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  • 1
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    Topological properties of the intersection graph of covers of n-dimensional surfaces (1995)
    Elsevier
    Details
    Publication Date: 1995-12-01
    Print ISSN: 0012-365X
    Electronic ISSN: 1872-681X
    Topics: Mathematics
    Published by Elsevier
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  • 2
    Electronic Resource
    Electronic Resource
    Graphical Dimension for Line Segments (1994)
    Oxford, UK : Blackwell Publishing Ltd
    Annals of the New York Academy of Sciences 728 (1994), S. 0 
    Details
    ISSN: 1749-6632
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Natural Sciences in General
    Notes: : The authors give a new definition of dimension for a graph. It turns out that the graph obtained from a line segment in Euclidean n-space by means of a construction involving n-cubes is always 1-dimensional.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1749-6632.1994.tb44142.x
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Dimensional properties of graphs and digital spaces (1996)
    Springer
    Journal of mathematical imaging and vision 6 (1996), S. 109-119 
    Details
    ISSN: 1573-7683
    Keywords: Alexandroff space ; computer graphics ; connected ordered topological space ; digital image ; digital picture ; digital space ; digital topology ; dimension ; graph ; locally finite space
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Many applications of digital image processing now deal with three- and higher-dimensional images. One way to represent n-dimensional digital images is to use the specialization graphs of subspaces of the Alexandroff topological space ℤn (where ℤ denotes the integers with the Khalimsky line topology). In this paper the dimension of any such graph is defined in three ways, and the equivalence of the three definitions is established. Two of the definitions have a geometric basis and are closely related to the topological definition of inductive dimension; the third extends the Alexandroff dimension to graphs. Diagrams are given of graphs that are dimensionally correct discrete models of Euclidean spaces, n-dimensional spheres, a projective plane and a torus. New characterizations of n-dimensional (digital) surfaces are presented. Finally, the local structure of the space ℤn is analyzed, and it is shown that ℤn is an n-dimensional surface for all n≥1.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00119834
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  • 4
    Electronic Resource
    Electronic Resource
    Dimension on discrete spaces (1994)
    Springer
    International journal of theoretical physics 33 (1994), S. 1553-1568 
    Details
    ISSN: 1572-9575
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract In this paper we develop some combinatorial models for continuous spaces. We study the approximations of continuous spaces by graphs, molecular spaces, and coordinate matrices. We define the dimension on a discrete space by means of axioms based on an obvious geometrical background. This work presents some discrete models ofn-dimensional Euclidean spaces,n-dimensional spheres, a torus, and a projective plane. It explains how to construct new discrete spaces and describes in this connection several three-dimensional closed surfaces with some topological singularities. It also analyzes the topology of (3+1)-space-time. We are also discussing the question by R. Sorkin about how to derive the system of simplicial complexes from a system of open coverings of a topological space.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00670697
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