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• 1.
Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
The boundary Harnack inequality for variable exponent p-Laplacian, Carleson estimates, barrier functions and p(⋅)-harmonic measures2016Ingår i: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 195, nr 2, s. 623-658Artikel i tidskrift (Refereegranskat)

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.

• 2.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för fysik.
Classification of spectral signatures in biological aerosols2013Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

I det här examensarbetet har multivariat dataanalys använts för att utvärdera förbehandlings-metoder, t.ex. normalisering, och klassificeringsmöjligheter för spektralt data insamlat med Laser Induced Breakdown Spectroscopy (LIBS). Systemet som FOI utvecklat under de senaste åren i syfte att analysera enstaka partiklar i luft har använts i laboratoriemiljö för att samla data från tio olika prover. En principalkomponentanalys (PCA) visar att det går att se skillnader mellan proverna utifrån de två datatyperna som erhålls från LIBS, 2D CCD bilder och spektra extraherat från dessa bilder. Vidare så visar resultat från partiell minstakvadrat-diskriminantanalys (PLS-DA) att normalisering av data endast har en visuell effekt i PCA score-plottar och inte har någon inverkan på modellernas prediktionsfömåga. Resultat visar även på möjligheter att beskära och binna pixlar i bilderna i viss utstreckning utan att förlora signifikant prediktionsförmåga.

• 3.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Adaptive finite element approximation of multiphysics problems: a fluid structure interaction model problem2010Ingår i: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 84, nr 12, s. 1451-1465Artikel i tidskrift (Refereegranskat)

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.

• 4.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Evaluating ∫0f(x)dx and ∫ab f(x)dx using residue calculus2014Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)

I den har uppsatsen använder vi komplex analys, då särskilt modern residykalkyl, for att beräkna vissa Riemann-integraler.

• 5.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Numerical Simulations of Linear Stochastic Oscillators: driven by Wiener and Poisson processes2017Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

Den huvudsakliga komponenten av uppsatsen är en numerisk analys av stokastiska differentialekvationer drivna av Wiener- och Poisson-processer. För att göra det så fokuserar vi på två modellproblem, den geometriska Brownska rörelsen samt den linjära stokastiska oscillatorn, studerade i litteratur för stokastiska differentialekvationer som bara drivs av en Wiener-process.Den här uppsatsen täcker teoretiska samt numeriska undersökningar av hopp - eller mer specifikt, Poisson - processer och hur de påverkar de ovan nämnda modellproblemen.

• 6.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Positive currents related to polynomial convexity1999Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
• 7.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för fysik.
GDP Growth Rate Nowcasting and Forecasting2017Självständigt arbete på avancerad nivå (yrkesexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

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.

• 8.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för fysik.
An analysis of a shared mating in V2.2014Självständigt arbete på avancerad nivå (magisterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

I detta examensarbete undersöker vi, från en topologisk synvinkel och utan applicering av Thurstons teorem, varför matchningen av det så kallade basilikapolynomet

$f_{-1}(z)=z^{2}-1$ och dendriten $f_{i}(z)=z^{2}+i$ är delad med matchningen av $f_{-1}$ och dendriten $f_{-i}(z)=z^{2}-i$.

Båda dessa matchningar är lika med den rationella avbildningen

$R_{3}(z)=\frac{3}{z^{2}+2z}$.

Definierat i examensarbetet är för båda matchningarna homoemorfa koordinatbyten$\psi_{-1}^{\pm}$ från mängden$L=\overset{\circ}{K}\left(f_{-1} \right)\cup\left(\cup_{n=0}^{\infty}f_{-1}^{\circ(-n)}(z_{\alpha})\right)$ till Fatou- och Juliamängden av $R_{3}$. Här är $K\left(f_{-1} \right)$ den ifyllda Juliamängden av avbildningen $f_{-1}$ och $z_{\alpha}$ är den $\alpha$-fixerade punkten i $K\left(f_{-1} \right)$.

• 9. Brännström, Niklas
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
A measure theoretic approach to linear inverse atmospheric dispersion problems2015Ingår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 31, nr 2, artikel-id 025009Artikel i tidskrift (Refereegranskat)

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.

