A Note on Element Centralizers in Finite Coxeter Groups

The normalizer $N_W(W_J)$ of a standard parabolic subgroup $W_J$ of a finite Coxeter group $W$ splits over the parabolic subgroup with complement $N_J$ consisting of certain minimal length coset representatives of $W_J$ in $W$. In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type $D_n$) the centralizer $C_W(w)$ of an element $w \in W$ is in a similar way a semidirect product of the centralizer of $w$ in a suitable small parabolic subgroup $W_J$ with complement isomorphic to the normalizer complement $N_J$.Comment: final version, 18 pages, to appear in J. Group Theor

Application of a Genetic Algorithm to Variable Selection in Fuzzy Clustering

In order to group the observations of a data set into a given number of clusters, an ?optimal? subset out of a greater number of explanatory variables is to be selected. The problem is approached by maximizing a quality measure under certain restrictions that are supposed to keep the subset most representative of the whole data. The restrictions may either be set manually, or generated from the data. A genetic optimization algorithm is developed to solve this problem. The procedure is then applied to a data set describing features of sub-districts of the city of Dortmund, Germany, to detect different social milieus and investigate the variables making up the differences between these. --

On real-time word problems

It is proved that the word problem of the direct product of two free groups of rank 2 can be recognised by a 2-tape real-time but not by a 1-tape real-time Turing machine. It is also proved that the Baumslag–Solitar groups B(1,r) have the 5-tape real-time word problem for all r != 0

Identification of Musical Instruments by means of the Hough-Transformation

In order to distinguish between the sounds of different musical instruments, certain instrument-specific sound features have to be extracted from the time series representing a given recorded sound. The Hough Transform is a pattern recognition procedure that is usually applied to detect specific curves or shapes in digital pictures (Shapiro, 1978). Due to some similarity between pattern recognition and statistical curve fitting problems, it may as well be applied to sound data (as a special case of time series data). The transformation is parameterized to detect sinusoidal curve sections in a digitized sound, the motivation being that certain sounds might be identified by certain oscillation patterns. The returned (transformed) data is the timepoints and amplitudes of detected sinusoids, so the result of the transformation is another ?condensed? time series. This specific Hough Transform is then applied to sounds played by different musical instruments. The generated data is investigated for features that are specific for the musical instrument that played the sound. Several classification methods are tried out to distinguish between the instruments and it turns out that RDA (a hybrid method combining LDA and QDA) (Friedman, 1989) performs best. The resulting error rate is better than those achieved by humans (Bruderer, 2003). --

Bayesian Random-Effects Meta-Analysis Using the bayesmeta R Package

The random-effects or normal-normal hierarchical model is commonly utilized in a wide range of meta-analysis applications. A Bayesian approach to inference is very attractive in this context, especially when a meta-analysis is based only on few studies. The bayesmeta R package provides readily accessible tools to perform Bayesian meta-analyses and generate plots and summaries, without having to worry about computational details. It allows for flexible prior specification and instant access to the resulting posterior distributions, including prediction and shrinkage estimation, and facilitating for example quick sensitivity checks. The present paper introduces the underlying theory and showcases its usage