Minimum Convex Partitions and Maximum Empty Polytopes

Let $S$ be a set of $n$ points in $\mathbb{R}^d$. A Steiner convex partition is a tiling of ${\rm conv}(S)$ with empty convex bodies. For every integer $d$, we show that $S$ admits a Steiner convex partition with at most $\lceil (n-1)/d\rceil$ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension $d\geq 3$. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any $n$ points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is $\omega(1/n)$. Here we give a $(1-\varepsilon)$-approximation algorithm for computing the maximum volume of an empty convex body amidst $n$ given points in the $d$-dimensional unit box $[0,1]^d$.Comment: 16 pages, 4 figures; revised write-up with some running times improve

Rain event properties at the source of the Blue Nile River

In the present study, spatial and temporal patterns of rain event properties are analysed. These event properties are rain event depth, event duration, mean event rain rate, peak rain rate and the time span between two consecutive rain events which is referred to as inter-event time (IET). In addition, we assessed how rain event properties change when the period over which rainfall data is aggregated changes from 1 to 6 min and when the minimum inter-event time (MIT) changes from 30 min to 8 h. Rainfall data is obtained from a field campaign in two wet seasons of June–August (JJA) of 2007 and 2008 in Gilgel Abbay watershed that is situated at the source basin of the Upper Blue Nile River in Ethiopia. The rainfall data was automatically recorded at eight stations. The results revealed that rain event depth is more related to peak rain rate than to event duration. At the start and towards the end of the wet season, the rain events have larger depth with longer duration and longer IET than those in mid-season. Event rain rate and IET are strongly related to terrain elevation. Sekela which is on a mountain area has the shortest IET while Bahir Dar which is at the south shore of Lake Tana has the longest IET. The period over which rainfall data is aggregated significantly affected the values of rain event properties that are estimated using relatively small value (30 min) of MIT but its effect diminished when the MIT is increased to 8 h. It is shown that increasing the value of MIT has the largest effect on rain event properties of mountain stations that are characterised by high rainfall intermittency

Developing Guidelines for Two-Dimensional Model Review and Acceptance

Two independent modelers ran two hydraulic models, SRH-2D and HEC-RAS 2D. The models were applied to the Lakina River (MP 44 McCarthy Road) and to Quartz Creek (MP 0.7 Quartz Creek Road), which approximately represent straight and bend flow conditions, respectively. We compared the results, including water depth, depth averaged velocity, and bed shear stress, from the two models for both modelers. We found that the extent and density of survey data were insufficient for Quartz Creek. Neither model was calibrated due to the lack of basic field data (i.e., discharge, water surface elevation, and sediment characteristics). Consequently, we were unable to draw any conclusion about the accuracy of the models. Concerning the time step and the equations used (simplified or full) to solve the momentum equation in the HEC-RAS 2D model, we found that the minimum time step allowed by the model must be used if the diffusion wave equation is used in the simulations. A greater time step can be used if the full momentum equation is used in the simulations. We developed a set of guidelines for reviewing model results, and developed and provided a two-day training workshop on the two models for ADOT&PF hydraulic engineers

Evaluation of the propulsion control system of a planetary rover and design of a mast for an elevation scanning laser/multi-detector system

Vertical wheel loads, wheel speeds, and torque relationships are considered in the design of a propulsion system capable of responding to steering, slope climbing, and irregular local terrains. The system developed is applied to the RPI Mars roving vehicle. The mechanical system required to implement the elevation laser scanning/multidetector principle was the design and construction of a mechanical system for implementing the elevation scanning/multidetector principle is also discussed