This paper investigates, how two fundamental measurement operations in Euclidean space, namely the construction of a straight line of given orientation angle and the measurement of distances along this line, are mapped to the non-Euclidean log-polar image plane. It is shown, how these two operations can be efficiently implemented for discrete log-polar images. The measurement operations are use...

Given a seven-element set X = {1, 2, 3, 4, 5, 6, 7}, there are 30 ways to define a Fano plane on it. Let us call a line of such a Fano plane—that is to say an unordered triple from X—ordinary or defective, according to whether the sum of two smaller integers from the triple is or is not equal to the remaining one, respectively. A point of the labeled Fano plane is said to be of the order s, 0 ≤...

We consider a class of two-stage stochastic integer programs with binary variables in the first stage and general integer variables in the second stage. We develop decomposition algorithms akin to the L-shaped or Benders’ methods by utilizing Gomory cuts to obtain iteratively tighter approximations of the second-stage integer programs. We show that the proposed methodology is flexible in that i...

For any connected abstract graph G, the tree graph T (G) is the graph that has one vertex for each spanning tree of G and an edge joining trees R and S whenever R is obtained from S by a single edge exchange. R. L. Cummings proved in [C] that T (G) is hamiltonian; see also [S] for a short proof. A geometric variation that has been studied is the following: For a set P of points in general posit...

The average number of (1) antichains, (2) maximal antichains, (3) chains, (4) infima closed sets, (5) connected sets, (6) independent sets, (7) maximal independent sets, (8) brooms, (9) matchings, (10) maximal matchings, (11) linear extensions, and (12) drawings in (of) a rooted plane tree on n vertices is investigated. Using generating functions we determine the asymptotics and give some expli...

Given a set of n disjoint line segments in the plane, we show that it is always possible to form a tree with the endpoints of the segments such that each line segment is an edge of the tree, the tree has no crossing edges, and the maximum vertex degree of the tree is 3. Furthermore, there exist conngurations of line segments where any such tree requires degree 3. We provide an O(n log n) time a...

Let T be a tree with at least four vertices, none of which has degree 2, embedded in the plane. A Halin graph is a plane graph constructed by connecting the leaves of T into a cycle. Thus the cycle C forms the outer face of the Halin graph, with the tree inside it. Let G be a Halin graph with order n. Denote by [Formula: see text] the Laplacian spectral radius of G. This paper determines all th...