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  • 51.
    Axén, Fredrik
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap.
    Laborativ matematik: Att variera undervisningen med alternativa metoder2009Självständigt arbete på avancerad nivå (yrkesexamen), 10 poäng / 15 hpStudentuppsats
    Abstract [sv]

    Denna rapport handlar om hur man som lärare kan variera undervisningen i matematik med en fokusering på laborativ matematik. Den riktar sig främst mot gymnasiet men är även aktuell för både för yngre och äldre åldrar. Rapporten behandlar lärares och elevers syn på undervisning samt ger konkreta förslag på hur sådan undervisning kan gå till. Syftet med rapporten är att undersöka vad det finns för alternativ till traditionell undervisning och presentera dessa. Metoden som använts är intervjuer där jag tagit kontakt med lärare som är kända för att använda alternativa undervisningsmetoder och laborativ matematik. Totalt fem intervjuer.

     

    Resultatet visar på att undervisningen i matematik är enformig och att den bör varieras för att bli lärorik och intressant. Rapporten ger konkreta förslag på hur sådan undervisning kan gå till. Flera lärare vill testa alternativ undervisning men tvekar eftersom de inte vet att den fungerar eller anser att det inte finns tid till sådan arbetsmetod. En varierad, laborativ undervisning verkar ge goda resultat men det finns inga vetenskapliga undersökningar som bevisar detta.

  • 52. Baba, S.
    et al.
    Granath, Håkan
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Genus 2 Curves with Quaternionic Multiplication2008Ingår i: Canadian Journal of Mathematics, Vol. 60 (4)Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We explicitly construct the canonical rational models of Shimuracurves, both analytically in terms of modular forms andalgebraically in terms of coefficients of genus 2 curves, in thecases of quaternion algebras of discriminant 6 and 10. Thisemulates the classical construction in the elliptic curvecase. We also give families of genus 2 QM curves, whose Jacobiansare the corresponding abelian surfaces on the Shimura curve, andwith coefficients that are modular forms of weight 12. We applythese results to show that our j-functions are supported exactlyat those primes where the genus 2 curve does not admitpotentially good reduction, and construct fields where thispotentially good reduction is attained. Finally, using j, weconstruct the fields of moduli and definition for some moduliproblems associated to the Atkin-Lehner group actions

  • 53. Baba, Srinath
    et al.
    Granath, Håkan
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Primes of superspecial reduction for QM abelian surfaces2008Ingår i: Bulletin of the London Mathematical Society, Vol. 40Artikel i tidskrift (Refereegranskat)
  • 54.
    Baba, Srinath
    et al.
    Champlain College.
    Granath, Håkan
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.
    Quaternionic modular forms and exceptional sets of hypergeometric functions2015Ingår i: International Journal of Number Theory, ISSN 1793-0421, Vol. 11, nr 2, s. 631-643Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We determine the exceptional sets of hypergeometric functions corresponding to the(2, 4, 6) triangle group by relating them to values of certain quaternionic modular formsat CM points. We prove a result on the number fields generated by exceptional values, and by using modular polynomials we explicitly compute some examples.

  • 55.
    Bai, Chengming
    et al.
    Republic of China.
    Fuchs, JürgenKarlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för ingenjörsvetenskap och fysik.Huang, Yi-ZhiUSA.Kong, LiangRepublic of China.Runkel, IngoGermany.Schweigert, ChristophGermany.
    Conformal Field Theories and Tensor Categories: Proceedings of a Workshop Held at Beijing International Center for Mathematical Research2014Proceedings (redaktörskap) (Refereegranskat)
    Abstract [en]

    The present volume is a collection of seven papers that are either based on the talks presented at the workshop "Conformal field theories and tensor categories" held June 13 to June 17, 2011 at theBeijing International Center for Mathematical Research, Peking University,  or are extensions of the material presented in the talks at the workshop. These papers present new developments beyond rational conformal field theories and modular tensor categories and new applications in mathematics and physics. The topics covered include tensor categories from representation categories of Hopf algebras, applications of conformal field theories and tensor categories to topological phases and gapped systems, logarithmic conformal field theories and the corresponding non-semisimple tensor categories, and new developments in the representation theory of vertex operator algebras. Some of the papers contain detailed introductory material that is helpful for graduate students and researchers looking for an introduction to these research directions. The papers also discuss exciting recent developments in the area of conformal field theories, tensor categories and their applications and will be extremely useful for researchers working in these areas.

