Metric Properties of Diestel-Leader Groups

In this paper we investigate metric properties of the groups $\Gamma_d(q)$ whose Cayley graphs are the Diestel-Leader graphs $DL_d(q)$ with respect to a given generating set $S_{d,q}$. These groups provide a geometric generalization of the family of lamplighter groups, whose Cayley graphs with respect to a certain generating set are the Diestel-Leader graphs $DL_2(q)$. Bartholdi, Neuhauser and Woess in \cite{BNW} show that for $d \geq 3$, $\Gamma_d(q)$ is of type $F_{d-1}$ but not $F_d$. We show below that these groups have dead end elements of arbitrary depth with respect to the generating set $S_{d,q}$, as well as infinitely many cone types and hence no regular language of geodesics. These results are proven using a combinatorial formula to compute the word length of group elements with respect to $S_{d,q}$ which is also proven in the paper and relies on the geometry of the Diestel-Leader graphs.Comment: 19 page

Automorphisms of Higher Rank Lamplighter Groups

Let $\Gamma_d(q)$ denote the group whose Cayley graph with respect to a particular generating set is the Diestel-Leader graph $DL_d(q)$, as described by Bartholdi, Neuhauser and Woess. We compute both $Aut(\Gamma_d(q))$ and $Out(\Gamma_d(q))$ for $d \geq 2$, and apply our results to count twisted conjugacy classes in these groups when $d \geq 3$. Specifically, we show that when $d \geq 3$, the groups $\Gamma_d(q)$ have property $R_{\infty}$, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when $d=2$ the lamplighter groups $\Gamma_2(q)=L_q = {\mathbb Z}_q \wr {\mathbb Z}$ have property $R_{\infty}$ if and only if $(q,6) \neq 1$.Comment: 28 page

Tame combing and almost convexity conditions

We give the first examples of groups which admit a tame combing with linear radial tameness function with respect to any choice of finite presentation, but which are not minimally almost convex on a standard generating set. Namely, we explicitly construct such combings for Thompson's group F and the Baumslag-Solitar groups BS(1, p) with p \ge 3. In order to make this construction for Thompson's group F, we significantly expand the understanding of the Cayley complex of this group with respect to the standard finite presentation. In particular we describe a quasigeodesic set of normal forms and combinatorially classify the arrangements of 2-cells adjacent to edges that do not lie on normal form paths.Comment: 36 pages, 9 figure

Pure braid subgroups of braided Thompson's groups

We describe pure braided versions of Thompson's group F. These groups, $BF$ and $\hat{BF}$, are subgroups of the braided versions of Thompson's group V, introduced by Brin and Dehornoy. Unlike V, elements of F are order-preserving self-maps of the interval and we use pure braids together with elements of F thus preserving order. We define these groups and give normal forms for elements and describe infinite and finite presentations of these groups.Comment: 26 pages, 6 figures, with updated bibliograph

ISOWN: accurate somatic mutation identification in the absence of normal tissue controls.

BackgroundA key step in cancer genome analysis is the identification of somatic mutations in the tumor. This is typically done by comparing the genome of the tumor to the reference genome sequence derived from a normal tissue taken from the same donor. However, there are a variety of common scenarios in which matched normal tissue is not available for comparison.ResultsIn this work, we describe an algorithm to distinguish somatic single nucleotide variants (SNVs) in next-generation sequencing data from germline polymorphisms in the absence of normal samples using a machine learning approach. Our algorithm was evaluated using a family of supervised learning classifications across six different cancer types and ~1600 samples, including cell lines, fresh frozen tissues, and formalin-fixed paraffin-embedded tissues; we tested our algorithm with both deep targeted and whole-exome sequencing data. Our algorithm correctly classified between 95 and 98% of somatic mutations with F1-measure ranges from 75.9 to 98.6% depending on the tumor type. We have released the algorithm as a software package called ISOWN (Identification of SOmatic mutations Without matching Normal tissues).ConclusionsIn this work, we describe the development, implementation, and validation of ISOWN, an accurate algorithm for predicting somatic mutations in cancer tissues in the absence of matching normal tissues. ISOWN is available as Open Source under Apache License 2.0 from https://github.com/ikalatskaya/ISOWN

Leistungsanforderungen und ihre Bedeutung bis zum Erwerb des Qualifizierten Sekundarabschlusses I in Rheinland-Pfalz

Die vorliegende Arbeit befasst sich, unter Einbeziehung der Ergebnisse aus der PISA Ergänzungsstudie (2000), mit der Bedeutung unterschiedlicher Leistungsanforderungen bis zum Erwerb des Qualifizierten Sekundarabschlusses I in den beiden ausgewählten Bundesländern Baden-Württemberg und Rheinland-Pfalz. Durch Interviews mit Auszubildenden, welche den Qualifizierten Sekundarabschluss I in einem der beiden Bundesländer absolviert haben, wurde erörtert, welche Faktoren schulischer Ausbildung die Auszubildenden, Berufsschullehrer/innen sowie die Ausbilder/innen für die Qualität dieses Bildungsabschlusses verantwortlich machen, in welchen Bereichen sie Defizite sehen, womit sie rückblickend zufrieden waren und welche Optimierungsvorschläge sie anbringen