4-Holes in point sets

We consider a variant of a question of Erdos on the number of empty k-gons (k-holes) in a set of n points in the plane, where we allow the k-gons to be non-convex. We show bounds and structural results on maximizing and minimizing the number of general 4-holes, and maximizing the number of non-convex 4-holes. In particular, we show that for n >= 9, the maximum number of general 4-holes is ((pi)(4)); the minimum number of general 4-holes is at least 5/2 n(2) - circle minus(n); and the maximum number of non-convex 4-holes is at least 1/2 n(3) - circle minus(n(2) logn) and at most 1/2 n(3) - circle minus(n(2)). 2014 (c) Elsevier B.V. All rights reserved.Postprint (author’s final draft

Relaxed Disk Packing

Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations.Comment: 8 pages => 5 pages of main text plus 3 pages in appendix. Submitted to CCCG 201

The Shadows of a Cycle Cannot All Be Paths

A "shadow" of a subset $S$ of Euclidean space is an orthogonal projection of $S$ into one of the coordinate hyperplanes. In this paper we show that it is not possible for all three shadows of a cycle (i.e., a simple closed curve) in $\mathbb R^3$ to be paths (i.e., simple open curves). We also show two contrasting results: the three shadows of a path in $\mathbb R^3$ can all be cycles (although not all convex) and, for every $d\geq 1$, there exists a $d$-sphere embedded in $\mathbb R^{d+2}$ whose $d+2$ shadows have no holes (i.e., they deformation-retract onto a point).Comment: 6 pages, 10 figure

Domino Tatami Covering is NP-complete

A covering with dominoes of a rectilinear region is called \emph{tatami} if no four dominoes meet at any point. We describe a reduction from planar 3SAT to Domino Tatami Covering. As a consequence it is NP-complete to decide whether there is a perfect matching of a graph that meets every 4-cycle, even if the graph is restricted to be an induced subgraph of the grid-graph. The gadgets used in the reduction were discovered with the help of a SAT-solver.Comment: 10 pages, accepted at The International Workshop on Combinatorial Algorithms (IWOCA) 201

District-Funded Common Core Collaboration Grants Used for Teacher Professional Development

With the adoption of the Common Core State Standards (CCSS) in English language arts and mathematics by the State of California in 2010, a shift in instructional practices along with the level of rigor and expectations for students began. As a result of these changes, a local school district sought a way through district-funded Common Core Collaboration Grants (CCCG) to provide professional development that supported 4th-6th grade teachers in their implementation of the CCSS. The purpose of this qualitative program evaluation case study was to examine teachers\u27 perceptions of the effectiveness of professional development funded by CCCG in supporting 4th-6th grade teachers in understanding and application of instructional strategies aligned with the CCSS. Weiss\u27s theory of change and Roy and Killion\u27s program evaluation framework guided the study. Data were collected from individual interviews of 7 teachers of 4th-6th grade who participated in the district CCCG professional development sessions. Interview data were coded and themes of choice, time, collaboration, and integration of the CCSS emerged. The results indicated that the use of CCCG for professional development is assisting teachers in successfully implementing the CCSS through increased collaboration and more opportunities to engage in learning within their own contexts. A program evaluation report and presentation to the district school board were developed. The results of this study may affect positive social change through suggestions of an alternative in the form of grants to schools and districts looking for innovative ways to support teachers and enhance student learning through professional development on the CCSS