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• 51.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Derivatan ur ett historiskt perspektiv2017Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

The derivative is a fundamental part of mathematics. This essay will be about historicaladvancements in mathematics, which led to the fact that the derivative has been definedas we are used to seeing it today.

• 52.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
An ordering of measures induced by plurisubharmonic functionsManuscript (preprint) (Other academic)

We study an ordering of measures induced by plurisubharmonic functions. This ordering arises naturally in connection with problems related to negative plurisubharmonic functions. We study maximality with respect to the ordering and a related notion of minimality for certain plurisubharmonic functions. The ordering is then applied to the problem of weak*-convergence of measures, in particular Monge-Ampère measures.

• 53.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Boundary values of plurisubharmonic functions and related topics2009Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of three papers concerning problems related to plurisubharmonic functions on bounded hyperconvex domains, in particular boundary values of such functions. The papers summarized in this thesis are:* Paper I Urban Cegrell and Berit Kemppe, Monge-Ampère boundary measures, Ann. Polon. Math. 96 (2009), 175-196.* Paper II Berit Kemppe, An ordering of measures induced by plurisubharmonic functions, manuscript (2009).* Paper III Berit Kemppe, On boundary values of plurisubharmonic functions, manuscript (2009).In the first paper we study a procedure for sweeping out Monge-Ampère measures to the boundary of the domain. The boundary measures thus obtained generalize measures studied by Demailly. A number of properties of the boundary measures are proved, and we describe how boundary values of bounded plurisubharmonic functions can be associated to the boundary measures.In the second paper, we study an ordering of measures induced by plurisubharmonic functions. This ordering arises naturally in connection with problems related to negative plurisubharmonic functions. We study maximality with respect to the ordering and a related notion of minimality for certain plurisubharmonic functions. The ordering is then applied to problems of weak*-convergence of measures, in particular Monge-Ampère measures.In the third paper we continue the work on boundary values in a more general setting than in Paper I. We approximate measures living on the boundary with measures on the interior of the domain, and present conditions on the approximation which makes the procedure suitable for defining boundary values of certain plurisubharmonic functions.

• 54.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
On boundary values of plurisubharmonic functionsManuscript (preprint) (Other academic)

In this paper we describe how boundary values of bounded plurisubharmonic functions can be defined in terms of measures with support on the boundary. We do this by approximating the boundary measures with measures on the interior of the domain, and present conditions on the approximation which makes the procedure suitable for defining boundary values of certain plurisubharmonic functions.

• 55.
Umeå University, Faculty of Science and Technology, Department of Physics. Institute of Applied Physics of the Russian Academy of Sciences, 603950 Nizhny Novgorod, Russia; Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany.
Two-component Superfluid Hydrodynamics of Neutron Star Cores2017In: Astrophysical Journal, ISSN 0004-637X, E-ISSN 1538-4357, Vol. 836, no 2, article id 203Article in journal (Refereed)

We consider the hydrodynamics of the outer core of a neutron star under conditions when both neutrons and protons are superfluid. Starting from the equation of motion for the phases of the wave functions of the condensates of neutron pairs and proton pairs, we derive the generalization of the Euler equation for a one-component fluid. These equations are supplemented by the conditions for conservation of neutron number and proton number. Of particular interest is the effect of entrainment, the fact that the current of one nucleon species depends on the momenta per nucleon of both condensates. We find that the nonlinear terms in the Euler-like equation contain contributions that have not always been taken into account in previous applications of superfluid hydrodynamics. We apply the formalism to determine the frequency of oscillations about a state with stationary condensates and states with a spatially uniform counterflow of neutrons and protons. The velocities of the coupled sound-like modes of neutrons and protons are calculated from properties of uniform neutron star matter evaluated on the basis of chiral effective field theory. We also derive the condition for the two-stream instability to occur.

