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  • 1.
    Andrist B., Rafael
    et al.
    Bergische Universität Wuppertal, Wuppertal, Germany.
    Kutzschebauch, Frank
    University of Bern, Bern, Switzerland.
    Lind, Andreas
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
    Holomorphic Automorphisms of Danielewski Surfaces II: Structure of the Overshear Group2015In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 25, no 3, p. 1859-1889Article in journal (Refereed)
    Abstract [en]

    We apply Nevanlinna theory for algebraic varieties to Danielewski surfaces and investigate their group of holomorphic automorphisms. Our main result states that the overshear group, which is known to be dense in the identity component of the holomorphic automorphism group, is a free product.

  • 2.
    Daghighi, Abtin
    Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
    Approach regions of Lebesgue measurable, locally bounded, quasi-continuous functions2012In: International Journal of Mathematical Analysis, ISSN 1312-8876, E-ISSN 1314-7579, Vol. 6, no 13, p. 659-680Article in journal (Refereed)
    Abstract [en]

    Quasi-continuity (in the sense of Kempisty) generalizes directional continuity of complex-valued functions on open subsets of ℝ n or ℂ n, and in particular provides certain approach regions at every point. We show that these can be used as a proof tool for proving several properties forLebesgue measurable, locally bounded, quasi-continuous functions e.g. that for such a function f the polynomial ring C(M,K)[f] (where K = ℝ or ℂ) satisfies that the equivalence classes under identification a.e. have cardinality one and an asymptotic maximum principle.

  • 3.
    Flodén, Liselott
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    G-Convergence and Homogenization of some Sequences of Monotone Differential Operators2009Doctoral thesis, monograph (Other academic)
    Abstract [en]

    This thesis mainly deals with questions concerning the convergence of some sequences of elliptic and parabolic linear and non-linear operators by means of G-convergence and homogenization. In particular, we study operators with oscillations in several spatial and temporal scales. Our main tools are multiscale techniques, developed from the method of two-scale convergence and adapted to the problems studied. For certain classes of parabolic equations we distinguish different cases of homogenization for different relations between the frequencies of oscillations in space and time by means of different sets of local problems. The features and fundamental character of two-scale convergence are discussed and some of its key properties are investigated. Moreover, results are presented concerning cases when the G-limit can be identified for some linear elliptic and parabolic problems where no periodicity assumptions are made.

  • 4.
    Flodén, Liselott
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Holmbom, Anders
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Olsson, Marianne
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    On the determination of effective properties of certain structures with non-periodic temporal oscillations2009In: MATHMOD 2009 - 6th Vienna International Conference on Mathematical Modelling, Wien: Vienna University Press (WUV), 2009, p. 2627-2630Conference paper (Refereed)
    Abstract [en]

    We investigate the homogenization of an evolution problem modelled by a parabolic equation, where the coefficient describing the structure is periodic in space but may vary in time in a non-periodic way. This is performed applying a generalization of two-scale convergence called λ-scale convergence. We give a result on the characterization of the λ-scale limit of gradients under certain boundedness assumptions. This is then applied to perform the homogenization procedure. It turns out that, under a certain condition on the rate of change of the temporal variations, the effective property of the given structure can be determined the same way as in periodic cases.

  • 5.
    Holmbom, Anders
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Lobkova, Tatiana
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics.
    Homogenization of a hyperbolic-parabolic problem in a perforated domain2016Conference paper (Other academic)
  • 6.
    Ivarsson, Björn
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
    Kutzschebauch, Frank
    Universität Bern.
    Holomorphic factorization of mappings into SL_n(C)2012In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 175, no 1, p. 45-69Article in journal (Refereed)
  • 7.
    Lind, Andreas
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Kutzschebauch, Frank
    Univ Bern, Inst Math, CH-3012 Bern, Switzerland.
    Holomorphic automorphisms of Danielewski surfaces I: Density of the group of overshears2011In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 139, no 11, p. 3915-3927Article in journal (Refereed)
    Abstract [en]

    We define the notion of shears and overshears on a Danielewski surface. We show that the group generated by shears and overshears is dense (in the compact open topology) in the path-connected component of the identity of the holomorphic automorphism group.

  • 8.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Homogenization of monotone parabolic problems with several temporal scales2012In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 57, no 3, p. 191-214Article in journal (Refereed)
    Abstract [en]

    In this paper we homogenize monotone parabolic problems with two spatial scales and any number of temporal scales. Under the assumption that the spatial and temporal scales are well-separated in the sense explained in the paper, we show that there is an H-limit defined by at most four distinct sets of local problems corresponding to slow temporal oscillations, slow resonant spatial and temporal oscillations (the “slow” self-similar case), rapid temporal oscillations, and rapid resonant spatial and temporal oscillations (the “rapid” self-similar case), respectively.

