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  • 1.
    Forbes, Anthony D.
    et al.
    School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, UK.
    Griggs, Terry S.
    School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, UK.
    Stokes, Klara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Existence results for pentagonal geometries2022In: The Australasian Journal of Combinatorics, ISSN 1034-4942, Vol. 82, no 1, p. 95-114Article in journal (Refereed)
    Abstract [en]

    New results on pentagonal geometries PENT(k, r) with block sizes k = 3 or k = 4 are given. In particular we completely determine the existence spectra for PENT(3, r) systems with the maximum number of opposite line pairs as well as those without any opposite line pairs. A wide-ranging result about PENT(3, r) with any number of opposite line pairs is proved. We also determine the existence spectrum of PENT(4, r) systems with eleven possible exceptions.

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  • 2.
    Leemans, Dimitri
    et al.
    Département de Mathématique, Université libre de Bruxelles, Algèbre et Combinatoire, Brussels, Belgium.
    Stokes, Klara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Incidence geometries with trialities coming from maps with Wilson trialities2023In: Innovations in Incidence Geometry, ISSN 2640-7337, Vol. 20, no 2-3, p. 325-340Article in journal (Refereed)
    Abstract [en]

    Triality is a classical notion in geometry that arose in the context of the Lie groups of type D4. Another notion of triality, Wilson triality, appears in the context of reflexible maps. We build a bridge between these two notions, showing how to construct an incidence geometry with a triality from a map that admits a Wilson triality. We also extend a result by Jones and Poulton, showing that for every prime power q, the group L2 (q3) has maps that admit Wilson trialities but no dualities.

  • 3.
    Lundqvist, Signe
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Stokes, Klara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Applying the pebble game algorithm to rod configurations2023In: EuroCG 2023: Book of abstracts, 2023, article id 41Conference paper (Refereed)
    Abstract [en]

    We present results on rigidity of structures of rigid rods connected in joints: rod configurations. The underlying combinatorial structure of a rod configuration is an incidence structure. Our aim is to find simple ways of determining which rod configurations admit non-trivial motions, using the underlying incidence structure.

    Rigidity of graphs in the plane is well understood. Indeed, there is a polynomial time algorithm for deciding whether most realisations of a graph are rigid. One of the results presented here equates rigidity of sufficiently generic rod configurations to rigidity of a related graph. As a consequence, itis possible to determine the rigidity of rod configurations using the previously mentioned polynomial time algorithm. We use this to show that all v3-configurations on up to 15 points and all triangle-free v3-configurations on up to 20 points are rigid in regular position, if such a realisation exists. We also conjecture that the smallest v3-configuration that is flexible in regular position is a previously known 283-configuration. 

  • 4.
    Lundqvist, Signe
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Stokes, Klara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Exploring the rigidity of planar configurations of points and rods2023In: Discrete Applied Mathematics, ISSN 0166-218X, E-ISSN 1872-6771, Vol. 336, p. 68-82Article in journal (Refereed)
    Abstract [en]

    In this article we explore the rigidity of realizations of incidence geometries consisting of points and rigid rods: rod configurations. We survey previous results on the rigidity of structures that are related to rod configurations, discuss how to find realizations of incidence geometries as rod configurations, and how this relates to the 2-plane matroid. We also derive further sufficient conditions for the minimal rigidity of k-uniform rod configurations and give an example of an infinite family of minimally rigid 3-uniform rod configurations failing the same conditions. Finally, we construct v3-configurations that are flexible in the plane, and show that there are flexible v3-configurations for all sufficiently large values of v.

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  • 5.
    Lundqvist, Signe
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Stokes, Klara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Öhman, Lars-Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    When is a planar rod configuration infinitesimally rigid?2023In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444Article in journal (Refereed)
    Abstract [en]

    We investigate the rigidity properties of rod configurations. Rod configurations are realizations of rank two incidence geometries as points (joints) and straight lines (rods) in the Euclidean plane, such that the lines move as rigid bodies, connected at the points. Note that not all incidence geometries have such realizations. We show that under the assumptions that the rod configuration exists and is sufficiently generic, its infinitesimal rigidity is equivalent to the infinitesimal rigidity of generic frameworks of the graph defined by replacing each rod by a cone over its point set. To put this into context, the molecular conjecture states that the infinitesimal rigidity of rod configurations realizing 2-regular hypergraphs is determined by the rigidity of generic body and hinge frameworks realizing the same hypergraph. This conjecture was proven by Jackson and Jordán in the plane, and by Katoh and Tanigawa in arbitrary dimension. Whiteley proved a version of the molecular conjecture for hypergraphs of arbitrary degree that have realizations as independent body and joint frameworks. Our result extends his result to hypergraphs that do not necessarily have realizations as independent body and joint frameworks, under the assumptions listed above.

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  • 6.
    Mc Glue, Ciaran
    et al.
    Maynooth University (NUI), Maynooth Co. Kildare, Ireland.
    Stokes, Klara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Maynooth University (NUI), Maynooth Co. Kildare, Ireland.
    Generating All Rigidity Circuits on at Most 10 Vertices and All Assur Graphs on at Most 11 Vertices2022In: Journal of Integer Sequences, E-ISSN 1530-7638, Vol. 25, no 1, article id 22.1.3Article in journal (Refereed)
    Abstract [en]

    We present an inventory and the enumeration of all non-isomorphic rigidity circuits on up to 10 vertices, as well as all non-isomorphic Assur graphs on up to 11 vertices. Assur graphs and Baranov trusses are closely related. We clarify the relation between Baranov trusses and (2, 3)-tight graphs on the one hand, and between Assur groups and Assur graphs on the other hand.

  • 7.
    Stokes, Klara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Geometric decoding of subspace codes with explicit Schubert calculus applied to spread codes2023In: Advances in mathematics of communicationsArticle in journal (Refereed)
1 - 7 of 7
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