  • 56.
    Barza, Ilie
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    On the structure of some irrotational vector fields2005Ingår i: Gen. Math. 13Artikel i tidskrift (Refereegranskat)
  • 57.
    Barza, Ilie
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Ghisa, D
    Blaschke produkt generated covering surfaces2009Ingår i: Math.Bohem., ISSN 0862-7959, Vol. 134, nr 2, s. 173-182Artikel i tidskrift (Refereegranskat)
  • 58.
    Barza, Ilie
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Ghisa, D.
    Blaschke self-mappings of the real projective plane2007Konferensbidrag (Refereegranskat)
  • 59.
    Barza, Ilie
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Ghisa, D.
    Measure and integration on nonorientable Klein surfaces2002Ingår i: Int. J.Pure Appl.Math 1Artikel i tidskrift (Refereegranskat)
  • 60.
    Barza, Ilie
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Ghisa, D.
    Nonorientable Compactifications of Riemann SurfacesI2003Ingår i: n the volume "Progress in Analysis", Worlds ScientificArtikel i tidskrift (Refereegranskat)
  • 61.
    Barza, Ilie
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Ghisa, D
    The geometry of Blaschke products mappings2009Konferensbidrag (Refereegranskat)
  • 62.
    Barza, Ilie
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Ghisa, D
    Vector Fields on Nonorientable Surfaces2003Ingår i: International Journal of Mathematics and Mathematical SciencesArtikel i tidskrift (Refereegranskat)
  • 63.
    Barza, Ilie
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Ghisa, Dorin
    Dynamics of Dianalytic Transformations of Klein Surfaces2004Ingår i: Mathematica BohemicaArtikel i tidskrift (Refereegranskat)
  • 64.
    Barza, Ilie
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Ghisa, Dorin
    The Normal Derivative to the Border of the Möbius Strip2002Ingår i: Revue Roumaine de Mathématiques Pures et AppliqueesArtikel i tidskrift (Refereegranskat)
  • 65.
    Barza, Ilie
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Vajaitu, V
    Cousin-I spaces and domains of holomorphy2009Ingår i: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 96, nr 1, s. 51-60Artikel i tidskrift (Refereegranskat)
  • 66.
    Barza, Sorina
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Multidimensional integral inequalities with homogeneous weights2005Ingår i: Gen. Math. Vol. 13, No. 1 (2005)Artikel i tidskrift (Refereegranskat)
  • 67.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Buse, Constantin
    Rumänien.
    Pecaric, Josep
    Kroatien.
    New characterizations of asymptotic stability for evolution families on Banach spaces2004Ingår i: Electronic Journal of Differential Equations, ISSN 1550-6150, E-ISSN 1072-6691, s. 1-9, artikel-id 38Artikel i tidskrift (Refereegranskat)
  • 68.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Heinig, Hans P.
    Persson, Lars- Erik
    Duality theorems over the cone of monotone functions and sequences in higher dimensions2002Ingår i: J. of Inequal. & and Appl., 2002, Vol.7 (1), 79-108Artikel i tidskrift (Refereegranskat)
  • 69.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Johansson, Maria
    Persson, Lars-Erik
    A Sawyer duality principle for radialy monotone functions in R^n2005Ingår i: JIPAM 6, (2005), electronic, 1-13Artikel i tidskrift (Refereegranskat)
  • 70.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Kaminska, A.
    Persson, L.-E.
    Soria, J.
    Mixed norm and multidimensional Lorentz spaces2006Ingår i: POsitivity, 10 (2006), no. 3Artikel i tidskrift (Refereegranskat)
  • 71.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Kolyada, Viktor
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Soria, Javier
    Department of Applied Mathematics and Analysis, University of Barcelona, Spain.
    Sharp constants related to the triangle inequality in Lorentz spaces2009Ingår i: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 361, nr 10, s. 5555-5574Artikel i tidskrift (Refereegranskat)
  • 72.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Kravvaritis, D.
    Popa, N.
    Matriceal Lebesgue spaces and Hölder inequality2005Ingår i: JFSA, 3(3)(2005)Artikel i tidskrift (Refereegranskat)
  • 73.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Lie, Victor
    Popa, Nicolae
    Approximation of infinite marices by matricial Haar polynomials2005Ingår i: Ark.Mat.,43(2005), 251-269Artikel i tidskrift (Refereegranskat)
  • 74.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Lind, Martin
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.
    A new variational characterization of Sobolev spaces2015Ingår i: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 25, nr 4, s. 2185-2195Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We obtain a new variational characterization of the Sobolev space $W_p^1(\Omega)$ (where $\Omega\subseteq\R^n$ and $p>n$). This is a generalization of a classical result of F. Riesz. We also consider some related results.

