liu.seSearch for publications in DiVA
Change search
Refine search result
123 1 - 50 of 148
CiteExportLink to result list
Permanent link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Rows per page
  • 5
  • 10
  • 20
  • 50
  • 100
  • 250
Sort
  • Standard (Relevance)
  • Author A-Ö
  • Author Ö-A
  • Title A-Ö
  • Title Ö-A
  • Publication type A-Ö
  • Publication type Ö-A
  • Issued (Oldest first)
  • Issued (Newest first)
  • Created (Oldest first)
  • Created (Newest first)
  • Last updated (Oldest first)
  • Last updated (Newest first)
  • Disputation date (earliest first)
  • Disputation date (latest first)
  • Standard (Relevance)
  • Author A-Ö
  • Author Ö-A
  • Title A-Ö
  • Title Ö-A
  • Publication type A-Ö
  • Publication type Ö-A
  • Issued (Oldest first)
  • Issued (Newest first)
  • Created (Oldest first)
  • Created (Newest first)
  • Last updated (Oldest first)
  • Last updated (Newest first)
  • Disputation date (earliest first)
  • Disputation date (latest first)
Select
The maximal number of hits you can export is 250. When you want to export more records please use the Create feeds function.
  • 1. Order onlineBuy this publication >>
    Achieng, Pauline
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Reconstruction of solutions of Cauchy problems for elliptic equations in bounded and unbounded domains using iterative regularization methods2023Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    Cauchy problems for elliptic equations arise in applications in science and engineering. These problems often involve finding important information about an elliptical system from indirect or incomplete measurements. Cauchy problems for elliptic equations are known to be disadvantaged in the sense that a small pertubation in the input can result in a large error in the output. Regularization methods are usually required in order to be able to find stable solutions. In this thesis we study the Cauchy problem for elliptic equations in both bounded and unbounded domains using iterative regularization methods. In Paper I and II, we focus on an iterative regularization technique which involves solving a sequence of mixed boundary value well-posed problems for the same elliptic equation. The original version of the alternating iterative technique is based on iterations alternating between Dirichlet-Neumann and Neumann-Dirichlet boundary value problems. This iterative method is known to possibly work for Helmholtz equation. Instead we study a modified version based on alternating between Dirichlet-Robin and Robin-Dirichlet boundary value problems. First, we study the Cauchy problem for general elliptic equations of second order with variable coefficients in a limited domain. Then we extend to the case of unbounded domains for the Cauchy problem for Helmholtz equation. For the Cauchy problem, in the case of general elliptic equations, we show that the iterative method, based on Dirichlet-Robin, is convergent provided that parameters in the Robin condition are chosen appropriately. In the case of an unbounded domain, we derive necessary, and sufficient, conditions for convergence of the Robin-Dirichlet iterations based on an analysis of the spectrum of the Laplacian operator, with boundary conditions of Dirichlet and Robin types.

    In the numerical tests, we investigate the precise behaviour of the Dirichlet-Robin iterations, for different values of the wave number in the Helmholtz equation, and the results show that the convergence rate depends on the choice of the Robin parameter in the Robin condition. In the case of unbounded domain, the numerical experiments show that an appropriate truncation of the domain and an appropriate choice of Robin parameter in the Robin condition lead to convergence of the Robin-Dirichlet iterations.

    In the presence of noise, additional regularization techniques have to implemented for the alternating iterative procedure to converge. Therefore, in Paper III and IV we focus on iterative regularization methods for solving the Cauchy problem for the Helmholtz equation in a semi-infinite strip, assuming that the data contains measurement noise. In addition, we also reconstruct a radiation condition at infinity from the given Cauchy data. For the reconstruction of the radiation condition, we solve a well-posed problem for the Helmholtz equation in a semi-infinite strip. The remaining solution is obtained by solving an ill-posed problem. In Paper III, we consider the ordinary Helmholtz equation and use seperation of variables to analyze the problem. We show that the radiation condition is described by a non-linear well-posed problem that provides a stable oscillatory solution to the Cauchy problem. Furthermore, we show that the ill–posed problem can be regularized using the Landweber’s iterative method and the discrepancy principle. Numerical tests shows that the approach works well.

    Paper IV is an extension of the theory from Paper III to the case of variable coefficients. Theoretical analysis of this Cauchy problem shows that, with suitable bounds on the coefficients, can iterative regularization methods be used to stabilize the ill-posed Cauchy problem.

    List of papers
    1. Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations
    Open this publication in new window or tab >>Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations
    2021 (English)In: Bulletin of the Iranian Mathematical Society, ISSN 1735-8515, Vol. 47, p. 1681-1699Article in journal (Refereed) Published
    Abstract [en]

    The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers k2, in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of k2 in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

    Place, publisher, year, edition, pages
    Springer, 2021
    Keywords
    Helmholtz equation, Cauchy problem, Inverse problem, Ill-posed problem
    National Category
    Mathematical Analysis
    Identifiers
    urn:nbn:se:liu:diva-170834 (URN)10.1007/s41980-020-00466-7 (DOI)000575739300001 ()2-s2.0-85092146699 (Scopus ID)
    Available from: 2020-10-26 Created: 2020-10-26 Last updated: 2024-02-22Bibliographically approved
    2. Robin-Dirichlet alternating iterative procedure for solving the Cauchy problem for Helmholtz equation in an unbounded domain
    Open this publication in new window or tab >>Robin-Dirichlet alternating iterative procedure for solving the Cauchy problem for Helmholtz equation in an unbounded domain
    2023 (English)In: Journal of Inverse and Ill-Posed Problems, ISSN 0928-0219, E-ISSN 1569-3945, Vol. 31, no 5Article in journal (Refereed) Published
    Abstract [en]

    We consider the Cauchy problem for the Helmholtz equation with a domain in with N cylindrical outlets to infinity with bounded inclusions in . Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Mazya proposed an alternating iterative method for solving Cauchy problems associated with elliptic, selfadjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Berntsson, Kozlov, Mpinganzima and Turesson (2018) for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in R-2 that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters mu(0) and mu(1), the Robin-Dirichlet alternating iterative procedure is convergent.

    Place, publisher, year, edition, pages
    WALTER DE GRUYTER GMBH, 2023
    Keywords
    Helmholtz equation; Cauchy problem; inverse problem ill-posed problem
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:liu:diva-192481 (URN)10.1515/jiip-2020-0133 (DOI)000940871600001 ()
    Available from: 2023-03-21 Created: 2023-03-21 Last updated: 2024-03-18Bibliographically approved
    3. Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment
    Open this publication in new window or tab >>Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment
    2023 (English)In: Computational Methods in Applied Mathematics, ISSN 1609-4840, E-ISSN 1609-9389Article in journal (Refereed) Epub ahead of print
    Abstract [en]

    We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well.

    Place, publisher, year, edition, pages
    WALTER DE GRUYTER GMBH, 2023
    Keywords
    Helmholtz Equation; Inverse Problem; Cauchy Problem; Ill-Posed Problem; Well-Posed Problem; Landweber Method
    National Category
    Fluid Mechanics and Acoustics
    Identifiers
    urn:nbn:se:liu:diva-196637 (URN)10.1515/cmam-2022-0244 (DOI)001035412500001 ()
    Available from: 2023-08-17 Created: 2023-08-17 Last updated: 2023-11-13
    Download full text (pdf)
    fulltext
    Download (png)
    presentationsbild
  • 2.
    Achieng, Pauline
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering. Univ Nairobi, Kenya.
    Berntsson, Fredrik
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment2023In: Computational Methods in Applied Mathematics, ISSN 1609-4840, E-ISSN 1609-9389Article in journal (Refereed)
    Abstract [en]

    We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well.

