We consider the problem of evaluating an expression over sets. The sets are preprocessed and are therefore sorted, and the operators can be any of union, intersection, difference, complement, and symmetric difference (exclusive union). Given the expression as a formula and the sizes of the input sets, we are interested in the worst-case complexity of evaluation (in terms of the size of the sets...

It is proved that certain incidence relations of hyperplanes and closed convex sets in a rf-polytope can be preserved while replacing these sets by suitable polytopal subsets. The purpose of this paper is to prove Theorem 1. IfP is a d-polytope in Ed and Cx, * • •, Cj are closed convex subsets of P, such that every hyperplane that meets P meets (J¿=i Q> tnen there exist poly topes Dx, • • ■ , D...

Karshon constructed the first counterexample to the log-concavity conjecture for the Duistermaat-Heckman measure: a Hamiltonian six manifold whose fixed points set is the disjoint union of two copies of T. In this article, for any closed symplectic four manifold N with b > 1, we show that there is a Hamiltonian circle manifold M fibred over N such that its DuistermaatHeckman function is not log...

We prove that for any partition of the plane into a closed set C and an open set O and for any configuration T of three points, there is a translated and rotated copy of T contained in C or in O. Apart from that, we consider partitions of the plane into two sets whose common boundary is a union of piecewise linear curves. We show that for any such partition and any configuration T which is a ve...

We encode arbitrary finite impartial combinatorial games in terms of lattice points in rational convex polyhedra. Encodings provided by these lattice games can be made particularly efficient for octal games, which we generalize to squarefree games. These additionally encompass all heap games in a natural setting, in which the Sprague–Grundy theorem for normal play manifests itself geometrically...

We call a family G ⊂ P[n] a k-generator of P[n] if every x ⊂ [n] can be expressed as a union of at most k disjoint sets in G. Frein, Lévêque and Seb˝ o [1] conjectured that for any n ≥ k, such a family must be at least as large as the k-generator obtained by taking a partition of [n] into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We generalize...

We call a family G ⊂ P[n] a k-generator of P[n] if every x ⊂ [n] can be expressed as a union of at most k disjoint sets in G. Frein, Lévêque and Sebő [1] conjectured that for any n ≥ k, such a family must be at least as large as the k-generator obtained by taking a partition of [n] into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We generalize a...

Gap-definability and the gap closure operator were defined by S. Fenner, L. Fortnow and S. Kurth (J. Comput. System Sci. 48, 116 148 (1994)). Few complexity classes were known at that time to be gapdefinable. In this paper, we give simple characterizations of both gapdefinability and the gap-closure operator, and we show that many complexity classes are gap-definable, including P, P, PSPACE, EX...

We s(ate the following conjecture and prove it for the case where q is a proper prime power: Let A be a nonsingular n by n matrix over the finite fieM GF~, q~-4, then there exists a vector x in (GFa) ~ such that both x and ~4x have no zero component. In this note we consider-the following conjecture: Conjecture 1. Let A be a nonsingular n by n matrix over the finite field GFa, q~_4, then there ...