• 56.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Generalized Maximum Spacing Estimation for Multivariate Observations2015In: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 42, no 4, p. 1092-1108Article in journal (Refereed)

In this paper, the maximum spacing method is considered for multivariate observations. Nearest neighbour balls are used as a multidimensional analogue to univariate spacings. A class of information-type measures is used to generalize the concept of maximum spacing estimators. Weak and strong consistency of these generalized maximum spacing estimators are proved both when the assigned model class is correct and when the true density is not a member of the model class. An example of the generalized maximum spacing method in model validation context is discussed.

• 57.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Adaptive finite element approximation of multiphysics problems2007In: Communications in Numerical Methods in Engineering, ISSN 1069-8299, E-ISSN 1099-0887, Vol. 24, no 6, p. 505-521Article in journal (Refereed)

Simulation of multiphysics problems is a common task in applied research and industry. Often a multiphysics solver is built by connecting several single-physics solvers into a network. In this paper, we develop a basic adaptive methodology for such multiphysics solvers. The adaptive methodology is based on a posteriori error estimates that capture the influence of the discretization errors in the different solvers on a given functional output. These estimates are derived using duality-based techniques.

• 58.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The Riemann-Stieltjes integral: and some applications in complex analysis and probability theory2014Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

The purpose of this essay is to prove the existence of the Riemann-Stieltjes integral. After doing so, we present some applications in complex analysis, where we define the complex curve integral as a special case of the Riemann- Stieltjes integral, and then focus on Cauchy’s celebrated integral theorem. To show the versatility of the Riemann-Stieltjes integral, we also present some applications in probability theory, where the integral generates a general formula for the expectation, regardless of its underlying distribution.

• 59.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the geometry of calibrated manifolds: with applications to electrodynamics2013Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

In this master thesis we study calibrated geometries, a family of Riemannian or Hermitian manifolds with an associated differential form, φ. We show that it isuseful to introduce the concept of proper calibrated manifolds, which are in asense calibrated manifolds where the geometry is derived from the calibration. In particular, the φ-Grassmannian is considered in the case of proper calibratedmanifolds. The impact of proper calibrated manifolds as a model is studied, aswell as the usefulness of pluripotential theory as tools for the model. The specialLagrangian calibration is an example of an important calibration introduced byHarvey and Lawson, which leads to the definition of the special Lagrangian differentialequation. This partial differential equation can be formulated in threeand four dimensions as det(H(u)) = Δu, where H(u) is the Hessian matrix of some potential u. We prove the existence of solutions and some other propertiesof this nonlinear differential equation and present the resulting 6- and 8-dimensional manifolds defined by the graph {x + i$\nabla$u(x)}. We also considerthe physical applications of calibrated geometry, which have so far largely beenrestricted to string theory. However, we consider the manifold (M,g,F), whichis calibrated by the scaled Maxwell 2-form. Some geometrical properties of relativisticand classical electrodynamics are translated into calibrated geometry.

• 60.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Modeling credit risk for an SME loan portfolio: An Error Correction Model approach2017Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

Sedan den globala finanskrisen 2008 har flera stora regelverk införts för att säkerställa att banker hanterar risker på sunt sätt. Bland dessa regelverk är Basel II som infört kapitalkrav för kreditrisk som baseras på Sannolikhet för Fallissemang och Förlust Givet Fallissemang. Basel II Advanced Internal-Based Approach ger banker möjligheten att skatta dessa riskmått för enskilda portföljer och göra interna kreditriskvärderingar. I överensstämmelse med Advanced Internal-Based-rating undersöker denna uppsats användningen av en Error Correction Model för modellering av Sannolikhet för Fallissemang. En modell som visat sin styrka inom stresstestning. Vidare implementeras en funktion för Förlust Givet Fallissemang som binder samman Sannolikhet för Fallissemang och Förlust Givet Fallissemang med systematisk risk.

Error Correction Modellen modellerar Sannolikhet för Fallissemang av en SME-portfölj från en av de "fyra stora" bankerna i Sverige. Modellen utvärderas och stresstestas med Europeiska Bankmyndighetens  stresstestscenario 2016  och analyseras, med lovande resultat.