  • 3.
    Achieng, Pauline
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering. Univ Nairobi, Kenya.
    Berntsson, Fredrik
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Robin-Dirichlet alternating iterative procedure for solving the Cauchy problem for Helmholtz equation in an unbounded domain2023In: Journal of Inverse and Ill-Posed Problems, ISSN 0928-0219, E-ISSN 1569-3945, Vol. 31, no 5Article in journal (Refereed)
    Abstract [en]

    We consider the Cauchy problem for the Helmholtz equation with a domain in with N cylindrical outlets to infinity with bounded inclusions in . Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Mazya proposed an alternating iterative method for solving Cauchy problems associated with elliptic, selfadjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Berntsson, Kozlov, Mpinganzima and Turesson (2018) for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in R-2 that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters mu(0) and mu(1), the Robin-Dirichlet alternating iterative procedure is convergent.

  • 4.
    Ahmad, Garmyan
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Komvuxelever med upplevda matematiksvårigheter: en studie om deras lärandesituation på komvuxkurser i matematik2023Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Syftet med detta arbete är att undersöka hur komvuxelever beskriver orsaker till sina upplevda matematiksvårigheter och hur de värderar sina lärares strategier för att hjälpa dem. Studien använde sig av semistrukturerade intervjuer för att undersöka matematiksvårigheter hos fem komvuxelever som har haft svårt med ämnet matematik under sin grundskole- eller gymnasietid. En gemensam nämnare visade sig vara att de under sin grundskoletid inte hade motivation för ämnet matematik och det ansåg de flesta av eleverna har haft en negativ inverkan på deras matematiklärande. Bristen på motivation ledde till att de missade väsentliga delar i den grundläggande matematiken trots att de fick hjälp och stöd av sina matematiklärare på grundskolan och gymnasiet. Samtliga elever berättade att deras otillräckliga grundkunskaper i matematik hade betydelse för deras svårigheter idag. Elevers brister från tidigare årskurser gör det svårt för eleverna att tillgodose sig utökade kunskaper. Alla ansåg också att det var svårt med matematikspråket. Med matematikspråk avsåg de begrepp, ord och symboler som har betydelse för att förstå matematik. Informanterna berättade att variation i undervisningen kunde ha påverkat deras motivation och inverkat positivt på deras intresse för matematik.Alla intervjuade elever sade sig idag ha varit missnöjda med den hjälp och det stöd de fick av sina Komvuxlärare, vilket betyder till att de nu hade mer svårigheter i matematik än tidigare.

    Download full text (pdf)
    fulltext
  • 5.
    Al Boraki, Nissren Mohammed
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Jugo, Susana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Ämnesintegrerade matematiklektioner för grundskolans högstadium: En designstudie om samverkan med lärare i teknik, träslöjd och hemkunskap2021Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Detta arbete utgår från tanken att utveckla undervisningen genom att integrera matematik med  andra ämnen. Projektet har genomförts genom att designa och utvärdera lektioner tillsammans med slöjd- och teknik- och hemkunskapslärare. Studien har utförts i åk 7–9, på en skola i Västmanland och en skola i Östergötland.

    Arbete syftar till att eleverna på högstadiet ska kunna dra nytta av sina matematiska kunskaper i reella situationer. Lektionerna bidrar till att utveckla elevernas tänkande och förmåga att kunna lösa olika problem.

    Vi har observerat lektioner, som de deltagande lärarna har genomfört, och då noterat moment som kan användas för vårt projekt. Efter det har vi intervjuat dem och de har fått bedöma sina kunskaper i den pedagogiska modellen, samt vilka möjligheter de har att genomföra integrering med andra ämnen.

    Enkätundersökning som vi genomfört, har visat deltagarnas intresse och engagemang för denna metod samt att de har motivation, att fortsätta och vidareutveckla sin undervisning enligt denna modell.

    Download full text (pdf)
    fulltext
  • 6.
    Albarracín, Lluís
    et al.
    Universitat Autònoma de Barcelona, Cerdanyola del Vallès, Spain.
    Bergman Ärlebäck, Jonas
    Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Characterizing mathematical activities promoted by Fermi problems2019In: For the Learning of Mathematics, ISSN 0228-0671, Vol. 39, no 3, p. 10-13Article in journal (Refereed)
  • 7.
    Albarracín, Lluís
    et al.
    Autonomous University of Barcelona, Spain.
    Bergman Ärlebäck, Jonas
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Esquemas de resolución de problemas de Fermi como herramienta de diseño y gestión para el profesor [Fermi problem activity templates as a design and managementtool for the teacher]2022In: Educación Matemática, E-ISSN 2448-8089, Vol. 34, no 2, p. 289-309Article in journal (Refereed)
    Abstract [en]

    This article presents a characterization of the possible resolutions toa Fermi problem from the Fermi problem Activity Templates (FpAT). The articlepresents the theoretical foundation based on previous research on Fermi prob-lems in Mathematics Education and other fields of research that allows theidentification of the four main activities that can be used when solving them.These activities are reasoned estimation, experimentation, searching for datafrom reliable sources, and statistical data collection and processing. From theseactivities, the FpAT are specified and their possibilities as tools for the manage-ment and design of classroom activities are discussed. Specific examples ofclassroom proposals are included to support both the work of students and theactivity design and management options enabled by the FpAT to promoteproblem solving and mathematical modelling.

  • 8.
    Albarracín, Lluís
    et al.
    Universitat Autònoma de Barcelona, Spain.
    Bergman Ärlebäck, Jonas
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Exploring the use of Fermi problems and the FRAT-framework with pre-service primary teachers to bring real-life contexts into classrooms2021In: The relation between mathematics education research and teachers’ professional development. Proceedings of MADIF13. The thirteenth research seminar of the Swedish Society for Research in Mathematics Education / [ed] L. Mattsson, J. Häggström, M. Carlsen, C. Kilhamn, H. Palmér, M. Perez, & K. Pettersson (Eds.), 2021, p. 81-91Conference paper (Refereed)
    Abstract [en]

    In this paper we analyse 11 pairs of pre-service primary teachers’ solutions to twoFermi problems. The solutions were expressed using the descriptive and analytic FPATframework (Fermi problem Activity Template), and our analysis focus on characterizingthe FPAT representations produced by the pre-service teachers. The results show thatalmost half of the produced FPATs in principle solved the problems but that only threesolutions provided enough detail to be practically implementable. Multiple keyconstructs were used to structure the solutions in the first problem, but not so in thesecond problem. The variation of different activities and ways of working suggested insolving the sub-problems in the two problems also differ in a significant way.

  • 9.
    Albarracín, Lluís
    et al.
    Autonomous University of Barcelona, Spain.
    López-Simó, Víctor
    Autonomous University of Barcelona, Spain.
    Bergman Ärlebäck, Jonas
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    REPENSANDO LOS PROBLEMAS DE FERMI PARA LA ENSEÑANZA Y APRENDIZAJE DE LAS CIENCIAS [Rethinking Fermi problems for science teaching and learning]2021In: Investigações em Ensino de Ciências, E-ISSN 1518-8795, Vol. 26, no 3, p. 56-56Article in journal (Refereed)
    Abstract [en]

    In this article we present a compilation of research results on the educational use of Fermi problems. While working on these problems, the key aspect is identifying the most relevant variables of a phenomenon and estimating them in order to obtain a solution to the problem. Fermi problems come from a long tradition, they have been used at all educational levels, but they have not yet established as a common activity in the classroom, especially in science classrooms. Given the broad didactic potential of Fermi problems identified in the literature, our contribution in this article is the connections with areas of interest for science teaching. We propose four didactic axes to which Fermi problems can make a relevant contribution in the science classroom: (a) To better understand what science is like, (b) To better participate in school science, (c) To better construct scientific ideas, (d) To better raise awareness of socio-scientific problems. To make the proposal more concrete, we show an example of a concrete solution with a problem of estimation of energy values, which allows us to exemplify classroom work and allows discussion with students on the relevance of science in social problems. 

