A geometric permutation induced by a transversal line of a finite family F of disjoint convex sets in Rd is the order in which the transversal meets the members of the family. We prove that for each natural k, each family of k permutations is realizable (as a family of geometric permutations of some F ) in Rd for d ≥ 2k − 1, but there is a family of k permutations which is non-realizable in Rd ...

We prove that S4 is complete with respect to Boolean combinations of countable unions of convex subsets of the real line, thus strengthening a 1944 result of McKinsey and Tarski (Ann. of Math. (2) 45 (1944) 141). We also prove that the same result holds for the bimodal system S4 + S5 + C, which is a strengthening of a 1999 result of Shehtman (J. Appl. Non-Classical Logics 9 (1999) 369). © 2004 ...

We investigate lattices that can be represented as sublattices of the lattice of all convex subsets of a linearly ordered set (X,≤) and as lattices of convex subsets of (X,≤).

For an algebra [Formula: see text] belonging to a quasivariety [Formula: see text], the quotient [Formula: see text] need not belong to [Formula: see text] for every [Formula: see text]. The natural question arises for which [Formula: see text]. We consider algebras [Formula: see text] of type (2, 0) where a partial order relation is determined by the operations [Formula: see text] and 1. Withi...

We prove a Desch-Schappacher type perturbation theorem for strongly continuous and locally equicontinuous one-parameter semigroups which are defined on a sequentially complete locally convex space.

We give two alternative proofs leading to di erent generalizations of the following theorem of [1]. Given n convex sets in the plane, such that the boundaries of each pair of sets cross at most twice, then the boundary of their union consists of at most 6n 12 arcs. (An arc is a connected piece of the boundary of one of the sets.) In the generalizations we allow pairs of boundaries to cross more...

To provide a Kolmogorov-type condition for characterizing a best approximation in a continuous complex-valued function space, it is usually assumed that the family of closed convex sets in the complex plane used to restrict the range satisfies a strong interior-point condition, and this excludes the interesting case when some t is a line-segment or a singleton. The main aim of the present paper...

We consider the problem of aggregating menus. Each agent submits a menu of options, which the social rule aggregates into a single menu. First, we study a general environment where menus are arbitrary subsets, and characterize a union rule which takes the union of individual menus. Second, we study a probabilistic environment where menus are closed convex subsets of lotteries over options, and ...

Kakeya sets in the affine plane AG(2, q) are point sets that are the union of lines, one through every point on the line at infinity. The finite field Kakeya problem asks for the size of the smallest Kakeya sets and the classification of these Kakeya sets. In this article we present a new example of a small Kakeya set and we give the classification of the smallest Kakeya sets up to weight q(q+2...