• 61. Ling, Zhi
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Turing pattern formation in a predator-prey system with cross diffusion2014In: Applied Mathematical Modelling, ISSN 0307-904X, E-ISSN 1872-8480, Vol. 38, no 21-22, p. 5022-5032Article in journal (Refereed)

The paper explores the impacts of cross-diffusion on the formation of spatial patterns in a ratio-dependent predator-prey system with zero-flux boundary conditions. Our results show that under certain conditions, cross-diffusion can trigger the emergence of spatial patterns which is however impossible under the same conditions when cross-diffusion is absent. We give a rigorous proof that the model has at least one spatially heterogenous steady state by means of the Leray-Schauder degree theory. In addition, numerical simulations are performed to visualize the complex spatial patterns.

• 62.
Umeå University, Faculty of Science and Technology, Department of mathematics.
Comparing two versions of Markov's inequality on compact sets1994In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 77, no 2, p. 202-211Article in journal (Refereed)

﻿We compare a local and a global version of Markov's inequality defined on compact subsets of C. As a main result we show that the local version implies the global one. The same result was also obtained independently by A. Volberg.﻿

• 63.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Two versions of Markov's inequality1993Doctoral thesis, comprehensive summary (Other academic)
• 64.
Umeå University, Faculty of Science and Technology, Department of mathematics.
University of Krakow.
A Note on Bernstein's Theorems1995In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 81, no 3, p. 316-322Article in journal (Refereed)

There is given a completion to Theorem 3.3 of [11] by showing that on compact subsets of R N (or C N) preserving Markov′s inequality, some speed of polynomial approximation leads to Lipschitz- and Zygmund-type classes of functions.

• 65.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Bimodal and Gaussian Ising spin glasses in dimension two2016In: Physical Review E, ISSN 2470-0045, Vol. 93, no 2, article id 022119Article in journal (Refereed)

An analysis is given of numerical simulation data to size L = 128 on the archetype square lattice Ising spin glasses (ISGs) with bimodal (+/- J) and Gaussian interaction distributions. It is well established that the ordering temperature of both models is zero. The Gaussian model has a nondegenerate ground state and thus a critical exponent. = 0, and a continuous distribution of energy levels. For the bimodal model, above a size-dependent crossover temperature T *(L) there is a regime of effectively continuous energy levels; below T *(L) there is a distinct regime dominated by the highly degenerate ground state plus an energy gap to the excited states. T *(L) tends to zero at very large L, leaving only the effectively continuous regime in the thermodynamic limit. The simulation data on both models are analyzed with the conventional scaling variable t = T and with a scaling variable tau(b) = T-2 /(1 + T 2) suitable for zero-temperature transition ISGs, together with appropriate scaling expressions. The data for the temperature dependence of the reduced susceptibility x(tau(b), L) and second moment correlation length xi(tau(b), L) in the thermodynamic limit regime are extrapolated to the tau(b) = 0 critical limit. The Gaussian critical exponent estimates from the simulations, eta= 0 and nu= 3.55(5), are in full agreement with the well-established values in the literature. The bimodal critical exponents, estimated from the thermodynamic limit regime analyses using the same extrapolation protocols as for the Gaussian model, are eta= 0.20(2) and nu= 4.8(3), distinctly different from the Gaussian critical exponents.

• 66.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Non-self-averaging in Ising spin glasses and hyperuniversality2016In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 93, no 1, article id 012118Article in journal (Refereed)

Ising spin glasses with bimodal and Gaussian near-neighbor interaction distributions are studied through numerical simulations. The non-self-averaging (normalized intersample variance) parameter U-22(T, L) for the spin glass susceptibility [and for higher moments Unn (T, L)] is reported for dimensions 2,3,4,5, and 7. In each dimension d the non-self-averaging parameters in the paramagnetic regime vary with the sample size L and the correlation length xi(T, L) as U-nn(beta, L) = [K-d xi (T, L)/L](d) and so follow a renormalization group law due to Aharony and Harris [Phys. Rev. Lett. 77, 3700 (1996)]. Empirically, it is found that the Kd values are independent of d to within the statistics. The maximum values [U-nn(T, L)](max) are almost independent of L in each dimension, and remarkably the estimated thermodynamic limit critical [U-nn (T, L)](max) peak values are also practically dimension-independent to within the statistics and so are " hyperuniversal." These results show that the form of the spin-spin correlation function distribution at criticality in the large L limit is independent of dimension within the ISG family. Inspection of published non-self-averaging data for three-dimensional Heisenberg and XY spin glasses the light of the Ising spin glass non-self-averaging results show behavior which appears to be compatible with that expected on a chiral-driven ordering interpretation but incompatible with a spin-driven ordering scenario.