    Download full text (pdf)
    fulltext
  • 10.
    Alkhutov, Yurij A.
    et al.
    AG & NG Stoletov Vladimir State Univ, Russia.
    Chechkin, Gregory A.
    Moscow MV Lomonosov State Univ, Russia; Russian Acad Sci, Russia.
    Mazya, Vladimir
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering. Inst Math & Math Modeling, Kazakhstan; RUDN Univ, Russia.
    Boyarsky-Meyers Estimate for Solutions to Zaremba Problem2022In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 245, no 2, p. 1197-1211Article in journal (Refereed)
    Abstract [en]

    The variational solution to the Zaremba problem for a divergent linear second order elliptic equation with measurable coefficients is considered. The problem is set in a local Lipschitz graph domain. An estimate in L2+δ, δ > 0, for the gradient of a solution, is proved. An example of the problem with the Dirichlet data supported by a fractal set of zero (n - 1)-dimensional measure and non-zero p-capacity, p > 1 is constructed.

  • 11.
    Alskog, Måns
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    The History of the Dirichlet Problem for Laplace’s Equation2023Independent thesis Basic level (degree of Bachelor), 14 HE creditsStudent thesis
    Abstract [en]

    This thesis aims to provide an introduction to the field of potential theory at an undergraduate level, by studying an important mathematical problem in the field, namely the Dirichlet problem. By examining the historical development of different methods for solving the problem in increasingly general contexts, and the mathematical concepts which were established to support these methods, the aim is to provide an overview of various basic techniques in the field of potential theory, as well as a summary of the fundamental results concerning the Dirichlet problem in Euclidean space.

    Download full text (pdf)
    fulltext
  • 12. Order onlineBuy this publication >>
    Andersson, Jonathan
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Bifurcations and Exchange of Stability with Density Dependence in a Coinfection Model and an Age-structured Population Model2022Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    In nature many pathogens and in particular strains of pathogens with negative effects on species coexists. This is for simplicity often ignored in many epidemiological models. It is however still of interest to get a deeper understanding how this coexistence affects the dynamics of the disease. There are several ways at which coexistence can influence the dynamics. Coinfection which is the simultaneous infection of two or more pathogens can cause increased detrimental health effects on the host. Pathogens can also limit each others growth by the effect of cross immunity  as well as promoting isolation. On the contrary one pathogen can also aid another by making the host more vulnerable to as well as more inclined to spread disease.

    Spread of disease is dependent on the density of the population. If a pathogen is able to spread or not, is strongly correlated with how many times individuals interact with each other. This in turn depends on how many individuals live in a given area. 

    The aim of papers I-III is to provide an understanding how different factors including the carrying capacity of the host population affect the dynamics of two coexisting diseases. 

    In papers I-III we investigate how the parameters effects the long term solution in the form of a stable equilibrium point. In particular we want to provide an understanding of how changes in the carrying capacity affects the long term existence of each disease as well as the occurrence of coinfection.

    The model that is studied in papers I-III is a generalization of the standard susceptible, infected, recovered (SIR) compartmental model. The SIR model is generalized by the introduction of the second infected compartment as well as the coinfection compartment. We also use a logistic growth term à la Verhulst with associated carrying capacity K. In paper I and II we make the simplifying assumption that a coinfected individual has to, if anything, transmit both of the disease and simply not just one of them. This restriction is relaxed in paper III. In all papers I-III however we do restrict ourselves by letting all transmission rates, that involves scenarios where the newly infected person does not move to same compartment as the infector, to be small. By small we here mean that the results at least hold when the relevant parameters are small enough.

    In all paper I-III it turns out that for each set of parameters excluding K there exist a unique branch of mostly stable equilibrium points depending continuously on K. We differentiate the equilibrium points of the branch by which compartments are non-zero which we refer to as the type of the equilibrium. The way that the equilibrium point changes its type with K is made clear with the use of transition diagrams together with graphs for the stable susceptible population over K.

    In paper IV we consider a model for a single age-structured population á la Mckendric-von-Foerster with the addition of differing density dependence on the birth and death rates. Each vital rate is a function of age as well as a weighing of the population also referred to as a size. The birth rate influencing size and the death rate influencing size can be weighted differently allowing us to consider different age-groups to influence the birth and death rate in different proportions compared to other age groups. 

    It is commonly assumed that an increase of population density is detrimental to the survival of each individual. However, for various reasons, it is know that for some species survival is positively correlated with population density when the population is small. This is called the Allee effect and our model includes this scenario.

    It is shown that the trivial equilibrium, which signifies extinction, is locally stable if the basic reproductive rate $R_0$ is less then 1. This implies global stability with certain extinction if no Allee effect is present. However if the Allee effect is present we show that the population can persist even if R0 < 1.

    List of papers
    1. Effect of density dependence on coinfection dynamics
    Open this publication in new window or tab >>Effect of density dependence on coinfection dynamics
    Show others...
    2021 (English)In: Analysis and Mathematical Physics, ISSN 1664-2368, E-ISSN 1664-235X, Vol. 11, no 4, article id 166Article in journal (Refereed) Published
    Abstract [en]

    In this paper we develop a compartmental model of SIR type (the abbreviation refers to the number of Susceptible, Infected and Recovered people) that models the population dynamics of two diseases that can coinfect. We discuss how the underlying dynamics depends on the carrying capacity K: from a simple dynamics to a more complex. This can also help in understanding the appearance of more complicated dynamics, for example, chaos and periodic oscillations, for large values of K. It is also presented that pathogens can invade in population and their invasion depends on the carrying capacity K which shows that the progression of disease in population depends on carrying capacity. More specifically, we establish all possible scenarios (the so-called transition diagrams) describing an evolution of an (always unique) locally stable equilibrium state (with only non-negative compartments) for fixed fundamental parameters (density independent transmission and vital rates) as a function of the carrying capacity K. An important implication of our results is the following important observation. Note that one can regard the value of K as the natural ‘size’ (the capacity) of a habitat. From this point of view, an isolation of individuals (the strategy which showed its efficiency for COVID-19 in various countries) into smaller resp. larger groups can be modelled by smaller resp. bigger values of K. Then we conclude that the infection dynamics becomes more complex for larger groups, as it fairly maybe expected for values of the reproduction number R0≈1. We show even more, that for the values R0>1 there are several (in fact four different) distinguished scenarios where the infection complexity (the number of nonzero infected classes) arises with growing K. Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra model considered previously in Ghersheen et al. (Math Meth Appl Sci 42(8), 2019).

    Place, publisher, year, edition, pages
    Basel, Switzerland: Birkhaeuser Science, 2021
    National Category
    Immunology Mathematical Analysis Other Mathematics
    Identifiers
    urn:nbn:se:liu:diva-179468 (URN)10.1007/s13324-021-00570-9 (DOI)000700279100001 ()34566882 (PubMedID)2-s2.0-85115265043 (Scopus ID)
    Note

    Funding: Swedish Research Council (VR)Swedish Research Council [2017-03837]

    Available from: 2021-09-21 Created: 2021-09-21 Last updated: 2022-05-09Bibliographically approved
    2. Effect of density dependence on coinfection dynamics: part 2
    Open this publication in new window or tab >>Effect of density dependence on coinfection dynamics: part 2
    Show others...
    2021 (English)In: Analysis and Mathematical Physics, ISSN 1664-2368, E-ISSN 1664-235X, Vol. 11, no 4, article id 169Article in journal (Refereed) Published
    Abstract [en]

    In this paper we continue the stability analysis of the model for coinfection with density dependent susceptible population introduced in Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We consider the remaining parameter values left out from Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We look for coexistence equilibrium points, their stability and dependence on the carrying capacity K. Two sets of parameter value are determined, each giving rise to different scenarios for the equilibrium branch parametrized by K. In both scenarios the branch includes coexistence points implying that both coinfection and single infection of both diseases can exist together in a stable state. There are no simple explicit expression for these equilibrium points and we will require a more delicate analysis of these points with a new bifurcation technique adapted to such epidemic related problems. The first scenario is described by the branch of stable equilibrium points which includes a continuum of coexistence points starting at a bifurcation equilibrium point with zero single infection strain #1 and finishing at another bifurcation point with zero single infection strain #2. In the second scenario the branch also includes a section of coexistence equilibrium points with the same type of starting point but the branch stays inside the positive cone after this. The coexistence equilibrium points are stable at the start of the section. It stays stable as long as the product of K and the rate γ¯γ¯ of coinfection resulting from two single infections is small but, after this it can reach a Hopf bifurcation and periodic orbits will appear.

