Compactness Properties of Weighted Summation Operators on Trees

We investigate compactness properties of weighted summation operators $V_{\alpha,\sigma}$ as mapping from $\ell_1(T)$ into $\ell_q(T)$ for some $q\in (1,\infty)$. Those operators are defined by $$(V_{\alpha,\sigma} x)(t) :=\alpha(t)\sum_{s\succeq t}\sigma(s) x(s)\,,\quad t\in T\;,$$ where $T$ is a tree with induced partial order $t \preceq s$ (or $s \succeq t$) for $t,s\in T$. Here $\alpha$ and $\sigma$ are given weights on $T$. We introduce a metric $d$ on $T$ such that compactness properties of $(T,d)$ imply two--sided estimates for $e_n(V_{\alpha,\sigma})$, the (dyadic) entropy numbers of $V_{\alpha,\sigma}$. The results are applied for concrete trees as e.g. moderate increasing, biased or binary trees and for weights with $\alpha(t)\sigma(t)$ decreasing either polynomially or exponentially. We also give some probabilistic applications for Gaussian summation schemes on trees

Compactness properties of weighted summation operators on trees - the critical case

The aim of this paper is to provide upper bounds for the entropy numbers of summation operators on trees in a critical case. In a recent paper [10] we elaborated a framework of weighted summation operators on general trees where we related the entropy of the operator with those of the underlying tree equipped with an appropriate metric. However, the results were left incomplete in a critical case of the entropy behavior, because this case requires much more involved techniques. In the present article we fill the gap left open in [10]. To this end we develop a method, working in the context of general trees and general weighted summation operators, which was recently proposed in [9] for a particular critical operator on the binary tree. Those problems appeared in natural way during the study of compactness properties of certain Volterra integral operators in a critical case

Small Deviations of Gaussian Random Fields in LqL_q--Spaces

We investigate small deviation properties of Gaussian random fields in the space $L_q(\R^N,\mu)$ where $\mu$ is an arbitrary finite compactly supported Borel measure. Of special interest are hereby "thin" measures $\mu$, i.e., those which are singular with respect to the $N$--dimensional Lebesgue measure; the so--called self--similar measures providing a class of typical examples. For a large class of random fields (including, among others, fractional Brownian motions), we describe the behavior of small deviation probabilities via numerical characteristics of $\mu$, called mixed entropy, characterizing size and regularity of $\mu$. For the particularly interesting case of self--similar measures $\mu$, the asymptotic behavior of the mixed entropy is evaluated explicitly. As a consequence, we get the asymptotic of the small deviation for $N$--parameter fractional Brownian motions with respect to $L_q(\R^N,\mu)$--norms. While the upper estimates for the small deviation probabilities are proved by purely probabilistic methods, the lower bounds are established by analytic tools concerning Kolmogorov and entropy numbers of H\"older operators

Kolmogorov numbers of Riemann–Liouville operators over small sets and applications to Gaussian processes

AbstractWe investigate compactness properties of the Riemann–Liouville operator Rα of fractional integration when regarded as operator from L2[0,1] into C(K), the space of continuous functions over a compact subset K in [0,1]. Of special interest are small sets K, i.e. those possessing Lebesgue measure zero (e.g. fractal sets). We prove upper estimates for the Kolmogorov numbers of Rα against certain entropy numbers of K. Under some regularity assumption about the entropy of K these estimates turn out to be two-sided. By standard methods the results are also valid for the (dyadic) entropy numbers of Rα. Finally, we apply these estimates for the investigation of the small ball behavior of certain Gaussian stochastic processes, as e.g. fractional Brownian motion or Riemann–Liouville processes, indexed by small (fractal) sets

Leadership Competencies for Managing Diversity

The new understanding of diversity involves more than increasing the number of different identity groups on the payroll. An important proposal is that the experience of diversity in an organisation results from pervasive styles of management. This article dealt with the specific paradigms of diversitymanagement and leadership style theory used to address the research problem in the empirical study, namely ‘Is diversity management experience related to leadership styles or competencies?’ The models of diversity and inclusion indicators are used to examine the experience of diversitymanagement. The population of this study into the experience of diversity management is two thousand six hundred and sixty nine (2669) respondents. Leadership styles were obtained from four hundred and forty (440) leaders. The Cronbach alpha values were determined in order to indicate internal validity and reliability.diversity management, engaging leadership style, experience, heroic leadership style, management, symbolic interactionism

Leadership competencies for managing diversity

The new understanding of diversity involves more than increasing the number of different identity groups on the payroll. An important proposal is that the experience of diversity in an organisation results from pervasive styles of management. This article dealt with the specific paradigms of diversity management and leadership style theory used to address the research problem in the empirical study, namely ‘Is diversity management experience related to leadership styles or competencies?’ The models of diversity and inclusion indicators are used to examine the experience of diversitymanagement. The population of this study into the experience of diversity management is two thousand six hundred and sixty nine (2669) respondents. Leadership styles were obtained from four hundred and forty (440) leaders. The Cronbach alpha values were determined in order to indicate internal validity and reliability

Intergroup conflict within a South African mining company

The uncertainty that accompanies organizational change heightens prospects for intra-organizational conflict. Notwithstanding this, the knowledge base on the sources (or causes) of organizational conflict is underdeveloped – largely as a result of a low incidence of empirical research, and in particular in South Africa. The current study explored the perceived sources of intergroup conflict in a South African mining company. The aim of this research is to investigate intergroup conflict and to look into how intergroup conflict influences employment relations throughout all levels of the organization. From a probability population of 1000 in the twelve departments a random sample of 200 employees was chosen to participate in the research. This number represents 20% of the total population. Intergroup conflict was analysed and evaluated. Conclusions were drawn and recommendations made on the occurrence of intergroup conflict based on the findings of the literature and empirical study. Data frequencies, correlations and a correlation analysis were performed through the SPSS programme

The robustness of preferences during a crisis: The case of COVID-19

We investigate how preferences have been affected by exposure to the COVID-19 crisis. Our main contributions are: first, our participant pool consists of a large general population sample; second, we elicited a wide range of preferences (risk, time, ambiguity, and social preferences) using different incentivized experimental tasks; third, we elicited preferences before the onset of the crises and in three additional waves during the crises over a time period of more than a year, allowing us to investigate both short-term and medium-term preference responses; fourth, besides the measurement of causal effects of the crisis, we also analyze within each wave during the crisis, how differential exposure to the crisis in the health and financial domain affects preferences. We find that preferences remain remarkably stable during the crisis. Comparing them before the start and during the crisis, we do not observe robust differences in any of the elicited preferences. Moreover, individual differences in the exposure to the crisis at best show only weak effects in the financial domain