The Random Division of an Interval and the Random Covering of a Circle

SIAM Review ◽  
1962 ◽  
Vol 4 (3) ◽  
pp. 211-222 ◽  
Author(s):  
Leopold Flatto ◽  
Alan G. Konheim
Keyword(s):  
Random Covering ◽  
Random Division

The random division of faces in a planar graph

1996 ◽  
Vol 28 (2) ◽  
pp. 331-331
Author(s):  
Richard Cowan ◽  
Simone Chen
Keyword(s):  
Planar Graph ◽  
Limiting Distribution ◽  
Face Type ◽  
Connected Planar Graph ◽  
Random Division

Consider a connected planar graph. A bounded face is said to be of type k, or is called a k-face, if the boundary of that face contains k edges. Under various natural rules for randomly dividing bounded faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

The random division of faces in a planar graph

1996 ◽  
Vol 28 (02) ◽  
pp. 331
Author(s):  
Richard Cowan ◽  
Simone Chen
Keyword(s):  
Planar Graph ◽  
Limiting Distribution ◽  
Face Type ◽  
Connected Planar Graph ◽  
Random Division

Consider a connected planar graph. A bounded face is said to be of type k, or is called a k-face, if the boundary of that face contains k edges. Under various natural rules for randomly dividing bounded faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

scholarly journals Multifractal spectra for random self-similar measures via branching processes

2011 ◽  
Vol 43 (01) ◽  
pp. 1-39
Author(s):  
J. D. Biggins ◽  
B. M. Hambly ◽  
O. D. Jones
Keyword(s):  
Random Number ◽  
Branching Processes ◽  
Multifractal Spectrum ◽  
Compact Set ◽  
Similar Measure ◽  
Random Mass ◽  
Random Fractal ◽  
Multifractal Spectra ◽  
Self Similar ◽  
Random Division

Start with a compact setK⊂Rd. This has a random number of daughter sets, each of which is a (rotated and scaled) copy ofKand all of which are insideK. The random mechanism for producing daughter sets is used independently on each of the daughter sets to produce the second generation of sets, and so on, repeatedly. The random fractal setFis the limit, asngoes to ∞, of the union of thenth generation sets. In addition,Khas a (suitable, random) mass which is divided randomly between the daughter sets, and this random division of mass is also repeated independently, indefinitely. This division of mass will correspond to a random self-similar measure onF. The multifractal spectrum of this measure is studied here. Our main contributions are dealing with the geometry of realisations inRdand drawing systematically on known results for general branching processes. In this way we generalise considerably the results of Arbeiter and Patzschke (1996) and Patzschke (1997).

Random division of an interval

1967 ◽  
Vol 21 (3-4) ◽  
pp. 231-244 ◽  
Author(s):  
F. W. Steutel
Keyword(s):  
Random Division

Partial and random covering times in one dimension

2001 ◽  
Vol 63 (6) ◽  
Author(s):  
Marcelo S. Nascimento ◽  
Maurício D. Coutinho-Filho ◽  
Carlos S. O. Yokoi
Keyword(s):  
One Dimension ◽  
Random Covering ◽  
Covering Times

scholarly journals Combinatorial and approximative analyses in a spatially random division process

2013 ◽  
Vol 392 (9) ◽  
pp. 2212-2225 ◽  
Author(s):  
Yukio Hayashi ◽  
Takayuki Komaki ◽  
Yusuke Ide ◽  
Takuya Machida ◽  
Norio Konno
Keyword(s):  
Division Process ◽  
Random Division

scholarly journals Cohesins Determine the Attachment Manner of Kinetochores to Spindle Microtubules at Meiosis I in Fission Yeast

2003 ◽  
Vol 23 (11) ◽  
pp. 3965-3973 ◽  
Author(s):  
Shihori Yokobayashi ◽  
Masayuki Yamamoto ◽  
Yoshinori Watanabe
Keyword(s):  
Fission Yeast ◽  
Chromosome Segregation ◽  
Sister Chromatid ◽  
Specific Requirement ◽  
Spindle Poles ◽  
Chromatid Cohesion ◽  
Spindle Microtubules ◽  
Yeast Meiosis ◽  
Meiosis I ◽  
Random Division

ABSTRACT During mitosis, sister kinetochores attach to microtubules that extend to opposite spindle poles (bipolar attachment) and pull the chromatids apart at anaphase (equational segregation). A multisubunit complex called cohesin, including Rad21/Scc1, plays a crucial role in sister chromatid cohesion and equational segregation at mitosis. Meiosis I differs from mitosis in having a reductional pattern of chromosome segregation, in which sister kinetochores are attached to the same spindle (monopolar attachment). During meiosis, Rad21/Scc1 is largely replaced by its meiotic counterpart, Rec8. If Rec8 is inactivated in fission yeast, meiosis I is shifted from reductional to equational division. However, the reason rec8Δ cells undergo equational rather than random division has not been clarified; therefore, it has been unclear whether equational segregation is due to a loss of cohesin in general or to a loss of a specific requirement for Rec8. We report here that the equational segregation at meiosis I depends on substitutive Rad21, which relocates to the centromeres if Rec8 is absent. Moreover, we demonstrate that even if sufficient amounts of Rad21 are transferred to the centromeres at meiosis I, thereby establishing cohesion at the centromeres, rec8Δ cells never recover monopolar attachment but instead secure bipolar attachment. Thus, Rec8 and Rad21 define monopolar and bipolar attachment, respectively, at meiosis I. We conclude that cohesin is a crucial determinant of the attachment manner of kinetochores to the spindle microtubules at meiosis I in fission yeast.

Analytical computer calculations in problems with random division of an interval

2012 ◽  
Vol 22 (2) ◽  
pp. 354-359 ◽  
Author(s):  
A. L. Reznik ◽  
V. M. Efimov ◽  
A. V. Torgov ◽  
A. A. Solov’ev
Keyword(s):  
Computer Calculations ◽  
Random Division

A fourth note on recent research in geometrical probability

1977 ◽  
Vol 9 (04) ◽  
pp. 824-860 ◽  
Author(s):  
Adrian Baddeley
Keyword(s):  
Integral Geometry ◽  
Random Sets ◽  
Euclidean Spaces ◽  
Line Segments ◽  
Geometrical Probability ◽  
Random Division

Recent research on topics related to geometrical probability is reviewed. The survey includes articles on random points, lines, line-segments and flats in Euclidean spaces, the random division of space, coverage, packing, random sets, stereology and probabilistic aspects of integral geometry.

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