The number of elementary connected regions arising from polygon overlay of two or more map layers is an important value to have in planning for data stoiage and in making processing time estimates for overlay applications. That number may be computed di rectly from the line graphs of the two (or more) layers and from the intersection graph(s) of those line graphs. A formula for that computation...

Denote by x a random innnite path in the graph of Pascal's triangle (left and right turns are selected independently with xed probabilities) and by d n (x) the binomial coeecient at the n'th level along the path x. Then for a dense G set of in the unit interval, fd n (x)g is almost surely dense but not uniformly distributed modulo 1.

Denote by x a random infinite path in the graph of Pascal’s triangle (left and right turns are selected independently with fixed probabilities) and by dn(x) the binomial coefficient at the n’th level along the path x. Then for a dense Gδ set of θ in the unit interval, {dn(x)θ} is almost surely dense but not uniformly distributed modulo 1.

(6) Let G be a column Y-constant line Y-increasing matrix over E2 T and i1, i2, j1, j2 be natural numbers. Suppose 〈i1, j1〉 ∈ the indices of G and 〈i2, j2〉 ∈ the indices of G. If (G◦ (i1, j1))2 = (G◦ (i2, j2))2, then j1 = j2. (7) Let G be a line X-constant column X-increasing matrix over E2 T and i1, i2, j1, j2 be natural numbers. Suppose 〈i1, j1〉 ∈ the indices of G and 〈i2, j2〉 ∈ the indices o...

Let X → B be a proper flat morphism between smooth quasiprojective varieties of relative dimension n, and L → X a line bundle which is ample on the fibers. We establish formulas for the first two terms in the Knudsen-Mumford expansion for det(π∗L ) in terms of Deligne pairings of L and the relative canonical bundle K. This generalizes the theorem of Deligne [1], which holds for families of rela...