*Pierre Calka*

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0005
- Subject:
- Mathematics, Geometry / Topology

Random tessellations and cellular structures occur in many domains of application, such as astrophysics, ecology, telecommunications, biochemistry and naturally cellular biology (see Stoyan, Kendall ...
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Random tessellations and cellular structures occur in many domains of application, such as astrophysics, ecology, telecommunications, biochemistry and naturally cellular biology (see Stoyan, Kendall and Mecke 1987 or Okabe, Boots, Sugihara and Chiu 2000 for complete surveys). The theoretical study of these objects was initiated in the second half of the twentieth century by D. G. Kendall, J. L. Meijering, E. N. Gilbert and R. E. Miles, notably. Two isotropic and stationary models have emerged as the most basic and useful: the Poisson hyperplane tessellation and the Poisson–Voronoi tessellation. Since then, a large majority of questions raised about random tessellations have concerned statistics of the population of cells (‘how many cells are triangles in the plane?’, ‘how many cells have a volume greater than one?’) or properties of a specific cell (typically the one containing the origin). Two types of results are presented below: exact distributional calculations and asymptotic estimations. In the first part, we describe the two basic constructions of random tessellations (i.e. by throwing random hyperplanes or by constructing Voronoi cells around random nuclei) and we introduce the fundamental notion of typical cell of a stationary tessellation. The second part is devoted to the presentation of exact distributional results on basic geometrical characteristics (number of hyperfaces, typical k‐face, etc.). The following part concerns asymptotic properties of the cells. It concentrates in particular on the well‐known D. G. Kendall conjecture which states that large planar cells in a Poisson line tessellation are close to the circular shape. In the last part, we present some recent models of iterated tessellations which appear naturally in applied fields (study of crack structures, telecommunications). Intentionally, this chapter does not contain an exhaustive presentation of all the models of random tessellations existing in the literature (in particular, dynamical constructions such as Johnson‐Mehl tessellations will be omitted). The aim of the text below is to provide a selective view of recent selected methods and results on a few specific models.Less

Random tessellations and cellular structures occur in many domains of application, such as astrophysics, ecology, telecommunications, biochemistry and naturally cellular biology (see Stoyan, Kendall and Mecke 1987 or Okabe, Boots, Sugihara and Chiu 2000 for complete surveys). The theoretical study of these objects was initiated in the second half of the twentieth century by D. G. Kendall, J. L. Meijering, E. N. Gilbert and R. E. Miles, notably. Two isotropic and stationary models have emerged as the most basic and useful: the Poisson hyperplane tessellation and the Poisson–Voronoi tessellation. Since then, a large majority of questions raised about random tessellations have concerned statistics of the population of cells (‘how many cells are triangles in the plane?’, ‘how many cells have a volume greater than one?’) or properties of a specific cell (typically the one containing the origin). Two types of results are presented below: exact distributional calculations and asymptotic estimations. In the first part, we describe the two basic constructions of random tessellations (i.e. by throwing random hyperplanes or by constructing Voronoi cells around random nuclei) and we introduce the fundamental notion of typical cell of a stationary tessellation. The second part is devoted to the presentation of exact distributional results on basic geometrical characteristics (number of hyperfaces, typical k‐face, etc.). The following part concerns asymptotic properties of the cells. It concentrates in particular on the well‐known D. G. Kendall conjecture which states that large planar cells in a Poisson line tessellation are close to the circular shape. In the last part, we present some recent models of iterated tessellations which appear naturally in applied fields (study of crack structures, telecommunications). Intentionally, this chapter does not contain an exhaustive presentation of all the models of random tessellations existing in the literature (in particular, dynamical constructions such as Johnson‐Mehl tessellations will be omitted). The aim of the text below is to provide a selective view of recent selected methods and results on a few specific models.

*Zoltan Szallasi, Jorg Stelling, and Vipul Periwal (eds)*

- Published in print:
- 2006
- Published Online:
- August 2013
- ISBN:
- 9780262195485
- eISBN:
- 9780262257060
- Item type:
- book

- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262195485.001.0001
- Subject:
- Mathematics, Mathematical Biology

Research in systems biology requires the collaboration of researchers from diverse backgrounds, including biology, computer science, mathematics, statistics, physics, and biochemistry. These ...
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Research in systems biology requires the collaboration of researchers from diverse backgrounds, including biology, computer science, mathematics, statistics, physics, and biochemistry. These collaborations, necessary because of the enormous breadth of background needed for research in this field, can be hindered by differing understandings of the limitations and applicability of techniques and concerns from different disciplines. The emerging area of systems level modeling in cellular biology has lacked a critical and thorough overview. The book provides the necessary critical comparison of concepts and approaches, with an emphasis on their possible applications. It presents key concepts and their theoretical background, including the concepts of robustness and modularity and their exploitation to study biological systems; the best-known modeling approaches, and their advantages and disadvantages; lessons from the application of mathematical models to the study of cellular biology; and available modeling tools and datasets, along with their computational limitations.Less

Research in systems biology requires the collaboration of researchers from diverse backgrounds, including biology, computer science, mathematics, statistics, physics, and biochemistry. These collaborations, necessary because of the enormous breadth of background needed for research in this field, can be hindered by differing understandings of the limitations and applicability of techniques and concerns from different disciplines. The emerging area of systems level modeling in cellular biology has lacked a critical and thorough overview. The book provides the necessary critical comparison of concepts and approaches, with an emphasis on their possible applications. It presents key concepts and their theoretical background, including the concepts of robustness and modularity and their exploitation to study biological systems; the best-known modeling approaches, and their advantages and disadvantages; lessons from the application of mathematical models to the study of cellular biology; and available modeling tools and datasets, along with their computational limitations.