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)

The relationship between leader-member exchange and work engagement in social work: A mediation analysis of job resources

Introduction: The working environment of social workers has become an important concern in research. Whereas studies typically focus on negative aspects of these environments, this study investigates the possibility of creating a positive environment by examining whether and how high-quality leader-member exchange (LMX) relationships are related to social workers' work engagement. It is hypothesized that particularly relevant job resources mediate the relation between LMX and social workers’ work engagement. Method: To identify those job resources that are particularly important to social workers, a qualitative preparatory study was first conducted with social workers and social work executives. Based on that, study data were collected by administering an online survey among social workers. Both parts of the study were realized at twenty different advice centers for pregnancy issues of a leading social organization in Germany. The dataset for statistical analyses comprised 43 social workers in total. Regression analysis with parallel mediation was used to test the hypothesized relationships. Results: Social workers in high-quality LMX relationships operate in working environments with more abundant social and structural job resources: they experience the team atmosphere more positively and have greater work control. Further, it was found that experiencing a positive team atmosphere was associated with higher work engagement, with team atmosphere fully mediating the relationship between LMX and work engagement. Notably, high-quality LMX relationships do not seem to promote work engagement directly. Discussion: The study underscores the relevance of high-quality LMX relationships in fostering a resourceful work environment. This, in turn, contributes significantly to a high level of work engagement. Organizations and managers in social work should therefore strengthen managerial relationships and create resourceful environments

Longitudinal determination of serum placental protein 13 during development of preeclampsia

Objective: To determine maternal serum placental protein 13 (PP13) in normal pregnancy and preeclampsia. Methods: A prospective, longitudinal study with 41 normal pregnant women, 18 cases with preterm delivery or cervix insufficiency and 4 with developing late-onset preeclampsia. Six hundred and sixty-six maternal blood samples were obtained every 2-4 weeks starting at 5-8 weeks gestation (10-12 samples/patient) and tested for serum PP13 by ELISA. Results: In normal pregnant women delivering at term, median maternal serum PP13 levels were growing from 166 to 202 pg/ml and 382 pg/ml in the first, second and third trimester, respectively. Preeclamptic women had significantly reduced PP13 levels in the first trimester (multiples of median of 0.14 at 7-8 weeks; p = 0.005 compared to normal). PP13 in the third trimester was significantly higher compared to normal at 35-36 weeks with PP13 multiples of median of 1.79. Conclusion: This preliminary study indicates that low levels of PP13 in early pregnancy identify at-risk pregnancies, whereas high levels precede the syndrome in late pregnancy and suggest syncytiotrophoblast necrosis. Copyright (C) 2008 S. Karger AG, Basel

Variances and Covariances in the Central Limit Theorem for the Output of a Transducer

We study the joint distribution of the input sum and the output sum of a deterministic transducer. Here, the input of this finite-state machine is a uniformly distributed random sequence. We give a simple combinatorial characterization of transducers for which the output sum has bounded variance, and we also provide algebraic and combinatorial characterizations of transducers for which the covariance of input and output sum is bounded, so that the two are asymptotically independent. Our results are illustrated by several examples, such as transducers that count specific blocks in the binary expansion, the transducer that computes the Gray code, or the transducer that computes the Hamming weight of the width-$w$ non-adjacent form digit expansion. The latter two turn out to be examples of asymptotic independence

ValiTex -- a uniform validation framework for computational text-based measures of social science constructs

Guidance on how to validate computational text-based measures of social science constructs is fragmented. Whereas scholars are generally acknowledging the importance of validating their text-based measures, they often lack common terminology and a unified framework to do so. This paper introduces a new validation framework called ValiTex, designed to assist scholars to measure social science constructs based on textual data. The framework draws on a long-established tradition within psychometrics while extending the framework for the purpose of computational text analysis. ValiTex consists of two components, a conceptual model, and a dynamic checklist. Whereas the conceptual model provides a general structure along distinct phases on how to approach validation, the dynamic checklist defines specific validation steps and provides guidance on which steps might be considered recommendable (i.e., providing relevant and necessary validation evidence) or optional (i.e., useful for providing additional supporting validation evidence. The utility of the framework is demonstrated by applying it to a use case of detecting sexism from social media data