  • 75.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Marcoci, A.N.
    Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, Romania.
    Persson, L.-E.
    Department of Mathematics, Luleå University of Technology.
    Best constants between equivalent norms in Lorentz sequence spaces2012Ingår i: Journal of Function Spaces and Applications, ISSN 0972-6802, E-ISSN 1758-4965, s. 1-19, artikel-id 713534Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We find the best constants in inequalities relating the standard norm, the dual norm, and the norm ‖𝑥‖(𝑝,𝑠)∑∶=inf{𝑘‖𝑥(𝑘)‖𝑝,𝑠}, where the infimum is taken over all finite representations ∑𝑥=𝑘𝑥(𝑘) in the classical Lorentz sequence spaces. A crucial point in this analysis is the concept of level sequence, which we introduce and discuss. As an application, we derive the best constant in the triangle inequality for such spaces.

  • 76.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
    Marcoci, Anca N.
    Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, Romania.
    Marcoci, Liviu G.
    Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, Romania.
    Factorizations of Weighted Hardy Inequalities2018Ingår i: Bulletin of the Brazilian Mathematical Society, ISSN 1678-7544, E-ISSN 1678-7714, Vol. 49, nr 4, s. 915-932Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present factorizations of weighted Lebesgue, Cesàro and Copson spaces, for weights satisfying the conditions which assure the boundedness of the Hardy’s integral operator between weighted Lebesgue spaces. Our results enhance, among other, the best known forms of weighted Hardy inequalities.

  • 77.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik. Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.
    Marcoci, Anca Nicoleta
    Technical University of Civil Engineering Bucharest, Department of Mathematics and Computer Science, Bucharest, Romania.
    Marcoci, Liviu Gabriel
    Technical University of Civil Engineering Bucharest, Department of Mathematics and Computer Science, Bucharest, Romania.
    Persson, Lars-Erik
    Luleå University of Technology, Department of Mathematics.
    Optimal estimates in Lorentz spaces of sequences with an increasing weight2013Ingår i: Proceedings of the Romanian Academy. Series A Mathematics, Physics, Technical Sciences, Information Science, ISSN 1454-9069, Vol. 14, nr 1, s. 20-27Artikel i tidskrift (Refereegranskat)
  • 78.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Niculescu, Constantin P.
    Strong and weak weighted norm inequalities for the geometric fractional maximal operator2012Ingår i: Bulletin of the Australian Mathematical Society, ISSN 0004-9727, E-ISSN 1755-1633, Vol. 86, nr 2, s. 205-215Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We characterise the strong- and weak-type boundedness of the geometric fractional maximal operator between weighted Lebesgue spaces in the case 0 < p ≤ q < ∞, generalising and improving some older results.

