For a compact, orientable, irreducible, ∂-irreducible, and an-annular 3-manifold, it is shown there are only finitely many boundary slopes for incompressible ∂-incompressible surfaces of bounded Euler characteristic. We use normal surface theory the inverse relationship crushing triangulation along [8] that inflating an ideal [12] to introduce study boundary-efficient triangulations end-efficie...

Time series of dilatometric measurements of relative displacements on rock cracks on stable and unstable sandstone slopes were analyzed. The inherent dynamics of rock slopes lack any significant nonlinearity. However, the residuals obtained by removing meteorological influences are fat-tailed non-Gaussian fluctuations, with short-range correlations in the case of stable slopes. The fluctuations...

It is proved that every series-parallel digraph whose maximum vertex-degree is ∆ admits an upward planar drawing with at most one bend per edge such that each edge segment has one of ∆ distinct slopes. This is shown to be worst-case optimal in terms of the number of slopes. Furthermore, our construction gives rise to drawings with optimal angular resolution π ∆ . A variant of the proof techniqu...

PURPOSE This study evaluated a variety of downhill slopes in an effort to determine the optimal slope for overspeed running. METHODS Thirteen NCAA Division III college athletes who participated in soccer, track, and football ran 40-yd (36.6-m) sprints, on downhill slopes of 2.1 degrees , 3.3 degrees , 4.7 degrees , 5.8 degrees , and 6.9 degrees in random order. All sprints were timed using th...

We study straight-line drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3-connected plane graph has a plane drawing ...