We show that the greedy spanner algorithm constructs a (1+ )-spanner of weight −O(d)w(MST) for a point set in metrics of doubling dimension d, resolving an open problem posed by Gottlieb [11]. Our result generalizes the result by Narasimhan and Smid [15] who showed that a point set in d-dimension Euclidean space has a (1+ )-spanner of weight at most −O(d)w(MST). Our proof only uses the packing ...

In this paper, we study the linear structure of sets A ⊂ Fn2 with doubling constant σ(A) < 2, where σ(A) := |A+A| |A| . In particular, we show that A is contained in a small affine subspace. We also show that A can be covered by at most four shifts of some subspace V with |V | ≤ |A|. Finally, we classify all binary sets with small doubling constant.

We study the properties of generalized solutions to the Monge– Ampère equation detD2u = ν, where the Borel measure ν satisfies a condition, introduced by Jerison, that is weaker than the doubling property. When ν = f dx, this condition, which we call D , admits the possibility of f vanishing or becoming infinite. Our analysis extends the regularity theory (due to Caffarelli) available when 0 < ...

The article focuses on the issue of influence historical conditions Polish-Russian relations at turn 20th and 21st centuries. history these two countries is linked by a long-lasting difficult past, thus numerous unresolved disputes, burdens, myths arose that have been felt to this day. In 1989, after fall Communist system in our country, new political elites managing process democratic changes ...

The results of an investigation on historical floods in four watersheds of the Aude region in France are presented herein. Using both, archive documents and field investigations, the discharges of the main historical floods could be estimated for a period ranging from one to two centuries. The use of this data, in addition to systematic discharge measurements (continuous series of about 30 year...