We present linear-time algorithms to construct tree-like Voronoi diagrams with disconnected regions after the sequence of their faces along an enclosing boundary (or at infinity) is known. We focus on the farthest-segment Voronoi diagram, however, our techniques are also applicable to constructing the order-(k+1) subdivision within an order-k Voronoi region of segments and updating a nearest-ne...

This conjecture can be traced to Chowla ([1], p. 96). It is closely related to the Bunyakovsky– Schinzel conjecture on primes represented by irreducible polynomials. The one-variable analogue of (1.1) is classical for deg f = 1 and quite hopeless for deg f > 1. We know (1.1) itself when deg f ≤ 2. (The main ideas of the proof go back to de la Vallée-Poussin ([3], [4]); see [11], §3.3, for an ex...

In 1960 Sierpiński proved that there are infinitely many odd positive integers k such that k · 2 + 1 is composite for all positive integers n. A polynomial variation of Sierpiński’s result has been investigated by several people. More specifically, the question has been asked, for which integers d does there exist a polynomial f(x) ∈ Z[x] with f(1) 6= −d such that f(x) ·x + d is reducible over ...

We present linear-time algorithms to construct tree-structured Voronoi diagrams, after the sequence of their regions at infinity or along a given boundary is known. We focus on Voronoi diagrams of line segments, including the farthest-segment Voronoi diagram, the order-(k+1) subdivision within a given order-k Voronoi region, and deleting a segment from a nearest-neighbor diagram. Although tree-...

C. Corrales-Rodrigáñez and R. Schoof answered the question and proved its analogue for number fields and for elliptic curves in [C-RS]. A. Schinzel proved the support problem for the pair of sets of positive integers in [S]. G. Banaszak, W. Gajda and P. Krasoń examined the support problem for abelian varieties for which the images of the l-adic representation is well controled and for K-theory ...

We investigate a kinetic version of point-set embeddability. Given a plane graph G(V,E) where |V | = n, and a set P of n moving points where the trajectory of each point is an algebraic function of constant maximum degree s, we maintain a point-set embedding of G on P with at most three bends per edge during the motion. This requires reassigning the mapping of vertices to points from time to ti...