The Adaptive Sampling Revisited

The problem of estimating the number $n$ of distinct keys of a large collection of $N$ data is well known in computer science. A classical algorithm is the adaptive sampling (AS). $n$ can be estimated by $R.2^D$, where $R$ is the final bucket (cache) size and $D$ is the final depth at the end of the process. Several new interesting questions can be asked about AS (some of them were suggested by P.Flajolet and popularized by J.Lumbroso). The distribution of $W=\log (R2^D/n)$ is known, we rederive this distribution in a simpler way. We provide new results on the moments of $D$ and $W$. We also analyze the final cache size $R$ distribution. We consider colored keys: assume that among the $n$ distinct keys, $n_C$ do have color $C$. We show how to estimate $p=\frac{n_C}{n}$. We also study colored keys with some multiplicity given by some distribution function. We want to estimate mean an variance of this distribution. Finally, we consider the case where neither colors nor multiplicities are known. There we want to estimate the related parameters. An appendix is devoted to the case where the hashing function provides bits with probability different from $1/2$

The largest missing value in a composition of an integer

AbstractIn this paper we find, asymptotically, the mean and variance for the largest missing value (part size) in a composition of an integer n. We go on to show that the probability that the largest missing value and the largest part of a composition differ by one is relatively high and we find the mean for the average largest value in compositions that have this property. The average largest value of compositions with at least one non-zero missing value is also found, and used to calculate how many distinct values exceed the largest missing value on average

On gaps and unoccupied urns in sequences of geometrically distributed random variables

info:eu-repo/semantics/publishe

THE MATERIALITY AND SPATIALITY OF GRAVES AND GRAVE MARKERS IN THE BORDER REGION BETWEEN LUXEMBOURG AND GERMANY

This PhD thesis is the partial result of the research project entitled "Material Culture and Spaces of Remembrance – A Study of Cemeteries in Luxembourg in the Context of the Greater Region", under the coordination and supervision of Dr. Thomas Kolnberger and Associate Professor Sonja Kmec at the University of Luxembourg, and funded by the Fonds National de la Recherche (FNR). Building on prior, seminal research, this thesis aims to address the following questions concerning a specific, predefined region between Luxembourg and Germany, and selected cemeteries: • Does the research approach demonstrated from Anglo-American literature also apply to the sample in the border region between Luxembourg and Germany? • Does the analysis of materiality within its spatial context provide indications of a neighbouring effect, i.e. do material characteristics appear in spatial clusters? • With regards to the materiality that can be observed at the selected cemeteries, what might explain the specific appearance of, especially, graves and grave markers; i.e. what factors, such as cemetery regulations or stonemasons, might have had an influence? Based on a pilot project at Walferdange (Luxembourg) cemetery, as well as a specifically developed and designed data collection approach and tool, the author of this thesis collected the data from full populations of grave and grave marker material culture at three additional, selected cemeteries in Luxembourg and Germany, i.e. Wormeldange, Wincheringen and Konz, in order to allow an analysis of the present assemblage, reaching back into the late 19th century. The data gained thus were analysed using statistical and geo-spatial methods. The results of this data collection and analysis indicate the following: similar methods compared to, for example, work in the Anglo-American context, can generally be applied; since materiality of funeral culture shows a certain level of fluctuation and volatility over time in this specific research context, the researcher has to be careful in order to ensure appropriate dating; results in a chronologically limited data set in which also accurate spatiality cannot be ensured. Whileclusters of materiality can be identified visually, they do not in all instances produce stable results during statistical testing. Thus, a neighbouring effect cannot in all cases be supported and needs to be critically questioned in the face of different tactical confidence intervals. Moreover, potential cultural differences and differences in cemetery management, manifested for example in cemetery regulations, are not enough to explain the actual materiality and spatiality that can be found on the researched cemeteries. The author uses additional literature from business studies and economics in order to highlight a different approach in historical archaeological research in understanding grave monument genesis, their explanatory power and studying related phenomena in the future, hypothesizing about business related aspects in the interrelationship between stonemasons and their customers

Abstract Book: Scales of Social, Environmental & Cultural Change in Past Societies

The interplay of environment, social relations, material culture, population dynamics, and human perception are the key factors of socio-environmental changes. The exploration of processes and parameters of societal change enable further exploration of transformations of human-environmental interactions. These processes and parameters are detectable in the development of, for example, settlement systems, material culture, or ritual sites, which link different socio-environmental components. Humans and environments deeply shaped each other, creating diverse social, environmental, and cultural constellations. On the one hand, examining the roots of social, environmental, and cultural phenomena and processes, which substantially marked past human development, can lead to a deeper understanding of the development of societies. On the other hand, a focus on transformation patterns within momentous developments of past societies opens up the possibility of identifying substantial and enduring re-organisation of socio-environmental interaction patterns