  • 79.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Niculescu, C.P.
    Integral inequalities for concave functions2006Ingår i: Publ. Math. Debrecen, 68(2006), no. 1-2Artikel i tidskrift (Refereegranskat)
  • 80.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.
    Persson, Lars-Erik
    Luleå tekniska universitet.
    Some new sharp limit Hardy-type  inequalities via convexity2014Ingår i: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, nr 6Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Some new limit cases of Hardy-type inequalities are proved, discussed and compared. In particular, some new refinements of Bennett’s inequalities are proved. Each of these refined inequalities contain two constants, and both constants are in fact sharp. The technique in the proofs is new and based on some convexity arguments of independent interest.

  • 81.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Persson, Lars-Erik
    Department of Mathematics, Luleå University of Technology.
    Popa, Emil C.
    epartment of Mathematics, "Lucian Blaga" University of Sibiu, Romania .
    Some multiplicative inequalities for inner products and of the Carlson type2008Ingår i: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242XArtikel i tidskrift (Refereegranskat)
    Abstract [en]

    We prove a multiplicative inequality for inner products, which enables us to deduce improvements of inequalities of the Carlson type for complex functions and sequences, and also other known inequalities.

  • 82.
    Barza, Sorina
    et al.
    Karlstads universitet, Institutionen för ingenjörsvetenskap, fysik och matematik.
    Persson, Lars-Erik
    Department of Mathematics, Luleå University.
    Soria, Javier
    Department of Applied Mathematics and Analysis, University of Barcelona, Spain.
    Multidimensional rearrangement and Lorentz spaces2004Ingår i: Acta Mathematica Hungarica, ISSN 0236-5294, E-ISSN 1588-2632, Vol. 104, nr 3, s. 203-224Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We define a multidimensional rearrangement, which is related to classical inequalities for functions that are monotone in each variable. We prove the main measure theoretical results of the new theory and characterize the functional properties of the associated weighted Lorentz spaces.

  • 83.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Persson, L.-E.
    Equivalence of Modular Inequalities of Hardy type on non-negative respective non-increasing functions2008Konferensbidrag (Refereegranskat)
  • 84.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Persson, L-E.
    Popa, N.
    A matriceal analogue of Fejer's theory2003Ingår i: Math NachrArtikel i tidskrift (Refereegranskat)
  • 85.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Popa, E. C.
    Weighted multiplicative integral inequalities2006Ingår i: Journal of inequal pure, ISSN 1443-5756, Vol. 7, nr 5, s. 169-175Artikel i tidskrift (Refereegranskat)
  • 86.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.
    Silvestre, Pilar
    Department of Mathematics and Systems Analysis, Aalto University, Finland.
    Functions of bounded second p-variation2014Ingår i: Revista Matemática Complutense, ISSN 1139-1138, E-ISSN 1988-2807, Vol. 27, s. 69-91Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The generalized functionals of Merentes type generate a scale of spaces connecting the class of functions of bounded second p -variation with the Sobolev space of functions with p-integrable second derivative. We prove some limiting relations for these functionals as well as sharp estimates in terms of the fractional modulus of order 2−1/p . These results extend the results in Lind (Math Inequal Appl 16:2139, 2013) for functions of bounded variation but are not consequence of the last.

  • 87.
    Barza, Sorina
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Soria, Javier
    Department of Applied Mathematics and Analysis, University of Barcelona, , Spain .
    Sharp constants between equivalent norms in weighted Lorentz spaces2010Ingår i: Journal of the Australian Mathematical Society, ISSN 1446-7887, E-ISSN 1446-8107, Vol. 88, nr 1, s. 19-27Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    For an increasing weight w in Bp (or equivalently in Ap), we find the best constants for the inequalities relating the standard norm in the weighted Lorentz space Λp(w) and the dual norm.