• 10. Burman, Erik
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Augmented Lagrangian finite element methods for contact problems2019Ingår i: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 53, nr 1, s. 173-195Artikel i tidskrift (Refereegranskat)

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.

• 11.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
An equivalence to the Gleason problem2010Ingår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 370, nr 2, s. 373-378Artikel i tidskrift (Refereegranskat)

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.

• 12.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Analytic properties in the spectrum of certain Banach algebras2009Ingår i: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 261, nr 1, s. 189-200Artikel i tidskrift (Refereegranskat)
• 13.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Nebenhülle and the Gleason problem2010Ingår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 138, nr 1, s. 267-273Artikel i tidskrift (Refereegranskat)
• 14.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Spectrum of certain Banach algebras and $\overline\partial$-problems2007Ingår i: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 90, nr 1, s. 51-58Artikel i tidskrift (Refereegranskat)
• 15.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Trivial generators for nontrivial fibres2008Ingår i: MATHEMATICA BOHEMICA, ISSN 0862-7959, Vol. 133, nr 2, s. 121-131Artikel i tidskrift (Refereegranskat)

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.

• 16.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
A note on B-envelope of holomorphy and B-extendable domains2008Ingår i: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 53, nr 4, s. 307-309Artikel i tidskrift (Refereegranskat)

Let   be a Banach Algebra on a Riemann domain X over  . We show that under certain conditions on   and X, all functions in   can be extended to functions in  where   is the  -envelope of holomorphy.

• 17. Cinti, Chiara
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
A Note on Harnack Inequalities and Propagation Sets for a Class of Hypoelliptic Operators2010Ingår i: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 33, nr 4, s. 341-354Artikel i tidskrift (Refereegranskat)

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.

• 18.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
Miniversal deformations of pairs of skew-symmetric matrices under congruence2016Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 506, s. 506-534Artikel i tidskrift (Refereegranskat)

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.

• 19.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
Miniversal deformations of pairs of symmetric matrices under congruence2019Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 568, s. 84-105Artikel i tidskrift (Refereegranskat)

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.

• 20.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
Generalization of Roth's solvability criteria to systems of matrix equations2017Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 527, s. 294-302Artikel i tidskrift (Refereegranskat)

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.

• 21.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap. Umeå Univ, HPC2N, SE-90187 Umeå, Sweden.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap. Umeå Univ, HPC2N, SE-90187 Umeå, Sweden.
Coupled Sylvester-type Matrix Equations and Block Diagonalization2015Ingår i: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 36, nr 2, s. 580-593Artikel i tidskrift (Refereegranskat)

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.

• 22.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Multiscale methods for problems with complex geometry2017Ingår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 321, s. 103-123Artikel i tidskrift (Refereegranskat)

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.

• 23.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Accumulation of Complex Eigenvalues of a Class of Analytic Operator FunctionsManuskript (preprint) (Övrigt vetenskapligt)
• 24.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
On equivalence and linearization of operator matrix functions with unbounded entries2017Ingår i: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 89, nr 4, s. 465-492Artikel i tidskrift (Refereegranskat)

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.

• 25.
Centre for Mathematical Science, Lund, Sweden.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Simplified a priori Estimate for the Time Periodic Burgers' Equation2010Ingår i: Proceedings of the Estonian Academy of Sciences, ISSN 1736-6046, E-ISSN 1736-7530, Vol. 59, nr 1, s. 34-41Artikel i tidskrift (Refereegranskat)

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.

• 26. Fontes, Magnus
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Time-Periodic Solutions of the Burgers Equation2009Ingår i: Journal of Mathematical Fluid Mechanics, ISSN 1422-6928, E-ISSN 1422-6952, Vol. 11, nr 2, s. 303-323Artikel i tidskrift (Refereegranskat)

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.

• 27.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Regularity in the obstacle problem for parabolic non-divergence operators of Hörmander typeManuskript (preprint) (Övrigt vetenskapligt)
• 28.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Topics on subelliptic parabolic equations structured on Hörmander vector fields2012Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
• 29.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Department of Mathematics, Purdue University, West Lafayette IN 47907-1968. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Department of Mathematics, Purdue University, West Lafayette IN 47907-1968. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Non-divergence form parabolic equations associated with non-commuting vector fields: Boundary behavior of nonnegative solutions2012Ingår i: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 11, nr 2, s. 437-474Artikel i tidskrift (Refereegranskat)

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.