• 67.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Estimates for p-harmonic functions vanishing on a flat2011In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, no 18, p. 6852-6860Article in journal (Refereed)

We study p-harmonic functions in a domain ΩCRn near an m-dimensional plane (an m-flat) Λm, where 0≤mn−1. In particular, let u be a positive p-harmonic function, with n<p, vanishing on a portion of Λm, and suppose that β=(pn+m)/(p−1), with β=1 if p=. We prove, using certain barrier functions, that

u ≈ d (x.Λm)8   near Λm.

The lower bound holds also in the range nm<p. Moreover, uC0,β near Λm and β is the optimal Hölder exponent of u.

• 68.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Phragmén-Lindelöf theorems and p-harmonic measures for sets near low-dimensional hyperplanes2016In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 44, no 2, p. 313-330Article in journal (Refereed)

We prove estimates of a p-harmonic measure, p∈(nm,], for sets in R n which are close to an m-dimensional hyperplane Λ⊂R n , m∈[0,n−1]. Using these estimates, we derive results of Phragmén-Lindelöf type in unbounded domains Ω⊂R n ∖Λ for p-subharmonic functions. Moreover, we give local and global growth estimates for p-harmonic functions, vanishing on sets in R n , which are close to an m-dimensional hyperplane.

• 69. Lundström, Niklas L.P.
Bifurcations and Strange Attractors in a Climate Related System2005In: Differentsial'nye Uravneniya i Protsessy Upravleniya / Differential Equations and Control Processes

, ISSN 1817-2172, E-ISSN 1817-2172, no 1, p. 1-53
Article in journal (Refereed)

R. V. Bekryaev derived a system for a horizontally baroclinic atmosphere consisting of six ordinary differential equations. We prove dissipativity and find estimates for the location of the global attractor. The evolution of a complicated attractor is analysed with a Poincar'e map showing difficult bifurcation behaviour. Investigations in bifurcation diagrams show a rich dynamical behaviour including a lot of known complicated bifurcations, of which a fold-Hopf bifurcation is examined in detail. Finally, we give some theory about the Lyapunov spectra and present a method for determining the exponents.

• 70.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
p-harmonic functions near the boundary2011Doctoral thesis, comprehensive summary (Other academic)
• 71.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Existence, uniqueness and regularity of solutions to systems of nonlocal obstacle problems related to optimal switching2019In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 475, no 1, p. 13-31Article in journal (Refereed)

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

• 72.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Decay of a p-harmonic measure in the plane2013In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, no 1, p. 351-366Article in journal (Refereed)

We study the asymptotic behaviour of a p-harmonic measure w(p), p is an element of (1, infinity], in a domain Omega subset of R-2, subject to certain regularity constraints. Our main result is that w(p) (B (w, delta) boolean AND partial derivative Omega, w(0)) approximate to delta(q) as delta -> 0(+), where q = q(v,p) is given explicitly as a function of v and p. Here, v is related to properties of Omega near w. If p = infinity, this extends to some domains in R-n. By a result due to Hirata, our result implies that the p-Green function for p is an element of (1, 2) is not quasi-symmetric in plane C-1,C-1-domains.

