On the maximal connected component of a hypercube with faulty vertices III

Hypercube is one of the most popular topologies for connecting processors in multicomputer systems. In this paper we address the maximum order of a connected component in a faulty cube. The results established include several known conclusions as special cases. We conclude that the hypercube structure is resilient as it includes a large connected component in the presence of large number of faulty vertices.

Fault tolerance of Mobius cubes under two forbidden fault set models

An n-dimensional Mobius cube, 0MQ(n) or 1MQ(n), is a variation of n-dimensional cube Q(n) which possesses many attractive properties such as significantly smaller communication delay and stronger graph-embedding capabilities. In some practical situations, the fault tolerance of a distributed memory multiprocessor system can be measured more precisely by the connectivity of the underlying graph under forbidden fault set models. This article addresses the connectivity of 0MQ(n)/1MQ(n), under two typical forbidden fault set models. We first prove that the connectivity of 0MQ(n)/1MQ(n) is 2n - 2 when the fault set does not contain the neighborhood of any vertex as a subset. We then prove that the connectivity of 0MQ(n)/1MQ(n) is 3n - 5 provided that the neighborhood of any vertex as well as that of any edge cannot fail simultaneously These results demonstrate that 0MQ(n)/1MQ(n) has the same connectivity as Q(n) under either of the previous assumptions.

Largest connected component of a star graph with faulty vertices

In order to make a full evaluation of an interconnection network, it is essential to estimate the minimum size of a largest connected component of this network provided the faulty vertices in the network may break its connectedness. Star graphs are recognized as promising candidates for interconnection networks. This article addresses the size of a largest connected component of a faulty star graph. We prove that, in an n-star graph (n >= 3) with up to 2n-4 faulty vertices, all fault-free vertices but at most two form a connected component. Moreover, all fault-free vertices but exactly two form a connected component if and only if the set of all faulty vertices is equal to the neighbourhood of a pair of fault-free adjacent vertices. These results show that star graphs exhibit excellent fault-tolerant abilities in the sense that there exists a large functional network in a faulty star graph.

Approximate Riemann solutions of the two-dimensional shallow-water equations

A finite-difference scheme based on flux difference splitting is presented for the solution of the two-dimensional shallow-water equations of ideal fluid flow. A linearised problem, analogous to that of Riemann for gasdynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearised problem. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second-order scheme which avoids non-physical, spurious oscillations. An extension to the two-dimensional equations with source terms, is included. The scheme is applied to a dam-break problem with cylindrical symmetry.

Exploring the relationship between listening development and strategy use

This paper reports on an investigation into the development of the listening proficiency and strategic behaviour of 15 lower-intermediate learners of French in England. We consider whether listeners remain in the same listening proficiency group after six months, and whether changes in strategy use are related to movement or non-movement between listening proficiency groups. We also examine whether learners’ strategic behaviour reflects their teachers’ approaches to listening. Data were gathered at two time points from a recall protocol which learners completed after listening to short passages and from verbal reports made by learners while they completed a multiple choice listening task. Teacher interviews provided information on how listening had been presented in learners' classrooms. We detected little movement by students across the listening proficiency groups between the two time points. In spite of some changes in frequency of strategy use, we also observed stability in manner of use by some learners. Differences in strategy use were more evident between groups (non-movers, improvers and decliners) than between uses from Time 1 to Time 2. We conclude by discussing the pedagogical implications of these findings.

Beyond the American century: Walter Lippmann and American grand strategy, 1943-1950

As the United States became a world Power, journalist and intellectual Walter Lippmann feared that it would become its own worst enemy. During and after the Second World War, he tried to steer the country towards coherent statecraft, to define the national interest and the limits of power, and give geopolitical expression to the role of the United States as the core of an Atlantic strategic system. But in response to world war, the Truman Doctrine, and the Korean War, he became pessimistic about the country's ability to conduct strategy effectively. In the prophetic tradition, he believed that a fatal symbiosis between America's growing strength and domestic politics led it towards crisis. Though at times ahistorical, Lippmann's concept of strategy deserves attention for its dialogue between power and identity, for its questioning of “ends” as well as means, and for its focus on the danger of self-defeating behaviour.

Wave scattering by an axisymmetric ice floe of varying thickness

The problem of water wave scattering by a circular ice floe, floating in fluid of finite depth, is formulated and solved numerically. Unlike previous investigations of such situations, here we allow the thickness of the floe (and the fluid depth) to vary axisymmetrically and also incorporate a realistic non-zero draught. A numerical approximation to the solution of this problem is obtained to an arbitrary degree of accuracy by combining a Rayleigh–Ritz approximation of the vertical motion with an appropriate variational principle. This numerical solution procedure builds upon the work of Bennets et al. (2007, J. Fluid Mech., 579, 413–443). As part of the numerical formulation, we utilize a Fourier cosine expansion of the azimuthal motion, resulting in a system of ordinary differential equations to solve in the radial coordinate for each azimuthal mode. The displayed results concentrate on the response of the floe rather than the scattered wave field and show that the effects of introducing the new features of varying floe thickness and a realistic draught are significant.