• 30.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
The obstacle problem for parabolic non-divergence form operators of Hörmander type2012Ingår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 252, nr 9, s. 5002-2041Artikel i tidskrift (Refereegranskat)

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.

• 31.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type2010Ingår i: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 234, nr 1, s. 146-164Artikel i tidskrift (Refereegranskat)

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.

• 32.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Optimal regularity in the obstacle problem for Kolmogorov operators related to American Asian options2010Ingår i: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 347, nr 4, s. 805-838Artikel i tidskrift (Refereegranskat)

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.

• 33.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Maximal idels in A(Omega)1999Ingår i: Conemporary Mathematics, Vol. 222, s. 173-179Artikel i tidskrift (Refereegranskat)
• 34.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
The Gleason problem for A(Omega)1995Ingår i: New Zealand J. Math., Vol. 24, nr 1, s. 17-22Artikel i tidskrift (Refereegranskat)
• 35.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
The Gleason property for Reinhardt domains1997Ingår i: Mathematische Annalen, Vol. 308, s. 85-91Artikel i tidskrift (Refereegranskat)
• 36.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Ändligt-genererade ideal och Gleason-problemet2008Ingår i: Nordan ÅttaArtikel i tidskrift (Övrigt vetenskapligt)
• 37. Fällström, Anders
A pseudoconvex domain with non schlicht H-infinity envelope1991Ingår i: Geometrical and algebraical aspects in several complex variables. Sem. Conf, Vol. 8, s. 13-18Artikel i tidskrift (Refereegranskat)
• 38.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Counterexamples to the Gleason Problem1998Ingår i: Ann. Scuola Norm. Sup. Pisa. Cl. Sci. (4), Vol. XXVI, s. 595-603Artikel i tidskrift (Refereegranskat)
• 39.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik. Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
The polynomial hull of unions of convex sets in Cn1996Ingår i: Colloquium Mathematicum, Vol. LXX, nr 1, s. 7-11Artikel i tidskrift (Refereegranskat)
• 40.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap. IBM T.J. Watson Research Center, New York, USA.
A Square Block Format for Symmetric Band Matrices2014Ingår i: Parallel Processing and Applied Mathematics: 10th International Conference, PPAM 2013 Warsaw, Poland, September 8–11, 2013, Revised Selected Papers, Part I / [ed] Wyrzykowski, R Dongarra, J Karczewski, K Wasniewski, J, Springer Berlin/Heidelberg, 2014, s. 683-689Konferensbidrag (Refereegranskat)

This contribution describes a Square Block, SB, format for storing a banded symmetric matrix. This is possible by rearranging "in place" LAPACK Band Layout to become a SB layout: store submatrices as a set of square blocks. The new format reduces storage space, provides higher locality of memory accesses, results in regular access patterns, and exposes parallelism.

• 41. Hansbo, Peter
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
STABILIZED FINITE ELEMENT APPROXIMATION OF THE MEAN CURVATURE VECTOR ON CLOSED SURFACES2015Ingår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 53, nr 4, s. 1806-1832Artikel i tidskrift (Refereegranskat)

The mean curvature vector of a surface is obtained by letting the Laplace-Beltrami operator act on the embedding of the surface in R-3. In this contribution we develop a stabilized finite element approximation of the mean curvature vector of certain piecewise linear surfaces which enjoys first order convergence in L-2. The stabilization involves the jump in the tangent gradient in the direction of the outer co-normals at each edge in the surface mesh. We consider both standard meshed surfaces and so-called cut surfaces that are level sets of piecewise linear distance functions. We prove a priori error estimates and verify the theoretical results numerically.

• 42. Hashorva, Enkelejd
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Boundary non-crossing probabilities for fractional Brownian motion with trend2015Ingår i: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 87, nr 6, s. 946-965Artikel i tidskrift (Refereegranskat)

In this paper, we investigate the boundary non-crossing probabilities of a fractional Brownian motion considering some general deterministic trend function. We derive bounds for non-crossing probabilities and discuss the case of a large trend function. As a by-product, we solve a minimization problem related to the norm of the trend function.

