Narrow Results

Let X and Y be two disjoint sets of points in the plane such that |X|=|Y| and no three points of X ∪ Y are on the same line. Then we can draw an alternating Hamilton cycle on X∪Y in the plane which passes through alternately points of X and those of Y, whose edges are straight-line segments, and which contains at most |X|-1 crossings. Our proof gives an O(n²logn) time algorithm for finding such an alternating Hamilton cycle, where n =|X|. Moreover we show that the above upper bound |X|-1 on crossing number is best possible for some configurations.

An ophthalmic hospital with wings.

Experimental investigations on wave reflections, run-up and run-down and wave pressures on plane, dentated and serrated seawalls were carried out for a wide range of wave heights, wave periods, seawall slopes and in a constant water depth of 0.7 m. Based on the measurements, predictive equations are proposed. A total performance evaluation is carried out on each seawall. The improved performance of the serrated seawall compared to plane seawall is brought out. It is found that closer to deep water condition, the reflection coefficients from the serrated and the dentated vertical seawalls are only about 0.5 and 0.6 respectively compared to the fully reflecting plane vertical wall. Sloped serrated seawall is capable of dissipating the wave energy almost two times that of the sloped plane seawall. Closer to still water level, the normalized wave pressure value (ratio of maximum wave pressure to static pressure due to the water column equal to incident wave height) for 2% exceedence is 0.9, 1.1 and 1.15 for the serrated, dentated and plane seawalls. The measured relative run-up value (ratio between the maximum run up to the incident wave height) for 2% exceedence is 1.30, 1.5 and 1.6 for serrated, dentated and plane seawall respectively. Hence, the design crest level of the serrated seawall can be at a lower level than that of the plane seawall. Overall, it is found that the serrated seawall is about 20% to 40% better than the plane seawall for reducing the wave pressure, reflection, run-up and rundown. The results can be used for better hydrodynamic design of coastal structures with vertical face (like quay walls in harbors, vertical seawalls and caisson type breakwaters) and sloped structures (like seawalls, dikes, sea side sloped caissons).