Diversities and the Geometry of Hypergraphs

The embedding of finite metrics in $\ell_1$ has become a fundamental tool for both combinatorial optimization and large-scale data analysis. One important application is to network flow problems in which there is close relation between max-flow min-cut theorems and the minimal distortion embeddings of metrics into $\ell_1$. Here we show that this theory can be generalized considerably to encompass Steiner tree packing problems in both graphs and hypergraphs. Instead of the theory of $\ell_1$ metrics and minimal distortion embeddings, the parallel is the theory of diversities recently introduced by Bryant and Tupper, and the corresponding theory of $\ell_1$ diversities and embeddings which we develop here.Comment: 19 pages, no figures. This version: further small correction

Constant distortion embeddings of Symmetric Diversities

Diversities are like metric spaces, except that every finite subset, instead of just every pair of points, is assigned a value. Just as there is a theory of minimal distortion embeddings of finite metric spaces into $L_1$, there is a similar, yet undeveloped, theory for embedding finite diversities into the diversity analogue of $L_1$ spaces. In the metric case, it is well known that an $n$-point metric space can be embedded into $L_1$ with $\mathcal{O}(\log n)$ distortion. For diversities, the optimal distortion is unknown. Here, we establish the surprising result that symmetric diversities, those in which the diversity (value) assigned to a set depends only on its cardinality, can be embedded in $L_1$ with constant distortion.Comment: 14 pages, 3 figure

Toward a Definition of High Expectations: Interpretive Case Studies of Selected Secondary Schools in West Virginia

The purpose of this study was to examine the curricular, instructional and administrative practices that teachers and principals used in establishing high expectations from interpretive case studies of selected West Virginia secondary schools. A qualitative research design was used to focus on the specific practices that teachers and principals used at the three sites. The three sites were chosen using purposeful sampling with a unique attribute. The unique attribute was that each school had to have met adequate yearly progress (AYP) for four consecutive years. Data collection included interviews where a semi-structured interview format was used. Included in the data collection were classroom observations, school observations, document collection and field notes. It was the teachers and principals in each building who created the practices of high expectations unique to their values and beliefs. Each building was a microcosm of the community the teachers and principals were members of. Therefore, one unified definition cannot exist because of the lack of unified communities throughout West Virginia. However, this research found the existence of seven commonalities in schools that promoted a culture of high academic achievement and hence high expectations in selected West Virginia secondary schools. The commonalities found in this research were: (1) master schedule, (2) strong professional learning community, (3) course curriculum aligned to the state assessment tool, (4) teachers teaching the prescribed curriculum, (5) strong offering of advanced placement courses, (6) strong parental involvement/support, and (7) time where extra help/extra time is provided

Mixed Data and Classification of Transit Stops

An analysis of the characteristics and behavior of individual bus stops can reveal clusters of similar stops, which can be of use in making routing and scheduling decisions, as well as determining what facilities to provide at each stop. This paper provides an exploratory analysis, including several possible clustering results, of a dataset provided by the Regional Transit Service of Rochester, NY. The dataset describes ridership on public buses, recording the time, location, and number of entering and exiting passengers each time a bus stops. A description of the overall behavior of bus ridership is followed by a stop-level analysis. We compare multiple measures of stop similarity, based on location, route information, and ridership volume over time