    Place, publisher, year, edition, pages
    Springer Basel AG, 2021
    Keywords
    Mathematical Physics, Algebra and Number Theory, Analysis
    National Category
    Mathematical Analysis Immunology
    Identifiers
    urn:nbn:se:liu:diva-179802 (URN)10.1007/s13324-021-00602-4 (DOI)000702411500001 ()
    Note

    Funding: Linkoping University

    Available from: 2021-10-03 Created: 2021-10-03 Last updated: 2022-05-09Bibliographically approved
    3. Density-Dependent Feedback in Age-Structured Populations
    Open this publication in new window or tab >>Density-Dependent Feedback in Age-Structured Populations
    Show others...
    2019 (English)In: Journal of Mathematical Sciences, ISSN 1072-3374, E-ISSN 1573-8795, Vol. 242, no 1, p. 2-24Article in journal (Refereed) Published
    Abstract [en]

    The population size has far-reaching effects on the fitness of the population, that, in its turn influences the population extinction or persistence. Understanding the density- and age-dependent factors will facilitate more accurate predictions about the population dynamics and its asymptotic behaviour. In this paper, we develop a rigourous mathematical analysis to study positive and negative effects of increased population density in the classical nonlinear age-structured population model introduced by Gurtin \& MacCamy in the late 1970s. One of our main results expresses the global stability of the system in terms of the newborn function only. We also derive the existence of a threshold population size implying the population extinction, which is well-known in population dynamics as an Allee effect.

    Place, publisher, year, edition, pages
    Springer Berlin/Heidelberg, 2019
    Keywords
    Age-Structured Populations
    National Category
    Mathematics Other Biological Topics
    Identifiers
    urn:nbn:se:liu:diva-157057 (URN)10.1007/s10958-019-04464-x (DOI)
    Available from: 2019-05-24 Created: 2019-05-24 Last updated: 2022-05-09Bibliographically approved
    Download full text (pdf)
    fulltext
    Download (png)
    presentationsbild
  • 13.
    Andersson, Jonathan
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Ghersheen, Samia
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Tkachev, Vladimir
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Wennergren, Uno
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Biology. Linköping University, Faculty of Science & Engineering.
    Effect of density dependence on coinfection dynamics2021In: Analysis and Mathematical Physics, ISSN 1664-2368, E-ISSN 1664-235X, Vol. 11, no 4, article id 166Article in journal (Refereed)
    Abstract [en]

    In this paper we develop a compartmental model of SIR type (the abbreviation refers to the number of Susceptible, Infected and Recovered people) that models the population dynamics of two diseases that can coinfect. We discuss how the underlying dynamics depends on the carrying capacity K: from a simple dynamics to a more complex. This can also help in understanding the appearance of more complicated dynamics, for example, chaos and periodic oscillations, for large values of K. It is also presented that pathogens can invade in population and their invasion depends on the carrying capacity K which shows that the progression of disease in population depends on carrying capacity. More specifically, we establish all possible scenarios (the so-called transition diagrams) describing an evolution of an (always unique) locally stable equilibrium state (with only non-negative compartments) for fixed fundamental parameters (density independent transmission and vital rates) as a function of the carrying capacity K. An important implication of our results is the following important observation. Note that one can regard the value of K as the natural ‘size’ (the capacity) of a habitat. From this point of view, an isolation of individuals (the strategy which showed its efficiency for COVID-19 in various countries) into smaller resp. larger groups can be modelled by smaller resp. bigger values of K. Then we conclude that the infection dynamics becomes more complex for larger groups, as it fairly maybe expected for values of the reproduction number R0≈1. We show even more, that for the values R0>1 there are several (in fact four different) distinguished scenarios where the infection complexity (the number of nonzero infected classes) arises with growing K. Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra model considered previously in Ghersheen et al. (Math Meth Appl Sci 42(8), 2019).

    Download full text (pdf)
    fulltext
  • 14.
    Andersson, Jonathan
    et al.
    Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Ghersheen, Samia
    Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Kozlov, Vladimir
    Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Tkachev, Vladimir
    Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Wennergren, Uno
    Linköping University, Department of Physics, Chemistry and Biology, Theoretical Biology. Linköping University, Faculty of Science & Engineering.
    Effect of density dependence on coinfection dynamics: part 22021In: Analysis and Mathematical Physics, ISSN 1664-2368, E-ISSN 1664-235X, Vol. 11, no 4, article id 169Article in journal (Refereed)
    Abstract [en]

    In this paper we continue the stability analysis of the model for coinfection with density dependent susceptible population introduced in Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We consider the remaining parameter values left out from Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We look for coexistence equilibrium points, their stability and dependence on the carrying capacity K. Two sets of parameter value are determined, each giving rise to different scenarios for the equilibrium branch parametrized by K. In both scenarios the branch includes coexistence points implying that both coinfection and single infection of both diseases can exist together in a stable state. There are no simple explicit expression for these equilibrium points and we will require a more delicate analysis of these points with a new bifurcation technique adapted to such epidemic related problems. The first scenario is described by the branch of stable equilibrium points which includes a continuum of coexistence points starting at a bifurcation equilibrium point with zero single infection strain #1 and finishing at another bifurcation point with zero single infection strain #2. In the second scenario the branch also includes a section of coexistence equilibrium points with the same type of starting point but the branch stays inside the positive cone after this. The coexistence equilibrium points are stable at the start of the section. It stays stable as long as the product of K and the rate γ¯γ¯ of coinfection resulting from two single infections is small but, after this it can reach a Hopf bifurcation and periodic orbits will appear.

    Download full text (pdf)
    fulltext
  • 15.
    Andersson, Philip
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Larsson, Jon
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Videoklipp i matematikundervisningen: En interventionsstudie om hur matematiska videoklipp påverkar elevers motivation och prestation i matematik2022Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    This intervention study aims to examine how mathematical videos effect students’ motivation and performance in mathematics. The study is based on our literature study Videos in Mathematics Education – A Systematic Literature Review on How Mathematical Videos Can Be Used and Influence Student’s Motivation (Larsson & Andersson, 2021). The study has one experimental group and one control group which contain participants from the seventh grade. The experimental group had mathematical videos as a complement during six mathematics lessons, while the control group had regular mathematic lessons during the same time. Both groups did a pre-test and a post-test. Also, the experimental group had to do one survey after the six lessons. The result of the study showed that mathematical videos affect students’ performance positively since the participants in the experimental group, as well as the students in the control group, showed a statistically significant difference between the pre- and post-test. However, there’s no statistically significant difference in performance between the groups. We were neither able to conclude if mathematical videos a sa complement affect students’ motivation positively. To analyze the result about students’ motivation, we used self-determination theory where all three requisites of autonomy, competence and relatedness need to be fulfilled to establish an increased motivation. In this study we only found data to support the claims of competence and autonomy. From the result of the study, teachers could work from the idea that mathematical videos as a complement to are an asset in teaching situations since the students’ performance was improved, which previous studies have shown as well.

