Hyperconvexity and Tight Span Theory for Diversities

The tight span, or injective envelope, is an elegant and useful construction that takes a metric space and returns the smallest hyperconvex space into which it can be embedded. The concept has stimulated a large body of theory and has applications to metric classification and data visualisation. Here we introduce a generalisation of metrics, called diversities, and demonstrate that the rich theory associated to metric tight spans and hyperconvexity extends to a seemingly richer theory of diversity tight spans and hyperconvexity.Comment: revised in response to referee comment

HAMSTER: visualizing microarray experiments as a set of minimum spanning trees

<p>Abstract</p> <p>Background</p> <p>Visualization tools allow researchers to obtain a global view of the interrelationships between the probes or experiments of a gene expression (<it>e.g. microarray</it>) data set. Some existing methods include hierarchical clustering and k-means. In recent years, others have proposed applying minimum spanning trees (MST) for microarray clustering. Although MST-based clustering is formally equivalent to the dendrograms produced by hierarchical clustering under certain conditions; visually they can be quite different.</p> <p>Methods</p> <p>HAMSTER (Helpful Abstraction using Minimum Spanning Trees for Expression Relations) is an open source system for generating a <b>set </b>of MSTs from the experiments of a microarray data set. While previous works have generated a single MST from a data set for data clustering, we recursively merge experiments and repeat this process to obtain a set of MSTs for data visualization. Depending on the parameters chosen, each tree is analogous to a snapshot of one step of the hierarchical clustering process. We scored and ranked these trees using one of three proposed schemes. HAMSTER is implemented in C++ and makes use of Graphviz for laying out each MST.</p> <p>Results</p> <p>We report on the running time of HAMSTER and demonstrate using data sets from the NCBI Gene Expression Omnibus (GEO) that the images created by HAMSTER offer insights that differ from the dendrograms of hierarchical clustering. In addition to the C++ program which is available as open source, we also provided a web-based version (HAMSTER⁺) which allows users to apply our system through a web browser without any computer programming knowledge.</p> <p>Conclusion</p> <p>Researchers may find it helpful to include HAMSTER in their microarray analysis workflow as it can offer insights that differ from hierarchical clustering. We believe that HAMSTER would be useful for certain types of gradient data sets (e.g time-series data) and data that indicate relationships between cells/tissues. Both the source and the web server variant of HAMSTER are available from <url>http://hamster.cbrc.jp/</url>.</p

Julius Paul Eppstein Collection 1932-1939

Letters from Max Warburg (May 1938) and Otto Hirsch (March 1939), thanking him for his contributions to Reichsvertretung and Hilfsverein. Also included are pages from a Jewish calendar for Wuerttemberg 1932/33.digitize