Size distributions in random triangles

Triangles in random graphs

Triangles in Random Graphs

OF THE DISSERTATION Triangles in Random Graphs by Robert DeMarco Dissertation Director: Jeff Kahn We prove four separate results. These results will appear or have appeared in various papers (see [10], [11], [12], [13]). For a gentler introduction to these results, the reader is directed to the first chapter of this thesis. Let G = Gn,p. With ξk = ξ n,p k the number of copies of Kk in G, p ≥ n−...

Random Triangles II

Let S denote the unit sphere in Euclidean 3-space. A spherical triangle T is a region enclosed by three great circles on S; a great circle is a circle whose center is at the origin. The sides of T are arcs of great circles and have length a, b, c. Each of these is ≤ π. The angle α opposite side a is the dihedral angle between the two planes passing through the origin and determined by arcs b, c...

Random Triangles III

Let Ω be a compact convex set in Euclidean n-space with nonempty interior. Random triangles are defined here by selecting three independent uniformly distributed points in Ω to be vertices. Generating such points for (n,Ω) = (2,unit square) or (n,Ω) = (3,unit cube) is straightforward. For (n,Ω) = (2,unit disk) or (n,Ω) = (3,unit ball), we use the following result [1]. Let X1, X2, X3, Y1, Y2, Y3...

Random Triangles V

We defined the d-dimensional Poisson(λ) point process in an earlier essay [1] and exhibited moment formulas for various cellular parameters of the corresponding Voronoi tessellation. Many of these formulas are analytically intractible; numerical integration is sometimes necessary. For example, when d = 2, the probability that a typical cell (a convex polygon) is a triangle is 0.01124001.... Mon...