Aggregate and fractal tessellations

Konstantin Tchoumatchenko, Sergei Zuev

    Research output: Contribution to journalArticle

    18 Citations (Scopus)

    Abstract

    Consider a sequence of stationary tessellations {‹n}, n=0,1,..., of  d consisting of cells {Cn(xin)}with the nuclei {xin}. An aggregate cell of level one, C01(xi0), is the result of merging the cells of ‹1 whose nuclei lie in C0(xi0). An aggregate tessellation ‹0n consists of the aggregate cells of level n, C0n(xi0), defined recursively by merging those cells of ‹n whose nuclei lie in Cnm1(xi0). We find an expression for the probability for a point to belong to atypical aggregate cell, and obtain bounds for the rate of itsexpansion. We give necessary conditions for the limittessellation to exist as nMX and provide upperbounds for the Hausdorff dimension of its fractal boundary and forthe spherical contact distribution function in the case ofPoisson-Voronoi tessellations {‹n}.
    LanguageEnglish
    Pages198-218
    Number of pages20
    JournalProbability Theory and Related Fields
    Volume121
    Issue number2
    DOIs
    Publication statusPublished - 31 Oct 2001

    Fingerprint

    Tessellation
    Fractal
    Cell
    Nucleus
    Merging
    Voronoi Tessellation
    Hausdorff Dimension
    Distribution Function
    Contact
    Necessary Conditions

    Keywords

    • probability
    • voronoï tessellation
    • pProbability distribution
    • telecommunications
    • poisson process
    • fractal

    Cite this

    Tchoumatchenko, Konstantin ; Zuev, Sergei. / Aggregate and fractal tessellations. In: Probability Theory and Related Fields. 2001 ; Vol. 121, No. 2. pp. 198-218.
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    Aggregate and fractal tessellations. / Tchoumatchenko, Konstantin; Zuev, Sergei.

    In: Probability Theory and Related Fields, Vol. 121, No. 2, 31.10.2001, p. 198-218.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - Aggregate and fractal tessellations

    AU - Tchoumatchenko, Konstantin

    AU - Zuev, Sergei

    PY - 2001/10/31

    Y1 - 2001/10/31

    N2 - Consider a sequence of stationary tessellations {‹n}, n=0,1,..., of  d consisting of cells {Cn(xin)}with the nuclei {xin}. An aggregate cell of level one, C01(xi0), is the result of merging the cells of ‹1 whose nuclei lie in C0(xi0). An aggregate tessellation ‹0n consists of the aggregate cells of level n, C0n(xi0), defined recursively by merging those cells of ‹n whose nuclei lie in Cnm1(xi0). We find an expression for the probability for a point to belong to atypical aggregate cell, and obtain bounds for the rate of itsexpansion. We give necessary conditions for the limittessellation to exist as nMX and provide upperbounds for the Hausdorff dimension of its fractal boundary and forthe spherical contact distribution function in the case ofPoisson-Voronoi tessellations {‹n}.

    AB - Consider a sequence of stationary tessellations {‹n}, n=0,1,..., of  d consisting of cells {Cn(xin)}with the nuclei {xin}. An aggregate cell of level one, C01(xi0), is the result of merging the cells of ‹1 whose nuclei lie in C0(xi0). An aggregate tessellation ‹0n consists of the aggregate cells of level n, C0n(xi0), defined recursively by merging those cells of ‹n whose nuclei lie in Cnm1(xi0). We find an expression for the probability for a point to belong to atypical aggregate cell, and obtain bounds for the rate of itsexpansion. We give necessary conditions for the limittessellation to exist as nMX and provide upperbounds for the Hausdorff dimension of its fractal boundary and forthe spherical contact distribution function in the case ofPoisson-Voronoi tessellations {‹n}.

    KW - probability

    KW - voronoï tessellation

    KW - pProbability distribution

    KW - telecommunications

    KW - poisson process

    KW - fractal

    U2 - 10.1007/PL00008802

    DO - 10.1007/PL00008802

    M3 - Article

    VL - 121

    SP - 198

    EP - 218

    JO - Probability Theory and Related Fields

    T2 - Probability Theory and Related Fields

    JF - Probability Theory and Related Fields

    SN - 0178-8051

    IS - 2

    ER -