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• 51.
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.

• 52.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Derivatan ur ett historiskt perspektiv2017Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)

Derivatan är en fundamental del av matematiken. Denna uppsats kommer att handla om historiska framstegen inom matematiken som lett fram till att derivatan har kunnat definierats så som vi är vana att se den idag.

• 53.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
An ordering of measures induced by plurisubharmonic functionsManuskript (preprint) (Övrigt vetenskapligt)

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.

• 54.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Boundary values of plurisubharmonic functions and related topics2009Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

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.

• 55.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
On boundary values of plurisubharmonic functionsManuskript (preprint) (Övrigt vetenskapligt)

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.

• 56.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för fysik. 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 Cores2017Ingår i: Astrophysical Journal, ISSN 0004-637X, E-ISSN 1538-4357, Vol. 836, nr 2, artikel-id 203Artikel i tidskrift (Refereegranskat)

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.

• 57.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Generalized Maximum Spacing Estimation for Multivariate Observations2015Ingår i: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 42, nr 4, s. 1092-1108Artikel i tidskrift (Refereegranskat)

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.

• 58.
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 problems2007Ingår i: Communications in Numerical Methods in Engineering, ISSN 1069-8299, E-ISSN 1099-0887, Vol. 24, nr 6, s. 505-521Artikel i tidskrift (Refereegranskat)

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.

• 59.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
The Riemann-Stieltjes integral: and some applications in complex analysis and probability theory2014Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)

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.

• 60.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
On the geometry of calibrated manifolds: with applications to electrodynamics2013Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

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.

• 61.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Modeling credit risk for an SME loan portfolio: An Error Correction Model approach2017Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

Since the global financial crisis of 2008, several big regulations have been implemented to assure that banks follow sound risk management. Among these are the Basel II Accords that implement capital requirements for credit risk. The core measures of credit risk evaluation are the Probability of Default and Loss Given Default. The Basel II Advanced Internal-Based-Rating Approach allows banks to model these measures for individual portfolios and make their own evaluations. This thesis, in compliance with the Advanced Internal-Based-rating approach, evaluates the use of an Error Correction Model when modeling the Probability of Default. A model proven to be strong in stress testing. Furthermore, a Loss Given Default function is implemented that ties Probability of Default and Loss Given Default to systematic risk.

The Error Correction Model is implemented on an SME portfolio from one of the "big four" banks in Sweden. The model is evaluated and stress tested with the European Banking Authority's 2016 stress test scenario and analyzed, with promising results.

• 62. Ling, Zhi
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Turing pattern formation in a predator-prey system with cross diffusion2014Ingår i: Applied Mathematical Modelling, ISSN 0307-904X, E-ISSN 1872-8480, Vol. 38, nr 21-22, s. 5022-5032Artikel i tidskrift (Refereegranskat)

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.

• 63.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Matematiska institutionen.
Comparing two versions of Markov's inequality on compact sets1994Ingår i: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 77, nr 2, s. 202-211Artikel i tidskrift (Refereegranskat)

﻿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.﻿

• 64.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Two versions of Markov's inequality1993Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
• 65.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Matematiska institutionen.
University of Krakow.
A Note on Bernstein's Theorems1995Ingår i: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 81, nr 3, s. 316-322Artikel i tidskrift (Refereegranskat)

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.

• 66.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Bimodal and Gaussian Ising spin glasses in dimension two2016Ingår i: Physical Review E, ISSN 2470-0045, Vol. 93, nr 2, artikel-id 022119Artikel i tidskrift (Refereegranskat)

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.

• 67.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Non-self-averaging in Ising spin glasses and hyperuniversality2016Ingår i: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 93, nr 1, artikel-id 012118Artikel i tidskrift (Refereegranskat)

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.

• 68.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Estimates for p-harmonic functions vanishing on a flat2011Ingår i: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, nr 18, s. 6852-6860Artikel i tidskrift (Refereegranskat)

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.

• 69.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Phragmén-Lindelöf theorems and p-harmonic measures for sets near low-dimensional hyperplanes2016Ingår i: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 44, nr 2, s. 313-330Artikel i tidskrift (Refereegranskat)

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.

• 70. Lundström, Niklas L.P.
Bifurcations and Strange Attractors in a Climate Related System2005Ingår i: Differentsial'nye Uravneniya i Protsessy Upravleniya / Differential Equations and Control Processes

, ISSN 1817-2172, E-ISSN 1817-2172, nr 1, s. 1-53
Artikel i tidskrift (Refereegranskat)

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.

• 71.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
p-harmonic functions near the boundary2011Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
• 72.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Existence, uniqueness and regularity of solutions to systems of nonlocal obstacle problems related to optimal switching2019Ingår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 475, nr 1, s. 13-31Artikel i tidskrift (Refereegranskat)

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.

• 73.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Decay of a p-harmonic measure in the plane2013Ingår i: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, nr 1, s. 351-366Artikel i tidskrift (Refereegranskat)

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.

• 74.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Estimates of size of cycle in a predator-prey system2018Ingår i: Differential Equations and Dynamical Systems, ISSN 0971-3514, E-ISSN 0974-6870Artikel i tidskrift (Refereegranskat)

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.

• 75.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Evolutionary dynamics in a Lotka–Volterra competition model with impulsive periodic disturbance2016Ingår i: Mathematical methods in the applied sciences, ISSN 0170-4214, E-ISSN 1099-1476, Vol. 39, nr 2, s. 177-188Artikel i tidskrift (Refereegranskat)

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.