• 43. Hashorva, Enkelejd
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Approximation of maximum of Gaussian random fields2018Ingår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 457, nr 1, s. 841-867Artikel i tidskrift (Refereegranskat)

This contribution is concerned with Gumbel limiting results for supremum M-n = sup(t epsilon[0,Tn])X(n)(t)vertical bar with X (n) ,n epsilon N-2 centered Gaussian random fields with continuous trajectories. We show first the convergence of a related point process to a Poisson point process thereby extending previous results obtained in [8] for Gaussian processes. Furthermore, we derive Gumbel limit results for M-n as n -> infinity and show a second-order approximation for E{M-n(p)}(1/p) for any p >= 1.

• 44.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
The plurisubharmonic Mergelyan property2012Doktorsavhandling, monografi (Övrigt vetenskapligt)

In this thesis, we study two different kinds of approximation of plurisubharmonic functions.

The first one is a Mergelyan type approximation for plurisubharmonic functions. That is, we study which domains in C^n have the property that every continuous plurisubharmonic function can be uniformly approximated with continuous and plurisubharmonic functions defined on neighborhoods of the domain. We will improve a result by Fornaess and Wiegerinck and show that domains with C^0-boundary have this property. We will also use the notion of plurisubharmonic functions on compact sets when trying to characterize those continuous and plurisubharmonic functions that can be approximated from outside. Here a new kind of convexity of a domain comes in handy, namely those domains in C^n that have a negative exhaustion function that is plurisubharmonic on the closure. For these domains, we prove that it is enough to look at the boundary values of a plurisubharmonic function to know whether it can be approximated from outside.

The second type of approximation is the following: we want to approximate functions u that are defined on bounded hyperconvex domains Omega in C^n and have essentially boundary values zero and bounded Monge-Ampère mass, with increasing sequences of certain functions u_j that are defined on strictly larger domains. We show that for certain conditions on Omega, this is always possible. We also generalize this to functions with given boundary values. The main tool in the proofs concerning this second approximation is subextension of plurisubharmonic functions.

• 45.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Matematiska institutionen.
The Diamond Lemma for Power Series Algebras2002Doktorsavhandling, monografi (Övrigt vetenskapligt)

The main result in this thesis is the generalisation of Bergman's diamond lemma for ring theory to power series rings. This generalisation makes it possible to treat problems in which there arise infinite descending chains. Several results in the literature are shown to be special cases of this diamond lemma and examples are given of interesting problems which could not previously be treated. One of these examples provides a general construction of a normed skew field in which a custom commutation relation holds.

There is also a general result on the structure of totally ordered semigroups, demonstrating that all semigroups with an archimedean element has a (up to a scaling factor) unique order-preserving homomorphism to the real numbers. This helps analyse the concept of filtered structure. It is shown that whereas filtered structures can be used to induce pretty much any zero-dimensional linear topology, a real-valued norm suffices for the definition of those topologies that have a reasonable relation to the multiplication operation.

The thesis also contains elementary results on degree (as of polynomials) functions, norms on algebras (in particular ultranorms), (Birkhoff) orthogonality in modules, and construction of semigroup partial orders from ditto quasiorders.

• 46. Johansson, A.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Adaptive finite element solution of multiscale PDE-ODE systems2015Ingår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 287, s. 150-171Artikel i tidskrift (Refereegranskat)

We consider adaptive finite element methods for a multiscale system consisting of a macroscale model comprising a system of reaction-diffusion partial differential equations coupled to a microscale model comprising a system of nonlinear ordinary differential equations. A motivating example is modeling the electrical activity of the heart taking into account the chemistry inside cells in the heart. Such multiscale models are computationally challenging due to the multiple scales in time and space that are involved. We describe a mathematically consistent approach to couple the microscale and macroscale models based on introducing an intermediate "coupling scale". Since the ordinary differential equations are defined on a much finer spatial scale than the finite element discretization for the partial differential equation, we introduce a Monte Carlo approach to sampling the fine scale ordinary differential equations. We derive goal-oriented a posteriori error estimates for quantities of interest computed from the solution of the multiscale model using adjoint problems and computable residuals. We distinguish the errors in time and space for the partial differential equation and the ordinary differential equations separately and include errors due to the transfer of the solutions between the equations. The estimate also includes terms reflecting the sampling of the microscale model. Based on the accurate error estimates, we devise an adaptive solution method using a "blockwise" approach. The method and estimates are illustrated using a realistic problem.