• 73.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Estimates of size of cycle in a predator-prey system2018In: Differential Equations and Dynamical Systems, ISSN 0971-3514, E-ISSN 0974-6870Article in journal (Refereed)

We consider a Rosenzweig–MacArthur predator-prey system which incorporates logistic growth of the prey in the absence of predators and a Holling type II functional response for interaction between predators and preys. We assume that parameters take values in a range which guarantees that all solutions tend to a unique limit cycle and prove estimates for the maximal and minimal predator and prey population densities of this cycle. Our estimates are simple functions of the model parameters and hold for cases when the cycle exhibits small predator and prey abundances and large amplitudes. The proof consists of constructions of several Lyapunov-type functions and derivation of a large number of non-trivial estimates which are of independent interest.

• 74.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Evolutionary dynamics in a Lotka–Volterra competition model with impulsive periodic disturbance2016In: Mathematical methods in the applied sciences, ISSN 0170-4214, E-ISSN 1099-1476, Vol. 39, no 2, p. 177-188Article in journal (Refereed)

In this paper, we develop a theoretical framework to investigate the influence of impulsive periodic disturbance on the evolutionary dynamics of a continuous trait, such as body size, in a general Lotka–Volterra-type competition model. The model is formulated as a system of impulsive differential equations. First, we derive analytically the fitness function of a mutant invading the resident populations when rare in both monomorphic and dimorphic populations. Second, we apply the fitness function to a specific system of asymmetric competition under size-selective harvesting and investigate the conditions for evolutionarily stable strategy and evolutionary branching by means of critical function analysis. Finally, we perform long-term simulation of evolutionary dynamics to demonstrate the emergence of high-level polymorphism. Our analytical results show that large harvesting effort or small impulsive harvesting period inhibits branching, while large impulsive harvesting period promotes branching.

• 75.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The Riesz-Thorin Interpolation Theorem2018Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

In this essay we present some elementary measure theory and some theory of Lp-spaces with the goal of proving the Riesz-Thorin interpolation theorem.

• 76.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The Riesz representation theorem for positive linear functionals2016Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 77.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Regularity near the initial state in the obstacle problem for a class of hypoelliptic ultraparabolic operators2010In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 249, no 8, p. 2044-2060Article in journal (Refereed)

This paper is devoted to a proof of regularity, near the initial state, for solutions to the Cauchy-Dirichlet and obstacle problem for a class of second order differential operators of Kolmogorov type. The approach used here is general enough to allow us to consider smooth obstacles as well as non-smooth obstacles.

• 78.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On Monte Carlo algorithms applied to Dirichlet problems for parabolic operators in the setting of time-dependent domains2009In: Monte Carlo Methods and Applications, ISSN 1569-3961, Vol. 15, no 1, p. 11-47Article in journal (Refereed)

Dirichlet problems for second order parabolic operators in space-time domains Ω⊂ Rn+1  are of paramount importance in analysis, partial differential equations and applied mathematics. These problems can be approached in many different ways using techniques from partial differential equations, potential theory, stochastic differential equations, stopped diffusions and Monte Carlo methods. The performance of any technique depends on the structural assumptions on the operator, the geometry and smoothness properties of the space-time domain Ω, the smoothness of the Dirichlet data and the smoothness of the coefficients of the operator under consideration. In this paper, which mainly is of numerical nature, we attempt to further understand how Monte Carlo methods based on the numerical integration of stochastic differential equations perform when applied to Dirichlet problems for uniformly elliptic second order parabolic operators and how their performance vary as the smoothness of the boundary, Dirichlet data and coefficients change from smooth to non-smooth. Our analysis is set in the genuinely parabolic setting of time-dependent domains, which in itself adds interesting features previously only modestly discussed in the literature. The methods evaluated and discussed include elaborations on the non-adaptive method proposed by Gobet [4] based on approximation by half spaces and exit probabilities and the adaptive method proposed in [3] for weak approximation of stochastic differential equations.

