On asymptotics of the beta-coalescents

We show that the total number of collisions in the exchangeable coalescent process driven by the beta $(1,b)$ measure converges in distribution to a 1-stable law, as the initial number of particles goes to infinity. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance $b=1$, which corresponds to the Bolthausen--Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta $(a,b)$-coalescents with $0<a<1$ leads to a simplified derivation of the known $(2-a)$-stable limit. We furthermore derive asymptotic expansions for the moments of the number of collisions and of the total branch length for the beta $(1,b)$-coalescent by exploiting the method of sequential approximations.Comment: 25 pages, submitted for publicatio

Isospectral Alexandrov Spaces

We construct the first non-trivial examples of compact non-isometric Alexandrov spaces which are isospectral with respect to the Laplacian and not isometric to Riemannian orbifolds. This construction generalizes independent earlier results by the authors based on Schueth's version of the torus method.Comment: 15 pages, no figures; minor clarification

Diversity and distribution of spiders (Arachnida: Araneae) in dry ecosystems of North Rhine-Westphalia (Germany)

The present study provides a robust data set for ecological planning and conservation of dry ecosystems in western Germany in general and North Rhine-Westphalia in particular. We summarised all available data from recent publications that dealt with spiders in dry ecosystems of North Rhine-Westphalia. Additionally, so far unpublished results of a detailed investigation regarding spiders in sand habitats of the Westphalian Bay that was conducted between 2006 and 2008 are presented. The analysis focussed on the habitat types according to Annex I of the EU Habitats Directive and related habitats. The investigation areas were scattered in the federal state of North Rhine-Westphalia. The data set comprised a total of 84436 individuals from 371 species and 28 families. Overall, an endangerment status is assigned to 68 species. Of these, 12 spiders are in imminent danger of becoming extinct. Two species, Erigonoplus globipes and Meioneta simplicitarsis, are believed to be extinct in North Rhine-Westphalia. Seven species (Dictyna major, Mastigusa arietina, Micaria formicaria, Styloctetor romanus, Thanatus striatus, Theridion uhligi and Xysticus ferrugineus) are new to the arachnofauna of North Rhine-Westphalia

Typical solution time for a vertex-covering algorithm on finite-connectivity random graphs

In this letter, we analytically describe the typical solution time needed by a backtracking algorithm to solve the vertex-cover problem on finite-connectivity random graphs. We find two different transitions: The first one is algorithm-dependent and marks the dynamical transition from linear to exponential solution times. The second one gives the maximum computational complexity, and is found exactly at the threshold where the system undergoes an algorithm-independent phase transition in its solvability. Analytical results are corroborated by numerical simulations.Comment: 4 pages, 2 figures, to appear in Phys. Rev. Let