• 47.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för fysik. FOI.
Boundary conditions for modeling deposition in a stochastic Lagrangian particle model2015Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

The Swedish defence agency (FOI) has developed a particle model (called Pello) that simulates the dispersion of aerosols and gases. At the boundaries, such as the ground, the particles can either reflect back into the domain (the atmosphere) or be absorbed. Which of the events that occurs is decided by a certain probability, which in the present model depends on mere physical properties. In this thesis we have investigated a newly proposed boundary behaviour which also depends on the time step used in the numerical simulations. We verified the accuracy of the new model by using a dispersion model with an explicit solution. To gain a better understanding of how important parameters at the boundary influence each other, we performed a sensitivity analysis.

Simulations showed an overall improving concentration profile as the time step became smaller and the new model working well. The convergence order of the simulations was found to be close to 0.5. In this thesis we have shown that there exist an upper limit for the time step, which depends on the specific model. The present used time step at FOI does not have this versatile property. But having this upper limit for the time step close to the boundary, and a uniform time step can be time demanding. This lead us to the conclusion that an adaptive time step should be implemented.

• 48. Kalisch, Henrik
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
A numerical study of nonlinear dispersive wave models with SpecTraVVave2017Ingår i: Electronic Journal of Differential Equations, ISSN 1550-6150, E-ISSN 1072-6691, s. 1-23, artikel-id 62Artikel i tidskrift (Refereegranskat)

In nonlinear dispersive evolution equations, the competing effects of nonlinearity and dispersion make a number of interesting phenomena possible. In the current work, the focus is on the numerical approximation of traveling-wave solutions of such equations. We describe our efforts to write a dedicated Python code which is able to compute traveling-wave solutions of nonlinear dispersive equations in a very general form. The Spec TraVVave code uses a continuation method coupled with a spectral projection to compute approximations of steady symmetric solutions of this equation. The code is used in a number of situations to gain an understanding of traveling-wave solutions. The first case is the Whitham equation, where numerical evidence points to the conclusion that the main bifurcation branch features three distinct points of interest, namely a turning point, a point of stability inversion, and a terminal point which corresponds to a cusped wave. The second case is the so-called modified Benjamin-Ono equation where the interaction of two solitary waves is investigated. It is found that two solitary waves may interact in such a way that the smaller wave is annihilated. The third case concerns the Benjamin equation which features two competing dispersive operators. In this case, it is found that bifurcation curves of periodic traveling-wave solutions may cross and connect high up on the branch in the nonlinear regime.

• 49.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Symplectic Automorphisms of C2n2018Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

Den här uppsatsen är en detaljerad undersökning av en artikel från 1996 publicerad av Franc Forstneric där han studerar symplektiska automorfismer av C2n. Visionen är att introducera täthetsegenskapen för holomorfa symplektiska mångfalder. Våran idé är som den av Dror Varolin när han 2001 introducerade täthetsegenskapen för Stein mångfalder. Huvudresultatet här är införandet av symplektiska skjuvningar på C2n med en holomorfisk symplektisk form och att visa att gruppen som genereras av ändliga sammansättningar av symplektiska skjuvningar är tät i gruppen av symplektiska automorfismer av C2n i den kompakt-öppna topologin. Vi ger en fullständig bakgrund av de verktyg från teorin om ordinära differentialekvationer, släta mångfalder och komplex och symplektisk geometri som behövs för att visa detta.

• 50.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
The Oka-Weil Theorem2017Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)

Vi ger ett bevis av Oka-Weil sats som säger att på kompakta och polynomkonvexa delmängder av Cn kan holomorfa funktioner approximeras likformigt med holomorfa polynom.

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