No Arabic abstract
Non-periodic systems have become more important in recent years, both theoretically and practically. Their description via Delone sets requires the extension of many standard concepts of crystallography. Here, we summarise some useful notions of symmetry and aperiodicity, with special focus on the concept of the hull of a Delone set. Our aim is to contribute to a more systematic and consistent use of the different notions.
We study densities of functionals over uniformly bounded triangulations of a Delaunay set of vertices, and prove that the minimum is attained for the Delaunay triangulation if this is the case for finite sets.
In this paper we consider an extremal problem in geometry. Let $lambda$ be a real number and $A$, $B$ and $C$ be arbitrary points on the unit circle $Gamma$. We give full characterization of the extremal behavior of the function $f(M,lambda)=MA^lambda+MB^lambda+MC^lambda$, where $M$ is a point on the unit circle as well. We also investigate the extremal behavior of $sum_{i=1}^nXP_i$, where $P_i, i=1,...,n$ are the vertices of a regular $n$-gon and $X$ is a point on $Gamma$, concentric to the circle circumscribed around $P_1...P_n$. We use elementary analytic and purely geometric methods in the proof.
A new class of plurisubharmonic functions on the octonionic plane O^2= R^{16} is introduced. An octonionic version of theorems of A.D. Aleksandrov and Chern- Levine-Nirenberg, and Blocki are proved. These results are used to construct new examples of continuous translation invariant valuations on convex subsets of O^2=R^{16}. In particular a new example of Spin(9)-invariant valuation on R^{16} is given.
The notion of a valuation on convex bodies is very classical. The notion of a valuation on a class of functions was recently introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translations invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite dimensional kernel. The proof uses the authors irreducibility theorem and few properties of the real Monge-Ampere operators due to A.D. Alexandrov and Z. Blocki. Fur- thermore we show how to use complex, quaternionic, and octonionic Monge-Ampere operators to construct more examples of continuous valuations on convex functions in an analogous way.
The set of points in a metric space is called an $s$-distance set if pairwise distances between these points admit only $s$ distinct values. Two-distance spherical sets with the set of scalar products ${alpha, -alpha}$, $alphain[0,1)$, are called equiangular. The problem of determining the maximum size of $s$-distance sets in various spaces has a long history in mathematics. We suggest a new method of bounding the size of an $s$-distance set in compact two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in $mathbb{R}^n$, $ngeq 7$, is $frac{n(n+1)}2$ with possible exceptions for some $n=(2k+1)^2-3$, $k in mathbb{N}$. We also prove the universal upper bound $sim frac 2 3 n a^2$ for equiangular sets with $alpha=frac 1 a$ and, employing this bound, prove a new upper bound on the size of equiangular sets in all dimensions. Finally, we classify all equiangular sets reaching this new bound.