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Random fields of segments and random mosaics on a plane
| Content Provider | Semantic Scholar |
|---|---|
| Author | Ambartsumian, R. V. |
| Copyright Year | 1972 |
| Abstract | The position ofan undirected segment of straight line of length r in a Euclidean plane is determined by the triple coordinate X = (x, y, (p), where x and y are the cartesian coordinates of the center of the segment and p is the angle made by the segment with the zero direction. Let X denote the phase space of segment coordinates, that is, the layer in the three dimensional Euclidean space defined by the inequalities -oe < x < oo, -o < y < cc, 0 < (o < 7r. Let X be a subset of the phase space X and let z be a positive real function defined for X E A. Assign a length c(X) to the segment which occupies the position X. This defines a certain set J of segments in the plane. We shall write J = [i; z(X)]. Call I the set of all those J for which the number of segments which intersects every bounded subset of the plane is finite. Moreover, ifJE I, then, by definition, any two segments of J either do not intersect or they intersect at a single point. Take a Borel set B in the phase space and a Borel set T c (0, x). Each such pair (B, T) defines a subset of I, namely, the set of those J = [E; c(X)] such that X rn B contains exactly one point and such that X0 e E rX B implies -r(X0) e T. Also, for each B, consider the subset of I formed by those J such that Xt n B = 0. The sets just introduced will be called cylindrical subsets of I. Let M denote the minimal a-algebra generated by the cylindrical subsets of I and let (Q. a, ,u) be a probability space. DEFINITION 1. A (X. a?) measurable map co ~ J(co)Q2 of Qi into I is called a random field of segments (r.f.s.) in the plane. If an r.f.s. J(co) is given, then a probability measure P will be induced in M. which we shall call the distribution of the r.f.s. J(co). The group of all Euclidean motions of a plane induces a group of transformations of A into itself (the group of motions of X). An r.f.s. is called homogeneous and isotropic (h.i.r.f.s.), if its distribution is invariant with respect to the group of motions of A. Only homogeneous and isotropic random fields of segments are examined herein. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.dtic.mil/dtic/tr/fulltext/u2/1025462.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