• 76.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
The Riesz-Thorin Interpolation Theorem2018Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)

I denna uppsats presenteras grundläggande måtteori och något kring teorin om Lp-rum med målet att bevisa Riesz-Thorins interpolationssats.

• 77.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
The Riesz representation theorem for positive linear functionals2016Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
• 78.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Regularity near the initial state in the obstacle problem for a class of hypoelliptic ultraparabolic operators2010Ingår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 249, nr 8, s. 2044-2060Artikel i tidskrift (Refereegranskat)

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.

• 79.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
On Monte Carlo algorithms applied to Dirichlet problems for parabolic operators in the setting of time-dependent domains2009Ingår i: Monte Carlo Methods and Applications, ISSN 1569-3961, Vol. 15, nr 1, s. 11-47Artikel i tidskrift (Refereegranskat)

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.

• 80.
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.1990Ingår i: Histochemistry, ISSN 0301-5564, Vol. 93, nr 3, s. 241-245, artikel-id 2312351Artikel i tidskrift (Refereegranskat)

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.

• 81.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
The Automorphism Groups on the Complex Plane2017Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)

Automorfigrupperna i det komplexa talplanet definieras och vi bevisar att de uppfyller gruppaxiomen. Automorfigruppen på några domän härleds. Genom att applicera Riemanns avbildningssats bevisas att varje automorfigrupp på enkelt sammanhängande, öppna och äkta delmängder av det komplexa talplanet är isomorf med automorfigruppen på enhetsdisken.

• 82.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
On Stein Neighborhood Bases and the Nebenhülle2010Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

Denna uppsats består av tre delar. Den första delen introducerar holomorfiområden genom teorin för konvexitet med avseende på holomorfa funktioner. Den andra delen är en introduktion till Steinomgivningsbaser i Cn och presenterar några smärre resultat rörande Nebenhüllet till en kompakt mängd i Cn. Den tredje och sista delen ger en överblick över en del resultat om existensen av Steinomgivningsbaser.

• 83.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
The Whitney embedding theorem2014Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)

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.

• 84.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
On the fundamental theorem of calculus2015Självständigt arbete på avancerad nivå (magisterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

Riemannintegralen har många brister. Vissa utav dessa ser man i integralkalkylens huvudsats. Huvudmålet med denna uppsats är att introducera gauge integralen och visa en mer lämplig version av huvudsatsen.

• 85.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
A generalization of a theorem of G. Freud on the differentiability of functions1985Ingår i: Acta Scientarum Mathematicarum, ISSN 0001-6969, Vol. 49, nr 1-4, s. 271-281Artikel i tidskrift (Refereegranskat)
• 86.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
A note on capacity and Hausdorff measure in homogeneous spaces1997Ingår i: Potential analysis, Vol. 6Artikel i tidskrift (Refereegranskat)
• 87.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
A note on the Carleman condition for determinacy of moment problems1987Ingår i: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 25, nr 2, s. 289-294Artikel i tidskrift (Refereegranskat)
• 88.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
A Short and Unified Proof of Kummer's Test2018Övrigt (Övrigt vetenskapligt)

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.

• 89.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Bernstein's analyticity theorem for binary differences1999Ingår i: Mathematische Annalen, ISSN 0025-5831 (paper), 1432-1807 (e), Vol. 315, nr 2, s. 251-261Artikel i tidskrift (Refereegranskat)
• 90.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Bernstein's analyticity theorem for quantum differences2007Ingår i: Czechoslovak Mathematical Journal, ISSN 0011-4642, E-ISSN 1572-9141, Vol. 57, nr 1, s. 67-73Artikel i tidskrift (Refereegranskat)

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.

• 91.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Bessel potentials and extension of continuous functione on compact sets1975Ingår i: Arkiv för matematik, ISSN 0004-2080 (p), 1871-2487 (e), Vol. 13, nr 2, s. 263-271Artikel i tidskrift (Refereegranskat)

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.

• 92.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Beurling's analyticity theorem for quantum differences2008Ingår i: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 127, nr 3, s. 369-380Artikel i tidskrift (Refereegranskat)

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.

• 93.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Capacities of compact sets in linear subspaces of R^n1978Ingår i: Pacific Journal of Mathematics, ISSN 0030-8730, Vol. 78, nr 1, s. 261-266Artikel i tidskrift (Refereegranskat)

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.

• 94.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Continuous functions and Riesz type potentials in homogeneous spaces2015Ingår i: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, nr 3, s. 495-511Artikel i tidskrift (Refereegranskat)

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.

• 95.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Explaining Kummer's test2015Manuskript (preprint) (Övrigt vetenskapligt)

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.

• 96.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Local and global weighted norm inequalities for thr sharp function and the Hardy-Littlewood maximal function1993Ingår i: Ricerche di Matamatica, Vol. 42, nr 2Artikel i tidskrift (Refereegranskat)
• 97.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Nonlinear potential theory in Lebesque spaces with mixed norm1988Ingår i: Potential Theory, Prague, 1987 / [ed] J. Kral, J. Lukes, I. Netuka, J. Vesely, New York: Plenum Publishing Corporation , 1988, s. 325-331Konferensbidrag (Refereegranskat)

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.

• 98.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
On Lp-differentiability and difference properties of functions1982Ingår i: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 74, nr 2, s. 153-168Artikel i tidskrift (Refereegranskat)
• 99.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
On ordinary differentiability of Bessel potentials1984Ingår i: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 44, nr 3, s. 325-352Artikel i tidskrift (Refereegranskat)
• 100.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
On properties of functions with conditions on their mean oscillation over cubes1982Ingår i: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 20, nr 2, s. 275-291Artikel i tidskrift (Refereegranskat)
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