    Download full text (pdf)
    fulltext
  • 16.
    Asekritova, Irina
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Kruglyak, Natan
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Mastylo, M.
    Adam Mickiewicz Univ, Poland.
    Stability of the inverses of interpolated operators with application to the Stokes system2023In: Revista Matemática Complutense, ISSN 1139-1138, E-ISSN 1988-2807, Vol. 36, p. 163-206Article in journal (Refereed)
    Abstract [en]

    We study the stability of isomorphisms between interpolation scales of Banach spaces, including scales generated by well-known interpolation methods. We develop a general framework for compatibility theorems, and our methods apply to general cases. As a by-product we prove that the interpolated isomorphisms satisfy uniqueness-of-inverses. We use the obtained results to prove the stability of lattice isomorphisms on interpolation scales of Banach function lattices and demonstrate their application to the Calderon product spaces as well as to the real method scales. We also apply our results to prove solvability of the Neumann problem for the Stokes system of linear hydrostatics on an arbitrary bounded Lipschitz domain with a connected boundary in , with data in some Lorentz spaces over the set ∂Ω equipped with a boundary surface measure.

    Download full text (pdf)
    fulltext
  • 17.
    Balci, Anna Kh.
    et al.
    Univ Bielefeld, Germany.
    Cianchi, Andrea
    Univ Firenze, Italy.
    Diening, Lars
    Univ Bielefeld, Germany.
    Maz'ya, Vladimir G.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering. RUDN Univ, Russia.
    A pointwise differential inequality and second-order regularity for nonlinear elliptic systems2022In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 383, no 3-4, p. 1775-1824Article in journal (Refereed)
    Abstract [en]

    A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic systems in domains in R-n are derived. Both local and global estimates are established. Minimal assumptions on the boundary of the domain are required for the latter. In the special case of the p-Laplace system, our conclusions broaden the range of the admissible values of the exponent p previously known.

    Download full text (pdf)
    fulltext
  • 18.
    Barletta, Giuseppina
    et al.
    Univ Mediterranea Reggio Calabria, Italy.
    Cianchi, Andrea
    Univ Firenze, Italy.
    Mazya, Vladimir
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Bounds for eigenfunctions of the Neumann p-Laplacian on noncompact Riemannian manifolds2022In: Advances in Calculus of Variations, ISSN 1864-8258, E-ISSN 1864-8266Article in journal (Refereed)
    Abstract [en]

    Eigenvalue problems for the p-Laplace operator in domains with finite volume, on noncompact Riemannian manifolds, are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed. Sharp assumptions ensuring L-q- or L-infinity-bounds for eigenfunctions are offered either in terms of the isoperimetric function or of the isocapacitary function of the domain.

    Download full text (pdf)
    fulltext
  • 19. Barquero, Berta
    et al.
    Bergman Ärlebäck, Jonas
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Carreira, Susana
    Greefrath, Gilbert
    Eyrich Jessen, Britta
    Vorhölter, Katrin
    Introduction to the papers of TW6: Application and modelling2022In: Proceedings of the Twelfth Congress of the European Society for Research in Mathematics Education (CERME12) / [ed] Hodgen, J., Geraniou, E., Bolondi,G. & Ferretti, F., Free University of Bozen-Bolzano, Italy and ERME , 2022, p. 999-1008Conference paper (Other academic)
  • 20.
    Barquero, Berta
    et al.
    Faculty of Education, University of Barcelona, Spain.
    Carreira, Susana
    University of Algarve, & UIDEF, Institute of Education, University of Lisbon, Portugal.
    Bergman Ärlebäck, Jonas
    Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Jessen, Britta
    Department of Science Education, University of Copenhagen, Denmark.
    Vorhölter, Katrin
    Faculty of Education, University of Hamburg, Germany.
    Wake, Geoff
    Faculty of Social Sciences, University of Nottingham, United Kingdom.
    Introduction to the papers of TWG6: Applications and modelling2019In: Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education, Utrecht, the Netherlands: Freudenthal Group & Freudenthal Institute, Utrecht University and ERME , 2019, p. 1104-1111Conference paper (Refereed)
  • 21.
    Bengtsson, Martin
    Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Aritmetik, algebra och funktionslära för grundlärare2022Book (Other academic)
  • 22.
    Bergfeldt, Aksel
    et al.
    Uppsala Univ, Sweden.
    Rodriguez-Lopez, Salvador
    Stockholm Univ, Sweden.
    Rule, David
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Staubach, Wolfgang
    Uppsala Univ, Sweden.
    MULTILINEAR OSCILLATORY INTEGRALS AND ESTIMATES FOR COUPLED SYSTEMS OF DISPERSIVE PDES2023In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850Article in journal (Refereed)
    Abstract [en]

    We establish sharp global regularity of a class of multilinear oscillatory integral operators that are associated to nonlinear dispersive equations with both Banach and quasi-Banach target spaces. As a consequence we also prove the (local in time) continuous dependence on the initial data for solutions of a large class of coupled systems of dispersive partial differential equations.

  • 23.
    Bergman Ärlebäck, Jonas
    Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    A case study of tensions and challenges arising as a Swedish upper secondary teacher designs and implements a Model Development Sequence on statistics2019In: Mathematical modelling education and sense-making / [ed] Gloria Ann Stillman, Gabriele Kaiser, & Christine Erna Lampen, Cham: Springer , 2019, p. 139-150Chapter in book (Refereed)
    Abstract [en]

    A study of an upper secondary teacher trying to transform her teaching by designing and implementing a model development sequence on statistics in a grade 11 mathematics class in Sweden is presented. Cultural-historical activity theory (CHAT) is used to analyse lesson plans, reflection notes, recordings from planning meetings, and student data to identify tensions and challenges arising as she tried to adopt this new perspective on teaching and learning mathematics. The results illustrate the need to renegotiate appropriate classroom norms and facilitate shifts in power, ownership, and responsibilities in the mathematics classroom if a modelling approach is to effectively impact and change the teaching and learning of mathematics using mathematical modelling.

  • 24.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Albarracin, Lluis
    Univ Autonomade Barcelona, Spain.
    Fermi problems as a hub for task design in mathematics and stem education2023In: Teaching Mathematics and its Applications, ISSN 0268-3679, E-ISSN 1471-6976Article in journal (Refereed)
    Abstract [en]

    In this paper, we draw on recent research on so-called Fermi problems and situate the fundamental principles underlying this type of tasks and their use from a task design perspective. We use the models and modelling perspective on teaching and learning to elaborate on aspects related to the design of single-use, as well as sequences of, Fermi problems. In addition, we discuss a framework (called the Fermi problem activity template [FPAT] framework) for supporting the design and use of Fermi problems to facilitate students learning within particular mathematics content areas and/or aimed at particular concepts or higher-order thinking skills. We also illustrate how the FPAT framework can be used to (i) facilitate interdisciplinary collaborations with other subjects such as the social sciences, but in particular with the other STEM subjects; and (ii) support teachers in adapting and implementing Fermi problems in their classrooms.

  • 25.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Albarracin, Lluis
    Univ Autonoma Barcelona, Spain.
    FPAT-A Framework for Facilitating the Teaching and Learning of Using Fermi Problems Originating in Mathematics Education Research2022In: Physics Teacher, ISSN 0031-921X, E-ISSN 1943-4928, Vol. 60, no 7, p. 569-571Article in journal (Refereed)
    Abstract [en]

    Enrico Fermi is remembered for his many contributions to theoretical and experimental physics, but from an educational point of view he also popularized the use of the kind of questions we now call "Fermi problems" (or "Fermi questions"). Fermi problems (FPs) are back-of-envelope problems that arose from the need to make order of magnitude calculations, and Fermi himself used them in his own research and physics classes. The procedure proposed by Fermi was to break the original problem down into simpler sub-problems, solve these by making reasonable estimates and educated guesses, and thereby reach a solution to the original question. In the literature, this way of working is known as the "Fermi (estimates) method."

