Operator FunctionsManuscript (preprint) (Other academic)
  • 36.
    Engström, Christian
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Torshage, Axel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On equivalence and linearization of operator matrix functions with unbounded entries2017In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 89, no 4, p. 465-492Article in journal (Refereed)
    Abstract [en]

    In this paper we present equivalence results for several types of unbounded operator functions. A generalization of the concept equivalence after extension is introduced and used to prove equivalence and linearization for classes of unbounded operator functions. Further, we deduce methods of finding equivalences to operator matrix functions that utilizes equivalences of the entries. Finally, a method of finding equivalences and linearizations to a general case of operator matrix polynomials is presented.

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  • 37.
    Eriksson, Sofia
    et al.
    Department of Mathematics, Linnaeus University, Växjö, Sweden.
    Wang, Siyang
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Summation-by-parts approximations of the second derivative: Pseudoinverse and revisitation of a high order accurate operator2021In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 59, no 5, p. 2669-2697Article in journal (Refereed)
    Abstract [en]

    We consider finite difference approximations of the second derivative, exemplified in Poisson's equation, the heat equation, and the wave equation. The finite difference operators satisfy a summation-by-parts (SBP) property, which mimics the integration-by-parts principle. Since the operators approximate the second derivative, they are singular by construction. When imposing boundary conditions weakly, these operators are modified using simultaneous approximation terms. The modification makes the discretization matrix nonsingular for most choices of boundary conditions. Recently, inverses of such matrices were derived. However, for problems with only Neumann boundary conditions, the modified matrices are still singular. For such matrices, we have derived an explicit expression for the Moore-Penrose inverse, which can be used for solving elliptic problems and some time-dependent problems. For this explicit expression to be valid, it is required that the modified matrix does not have more than one zero eigenvalue. This condition holds for the SBP operators with second and fourth order accurate interior stencil. For the sixth order accurate case, we have reconstructed the operator with a free parameter and show that there can be more than one zero eigenvalue. We have performed a detailed analysis on the free parameter to improve the properties of the second derivative SBP operator. We complement the derivations by numerical experiments to demonstrate the improvements.

  • 38.
    Eskandari, Setareh
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran.
    Abkar, Ali
    Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran.
    Åhag, Per
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Perälä, Antti
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Sub-Hilbert relation for Fock-Sobolev type spaces2022In: New York Journal of Mathematics, E-ISSN 1076-9803, Vol. 28, p. 958-969Article in journal (Refereed)
    Abstract [en]

    In this paper, two specific sub-Hilbert spaces are studied. They arise from the action of a Toeplitz operator on Fock–Sobolev type spaces, in-duced by a general Gaussian type weight. The argument is based on analysing the reproducing kernel of the corresponding sub-Hilbert space.

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  • 39.
    Fontes, Magnus
    et al.
    Centre for Mathematical Science, Lund, Sweden.
    Verdier, Olivier
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Simplified a priori Estimate for the Time Periodic Burgers' Equation2010In: Proceedings of the Estonian Academy of Sciences, ISSN 1736-6046, E-ISSN 1736-7530, Vol. 59, no 1, p. 34-41Article in journal (Refereed)
    Abstract [en]

    We present here a version of the existence and uniqueness result of time periodic solutions to the viscous Burgers’ equation with irregular forcing terms (with Sobolev regularity –1 in space). The key result here is an a priori estimate which is simpler than the previously treated case of forcing terms with regularity –½ in time.

  • 40. Fontes, Magnus
    et al.
    Verdier, Olivier
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Time-Periodic Solutions of the Burgers Equation2009In: Journal of Mathematical Fluid Mechanics, ISSN 1422-6928, E-ISSN 1422-6952, Vol. 11, no 2, p. 303-323Article in journal (Refereed)
    Abstract [en]

    Abstract: We investigate the time periodic solutions to the viscous Burgers equation u(t) - mu u(xx) + uu(x) = f for irregular forcing terms. We prove that the corresponding Burgers operator is a diffeomorphism between appropriate function spaces.

