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  • 1.
    Cutler, Jonathan
    et al.
    Umeå University, Faculty of Science and Technology, Department of mathematics. Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA.
    Montagh, Balazs
    Unavoidable subgraphs of colored graphs2008In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 308, no 19, p. 4396-4413Article in journal (Refereed)
    Abstract [en]

    A natural generalization of graph Ramsey theory is the study of unavoidable sub-graphs in large colored graphs. In this paper. we find a minimal family of unavoidable graphs in two-edge-colored graphs. Namely, for a positive even integer k, let y(k) be the family of two-edge-colored graphs on k vertices such that one of the colors forms either two disjoint K-k/2's or simply one K-k/2. Bollobas conjectured that for all k and epsilon > 0, there exists an n(k, epsilon) such that if n >= n(k, epsilon) then every two-edge-coloring of K-n, in which the density of each color is at least epsilon, contains a member of this family. We solve this conjecture and present a series of results bounding it (k, s) for different ranges of epsilon. In particular, if epsilon is sufficiently close to 1/2, the gap between out upper and lower bounds for n(k, epsilon) is smaller than those for the classical Ramsey number R(k, k).

  • 2.
    Gelfgren, Jan
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Multipoint Padé approximants used for piecewise rational interpolation and for interpolation to functions of Stieltjes' type1978Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    A multipoint Padë approximant, R, to a function of Stieltjes1 type is determined.The function R has numerator of degree n-l and denominator of degree n.The 2n interpolation points must belong to the region where f is analytic,and if one non-real point is amongst the interpolation points its complex-conjugated point must too.The problem is to characterize R and to find some convergence results as n tends to infinity. A certain kind of continued fraction expansion of f is used.From a characterization theorem it is shown that in each step of that expansion a new function, g, is produced; a function of the same type as f. The function g is then used,in the second step of the expansion,to show that yet a new function of the same type as f is produced. After a finite number of steps the expansion is truncated,and the last created function is replaced by the zero function.It is then shown,that in each step upwards in the expansion a rational function is created; a function of the same type as f.From this it is clear that the multipoint Padê approximant R is of the same type as f.From this it is obvious that the zeros of R interlace the poles, which belong to the region where f is not analytical.Both the zeros and the poles are simple. Since both f and R are functions of Stieltjes ' type the theory of Hardy spaces can be applied (p less than one ) to show some error formulas.When all the interpolation points coincide ( ordinary Padé approximation) the expected error formula is attained. From the error formula above it is easy to show uniform convergence in compact sets of the region where f is analytical,at least wien the interpolation points belong to a compact set of that region.Convergence is also shown for the case where the interpolation points approach the interval where f is not analytical,as long as the speed qî approach is not too great.

  • 3.
    Gelfgren, Jan
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Piecewise rational interpolation1975Report (Other academic)
  • 4.
    Gelfgren, Jan
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Rational interpolation to functions of Stieltjes' type1978Report (Other academic)
  • 5.
    Hellström, Lars
    Umeå University, Faculty of Science and Technology, Department of mathematics. Mittag Leffler Inst, Djursholm, Sweden; Lund Univ, Ctr Math Sci, S-22100 Lund, Sweden .
    Rewriting in operads and PROPs2006In: Journal of nonlinear mathematical physics, Luleå: Norbert Euler , 2006, Vol. 13 S, p. 66-75Conference paper (Refereed)
    Abstract [en]

    This paper is an informal collection of observations on how established rewriting techniques can be applied to or need to be adapted for use in non-associative algebras, operads, and PROPs.

  • 6.
    Hellström, Lars
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    The Diamond Lemma for Power Series Algebras2002Doctoral thesis, monograph (Other academic)
    Abstract [en]

    The main result in this thesis is the generalisation of Bergman's diamond lemma for ring theory to power series rings. This generalisation makes it possible to treat problems in which there arise infinite descending chains. Several results in the literature are shown to be special cases of this diamond lemma and examples are given of interesting problems which could not previously be treated. One of these examples provides a general construction of a normed skew field in which a custom commutation relation holds.

    There is also a general result on the structure of totally ordered semigroups, demonstrating that all semigroups with an archimedean element has a (up to a scaling factor) unique order-preserving homomorphism to the real numbers. This helps analyse the concept of filtered structure. It is shown that whereas filtered structures can be used to induce pretty much any zero-dimensional linear topology, a real-valued norm suffices for the definition of those topologies that have a reasonable relation to the multiplication operation.

    The thesis also contains elementary results on degree (as of polynomials) functions, norms on algebras (in particular ultranorms), (Birkhoff) orthogonality in modules, and construction of semigroup partial orders from ditto quasiorders.

  • 7.
    Jonsson, Alf
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Dirichlet forms and Brownian motion penetrating fractals2000In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 13, no 1, p. 69-80Article in journal (Refereed)
    Abstract [en]

    Can a Brownian motion penetrate the two-dimensional Sierpinski gasket? This question was studied in [8], and an affirmative answer was given. In this paper, the problem is studied with a different approach, using Dirichlet forms and function space theory. The results obtained are somewhat different from, and from certain aspects more general than, the results in [8].