• 79.
Departmen of Anatomy,University of Kuopio, Kuopio, Finland.
Departmen of Anatomy, University of Kuopio, Kuopio, Finland. Departmen of Computer Science and Applied Mathematics, University of Kuopio, Kuopio, Finland. Departmen of Anatomy, University of Kuopio, Kuopio, Finland. Departmen of Anatomy, University of Kuopio, Kuopio, Finland. Departmen of Anatomy, University of Kuopio, Kuopio, Finland. Departmen of Anatomy, University of Kuopio, Kuopio, Finland.
Quantitation of autoradiographic grains in different zones of articular cartilage with image analyzer.1990In: Histochemistry, ISSN 0301-5564, Vol. 93, no 3, p. 241-245, article id 2312351Article in journal (Refereed)

A novel method is introduced for the estimation of grain numbers in autoradiographic sections of articular cartilage with an image analyzer. It is based on separation of grains from the underlying structures by gray level thresholding and determination of the percentage of total area occupied by grains in a relatively large measuring field. The mean grain size is used as a reference to calculate grain numbers per cell profile and per unit area of tissue in various zones of bovine articular cartilage labelled with 35S-sulphate in tissue culture. The results demonstrate considerable zonal differences as well as site related topographic variation in the rate of 35S-sulphate incorporation. The largest site-related variation in the grain counts was observed in the superficial zone, suggesting a delicate control of proteoglycan synthesis in this zone.

• 80.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The Automorphism Groups on the Complex Plane2017Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

The automorphism groups in the complex plane are defined, and we prove that they satisfy the group axioms. The automorphism group is derived for some domains. By applying the Riemann mapping theorem, it is proved that every automorphism group on simply connected domains that are proper subsets of the complex plane, is isomorphic to the automorphism group on the unit disc.

• 81.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
On Stein Neighborhood Bases and the Nebenhülle2010Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

This thesis consists of three parts. The first part is a introduction to the theory of domains of holomorphy through holomorphic convexity. The second part gives a introduction to Stein neighborhood bases in Cn and presents some minor results on the Nebenhülle of a compact set in Cn. The third and final part reviews some results on the existence of Stein neighborhood bases.

• 82.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The Whitney embedding theorem2014Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

A fundamental theorem in differential geometry is proven in this essay. It is the embedding theorem due to Hassler Whitney, which shows that the ever so general and useful topological spaces called manifolds, can all be regarded as subspaces of some Euclidean space. The version of the proof given in this essay is very similar to the original from 1944. Modern definitions are used, however, and many illustrations have been made, wherever it helps the understanding.

• 83.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the fundamental theorem of calculus2015Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis

The Riemann integral has many flaws, some that becomes visible in the fundamental theorem of calculus. The main point of this essay is to introduce the gauge integral, and prove a much more suitable version of that theorem.

• 84.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
A generalization of a theorem of G. Freud on the differentiability of functions1985In: Acta Scientarum Mathematicarum, ISSN 0001-6969, Vol. 49, no 1-4, p. 271-281Article in journal (Refereed)
• 85.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
A note on capacity and Hausdorff measure in homogeneous spaces1997In: Potential analysis, Vol. 6Article in journal (Refereed)
• 86.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
A note on the Carleman condition for determinacy of moment problems1987In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 25, no 2, p. 289-294Article in journal (Refereed)
• 87.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
A Short and Unified Proof of Kummer's Test2018Other (Other academic)

Kummer’s test from 1835 states that the positive series ∑n=1an is convergent if and only if there is a sequence {Bn}1 of positive numbers such that Bn·an/an+1Bn+1 ≥ 1, for all sufficiently large n. We present an exact analysis and a short and unified proof of Kummer’s test. The test has been applied to differential equations and studied in mathematical philosophy.

• 88.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Bernstein's analyticity theorem for binary differences1999In: Mathematische Annalen, ISSN 0025-5831 (paper), 1432-1807 (e), Vol. 315, no 2, p. 251-261Article in journal (Refereed)
• 89.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Bernstein's analyticity theorem for quantum differences2007In: Czechoslovak Mathematical Journal, ISSN 0011-4642, E-ISSN 1572-9141, Vol. 57, no 1, p. 67-73Article in journal (Refereed)

We consider real valued functions f defined on a subinterval I of the positive real axis and prove that if all of f's quantum differences are nonnegative then f has a power series representation on I. Further, if the quantum differences have fixed sign on I then f is analytic on I.