  • 41.
    Forsberg, Joel
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Swaptions from a Clearinghouse perspective: Hedging swaptions, an option on interest rate swaps, using compression2022Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    With the increasing popularity of interest rate swaps the need to understandswaptions, an option of an interest rate swap, is of great importance. A swap-tion can be used in both speculative purposes and to hedge against changesin interest rates. The most important thing to understand is the pricing for-mula. By starting at the basic rate instrument, bonds, we will work our waytowards the pricing formula for a swaption, the Black76 model. The Black76model is a variant of the Nobel prize winning formula Black-Scholes-Merton.With the pricing model we can start looking at the main scope of this thesis,a hedging strategy against swaptions from a clearinghouse perspective.Clearinghouses are central to the modern financial market. They act asa middleman in order to clear trades from clearing members and have anoversight of the financial market. In case a clearing member defaults, theclearinghouse will gain control over the defaulted portfolio. The clearing-house will host an auction of the portfolio which they strive to hold after5 to 15 days. When they hold the portfolio, they are exposed to the risksand therefore it’s of great importance to be able to hedge the assets in theportfolio. In this thesis a strategy and algorithm have been developed todelta-hedge swaptions in order to be delta-neutral under stable market con-ditions.In the thesis we will consider two cases. The first case is when the clear-inghouse receives the portfolio long before the swaptions maturity. In thiscase forward swaps are used to hedge and in order to reduce the number offorward swaps obtained, compression is used. The second case is when theswaption maturity will be reached within the period the clearinghouse holdsthe portfolio. For the days before maturity is reached, forward swaps andcompression is used. After maturity is reached interest rate swaps is used tohedge.For both cases the result is very close to achieving delta-neutrality. Withnormalized deltas with respect to the notional amount the mean delta ex-posure is of the magnitude 10−4 for the first case and 10−6 for the second.However, one thing to keep in mind is that everything is based on simu-lated values under some simplifying assumptions. This thesis should be asolid ground for future studies where more extreme scenarios are considered.With more extreme scenarios one could investigate the possibility to hedgewith Gamma or another Greek such as Vega.

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  • 42.
    Frentz, Marie
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Regularity in the obstacle problem for parabolic non-divergence operators of Hörmander typeManuscript (preprint) (Other academic)
  • 43.
    Frentz, Marie
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Topics on subelliptic parabolic equations structured on Hörmander vector fields2012Doctoral thesis, comprehensive summary (Other academic)
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  • 44.
    Frentz, Marie
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Garofalo, Nicola
    Department of Mathematics, Purdue University, West Lafayette IN 47907-1968.
    Götmark, Elin
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Munive, Isidro
    Department of Mathematics, Purdue University, West Lafayette IN 47907-1968.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Non-divergence form parabolic equations associated with non-commuting vector fields: Boundary behavior of nonnegative solutions2012In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 11, no 2, p. 437-474Article in journal (Refereed)
    Abstract [en]

    In a cylinder Omega(T) = Omega x (0, T) subset of R-+(n+1) we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form

    H u = Sigma(m)(i,j=1) a(ij)(x, t)XiX (j)u - partial derivative(t)u = 0, (x, t) is an element of R-+(n+1),

    where X = {X-l, . . . , X-m} is a system of C-infinity vector fields inR(n) satisfying Hormander's rank condition (1.2), and Omega is a non-tangentially accessible domain with respect to the Carnot-Caratheodory distance d induced by X. Concerning the matrix-valued function A = {a(ij)}, we assume that it is real, symmetric and uniformly positive definite. Furthermore, we suppose that its entries a(ij) are Holder continuous with respect to the parabolic distance associated with d. Our main results are: I) a backward Harnack inequality for nonnegative solutions vanishing on the lateral boundary (Theorem 1.1); 2) the Holder continuity up to the boundary of the quotient of two nonnegative solutions which vanish continuously on a portion of the lateral boundary (Theorem 1.2); 3) the doubling property for the parabolic measure associated with the operator H (Theorem 1.3). These results generalize to the subelliptic setting of the present paper, those in Lipschitz cylinders by Fabes, Safonov and Yuan in [20, 39]. With one proviso: in those papers the authors assume that the coefficients a(ij) be only bounded and measurable, whereas we assume Holder continuity with respect to the intrinsic parabolic distance.