  • 8.
    Jonsson, Alf
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Haar wavelets of higher order on fractals and regularity of functions2004In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 290, no 1, p. 86-104Article in journal (Refereed)
    Abstract [en]

    Wavelets of Haar type of higher order m on self-similar fractals were introduced by the author in J. Fourier Anal. Appl. 4 (1998) 329–340. These are piecewise polynomials of degree m instead of piecewise constants. It was shown that for certain totally disconnected fractals, spaces of functions defined on the fractal may be characterized by means of the magnitude of the wavelet coefficients of the functions. In this paper, the study of these wavelets is continued. It is shown that also in the case when the fractals are not totally disconnected, the wavelets can be used to study regularity properties of functions. In particular, the self-similar sets considered can be, e.g., an interval in R or a cube in Rn. It turns out that it is natural to use Haar wavelets of higher order also in these classical cases, and many of the results in the paper are new also for these sets.

  • 9.
    Jonsson, Alf
    et al.
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Kamont, Anna
    Piecewise linear bases and Besov spaces on fractal sets2001In: Analysis Mathematica, ISSN 0133-3852, E-ISSN 1588-273X, Vol. 27, no 2, p. 77-117Article in journal (Refereed)
    Abstract [en]

    For a class of closed sets FRn admitting a regular sequence of triangulations or generalized triangulations, the analogues on F of the Faber—Schauder and Franklin bases are discussed. The characterizations of the Besov spaces on F in the terms of coefficients of functions with respect to these bases are proved. As a consequence, analogous characterizations of the Besov spaces on some fractal domains (including the Sierpinski gasket and the von Koch curve) by coefficients of functions with respect to the wavelet bases constructed in [26] are obtained.

  • 10. Lewis, John
    et al.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Boundary behavior and the Martin boundary problem for p harmonic functions in Lipschitz domains2010In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 172, no 3, p. 1907-1948Article in journal (Refereed)
    Abstract [en]

    In a previous article, we proved a boundary Harnack inequality for the ratio of two positive p harmonic functions, vanishing on a portion of the boundary of a Lipschitz domain. In the current paper we continue our study by showing that this ratio is Holder continuous up to the boundary. We also consider the Martin boundary of certain domains and the corresponding question of when a minimal positive p harmonic function (with respect to a given boundary point w) is unique up to constant multiples. In particular we show that the Martin boundary can be identified with the topological boundary in domains that are convex or C(1). Minimal positive p harmonic functions relative to a boundary point w in a Lipschitz domain are shown to be unique, up to constant multiples, provided the boundary is sufficiently flat at w.

  • 11.
    Lithner, Johan
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Comparing two versions of Markov's inequality on compact sets1994In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 77, no 2, p. 202-211Article in journal (Refereed)
    Abstract [en]

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

  • 12.
    Lithner, Johan
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Lusten att Lära, Luleå kommun: Nationella kvalitetsgranskningar 2001/20022002Report (Other (popular science, discussion, etc.))
  • 13.
    Lithner, Johan
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Lusten att Lära, Osby kommun: Nationella kvalitetsgranskningar 2001/20022002Report (Other (popular science, discussion, etc.))
  • 14.
    Lithner, Johan
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Mathematical reasoning and familiar procedures2000In: International journal of mathematical education in science and technology, ISSN 0020-739X, E-ISSN 1464-5211, Vol. 31, no 1, p. 83-95Article in journal (Refereed)
    Abstract [en]

    Four first-year undergraduate students are working with two tasks. The underlying question treated is "what are the characteristics and background causes of their difficulties when trying to solve these tasks?" The purpose is to give a general survey of their main difficulties, rather than to go deeply into details. It seems like one of the common characteristics is that the students are more focused on what is familiar and remembered, than on (even elementary) mathematical reasoning and accuracy.

  • 15.
    Lithner, Johan
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Mathematical reasoning in calculus textbook exercises2000Report (Other academic)
  • 16.
    Lithner, Johan
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Mathematical reasoning in calculus textbook exercises2004In: Journal of Mathematical Behavior, ISSN 0732-3123, E-ISSN 1873-8028, Vol. 23, no 4, p. 405-427Article in journal (Refereed)
    Abstract [en]

    The aim of this paper is to study some of the strategies that are possible to use in order to solve the exercises in undergraduate calculus textbooks. It is described in detail how most exercises may be solved by mathematically superficial strategies, often without actually considering the core mathematics of the book section in question.

