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Embedding undirected graphs in a Euclidean space has many computational benefits. FastMap is an efficient embedding algorithm that facilitates a geometric interpretation of problems posed on undirected graphs. However, Euclidean distances are inherently symmetric and, thus, Euclidean embeddings cannot be used for directed graphs. In this paper, we present FastMap-D, an efficient generalization of FastMap to directed graphs. FastMap-D embeds vertices using a potential field to capture the asymmetry between the pairwise distances in directed graphs. FastMap-D learns a potential function to define the potential field using a machine learning module. In experiments on various kinds of directed graphs, we demonstrate the advantage of FastMap-D over other approaches. Errata: This version of the paper corrects a programming mistake, resulting in even better experimental results than those reported in the original paper.

We present a numerical investigation of the processes that influenced the contrasting rapid intensity changes in Tropical Cyclones (TC) Phailin and Lehar (2013) over the Bay of Bengal. Our emphasis is on the significant differences in the environments experienced by the TCs within a few weeks and the consequent differences in their organization of vortex-scale convection that resulted in their different rapid intensity changes. The storm-relative proximity, intensity, and depth of the subtropical ridge resulted in the establishment of a low-sheared environment for Phailin and a high-sheared environment for Lehar. Our primary finding here is that in Lehar’s sheared vortex, the juxtaposition in the azimuthal phasing of the asymmetrically distributed downward eddy flux of moist-entropy through the top of the boundary layer, and the radial eddy flux of moist-entropy within the boundary layer in the upshear left-quadrant of Lehar (40–80 km radius) establishes a pathway for the low moist-entropy air to intrude into the vortex from the environment. Conversely, when the azimuthal variations in boundary layer moist-entropy, inflow, and convection are weak in Phailin’s low-sheared environment, the inflow magnitude and radial location of boundary layer convergence relative to the radius of maximum wind dictated the rapid intensification.