The Planar Tree Packing Theorem

Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 and T2. A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of Garc\'ia et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.Comment: Full version of our SoCG 2016 pape

The Effect of Gas Loss on the Formation of Bound Stellar Clusters

The effect of gas ejection on the structure and binding energy of newly formed stellar clusters is investigated. The star formation efficiency (SFE), necessary for forming a gravitationally bound stellar cluster, is determined. Two sets of numerical N-body simulations are presented: As a first simplified approach we treat the residual gas as an external potential. The gas expulsion is approximated by reducing the gas mass to zero on a given timescale, which is treated as a free parameter. In a second set of simulations we use smoothed particle hydrodynamics (SPH) to follow the dynamics of the outflowing residual gas self-consistently. We investigate cases where gas outflow is induced by an outwards propagating shock front and where the whole gas cloud is heated homogeneously, leading to ejection. If the stars are in virial equilibrium with the gaseous environment initially, bound clusters only form in regions where the local SFE is larger than 50% or where the gas expulsion timescale is long compared to the dynamical timescale. A small initial velocity dispersion of the stars leads to a compaction of the cluster during the expulsion phase and reduces the SFE needed to form bound clusters to less than 10%.Comment: 6 pages, 7 figures, submitted to MNRA

Recent Advances in Modularity Optimization and Their Application in Retailing

In this contribution we report on three recent advances in modularity optimization, namely: 1. The randomized greedy (RG) family of modularity optimization algorithms are state-of-the-art graph clustering algorithms which are near optimal, fast, and scalable. 2. The extension of the RG family to multi-level clustering. 3. A new entropy based cluster index which allows the detection of the proper clustering levels and of stable core clusters at each level. Last, but not least, several marketing applications of these algorithms for customer enablement and empowerment are discussed: e.g. the detection of low-level cluster structures from retail purchase data, the analysis of the co-usage structure of scientific documents for detecting multilevel category structures for scientific libraries, and the analysis of social groups from the friend relation of social network sites

The future of the German past: transatlantic reflections for the 1990s

"The comparative reluctance of German historians to engage the postmodern challenge suggested the need for a reflection on what post-structuralist impulses might have to offer for analyzing the Central European past. In the United States the criticism voiced by Geoff Eley and David Blackbourn had already undercut the hegemony of the 'Sonderweg' paradigm, promoted by the 'societal historians' of the Bielefeld school which was slow to respond to feminist and everyday history approaches. The authors therefore set out to initiate a discussion about the deconstruction of 'grand narratives' about the German past, in order to create more interpretative space for stories that did not fit into the model of 'historical social science'. In the American intellectual climate this objectivist and modernist outlook seemed no longer persuasive enough, since various minorities promoted views that emphasized the constructivist character of historical understanding. Our joint programmatic essay therefore tried to open space for recovering a greater plurality of experiences." (author's abstract

Foreword

Early Middle Cambrian bituminous coquinoid limestones from a tectonically isolated outcrop in southwestern Kyrgyzstan yield a remarkably diverse fauna, with stem-group cnidarians, trilobites, rhynchonelliformean brachiopods, and other shelly fossils. The fossil site is in the northern foothills of the Turkestan Range and thus forms part of the westernmost extension of the South Tien Shan. The fauna includes two fairly well known trilobite species, Glabrella ventrosa Lermontova, 1940 and Dorypyge richthofeniformis Lermontova, 1940, that provide confident support for an Amgan age of the rocks. New described taxa include the stem-group cnidarian Cambroctoconus kyrgyzstanicus Peel sp. nov., the trilobite Olenoides sagittatus Geyer sp. nov., and the helcionelloid Manasoconus bifrons Peel gen. et sp. nov. Additional fossils within the samples include the trilobites Olenoides sp. A, Kootenia sp., and Pseudoeteraspis? sp.; the rhynchonelliform brachiopods Narynella cf. ferganensis (Andreeva, 1962), Narynella? sp., Austrohedra? sp. nov., and two species of uncertain generic affinity; the tommotiid Tesella sp.; the hyolithelminth Hyolithellus sp.; and the palaeoscolecid Hadimopanella oezgueli Gedik, 1977. Of particular interest is Cambroctoconus kyrgyzstanicus with an octagonal corallum and a sparsely septate calyx