A note on invariant sets of iterated function systems

We prove that the family of all invariant sets of iterated systems of contractions R ^N → R ^N is a nowhere dense F _σ type subset in the space of the nonempty compact subsets of R ^N equipped with the Hausdorff metric.     

A note on multistep methods and attracting sets of dynamical systems

Summary We consider a dynamical system described by an autonomous ODE with an asymptotically stable attractor, a compact set of orbitrary shape, for which the stability can be characterized by a Lyapunov function. Using recent results of Eirola and Nevanlinna [1], we establish a uniform estimate for the change in value of this Lyapunov function on discrete trajectories of a consistent, strictly stable multistep method approximating the dynamical system. This estimate can then be used to determine nearby attracting sets and attractors for the discretized system as done in Kloeden and Lorenz [3, 4] for 1-step methods.This work was supported by the U.S. Department of Energy Contract DE-A503-76 ER72012     

A note on planar difference sets

It is proven that if G is an abelian regular automorphism group of a projective plane of order n and if p is a prime dividing n exactly once, then a certain identity holds in the group algebra $\text{F}$_p|G|. As a corollary we obtain that n = p if p = 2 or 3.     

A note on maximized likelihood sets

The purpose of this note is to show that the posterior measure under a partial prior information, which is constructed on the maximized likelihood function,is compatible with the Bayesian properties of the likelihood sets.     

A note on DOL length sets

Length sets of polynomially bounded DOL systems are considered by analyzing sets K ∩ {n : n ? q} where K is a DOL length set. It turns out that DOL and PDOL systems behave in different ways with respect to these sets. Especially, it follows that all DOL length sets— contrary to PDOL length sets—are not generated unambiguously, i.e. without repetitions, by DOL systems. Moreover it is proved that the number of axioms used determines an infinite hierarchy of length set families both in DOL and PDOL case.     

A note on maximal progression-free sets

Erd?s et al [Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Math. 200 (1999) 119-135.] asked whether there exists a maximal set of positive integers containing no three-term arithmetic progression and such that the difference of its adjacent elements approaches infinity. This note answers the question affirmatively by presenting such a set in which the difference of adjacent elements is strictly increasing. The construction generalizes to arithmetic progressions of any finite length.     

A note on orthogonal sets of hypercubes

Mutually orthogonal sets of hypercubes are higher dimensional generalizations of mutually orthogonal sets of Latin squares. For Latin squares, it is well known that the Cayley table of a group of order n is a Latin square, which has no orthogonal mate if n is congruent to 2 modulo 4. We will prove an analogous result for hypercubes. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 231–233, 1997     

A note on diameters of point sets

Relationships between the diameter of a set of n points in the plane at mutual distance at least one, the diameter of an equilateral n-gon and the radius of a circle including n unit disks are explored. Upper bounds on the minimal diameter of a point set at mutual distance at least one are presented for up to 30 points.     

A note on small sets of reals

We construct an example of a combinatorially large measure-zero set.