Detecting trends and shocks in terrorist activities.

Although there are some techniques for dealing with sparse and concentrated discrete data, standard time-series analyses appear ill-suited to understanding the temporal patterns of terrorist attacks due to the sparsity of the events. This article addresses these issues by proposing a novel technique for analysing low-frequency temporal events, such as terrorism, based on their cumulative curve and corresponding gradients. Using an iterative algorithm based on a piecewise linear function, our technique detects trends and shocks observed in the events associated with terrorist groups that would not necessarily be visible using other methods.

Occupancy fraction, fractional colouring, and triangle fraction.

Given , there exists such that, if , then for any graph on vertices of maximum degree in which the neighbourhood of every vertex in spans at most edges, (i)an independent set of drawn uniformly at random has at least vertices in expectation, and(ii)the fractional chromatic number of is at most . These bounds cannot in general be improved by more than a factor 2 asymptotically. One may view these as stronger versions of results of Ajtai, Komlós and Szemerédi and Shearer.

Coloring triangle-free graphs with local list sizes.

We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow-up work of Bernshteyn) on the (list) chromatic number of triangle-free graphs. In both our results, we permit the amount of color made available to vertices of lower degree to be accordingly lower. One result concerns list coloring and correspondence coloring, while the other concerns fractional coloring.