  • 88. Beentjes, Casper
    et al.
    Di Bucchianico, Alessandro
    Hamster, Christian
    Kadu, Ajinkya
    Man, Irene
    Myerscough, Keith
    Regis, Marta
    Richardson, Omar
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
    Equalizing the Cost of Health Insurance2017Ingår i: Proceedings of the 126th European Study GroupMathematics with Industry / [ed] Daan Crommelin, Stella Kapodistria, Guus Regts, Chris Stolk, Peter van de Ven, SWI , 2017, s. 29-49Konferensbidrag (Övrigt vetenskapligt)
    Abstract [en]

    The Dutch government compensates health insurance companies when insuringindividuals who are estimated to have high health care costs. This is necessaryto avoid insurers not offering services to certain groups or not providing themwith a high quality of service. It is, however, unknown to what extent thedifferences in health care expenses by different groups of people are truly due toa poorer or better health status. We explore several statistical approaches thatfacilitate explaining the cause of these differences.

  • 89.
    Beilina, Larissa
    et al.
    Chalmers University, Gothenburg University.
    Shestopalov, YuriKarlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.
    Inverse problems and large-scale computations2013Proceedings (redaktörskap) (Refereegranskat)
  • 90.
    Berg, Marcus
    et al.
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013).
    Buchberger, Igor
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013).
    Enander, Jonas
    Stockholm University.
    Mörtsell, Edvard
    Stockholm University.
    Sjörs, Stefan
    Stockholm University.
    Growth Histories in Bimetric Massive Gravity2012Ingår i: Journal of Cosmology and Astroparticle Physics, ISSN 1475-7516, E-ISSN 1475-7516, nr 12, artikel-id 021Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We perform cosmological perturbation theory in Hassan-Rosen bimetric gravity for general homogeneous and isotropic backgrounds. In the de Sitter approximation, we obtain decoupled sets of massless and massive scalar gravitational fluctuations. Matter perturbations then evolve like in Einstein gravity. We perturb the future de Sitter regime by the ratio of matter to dark energy, producing quasi-de Sitter space. In this more general setting the massive and massless fluctuations mix. We argue that in the quasi-de Sitter regime, the growth of structure in bimetric gravity differs from that of Einstein gravity.

  • 91.
    Bergh, Daniel
    Karlstads universitet, Fakulteten för samhälls- och livsvetenskaper, Avdelningen för hälsa och miljö. Karlstads universitet, Fakulteten för humaniora och samhällsvetenskap (from 2013), Centrum för forskning om barns och ungdomars psykiska hälsa.
    Chi-Squared Test of Fit and Sample Size-A Comparison between a Random Sample Approach and a Chi-Square Value Adjustment Method2015Ingår i: Journal of Applied Measurement, ISSN 1529-7713, Vol. 16, nr 2, s. 204-217Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Chi-square statistics are commonly used for tests of fit of measurement models. Chi-square is also sensitive to sample size, which is why several approaches to handle large samples in test of fit analysis have been developed. One strategy to handle the sample size problem may be to adjust the sample size in the analysis of fit. An alternative is to adopt a random sample approach. The purpose of this study was to analyze and to compare these two strategies using simulated data. Given an original sample size of 21,000, for reductions of sample sizes down to the order of 5,000 the adjusted sample size function works as good as the random sample approach. In contrast, when applying adjustments to sample sizes of lower order the adjustment function is less effective at approximating the chi-square value for an actual random sample of the relevant size. Hence, the fit is exaggerated and misfit under-estimated using the adjusted sample size function. Although there are big differences in chi-square values between the two approaches at lower sample sizes, the inferences based on the p-values may be the same.

  • 92.
    Bergh, Daniel
    Karlstads universitet, Fakulteten för humaniora och samhällsvetenskap (from 2013), Centrum för forskning om barns och ungdomars psykiska hälsa.
    Sample Size and Chi-Squared Test of Fit: A comparison Between a Random Sample Approach and a Chi-Square Value Adjustment Method Using Swedish Adolescent Data2015Ingår i: Pacific Rim Objective Measurement Symposium (PROMS) 2014 Conference Proceedings: Rasch and the Future / [ed] Zhang, Quan., & Yang, Hong. (Eds.), Springer Berlin/Heidelberg, 2015, s. 197-211Konferensbidrag (Refereegranskat)
    Abstract [en]

    Background: Significance tests are commonly sensitive to sample size, and Chi-Squared statistics is not an exception. Nevertheless, Chi-Squared statistics are commonly used for test of fit of measurement models. Thus, for analysts working with very large (or very small) sample sizes this may require particular attention. However, several different approaches to handle a large sample size in test of fit analysis have been developed. Thus, one strategy may be to adjust the fit statistic to correspond to an equivalent sample of different size. This strategy has been implemented in the RUMM2030 software. Another strategy may be to adopt a random sample approach.