  • 26.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Albarracín, Lluís
    Universitat Autònoma de Barcelona, Barcelona, Spain.
    An extension of the MAD framework and its possible implications for research2019In: Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education / [ed] U. T. Jankvist, M. Van den Heuvel-Panhuizen, & M. Veldhuis, Utrecht, the Netherlands: Freudenthal Group & Freudenthal Institute, Utrecht University and ERME , 2019, p. 1128-1135Conference paper (Refereed)
    Abstract [en]

    In this paper we investigate the potential in extending Model Activity Diagrams (MADs) as an analytical tool for studying the activities students engage in when working on modelling problems. By including a sub-division of the task students work on in the MAD framework, we are able to make a more nuanced analysis that in a more explicit way reveals the structure of the students’ modelling process. The framework is applied to groups of students engaged in solving a Fermi problem using a two-level analysis, and the result is contrasted and discussed by comparing the outcome of the analysis with the corresponding analysis made using the modelling cycle. In addition, possible applications of the extended MAD framework in research and the training of teachers are discussed.

  • 27.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Frejd, Peter
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Educational Sciences. Linköping University, Faculty of Science & Engineering.
    Developing and testing a framework for analysing, comparing and visualizing content matter in and across mathematics textbooks2022In: The relation between mathematics education research and teachers’ professional development.: Proceedings of MADIF 13The thirteenth research conference ofthe Swedish Society for Researchin Mathematics EducationVäxjö, March 29–30, 2022 / [ed] Linda Mattsson, Johan Häggström, Martin Carlsen, Cecilia Kilhamn, Hanna Palmér, Miguel Perez, Kerstin Pettersson, 2022, p. 105-115Conference paper (Refereed)
    Abstract [en]

    In this methodological paper we discuss the development and testing of a framework for analysing and visualizing the overall structure of the content and tasks in mathe-matics textbooks. The aim of the framework is to provide a lens on mathematics textbooks at the secondary level that is mathematics topic- and content-area-unbiased to facilitate an as objective as possible comparisons between both (a) different contents, topics, and tasks within a textbook; and (b) contents, topics, and tasks across different series of textbooks or within a textbook series. The paper explores the results of the framework applied to four mathematical textbook chapters visualized in terms of so-called hierarchy charts. Inter-rater coding reliability with respect to a developed coding manual, and future development and applications of the framework is also discussed.

  • 28.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Frejd, Peter
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Developing and testing a framework for analysing, comparing and visualizing content matter in and across mathematics textbooks2022In: The relation between mathematics education research and teachers’ professional development. Proceedings of MADIF13. The thirteenth research seminar of the Swedish Society for Research in Mathematics Education, Växjö, 2022 / [ed] L. Mattsson, J. Häggström, M. Carlsen, C. Kilhamn, H. Palmér, M. Perez, M. & K. Pettersson, 2022, p. 105-115Conference paper (Refereed)
  • 29.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Frejd, Peter
    Linköping University, Faculty of Educational Sciences. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Eliciting pre-service secondary teachers’ initial ideas of sampling2021In: Sustainable mathematics education in a digitalized world. : Proceedings of MADIF12. The twelfth research seminar of the Swedish Society for Research in Mathematics Education, Växjö, January 14–15, 2020 / [ed] Y. Liljekvist, L. Björklund Boistrup, J. Häggström, L. Mattsson, O. Olande, & H. Palmér, H., Göteborg, 2021, p. 234-244Conference paper (Other academic)
    Abstract [en]

    This paper investigates secondary pre-service teachers’ strategies of sampling. The work of eight groups of pre-service secondary teachers asked to devise and imple- ment a sampling strategy to answer two questions about three given populations of different sizes are analyzed. The result presented highlights the models devised and used by the groups. The paper especially discusses the pre-service teachers’ models of samplings in term of how the number of samples and sample size used were decided on in the model as fixed, interval-static, or dynamic.

  • 30.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Frejd, Peter
    Linköping University, Faculty of Educational Sciences. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    On pre-service secondary teachers’ mathematical content knowledge in statistics2021In: Sustainable mathematics education in a digitalized world. : Proceedings of MADIF12. The twelfth research seminar of the Swedish Society for Research in Mathematics Education, Växjö, January 14–15, 2020. / [ed] Y. Liljekvist, Y., L. Björklund Boistrup, J. Häggström, L. Mattsson, O. Olande, & H. Palmér, H. (, Göteborg: SMDF - Svensk Förening för Matematikdidaktisk Forskning , 2021, p. 203-212Conference paper (Other academic)
    Abstract [en]

    In this paper we report on the use of a pre-post-test design to study pre-service secondary mathematics teachers’ mathematical content knowledge in statistics before and after their first university course in statistics. The results show that the participants were successful in the pre-test on items related to sampling, probabi- lity and the general logic of making formal statistical inferences, but struggled with items concerning distributions. Comparing the pre- and post-test reveals an increas- ing average of the participants’ scores in most statistical areas, but that topics like informal statistical inferences and distributions remain challenging for the majority of the participants.

  • 31.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Frejd, Peter
    Linköping University, Faculty of Educational Sciences. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    The clash of cultural affordances: Using cultural affordances to analyze students’ work in an activity using a mobile game app to simulate a pandemic outbreak2020In: Journal of Mathematics and Culture, E-ISSN 1558-5336, Vol. 14, no 2, p. 111-131Article in journal (Refereed)
    Abstract [en]

    In this paper we use the framework of cultural affordances to analyze students’ work in an activity using a commercial mobile game app for smartphones and tablets that provided a context and simulation of a pandemic outbreak. The written work carried out by 22 upper secondary students working in pairs is analyzed, with a focus on the extent to which the students’ answers and produced graphs illustrated i) the students’ gameplay, ii) the context of the phenomena (the development and spread of a pandemic), or iii) the mathematical characteristics of the graphs. The results highlight the challenges of exploiting, leveraging, and aligning students’ gameplay with curricula learning goals in what can be seen as a clash of norms and behavior associated with the game and gameplay on the one hand, and the teaching and learning of mathematics on the other.

  • 32.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Frejd, Peter
    Linköping University, Faculty of Educational Sciences. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    The red book activity: a model eliciting activity to introduce and initiate a section on statistics focusing on variability and sampling2021In: Mathematical Modelling Education in East and West / [ed] F. Leung, G. A. Stillman, G. Kasier, & K. L. Wong, Cham: Springer, 2021, p. 595-605Chapter in book (Refereed)
    Abstract [en]

    This chapter analyses and discusses nine groups of upper secondary students’ work on the question “How many red books are there in the library?”. The students devised and implemented a plan, collected data, calculated estimates and reflected on what aspects and factors might have influenced their results caused by their adopted strategy. The analysis of the students’ work focused on reconstructing and categorizing the models the students devised and implemented, as well as the sources and types of variability that the activity elicited. The results show how the central statistical idea of variability is manifested in the models developed and implemented by the students, and how these can be further explored and applied as central and bearing ideas for organizing a whole section of statistics at the upper secondary level.

  • 33.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Frejd, Peter
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Towards a dual integrated modeling approach to the teaching and learning of mathematics: Challenges and tensions in implementation2022In: Proceedings of the Twelfth Congress of the European Society for Research in Mathematics Education February, 2-6, 2022. Bozen-Bolzano, Italy., 2022Conference paper (Refereed)
    Abstract [en]

    This paper elaborates on the theoretical conceptualization of a so-called dual integrated modeling approach to the teaching and learning of modeling and the teaching and learning of mathematics through modeling. Departing from a sociocultural perspective on teaching, the models and modeling perspective, and using the previously developed theoretical constructs of the Teaching Triad and the Expanded Mediated Triangle, we discuss modeling in terms of (i) micro modeling in which modeling is used as a vehicle to teach and learning other curricula content; and (ii) macro modeling in which modeling is a self-standing mathematics curricula goal. We illustrate this dual integrated approach by analyzing and discussing a lower secondary teachers’ implementation of a modeling activity focusing on a simple statistical investigation and measures of central tendency in statistics.

