An intersecting system of type (∃,∀, k, n) is a collection F = {F1, . . . ,Fm} pf pairwise disjoint families of k–subsets of an n–element set satisfying the following condition. For every ordered pair Fi and Fj of distinct members of F there exists an A ∈ Fi that intersects every B ∈ Fj . Let In(∃,∀, k) denote the maximum possible cardinality of an intersecting system of type (∃,∀, n). Ahlswede...

Second?degree dominance has become a widely accepted criterion for ordering distribution functions according to social welfare. However, it provides only partial ordering, and may fail rank distributions that intersect. To intersecting functions, we propose general approach based on rank?dependent theory. This avoids making arbitrary restrictions or parametric assumptions about welfare allows r...

Non-self-intersection is both a topological and a geometric property. Recent theorems show how non-self-intersecting Bézier curves also have non-self-intersecting control polygons, after sufficiently many uniform subdivisions. As a partial generalization, a sufficient condition is given within R for a non-self-intersecting, C cubic Bézier curve to be ambient isotopic to its control polygon form...

We present a uniform approach to problems involving lines in 3-space. This approach is based on mapping lines in R 3 into points and hyperplanes in 5-dimensional projective space (Pl ucker space). We obtain new results on the following problems: 1. Preprocess n triangles so as to eeciently answer the query: \Given a ray, which is the rst triangle hit?" (Ray-shooting problem). We discuss the ray...

We show that the Erdős–Ko–Rado inequality for t-intersecting families of subsets can be easily extended to an inequality for cross t-intersecting families by using the eigenvalue method. The same applies to the case of t-intersecting families of subspaces. The eigenvalue method is one of the proof techniques to get Erdős–Ko–Rado type inequalities for t-intersecting families, for example, a proo...