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Boolean Model

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Dietrich Stoyan – 1st expert on this subject based on the ideXlab platform

  • stochastic geometry and its applications
    , 2009
    Co-Authors: Sung Nok Chiu, Dietrich Stoyan, Wilfrid S Kendall, Joseph Mecke

    Abstract:

    Mathematical Foundation. Point Processes I–The Poisson Point Process. Random Closed Sets I–The Boolean Model. Point Processes II–General Theory. Point Processes III–Construction of Models. Random Closed Sets II–The General Case. Random Measures. Random Processes of Geometrical Objects. Fibre and Surface Processes. Random Tessellations. Stereology. References. Indexes.

  • On Some Qualitative Properties of the Boolean Model of Stochastic Geometry
    Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik, 2006
    Co-Authors: Dietrich Stoyan

    Abstract:

    For the Boolean Model, one of the basic Models of stochastic geometry and stereology, monotonicity and continuity properties are proved. In particular, for Boolean Models on the line the relation between primary and number granulometry is discussed, and monotonicity properties of the covariance function are studied. Finally, the probability of hitting of an arbitrary primary grain by one of the other ones is considered.

    Fur das Boolesche Modell, eines der GrundModelle der stochastischen Geometrie und Stereologie, werden Monotonie- und Stetigkeitseigenschaften bewiesen. Insbesondere werden die Beziehung zwischen der primary und number granulometry fur Boolesche Modelle auf der Geraden diskutiert und Monotonieeigenschaften der Kovarianzfunktion studiert. Schlieslich wird die Wahrscheinlichkeit dafur betrachtet, das ein beliebiges primary grain von einem der anderen beruhrt wird.

  • the Boolean Model from matheron till today
    , 2005
    Co-Authors: Dietrich Stoyan, Klaus Mecke

    Abstract:

    Until the 1970s random sets were only a marginal or exotic part of probability theory. This situation has changed completely since the publication of the fundamental and seminal book by Matheron [43]. This book has laid the fundamentals of the theory of random closed sets, provided the suitable measure-theoretic machinery and offered the fundamental theorems. It also spresented an excellent introduction to the theory of the Boolean Model.

Ilya Molchanov – 2nd expert on this subject based on the ideXlab platform

  • Set-Valued Estimators for Mean Bodies Related to Boolean Models
    Statistics, 2007
    Co-Authors: Ilya Molchanov

    Abstract:

    The stationary Boolean Model in the Euclidean space with a convex typical grain is considered. Relations between the mean (or the Aumann expectation) of the difference body of the grain and other characteristics of the Boolean Model are discussed, and set-valued estimators for the mean difference body are suggested.

  • statistics of the Boolean Model for practitioners and mathematicians
    , 1997
    Co-Authors: Ilya Molchanov

    Abstract:

    The Boolean Model. Estimation of Aggregate Parameters. Estimation of Functional Aggregate Parameters. Estimation of Numerical Individual Parameters. Estimation of Set-Valued Individual Parameters. Individual Parameters: Distributions. Other Sampling Schemes. Testing the Boolean Model Assumption. An Example. Concluding Remarks. References. Index.

  • A limit theorem for scaled vacancies of the Boolean Model
    Stochastics and Stochastics Reports, 1996
    Co-Authors: Ilya Molchanov

    Abstract:

    The vacancy of the high-intensity Boolean Model (or mosaic process) is considered. The main result gives a limit distribution for the scaled elementary connected vacant component. It is shown that the limiting set is the volume law polyhedron generated by a (in general anisouopic) Poisson network of hyperplanes driven by the expected surface measure of the typical grain of the Boolean Model. The associated zonoid of the network is equal to the projection body of the so-called Blaschke expectation of the typical grain. In difference to earlier results of P. Hall, few geometrical and no isotropy assumptions are imposed on the grain and the proofs are based on the translative integral geometric formula instead of direct analytical computations

Laleh Kazemzadeh – 3rd expert on this subject based on the ideXlab platform

  • Boolean Model of yeast apoptosis as a tool to study yeast and human apoptotic regulations
    Frontiers in Physiology, 2012
    Co-Authors: Laleh Kazemzadeh, Marija Cvijovic, Dina Petranovic

    Abstract:

    Programmed cell death (PCD) is an essential cellular mechanism that is evolutionary conserved, mediated through various pathways and acts by integrating different stimuli. Many diseases such as neurodegenerative diseases and cancers are found to be caused by, or associated with, regulations in the cell death pathways. Yeast Saccharomyces cerevisiae, is a unicellular eukaryotic organism that shares with human cells components and pathways of the PCD and is therefore used as a Model organism. Boolean Modeling is becoming promising approach to capture qualitative behavior and describe essential properties of such complex networks. Here we present large literature-based and to our knowledge first Boolean Model that combines pathways leading to apoptosis (a type of PCD) in yeast. Analysis of the yeast Model confirmed experimental findings of anti-apoptotic role of Bir1p and pro-apoptotic role of Stm1p and revealed activation of the stress protein kinase Hog proposing the maximal level of activation upon heat stress. In addition we extended the yeast Model and created an in silico humanized yeast in which human pro- and anti-apoptotic regulators Bcl-2 family and Valosin-contain protein (VCP) are included in the Model. We showed that accumulation of Bax in silico humanized yeast shows apoptotic markers and that VCP is essential target of Akt Signaling. The presented Boolean Model provides comprehensive description of yeast apoptosis network behavior. Extended Model of humanized yeast gives new insights of how complex human disease like neurodegeneration can initially be tested.