    Aims: The RUMM2030 Chi-Square value adjustment facility has been available for a long time, but still there are few studies describing the empirical consequences of adjusting the sample to correspond to a smaller effective sample size in the statistical analysis of fit. Alternatively a random sample approach could be adopted in order to handle the large sample size problem. The purpose of this study was to analyze and compare these two strategies as test of fit approximations, using Swedish adolescent data. 

    Sample:The analysis is based on the survey Young in Värmland which is a paper-and-pencil based survey conducted recurrently since 1988 targeting all adolescent in school year 9 residing the county of Värmland, Sweden. So far, more than 20,000 individuals have participated in the survey. In the analysis presented here, seven items based on the adolescents, experiences of the school environment were subjected to analysis, in total 21,088 individuals.

    Methods: For the purposes of this study, the original sample size was adjusted to several different effective samples using the RUMM2030 adjustment function, in the test of fit analysis. In addition, 10 random samples for each sample size were drawn from the original sample, and averaged Chi-Square values calculated. The Chi-Square values obtained using the two strategies were compared.

    Results: Given the original sample of 21,000, adjusting to samples 5,000 or larger, the RUMM2030 adjustment facility work as well as a random sample approach. In contrast, when adjusting to lower samples the adjustment function is less effective in approximating the Chi-Square value for an actual random sample of the relevant size. Hence, fit is exaggerated and misfit under estimated using the adjustment function, in particular is that true for fitting but not misfitting items.  

    Conclusion: Even though the inferences based on p-values may be the same despite big Chi-Square value differences between the two approaches, the danger of using fit statistics mechanically cannot be enough stressed. Neither the adjustment function, nor the random sample approach is sufficient in evaluating model fit, instead several complementing methods should be used.

     

  • 93.
    Bergman, Ivar
    Karlstads universitet, Institutionen för ingenjörsvetenskap, fysik och matematik.
    Baire category theorem2009Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)
    Abstract [en]

    In this thesis we give an exposition of the notion of category and the Baire category theorem as a set theoretical method for proving existence. The category method was introduced by René Baire to describe the functions that can be represented by a limit of a sequence of continuous real functions. Baire used the term functions of the first class to denote these functions.

    The usage of the Baire category theorem and the category method will be illustrated by some of its numerous applications in real and functional analysis. Since the usefulness, and generality, of the category method becomes fully apparent in Banach spaces, the applications provided have been restricted to these spaces.

    To some extent, basic concepts of metric topology will be revised, as the Baire category theorem is formulated and proved by these concepts. In addition to the Baire category theorem, we will give proof of equivalence between different versions of the theorem.

    Explicit examples, of first class functions will be presented, and we shall state a theorem, due to Baire, providing a necessary condition on the set of points of continuity for any function of the first class.

     

  • 94.
    Bergström, Sofia
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013).
    Kommunikativ matematik: En läromedelsanalys av kommunikativa matematikuppgifter i årskurs tre2018Självständigt arbete på avancerad nivå (yrkesexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)
    Abstract [sv]

    Sammanfattning

    Syftet med studien är att undersöka förekomsten och placeringen av samarbets- och kommunikationsuppgifter i matematikläromedel för årskurs tre. Vidare är syftet att granska dessa uppgifter djupare för att ta reda på vilken typ av kommunikation de uppmanar till samt vilka kognitiva nivåer uppgifterna har. För att svara på mitt syfte har jag gjort en läromedelsanalys av fyra olika matematikläromedel i årskurs tre. Resultatet visar att i alla analyserade läromedel tillkom över hälften av antalet uppgifter genom att använda lärarhandledningen jämfört med att endast använda läroboken. Studiens resultat visar därmed vikten av att använda sig att lärarhandledningen för att till fullo kunna utnyttja läromedlet som resurs. Vidare visar studien att arbete i par är den vanligaste kommunikationsformen som det uppmanas till genom läromedlen. Resultatet visar också att störst antal av samarbets- och kommunikationsuppgifterna befinner sig på låga kognitiva nivåer, vilket enligt forskning inte gynnar en kommunikativ matematikundervisning. 

