Aperiodic Linearly Repetitive Delone Sets Are Densely Repetitive

We show that aperiodic linearly repetitive Delone sets are densely repetitive. This confirms a conjecture of Lagarias and Pleasants.     

Asymptotics of best-packing on rectifiable sets

We investigate the asymptotic behavior, as grows, of the largest minimal pairwise distance of points restricted to an arbitrary compact rectifiable set embedded in Euclidean space, and we find the limit distribution of such optimal configurations. For this purpose, we compare best-packing configurations with minimal Riesz -energy configurations and determine the -th root asymptotic behavior (as of the minimal energy constants.

We show that the upper and the lower dimension of a set defined through the Riesz energy or best-packing coincides with the upper and lower Minkowski dimension, respectively.

For certain sets in of integer Hausdorff dimension, we show that the limiting behavior of the best-packing distance as well as the minimal -energy for large is different for different subsequences of the cardinalities of the configurations.

     

Random processes of Hausdorff rectifiable closet sets

Problems of intersections of processes of HAUSDORFF rectifiable closed sets in Rⁿ are studied. The first part deals with integral geometric fundamentals and questions of measurability. In the second part the stochastic approach is given.     

Planar rectifiable curves are determined by their projections

In 1954 Steinhaus raised the question of whether a rectifiable curve is characterized by its projections. A projection onto a lineG at the pointp counts the number of points in the set which lie on the line which is perpendicular toG and passes throughp. We prove this is so, and give a method to reconstruct a closed connected rectifiable set from its projections.     

Linearly ordered compact sets and co-Namioka spaces

We prove that, for an arbitrary Baire space X, a linearly ordered compact set Y, and a separately continuous mapping ƒ: X × Y → R, there exists a G _δ-set A ⊆ X dense in X and such that the function ƒ is jointly continuous at every point of the set A × Y, i.e., any linearly ordered compact set is a co-Namioka space. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 1001–1004, July, 2007.     

Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets

Given a closed -rectifiable set embedded in Euclidean space, we investigate minimal weighted Riesz energy points on ; that is, points constrained to and interacting via the weighted power law potential , where is a fixed parameter and is an admissible weight. (In the unweighted case () such points for fixed tend to the solution of the best-packing problem on as the parameter .) Our main results concern the asymptotic behavior as of the minimal energies as well as the corresponding equilibrium configurations. Given a distribution with respect to -dimensional Hausdorff measure on , our results provide a method for generating -point configurations on that are ``well-separated' and have asymptotic distribution as .

     

Periodic Delone Tilings

Given a periodic point set in 3-dimensional Euclidean space, an algorithm is described for computing the corresponding Delone tiling (and its Delaney symbol). Examples of applications in tiling theory and crystallography are discussed.     

An intersection formula for Hausdorff rectifiable sets using mixed volumes of the associated zonoids

A general Poincaré formula for a finite number of Hausdorff rectifiable sets in ^d is derived. The intersection volume intensity is expressed by means of the mixed volume of associated zonoids.     

Substitution Delone Sets

   Abstract. Substitution Delone set families are families of Delone sets X =(X ₁ , . . ., X _n ) which satisfy the inflation functional equation

Remainders of rectifiable spaces

We prove a Dichotomy Theorem: for any Hausdorff compactification bG of an arbitrary rectifiable space G, the remainder bG?G is either pseudocompact or Lindelöf. This theorem generalizes a similar theorem on topological groups obtained earlier in A.V. Arhangel'skii (2008) [6], but the proof for rectifiable spaces is considerably more involved than in the case of topological groups. It follows that if a remainder of a rectifiable space G is paracompact or Dieudonné complete, then the remainder is Lindelöf and that G is a p-space. We also present an example showing that the Dichotomy Theorem does not extend to all paratopological groups. Some other results are obtained, and some open questions are formulated.     

Global curvature for rectifiable loops

We study in detail the notion of global curvature defined on rectifiable closed curves, a concept which has been successfully applied in existence and regularity investigations regarding elastic self-contact problems in nonlinear elasticity. A bound on this purely geometric quantity serves as an excluded volume constraint to prevent selfintersections of slender elastic bodies modeled as elastic rods. Moreover, a finite global curvature characterizes simple closed curv es, whose arc length parameterizations possess a Lipschitz continuous tangent field. The investigation of local and non-local properties of global curvature motivates, in particular, an extended definition of local curvature at any point of a rectifiable loop. Finally we show how a bound on global curvature can be used to define and control topological constraints such as a given knot type for closed loops or a prescribed linking number for closed framed curves, suitable to describe, e.g., supercoiling phen omena of biomolecules. Received: 10 December 2001 / Published online: 8 November 2002 Both authors were supported by the Max-Planck Institute for Mathematics in the Sciences in Leipzig and the Sonderforschungsbereich 256 at the University of Bonn. The second author enjoyed the hospitality of the Forschungsinstitut für Mathematik at the ETH Zürich during the spring term 2001.     

BV Functions over rectifiable currents

Summary Via a convenient integration by parts formula and a suitable definition of graph, we introduce two notions of function of bounded variation over an integer rectifiable current. Comparing them, one finds out that the first notions is weaker than the second one. For the strong BV functions, i.e. the ones corresponding to graphs, we are able to obtain some interesting results. In particular we prove their approximate differentiability. Finally, the possible limits of graphs with respect to suitable convergences are studied.     

Some remarks on Legendrian rectifiable currents

A rectifiable current of dimension n−1 in the sphere bundle Sℝⁿ≃ℝⁿ×S ^{n −1} for euclidean space is Legendrian if it annihilates the contact 1-form α (i.e. T(α∧φ)=0 for all forms φ of degree n−2). Such a current may be naturally associated to any convex set or to any singular real analytic variety, and induces the curvature measures of such a set. We prove that the projection to ℝⁿ of a carrier of a general such T is C ²-rectifiable in the sense of Anzellotti–Serapioni. We deduce that the boundary of a set with positive reach, as well as its singular skeleta, are C ²-rectifiable. In case ∂T= 0 we prove also that the curvature measures associated to T satisfy the analogues of the classical variational formulas for curvature integrals. It follows that such formulas are valid for the curvature measures of subsets of space forms. Received: 3 December 1997/ Revised version: 25 May 1998     

Intersection homology theory via rectifiable currents

Here is given a rectifiable currents version of intersection homology theory on stratified subanalytic pseudomanifolds. This new version enables one to study some variational problems on stratified subanalytic pseudomanifolds. We first achieve an isomorphism between this rectifiable currents version and the version using subanalytic chains. Then we define a suitably modified mass on the complex of rectifiable currents to ensure that each sequence of subanalytic chains with bounded modified mass has a convegent subsequence and the limit rectifiable current still satisfies the perversity condition of the approximating chains. The associated mass minimizers turn out to be almost minimal currents and this fact leads to some regularity results.Received: 21 April 2001, Accepted: 15 April 2003, Published online: 1 July 2003     

Not all Julia sets are quasi-self-similar

We show that there exist rational functions, whose Julia set fails to be quasi-self-similar.