• 90.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Bessel potentials and extension of continuous functione on compact sets1975In: Arkiv för matematik, ISSN 0004-2080 (p), 1871-2487 (e), Vol. 13, no 2, p. 263-271Article in journal (Refereed)

We characterize the compact sets K in the n-dimensional Euclidean space with capacity zero relative to a certain kernel as exactly those sets for which every continous function on K has an extension to a continuous potential in the full space. A special case in the Bessel kernel and the related capacity.

• 91.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Beurling's analyticity theorem for quantum differences2008In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 127, no 3, p. 369-380Article in journal (Refereed)

A theorem of Beurling states that if f satisfies , n = 1, 2,..., for some 0 < ρ < 2, on a real interval I, then f is analytic in a rhombus containing I. We study the corresponding problem for the quantum differences Δ n f (q, x), q > 1, n = 1, 2,..., for functions defined on (0, ∞) and prove quantitative and qualitative analogues of Beurling’s result. We also characterize the analyticity of f on subintervals of (0, ∞) in q-analytic terms.

• 92.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Capacities of compact sets in linear subspaces of R^n1978In: Pacific Journal of Mathematics, ISSN 0030-8730, Vol. 78, no 1, p. 261-266Article in journal (Refereed)

We characterize the exceptional sets for Besov spaces in the n-dimensional Euclidean space by an extension property for continuous functions and prove an inequality between Bessel and Besov capacities.

• 93.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Continuous functions and Riesz type potentials in homogeneous spaces2015In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, no 3, p. 495-511Article in journal (Refereed)

We develop a potential theory for a Riesz type kernel in a homogeneous space and characterize the compact sets K with capacity zero as the sets K for which every continous function f on K is the restriction to K of a continuous potential Uσfk of an absolutely continuous measure σ f supported in an arbitrarily small neighbourhood of K. The measure σ f can be choosen as a suitable restriction of a single measure σ that only depends on the set K and the kernel k.

• 94.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Explaining Kummer's test2015Manuscript (preprint) (Other academic)

Mathematical theorems and their proofs are often described as being beautyful, natural or explanatory. This paper treats the last case  and discusses explanatory versus non explanatory proofs using  a convergence theorem for positive series due to Kummer, called Kummer's test. This example was studied by Pringsheim already in the beginning in the last century and has recently been discussed by Hafner and Mancuso in relation to Steiner's criteria for an explanatory proof. We provide a mathematical analysis of Kummer's test and give a proof that we claim is explanatory in this sense. Besides this we answer a number of questions that are frequently asked about the test.

• 95.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Local and global weighted norm inequalities for thr sharp function and the Hardy-Littlewood maximal function1993In: Ricerche di Matamatica, Vol. 42, no 2Article in journal (Refereed)
• 96.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Nonlinear potential theory in Lebesque spaces with mixed norm1988In: Potential Theory, Prague, 1987 / [ed] J. Kral, J. Lukes, I. Netuka, J. Vesely, New York: Plenum Publishing Corporation , 1988, p. 325-331Conference paper (Refereed)

The classical potential theory for the Riesz kernel in the d-dimentional Euclidean space has been generalized in many different ways. We are here concerned with the L^p-potential theory which appeared around 1970 in papers by V.G. Maz'ya, V. P, Havin and N. G. Mayers. In this paper we extend that theory to the so called Lebesgue spaces with mixed norm.

• 97.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
On Lp-differentiability and difference properties of functions1982In: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 74, no 2, p. 153-168Article in journal (Refereed)
• 98.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
On ordinary differentiability of Bessel potentials1984In: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 44, no 3, p. 325-352Article in journal (Refereed)
• 99.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
On properties of functions with conditions on their mean oscillation over cubes1982In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 20, no 2, p. 275-291Article in journal (Refereed)
• 100.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
On s-sets and mutual absolute continuity of measures on homogeneous spaces1997In: Manuscripta Mathematica, ISSN 0025-2611 (p), 1432-1785 (e), Vol. 94, p. 169-186Article in journal (Refereed)
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