  • 95.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
    Boundary Layers and Shock Profiles for the Broadwell Model2016Ingår i: International Journal of Differential Equations, ISSN 1687-9643, E-ISSN 1687-9651, Vol. 2016, s. 1-8, artikel-id 5801728Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the existence of nonlinear boundary layers and the typically nonlinear problem of existence of shock profiles for the Broadwell model, which is a simplified discrete velocity model for the Boltzmann equation. We find explicit expressions for the nonlinear boundary layers and the shock profiles. In spite of the few velocities used for the Broadwell model, the solutions are (at least partly) in qualitatively good agreement with the results for the discrete Boltzmann equation, that is the general discrete velocity model, and the full Boltzmann equation.

  • 96.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Boundary Layers and Shock Profiles for the Discrete Boltzmann Equation for Mixtures2012Ingår i: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 5, nr 1, s. 1-19Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the discrete Boltzmann equation for binary gas mixtures. Some known results for half-space problems and shock profile solutions of the discrete Boltzmann for single-component gases are extended to the case of two-component gases. These results include well-posedness results for half-space problems for the linearized discrete Boltzmann equation, existence results for half-space problems for the weakly non-linear discrete Boltzmann equation, and existence results for shock profile solutions of the discrete Boltzmann equation. A characteristic number, corresponding to the speed of sound in the continuous case, is calculated for a class of symmetric models. Some explicit calculations are also made for a simplified 6+4 -velocity model.

  • 97.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
    Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations2017Ingår i: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 10, nr 4, s. 925-955Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider some extensions of the classical discrete Boltzmann equation to the cases of multicomponent mixtures, polyatomic molecules (with a finite number of different internal energies), and chemical reactions, but also general discrete quantum kinetic Boltzmann-like equations; discrete versions of the Nordheim-Boltzmann (or Uehling-Uhlenbeck) equation for bosons and fermions and a kinetic equation for excitations in a Bose gas interacting witha Bose-Einstein condensate. In each case we have an H-theorem and so for the planar stationary half-space problem, we have convergence to an equilibrium distribution at infinity (or at least a manifold of equilibrium distributions). In particular, we consider the nonlinear half-space problem of condensation and evaporation for these discrete Boltzmann-like equations. We assume that the flow tends to a stationary point at infinity and that the outgoing flow is known at the wall, maybe also partly linearly depending on the incoming flow. We find that the systems we obtain are of similar structures as for the classical discrete Boltzmann equation (for single species), and that previously obtained results for the discrete Boltzmann equation can be applied after being generalized. Then the number of conditions on the assigned data at the wall needed for existence of a unique solution is found. The number of parameters to be specified in the boundary conditions depends on if we have subsonic or supersonic condensation or evaporation. All our results are valid for any finite number of velocities.

  • 98.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Boundary layers for the nonlinear discrete Boltzmann equation: Condensing vapor flow in the presence of a non-condensable gas2012Ingår i: Proceedings of 28th International Symposium on Rarefied Gas Dynamics 2012 / [ed] Michel Mareschal, Andrés Santos, Melville, New York: American Institute of Physics (AIP), 2012, 1, s. 223-230Konferensbidrag (Refereegranskat)
    Abstract [en]