  • 34.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Frejd, Peter
    Linköping University, Faculty of Educational Sciences. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Doerr, Helen
    Department of Mathematics, Syracuse University, United States of America.
    Exploring students’ models of sampling and inference with nominal variables2021In: Quadrante: Revista de Investigação em Educação Matemática, ISSN 0872-3915, Vol. 30, no 1, p. 158-177Article in journal (Refereed)
    Abstract [en]

    Making inferences about unknown populations is central in statistical reasoning. However, little attention has been paid to empirical investigations of how and why students develop sampling models when investigating a categorical variable whose values are nominal. This paper reports on an intervention that draws on the models and modeling perspective, where 25 pre-service teachers were asked to develop a sampling model that could be used to make inferences about the number of different colored beads and the distribution of different colored beads in different sized populations. Using a thematic analysis, three main results about the characteristics of the students’ models of sampling and inference with nominal variables were identified: to catch all the low frequency colors in the population; not to overestimate the low frequency colors in the population; and, to formalize the relationships used in making inferences. These results highlight several issues about students’ understanding of the relationship between sample representativeness and sample variability and its consequences for making inferences.

  • 35.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Kawakami, Takashi
    Cooperative Faculty of Education, Utsunomiya University, Utsunomiya, Japan.
    The relationship between mathematical modelling, statistics, and statistical modelling2023In: Advancing and Consolidating Research on Applications and Modelling in Mathematics Education – Approaches from ICME 14 / [ed] Gilbert Greefrath, Susana Carreira, Gloria Ann Stillman, Springer, 2023, p. 293-309Chapter in book (Refereed)
    Abstract [en]

    This chapter presents arguments and examples highlighting the similarities and differences between statistics, statistical modelling and mathematical modelling. Based on educational research that focuses on statistical modelling and mathematical modelling, we elaborate on the potentially productive connections for the development of, and research on, the teaching and learning of statistics, statistical modelling and mathematical modelling. We outline the development of an ongoing research agenda that pursues a framework for how to conceptualise the connection between statistics and statistical modelling on the one hand, and mathematical modelling on the other, as well as suggest how to further develop this emerging framework in order to provide a richer, more nuanced and useful picture of the relationship between statistics, statistical modelling and mathematical modelling.

  • 36.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Analysis and Mathematics Education.
    Lindgren, Anna
    Ebersteinska gymnasiet, Norrköping, Sweden.
    Lindroos, Åsa
    Ebersteinska gymnasiet, Norrköping, Sweden.
    Mattsson, Jonny
    Ebersteinska gymnasiet, Norrköping, Sweden.
    5-minutare: ett arbetssätt för att öva elevernas grundläggande färdigheter och öka elevernas säkerhet och självkänsla i matematik2019Report (Other academic)
    Download full text (pdf)
    fulltext
    Download (png)
    presentationsbild
  • 37.
    Berntsson, Fredrik
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Chepkorir, Chepkorir
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering. Univ Nairobi, Kenya.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Accelerated Dirichlet-Robin alternating algorithms for solving the Cauchy problem for the Helmholtz equation2021In: IMA Journal of Applied Mathematics, ISSN 0272-4960, E-ISSN 1464-3634, Vol. 86, no 6, p. 1181-1203Article in journal (Refereed)
    Abstract [en]

    The Cauchy problem for Helmholtz equation, for moderate wave number k(2), is considered. In the previous paper of Achieng et al. (2020, Analysis of Dirichlet-Robin iterations for solving the Cauchy problem for elliptic equations. Bull. Iran. Math. Soc.), a proof of convergence for the Dirichlet-Robin alternating algorithm was given for general elliptic operators of second order, provided that appropriate Robin parameters were used. Also, it has been noted that the rate of convergence for the alternating iterative algorithm is quite slow. Thus, we reformulate the Cauchy problem as an operator equation and implement iterative methods based on Krylov subspaces. The aim is to achieve faster convergence. In particular, we consider the Landweber method, the conjugate gradient method and the generalized minimal residual method. The numerical results show that all the methods work well. In this work, we discuss also how one can approach non-symmetric differential operators by similar operator equations and model problems which are used for symmetric differential operators.

  • 38.
    Björn, Anders
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Removable singularities for bounded A-(super)harmonic and quasi(super)harmonic functions on weighted Rn2022In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 222, article id 112907Article in journal (Refereed)
    Abstract [en]

    It is well known that sets of p-capacity zero are removable for bounded p-harmonic functions, but on metric spaces there are examples of removable sets of positive capacity. In this paper, we show that this can happen even on unweighted when n > p, although only in very special cases. A complete characterization of removable singularities for bounded A-harmonic functions on weighted , is also given, where the weight is p-admissible. The same characterization is also shown to hold for bounded quasiharmonic functions on weighted , as well as on unweighted . For bounded A-superharmonic functions and bounded quasisuperharmonic functions on weighted , we show that relatively closed sets are removable if and only if they have zero capacity.

    Download full text (pdf)
    fulltext
  • 39.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Sharp Besov capacity estimates for annuli in metric spaces with doubling measures2023In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 305, no 3, article id 41Article in journal (Refereed)
    Abstract [en]

    We obtain precise estimates, in terms of the measure of balls, for the Besov capacity of annuli and singletons in complete metric spaces. The spaces are only assumed to be uniformly perfect with respect to the centre of the annuli and equipped with a doubling measure.

    Download full text (pdf)
    fulltext
  • 40.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Lahti, Panu
    Chinese Acad Sci, Peoples R China.
    Removable sets for Newtonian Sobolev spaces and a characterization of p-path almost open sets2023In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 39, no 3, p. 1143-1180Article in journal (Refereed)
    Abstract [en]

    We study removable sets for Newtonian Sobolev functions in metric measure spaces satisfying the usual (local) assumptions of a doubling measure and a Poincare inequality. In particular, when restricted to Euclidean spaces, a closed set E c Rn with zero Lebesgue measure is shown to be removable for W1;p.Rn \ E/ if and only if Rn \ E supports a p-Poincare inequality as a metric space. When p &gt; 1, this recovers Koskelas result (Ark. Mat. 37 (1999), 291-304), but for p = 1, as well as for metric spaces, it seems to be new. We also obtain the corresponding characterization for the Dirichlet spaces L1;p. To be able to include p = 1, we first study extensions of Newtonian Sobolev functions in the case p = 1 from a noncom-plete space X to its completion Xy. In these results, p-path almost open sets play an important role, and we provide a characterization of them by means of p-path open, p-quasiopen and p-finely open sets. We also show that there are nonmeasurable p -path almost open subsets of Rn, n &gt; 2, provided that the continuum hypothesis is assumed to be true. Furthermore, we extend earlier results about measurability of functions with Lp-integrable upper gradients, about p-quasiopen, p-path open and p-finely open sets, and about Lebesgue points for N1;1-functions, to spaces that only satisfy local assumptions.

    Download full text (pdf)
    fulltext
  • 41.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Latvala, Visa
    Univ Eastern Finland, Finland.
    Convergence and local-to-global results for p-superminimizers on quasiopen sets2023In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 365, p. 812-831Article in journal (Refereed)
    Abstract [en]

    In this paper, several convergence results for fine p-(super)minimizers on quasiopen sets in metric spaces are obtained. For this purpose, we deduce a Caccioppoli-type inequality and local-to-global principles for fine p-(super)minimizers on quasiopen sets. A substantial part of these considerations is to show that the functions belong to a suitable local fine Sobolev space. We prove our results for a complete metric space equipped with a doubling measure supporting a p-Poincare inequality with 1 &lt; p &lt; & INFIN;. However, most of & COPY; 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).

