Exponential sum approximations for t−βt^{-\beta}

Given $\beta>0$ and $\delta>0$, the function $t^{-\beta}$ may be approximated for $t$ in a compact interval $[\delta,T]$ by a sum of terms of the form $we^{-at}$, with parameters $w>0$ and $a>0$. One such an approximation, studied by Beylkin and Monz\'on, is obtained by applying the trapezoidal rule to an integral representation of $t^{-\beta}$, after which Prony's method is applied to reduce the number of terms in the sum with essentially no loss of accuracy. We review this method, and then describe a similar approach based on an alternative integral representation. The main difference is that the new approach achieves much better results before the application of Prony's method; after applying Prony's method the performance of both is much the same.Comment: 18 pages, 5 figures. I have completely rewritten this paper because after uploading the previous version I realised that there is a much better approach. Note the change to the title. Have included minor corrections following revie

Spectral partitioning with multiple eigenvectors

AbstractThe graph partitioning problem is to divide the vertices of a graph into disjoint clusters to minimize the total cost of the edges cut by the clusters. A spectral partitioning heuristic uses the graph's eigenvectors to construct a geometric representation of the graph (e.g., linear orderings) which are subsequently partitioned. Our main result shows that when all the eigenvectors are used, graph partitioning reduces to a new vector partitioning problem. This result implies that as many eigenvectors as are practically possible should be used to construct a solution. This philosophy is in contrast to that of the widely used spectral bipartitioning (SB) heuristic (which uses only a single eigenvector) and several previous multi-way partitioning heuristics [8, 11, 17, 27, 38] (which use k eigenvectors to construct k-way partitionings). Our result motivates a simple ordering heuristic that is a multiple-eigenvector extension of SB. This heuristic not only significantly outperforms recursive SB, but can also yield excellent multi-way VLSI circuit partitionings as compared to [1, 11]. Our experiments suggest that the vector partitioning perspective opens the door to new and effective partitioning heuristics. The present paper updates and improves a preliminary version of this work [5]

Obstacle Numbers of Planar Graphs

Given finitely many connected polygonal obstacles $O_1,\dots,O_k$ in the plane and a set $P$ of points in general position and not in any obstacle, the {\em visibility graph} of $P$ with obstacles $O_1,\dots,O_k$ is the (geometric) graph with vertex set $P$, where two vertices are adjacent if the straight line segment joining them intersects no obstacle. The obstacle number of a graph $G$ is the smallest integer $k$ such that $G$ is the visibility graph of a set of points with $k$ obstacles. If $G$ is planar, we define the planar obstacle number of $G$ by further requiring that the visibility graph has no crossing edges (hence that it is a planar geometric drawing of $G$). In this paper, we prove that the maximum planar obstacle number of a planar graph of order $n$ is $n-3$, the maximum being attained (in particular) by maximal bipartite planar graphs. This displays a significant difference with the standard obstacle number, as we prove that the obstacle number of every bipartite planar graph (and more generally in the class PURE-2-DIR of intersection graphs of straight line segments in two directions) of order at least $3$ is $1$.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Train‐the‐trainer: Methodology to learn the cognitive interview

Research has indicated that police may not receive enough training in interviewing cooperative witnesses, specifically in use of the cognitive interview (CI). Practically, for the CI to be effective in real‐world investigations, police investigators must be trained by law enforcement trainers. We conducted a three‐phase experiment to examine the feasibility of training experienced law enforcement trainers who would then train others to conduct the CI. We instructed Federal Bureau of Investigation and local law enforcement trainers about the CI (Phase I); law enforcement trainers from both agencies (n = 4, 100% male, mean age = 50 years) instructed university students (n = 25, 59% female, mean age = 21 years) to conduct either the CI or a standard law enforcement interview (Phase II); the student interviewers then interviewed other student witnesses (n = 50, 73% female, mean age = 22 years), who had watched a simulated crime (phase III). Compared with standard training, interviews conducted by those trained by CI‐trained instructors contained more information and at a higher accuracy rate and with fewer suggestive questions.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/147804/1/jip1518_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/147804/2/jip1518.pd