    Half-space problems for the Boltzmann equation are of great importance in the study of the asymptotic behaviorof the solutions of boundary value problems of the Boltzmann equation for small Knudsen numbers. Half-space problems provide the boundary conditions for the fluid-dynamic-type equations and Knudsen-layer corrections to the solution of the fluid-dynamic-type equations in a neighborhood of the boundary. Here we consider a half-space problem of condensation for apure vapor in the presence of a non-condensable gas by using discrete velocity models (DVMs) of the Boltzmann equation. The Boltzmann equation can be approximated by DVMs up to any order, and these DVMs can be applied for numerical methods,but also for mathematical studies to bring deeper understanding and new ideas. For one-dimensional half-space problems,the discrete Boltzmann equation (the general DVM) reduces to a system of ODEs. We obtain that the number of parametersto be specified in the boundary conditions depends on whether the condensing vapor flow is subsonic or supersonic. Thisbehavior has earlier been found numerically. We want to stress that our results are valid for any finite number of velocities.This is an extension of known results for single-component gases (and for binary mixtures of two vapors) to the case when a non-condensable gas is present. The vapor is assumed to tend to an assigned Maxwellian, with a flow velocity towards thecondensed phase, at infinity, while the non-condensable gas tends to zero at infinity. Steady condensation of the vapor takes place at the condensed phase, which is held at a constant temperature. We assume that the vapor is completely absorbed, that the non-condensable gas is diffusively reflected at the condensed phase, and that vapor molecules leaving the condensed phase are distributed according to a given distribution. The conditions, on the given distribution at the condensed phase, needed for the existence of a unique solution of the problem are investigated, assuming that the given distribution at the condensed phase is sufficiently close to the Maxwellian at infinity and that the total mass of the non-condensable gas is sufficiently small. Exact solutions and solvability conditions are found for a specific simplified discrete velocity model (with few velocities).

  • 99.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
    Discrete quantum Boltzmann equation2019Ingår i: AIP Conference Proceedings, American Institute of Physics (AIP), 2019, Vol. 2132, s. 1-9, artikel-id 130011Konferensbidrag (Refereegranskat)
    Abstract [en]

    In this work, we consider a Boltzmann equation for anyons. In particular, we study a general discrete velocity model of the equation, where the velocity variable is assumed to only take values from a given finite-such that the (finite) number of velocities is arbitrary-set of velocities. Included, as two limiting cases, is the discrete quantum Boltzmann equation (Nordheim-Boltzmann/Uehling-Uhlenbeck equation) for bosons and fermions. Mass, momentum, and energy are assumed to be conserved during collisions, and considering suitable discrete velocity models, they will also be the only collision invariants. The equilibrium distributions will be given by a transcendental equation, and only in some few cases-including the two limiting cases where they are Planckians-they will be explicitly expressed. However, there is an H-theorem, and therefore one can prove that for the spatially homogeneous equation, as time tends to infinity, as well as, for the steady equation in a half-space with slab-symmetry, as the space variable tends to infinity, the distribution function converges to an equilibrium distribution. Linearizing around an equilibrium distribution in a suitable way, we find that the obtained linearized operator has similar properties as the corresponding linearized operator for the discrete Boltzmann equation: E.g. it is symmetric and positive semi-definite. Hence, previously obtained results for the spatially homogeneous Cauchy problem and the steady half-space problem in a slab symmetry for the discrete Boltzmann equation, can be applied also in the considered quantum case.

  • 100.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Discrete Velocity Models and Half-Space Problems2003Licentiatavhandling, monografi (Övrigt vetenskapligt)
    Abstract [en]

    We study some questions related to discrete velocity models (DVMs) of the Boltzmann equation. In the case of plane stationary problems the typical DVM reduces to a dynamical system (system of ODEs). Properties of such systems are studied in this paper in the most general case. In particular, a topological classification of their singular points is made and dimensions of the corresponding stable, unstable and center manifolds are computed. These results are applied to typical half-space problems of rarefied gas dynamics. A classification of well-posed half-space problems for linearized DVMs is made. Exact solutions of a (simplified) linearized kinetic model of BGK type are found as limiting case of corresponding discrete models. The main results of the paper can be also used for moment approximations and other versions of discretizised kinetic equations

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