    Download full text (pdf)
    fulltext
  • 42.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Latvala, Visa
    Univ Eastern Finland, Finland.
    Correction: The Dirichlet Problem for p-minimizers on Finely Open Sets in Metric Spaces (May, 10.1007/s11118-022-09996-7, 2022)2022In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929XArticle in journal (Other academic)
    Download full text (pdf)
    fulltext
  • 43.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Latvala, Visa
    Univ Eastern Finland, Finland.
    The Dirichlet Problem for p-minimizers on Finely Open Sets in Metric Spaces2023In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 59, p. 1117-1140Article in journal (Refereed)
    Abstract [en]

    We initiate the study of fine p-(super)minimizers, associated with p-harmonic functions, on finely open sets in metric spaces, where infinity. After having developed their basic theory, we obtain the p-fine continuity of the solution of the Dirichlet problem on a finely open set with continuous Sobolev boundary values, as a by-product of similar pointwise results. These results are new also on unweighted . We build this theory in a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality.

    Download full text (pdf)
    fulltext
  • 44.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Lehrback, Juha
    Univ Jyvaskyla, Finland.
    Volume growth, capacity estimates, p-parabolicity and sharp integrability properties of p-harmonic Green functions2023In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 150, p. 159-214Article in journal (Refereed)
    Abstract [en]

    In a complete metric space equipped with a doubling measure supporting a p-Poincare inequality, we prove sharp growth and integrability results for p-harmonic Green functions and their minimal p-weak upper gradients. We show that these properties are determined by the growth of the underlying measure near the singularity. Corresponding results are obtained also for more general p-harmonic functions with poles, as well as for singular solutions of elliptic differential equations in divergence form on weighted R-n and on manifolds.The proofs are based on a new general capacity estimate for annuli, which implies precise pointwise estimates for p-harmonic Green functions. The capacity estimate is valid under considerably milder assumptions than above. We also use it, under these milder assumptions, to characterize singletons of zero capacity and the p-parabolicity of the space. This generalizes and improves earlier results that have been important especially in the context of Riemannian manifolds.

  • 45.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Mwasa, Abubakar
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering. Busitema Univ, Uganda.
    Resolutivity and invariance for the Perron method for degenerate equations of divergence type2022In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 509, no 1, article id 125937Article in journal (Refereed)
    Abstract [en]

    We consider Perron solutions to the Dirichlet problem for the quasilinear elliptic equation in a bounded open set . The vector-valued function A satisfies the standard ellipticity assumptions with and a p-admissible weight w. We show that arbitrary perturbations on sets of (p, w)-capacity zero of continuous (and certain quasicontinuous) boundary data f are resolutive and that the Perron solutions for f and such perturbations coincide. As a consequence, we prove that the Perron solution with continuous boundary data is the unique bounded solution that takes the required boundary data outside a set of (p, w) capacity zero.

    Download full text (pdf)
    fulltext
  • 46.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Shanmugalingam, Nageswari
    Department of Mathematical Sciences, University of Cincinnati.
    Classification of metric measure spaces and their ends using p-harmonic functions2022In: Annales Fennici Mathematici, ISSN 2737-0690, Vol. 47, no 2, p. 1025-1052Article in journal (Refereed)
    Abstract [en]

    By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite p-energy p-harmonic and p-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local p-Poincare inequality. Similar classifications have earlier been obtained for Riemann surfaces and Riemannian manifolds. We study the inclusions between these classes of metric measure spaces, and their relationship to the p-hyperbolicity of the metric space and its ends. In particular, we characterize spaces that carry nonconstant p-harmonic functions with finite p-energy as spaces having at least two well-separated p-hyperbolic sequences of sets towards infinity. We also show that every such space X has a function f is an element of/ LP(X) + R with finite p-energy.

  • 47.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Shanmugalingam, Nageswari
    Univ Cincinnati, OH 45221 USA.
    Extension and trace results for doubling metric measure spaces and their hyperbolic fillings2022In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, E-ISSN 1776-3371, Vol. 159, p. 196-249Article in journal (Refereed)
    Abstract [en]

    In this paper we study connections between Besov spaces of functions on a compactmetric space Z, equipped with a doubling measure, and the Newton–Sobolev spaceof functions on a uniform domain Xε. This uniform domain is obtained as auniformization of a (Gromov) hyperbolic filling of Z. To do so, we construct afamily of hyperbolic fillings in the style of Bonk–Kleiner [9] and Bourdon–Pajot [13]. Then for each parameter β > 0 we construct a lift μβ of the doubling measure νon Z to Xε, and show that μβ is doubling and supports a 1-Poincaré inequality.We then show that for each θ with 0 < θ < 1 and p ≥ 1 there is a choice of β = p(1 − θ)ε such that the Besov space is the trace space of the Newton–Sobolev space N1,p(Xε, μβ). Finally, we exploit the tools of potential theory on Xεto obtain fine properties of functions in , such as their quasicontinuity andquasieverywhere existence of Lq-Lebesgue points with q = sνp/(sν − pθ), where sν is a doubling dimension associated with the measure ν on Z. Applying this tocompact subsets of Euclidean spaces improves upon a result of Netrusov [43] in

    Download full text (pdf)
    fulltext
  • 48.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Hansevi, Daniel
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Semiregular and strongly irregular boundary points for p-harmonic functions on unbounded sets in metric spaces2022In: Collectanea Mathematica (Universitat de Barcelona), ISSN 0010-0757, E-ISSN 2038-4815, Vol. 73, no 2, p. 253-270Article in journal (Refereed)
    Abstract [en]

    The trichotomy between regular, semiregular, and strongly irregular boundary points for p-harmonic functions is obtained for unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincare inequality, 1 &lt; p &lt; infinity. We show that these are local properties. We also deduce several characterizations of semiregular points and strongly irregular points. In particular, semiregular points are characterized by means of capacity, p-harmonic measures, removability, and semibarriers.

    Download full text (pdf)
    fulltext
  • 49.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Boundary Estimates and a Wiener Criterion for the Fractional Laplacian2023In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826Article in journal (Refereed)
    Abstract [en]

    Using the Caffarelli–Silvestre extension, we show for a general open set Ω⊂Rn that a boundary point x0 is regular for the fractional Laplace equation (−Δ)s⁢u=0, 0<s<1, if and only if (x0,0) is regular for the extended weighted equation in a subset of Rn+1. As a consequence, we characterize regular boundary points for (−Δ)s⁢u=0 by a Wiener criterion involving a Besov capacity. A decay estimate for the solutions near regular boundary points and the Kellogg property are also obtained.

  • 50.
    Björn, Jana
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Kalamajska, Agnieszka
    Univ Warsaw, Poland.
    Poincaré inequalities and compact embeddings from Sobolev type spaces into weighted Lq spaces on metric spaces2022In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 282, no 11, article id 109421Article in journal (Refereed)
    Abstract [en]

    We study compactness and boundedness of embeddings from Sobolev type spaces on metric spaces into L-q spaces with respect to another measure. The considered Sobolev spaces can be of fractional order and some statements allow also nondoubling measures. Our results are formulated in a general form, using sequences of covering families and local Poincare type inequalities. We show how to construct such suitable coverings and Poincare inequalities. For locally doubling measures, we prove a self-improvement property for two weighted Poincare inequalities, which applies also to lower dimensional measures.& nbsp;We simultaneously treat various Sobolev spaces, such as the Newtonian, fractional Hajlasz and Poincare type spaces, for rather general measures and sets, including fractals and domains with fractal boundaries. By considering lower dimensional measures on the boundaries of such domains, we obtain trace embeddings for the above spaces. In the case of Newtonian spaces we exactly characterize when embeddings into L-q spaces with respect to another measure are compact. Our tools are illustrated by concrete examples. For measures satisfying suitable dimension conditions, we recover several classical embedding theorems on domains and fractal sets in .

    Download full text (pdf)
    fulltext
123 1 - 50 of 148
CiteExportLink to result list
Permanent link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf