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We say that a graph is k-wide if for any partition of its vertex set into two subsets, one can choose vertices at distance at least k in these subsets (i.e., the complement of the kth power of this graph is connected). We say that a graph is k-mono-wide if for any partition of its vertex set into two subsets, one can choose vertices at distance exactly k in these subsets. We prove that the complement of a 3-wide graph on n vertices has at least 3n − 7 edges, and the complement of a 3-mono-wide graph on n vertices has at least 3n − 8 edges. We construct infinite series of graphs for which these bounds are attained. We also prove an asymptotically tight bound for the case k ≥ 4: the complement of a k-wide graph contains at least (n − 2k)(2k − 4[log2k] − 1) edges.

Terrorist threat prevention and counteraction require timely detection of hostile plans. However, adversary efforts at concealment and other challenges involved in monitoring terrorist organizations may impede timely intelligence acquisition or interpretation. This study develops an approach to analyzing technological means rather than content of communications produced within the social networks comprising covert organizations, and shows how it can be applied towards detecting terrorist attack precursors. We find that differential usage patterns of hi-tech versus low-tech communication solutions could reveal significant information about organizational activities, which may be further used to detect signals of impending terrorist attacks. (Such potential practical utility of our method is supported by the detailed empirical analysis of available al Qaeda communications.) The described approach thus provides a common framework for utilizing diverse activity records from heterogeneous sources as well as contributes new tools for their rapid analysis aimed at better informing operational and policy decision-making.

In biological networks, any manifestations of behaviors substantially 'deviant' from the predictions of continuous-deterministic classical chemical kinetics (CCK) are typically ascribed to systems with complex dynamics and/or a small number of molecules. Here we show that in certain cases such restrictions are not obligatory for CCK to be largely incorrect. By systematically identifying properties that may cause significant divergences between CCK and the more accurate discrete-stochastic chemical master equation (CME) system descriptions, we comprehensively characterize potential CCK failure patterns in biological settings, including consequences of the assertion that CCK is closer to the 'mode' rather than the 'average' of stochastic reaction dynamics, as generally perceived. We demonstrate that mechanisms underlying such nonclassical effects can be very simple, are common in cellular networks and result in often unintuitive system behaviors. This highlights the importance of deviant effects in biotechnologically or biomedically relevant applications, and suggests some approaches to diagnosing them in situ.

1. X-ray diffraction patterns of natural and heat-treated anthracites, graphites, pyrolytic carbon, schungite, and polyacetylene contain, besides graphitic reflections, a significant number of other, nongraphitic, reflections. A portion of them is caused by heteroatomic fragments and disappears upon heating, but other reflections, on the other hand, grow during heat treatment. 2. The basic mass of carbonaceous material in the original anthracites, polyacetylene, and schungite exists not in the form of graphitoid substances, but in the form of partially crosslinked, spatially branched carbon chains, containing, perhaps, secondary fragments. 3. During heating, the processes of aromatization and graphitation spread, and the lamellar structure is formed. In a parallel manner, crystalline fragments of nongraphitic nature are formed. 4. Upon heating, anthracites twice transform into the thermoplastic state.

An approximate solution is given for the equations of motion and energy in the main part of a turbulent vortex jet of incompressible fluid in a space full of the same fluid.