  • 17.
    Lithner, Johan
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Students’ mathematical reasoning in university textbook exercises2003In: Educational Studies in Mathematics, ISSN 0013-1954, E-ISSN 1573-0816, Vol. 52, no 1, p. 29-55Article in journal (Refereed)
    Abstract [en]

    Video recordings of three undergraduate students’ textbook-based home- work are analysed. A focus is on the ways their exercise reasoning is mathematically well- founded or superficial. Most strategy choices and implementations are carried out without considering the intrinsic mathematical properties of the components involved in their work. It is essential in their strategies to find procedures to mimick and few constructive reasoning attempts are made.

     

  • 18.
    Lithner, Johan
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Undergraduate learning difficulties and mathematical reasoning: a literature survey and project overview2001Report (Other academic)
  • 19.
    Lithner, Johan
    et al.
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Woijcik, Adam
    University of Krakow.
    A Note on Bernstein's Theorems1995In: Journal of Approximation Theory, ISSN 0021-9045, E-ISSN 1096-0430, Vol. 81, no 3, p. 316-322Article in journal (Refereed)
    Abstract [en]

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

  • 20.
    Lundow, Per-Håkan
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Compression of transfer matrices2001In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 231, no 1-3, p. 321-329Article in journal (Refereed)
    Abstract [en]

    We present a method for reducing the size of transfer matrices by exploiting symmetry. For example, the transfer matrix for enumeration of matchings in the graph C-4 x C-4 x P-n can be reduced from order 65536 to 402 simply due to the 384 automorphisms of C-4 x C-4. The matrix for enumeration of perfect matchings can be still further reduced to order 93, all in a straightforward and mechanical way. As an application we report an improved upper bound for the three-dimensional dimer problem. (C) 2001 Elsevier Science B.V. All rights reserved.

  • 21.
    Palm, Torulf
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Authentic assessment and performance assessment: A conceptual analysis of the literature2008In: Practical Assessment, Research & Evaluation, ISSN 1531-7714, E-ISSN 1531-7714, Vol. 13, no 4Article in journal (Refereed)
    Abstract [en]

     Performance assessment and authentic assessment are recurrent terms in the literature on education and educational research. They have both been given a number of different meanings and unclear definitions and are in some publications not defined at all. Such uncertainty of meaning causes difficulties in interpretation and communication and can cause clouded or misleading research conclusions. This paper reviews the meanings attached to these concepts in the literature and describes the similarities and wide range of differences between the meanings of each concept.

  • 22.
    Palm, Torulf
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Impact of authenticity on sense making in word problem solving2008In: Educational Studies in Mathematics, ISSN 0013-1954, E-ISSN 1573-0816, Vol. 67, no 1Article in journal (Refereed)
  • 23.
    Palm, Torulf
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    The realism of mathematical school tasks: Features and consequences2002Doctoral thesis, comprehensive summary (Other academic)
  • 24.
    Palm, Torulf
    et al.
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Burman, Lars
    Reality in mathematics assessment: An analysis of task-reality concordance in Finnish and Swedish national assessment2004In: Nordic Studies in Mathematics Education, Vol. 9, no 3Article in journal (Refereed)
  • 25. Pratt, Stephen C
    et al.
    Sumpter, David JT
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    Mallon, Eamonn B
    Franks, Nigel R
    An agent-based model of collective nest choice by the ant Temnothorax albipennis2005In: Animal Behaviour, ISSN 0003-3472, E-ISSN 1095-8282, Vol. 70, no 5, p. 1023-1036Article in journal (Refereed)
    Abstract [en]

    Colonies of the ant Temnothorax (formerly Leptothorax) albipennis can collectively choose the best of several nest sites, even when many of the active ants who organize the move visit only one site. Previous studies have suggested that this ability stems from the ants' strategy of graded commitment to a potential home. On finding a site, an ant proceeds from independent assessment, to recruiting fellow active ants via slow tandem runs, to bringing the passive bulk of the colony via rapid transports. Assessment duration varies inversely with site quality, and the switch from tandem runs to transports requires that a quorum of ants first be summoned to the site. These rules may generate a collective decision, by creating and amplifying differential population growth rates among sites. We test the importance of these and other known behavioural rules by incorporating them into an agent-based model. All parameters governing individual behaviour were estimated from videotaped emigrations of individually marked ants given a single nest option of either good or mediocre quality. The time course of simulated emigrations and the distribution of behaviour across ants largely matched these observations, except for the speed with which the final transport phase was completed, and the overall emigration speed of one particularly large colony. The model also predicted the prevalence of splitting between sites when colonies had to choose between two sites of different quality, although it correctly predicted the degree of splitting in only four of six cases. It did not fully capture variance in colony performance, but it did predict the emergence of variation in individual behaviour, despite the use of identical parameter values for all ants. The model shows how, with adequate empirical data, the algorithmic form of a collective decision-making mechanism can be captured.

  • 26.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of mathematics.
    List colorings of graphs: a short survey2000Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    We present an overview of the current state of the List Chromatic Conjecture (LCC) and results on the choice number of planar graphs, as well as of general progress on the subject of list colorings.

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