  • 45.
    Frentz, Marie
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Götmark, Elin
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The obstacle problem for parabolic non-divergence form operators of Hörmander type2012In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 252, no 9, p. 5002-2041Article in journal (Refereed)
    Abstract [en]

    In this paper we establish the existence and uniqueness of strong solutions to the obstacle problem for a class of parabolic sub-elliptic operators in non-divergence form structured on a set of smooth vector fields in Rn, X={X1,…,Xq}X={X1,…,Xq}, q⩽n, satisfying Hörmanderʼs finite rank condition. We furthermore prove that any strong solution belongs to a suitable class of Hölder continuous functions. As part of our argument, and this is of independent interest, we prove a Sobolev type embedding theorem, as well as certain a priori interior estimates, valid in the context of Sobolev spaces defined in terms of the system of vector fields.

  • 46.
    Frentz, Marie
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type2010In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 234, no 1, p. 146-164Article in journal (Refereed)
    Abstract [en]

    Degenerate parabolic equations of Kolmogorov type occur in many areas of analysis and applied mathematics. In their simplest form these equations were introduced by Kolmogorov in 1934 to describe the probability density of the positions and velocities of particles but the equations are also used as prototypes for evolution equations arising in the kinetic theory of gases. More recently equations of Kolmogorov type have also turned out to be relevant in option pricing in the setting of certain models for stochastic volatility and in the pricing of Asian options. The purpose of this paper is to numerically solve the Cauchy problem, for a general class of second order degenerate parabolic differential operators of Kolmogorov type with variable coefficients, using a posteriori error estimates and an algorithm for adaptive weak approximation of stochastic differential equations. Furthermore, we show how to apply these results in the context of mathematical finance and option pricing. The approach outlined in this paper circumvents many of the problems confronted by any deterministic approach based on, for example, a finite-difference discretization of the partial differential equation in itself. These problems are caused by the fact that the natural setting for degenerate parabolic differential operators of Kolmogorov type is that of a Lie group much more involved than the standard Euclidean Lie group of translations, the latter being relevant in the case of uniformly elliptic parabolic operators.

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  • 47.
    Frentz, Marie
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Pascucci, Andrea
    Polidoro, Sergio
    Optimal regularity in the obstacle problem for Kolmogorov operators related to American Asian options2010In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 347, no 4, p. 805-838Article in journal (Refereed)
    Abstract [en]

    In this paper we prove optimal interior regularity for solutions to the obstacle problem for a class of second order differential operators of Kolmogorov type. We treat smooth obstacles as well as non-smooth obstacles. All our proofs follow the same line of thought and are based on blow-ups, compactness, barriers and arguments by contradiction. The problem considered arises in financial mathematics, when considering path-dependent derivative contracts with early exercise feature.

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  • 48.
    Fällström, Anders
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    Maximal idels in A(Omega)1999In: Conemporary Mathematics, Vol. 222, p. 173-179Article in journal (Refereed)
  • 49.
    Fällström, Anders
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    The Gleason problem for A(Omega)1995In: New Zealand J. Math., Vol. 24, no 1, p. 17-22Article in journal (Refereed)
  • 50.
    Fällström, Anders
    Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
    The Gleason property for Reinhardt domains1997In: Mathematische Annalen, Vol. 308, p. 85-91Article in journal (Refereed)
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