  • 9.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Homogenization of Some Selected Elliptic and Parabolic Problems Employing Suitable Generalized Modes of Two-Scale Convergence2010Licentiate thesis, monograph (Other academic)
    Abstract [en]

    The present thesis is devoted to the homogenization of certain elliptic and parabolic partial differential equations by means of appropriate generalizations of the notion of two-scale convergence. Since homogenization is defined in terms of H-convergence, we desire to find the H-limits of sequences of periodic monotone parabolic operators with two spatial scales and an arbitrary number of temporal scales and the H-limits of sequences of two-dimensional possibly non-periodic linear elliptic operators by utilizing the theories for evolution-multiscale convergence and λ-scale convergence, respectively, which are generalizations of the classical two-scale convergence mode and custom-made to treat homogenization problems of the prescribed kinds. Concerning the multiscaled parabolic problems, we find that the result of the homogenization depends on the behavior of the temporal scale functions. The temporal scale functions considered in the thesis may, in the sense explained in the text, be slow or rapid and in resonance or not in resonance with respect to the spatial scale function. The homogenization for the possibly non-periodic elliptic problems gives the same result as for the corresponding periodic problems but with the exception that the local gradient operator is everywhere substituted by a differential operator consisting of a product of the local gradient operator and matrix describing the geometry and which depends, effectively, parametrically on the global variable.

  • 10.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Selected Topics in Homogenization2012Doctoral thesis, monograph (Other academic)
    Abstract [en]

    The main focus of the present thesis is on the homogenization of some selected elliptic and parabolic problems. More precisely, we homogenize: non-periodic linear elliptic problems in two dimensions exhibiting a homothetic scaling property; two types of evolution-multiscale linear parabolic problems, one having two spatial and two temporal microscopic scales where the latter ones are given in terms of a two-parameter family, and one having two spatial and three temporal microscopic scales that are fixed power functions; and, finally, evolution-multiscale monotone parabolic problems with one spatial and an arbitrary number of temporal microscopic scales that are not restricted to be given in terms of power functions. In order to achieve homogenization results for these problems we study and enrich the theory of two-scale convergence and its kins. In particular the concept of very weak two-scale convergence and generalizations is developed, and we study an application of this convergence mode where it is employed to detect scales of heterogeneity.

  • 11.
    Persson, Jens
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    λ-scale convergence applied to the stationary heat conduction equation with non-periodic thermal conductivity matrix2009In: MATHMOD 2009 - 6th Vienna International Conference on Mathematical Modelling, Wien: Vienna University Press (WUV), 2009, p. 2720-2723Conference paper (Refereed)
    Abstract [en]

    In this contribution we study the homogenization of non-periodic stationary heat conduction problems with homogeneous Dirichlet boundary data by applying the recently developed λ-scale convergence technique developed by Holmbom and Silfver. λ-scale convergence can be seen as either being a special case of scale convergence (developed by Mascarenhas and Toader) or of “generalized” two-scale convergence (developed by Holmbom, Silfver, Svanstedt and Wellander). From either viewpoint, it is a possibly powerful generalization of Nguetseng’s classical, periodic two-scale convergence method. We give a definition of the concept of λ-scale convergence, which is then used to claim a main theorem on homogenization of certain non-periodic stationary heat conduction problems. The original part of the contribution starts by defining a two-dimensional “toy model”. We show that the “toy model” satisfies the right conditions such that the aforementioned main theorem on the homogenization can be employed. In this way we derive the homogenized problem, i.e. the homogenized thermal conductivity matrix, and the local problem. The contribution is concluded by giving a numerical example where we explicitly compute the homogenized thermal conductivity matrix.

  • 12.
    Porten, Egmont
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
    Boggess, Al
    Arizona State University, Tempe, AZ, U.S.A.
    Dwilewicz, Roman
    Cardinal Stefan Wyszy´nski University, Warsaw, Poland.
    On the Hartogs extension theorem for unbounded domains in Cn2018Report (Other academic)
    Abstract [en]

    Let Ω ⊂ Cn, n ≥ 2, be a domain with smooth connected boundary. IfΩ is relatively compact, the Hartogs-Bochner theorem ensures that everyCR distribution on ∂Ω has a holomorphic extension to Ω. For unboundeddomains this extension property may fail, for example if Ω contains a complex hypersurface. The main result in this paper tells that the extensionproperty holds if and only if the envelope of holomorphy of Cn\Ω is Cn.It seems that it is a first result in the literature which gives a geometriccharacterization of unbounded domains in Cnfor which the Hartogs phenomenon holds. Comparing this to earlier work by the first two authorsand Z. S lodkowski, one observes that the extension problem sensitively depends on a finer geometry of the contact of a complex hypersurface andthe boundary of the domain.

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