The height of multiple edge plane trees

Multi-edge trees as introduced in a recent paper of Dziemia\'nczuk are plane trees where multiple edges are allowed. We first show that $d$-ary multi-edge trees where the out-degrees are bounded by $d$ are in bijection with classical $d$-ary trees. This allows us to analyse parameters such as the height. The main part of this paper is concerned with multi-edge trees counted by their number of edges. The distribution of the number of vertices as well as the height are analysed asymptotically

Canonical Trees, Compact Prefix-free Codes and Sums of Unit Fractions: A Probabilistic Analysis

For fixed $t\ge 2$, we consider the class of representations of $1$ as sum of unit fractions whose denominators are powers of $t$ or equivalently the class of canonical compact $t$-ary Huffman codes or equivalently rooted $t$-ary plane "canonical" trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of distinct summands (normal distribution), the path length (normal distribution), the width (main term of the expectation and concentration property) and the number of leaves at maximum distance from the root (discrete distribution)

Introduction to special issue: quantitative studies of science in Germany

Deutsches Zentrum für Hochschul- und Wissenschaftsforschung GmbH (DZHW) (5308)Peer Reviewe

On the Impact of the Degree of Fluorination on the ORR Limiting Processes within Iron Based Catalysts: A Model Study on Symmetrical Films of Barium Ferrate

In this study, symmetrical films of BaFeO$_{2.67}$, BaFeO$_{2.33}$F$_{0.33}$ and BaFeO$_{2}$F were synthesized and the oxygen uptake and conduction was investigated by high temperature impedance spectroscopy under an oxygen atmosphere. The data were analyzed on the basis of an impedance model designed for highly porous mixed ionic electronic conducting (MIEC) electrodes. Variable temperature X-ray diffraction experiments were utilized to estimate the stability window of the oxyfluoride compounds, which yielded a degradation temperature for BaFeO$_{2.33}$F$_{0.33}$ of 590 °C and a decomposition temperature for BaFeO$_{2}$F of 710 °C. The impedance study revealed a significant change of the catalytic behavior in dependency of the fluorine content. BaFeO$_{2.67}$ revealed a bulk-diffusion limited process, while BaFeO$_{2.33}$F$_{0.33}$ appeared to exhibit a fast bulk diffusion and a utilization region δ larger than the electrode thickness L (8 μm). In contrast, BaFeO$_{2}$F showed very area specific resistances due to the lack of oxygen vacancies. The activation energy for the uptake and conduction process of oxygen was found to be 0.07/0.29 eV (temperature range-dependent), 0.33 eV and 0.67 eV for BaFeO$_{2.67}$, BaFeO$_{2.33}$F$_{0.33}$ and BaFeO$_{2}$F, respectively