We present a generalisation of the classical Lanchester model for directed fire between two combat forces but now employing networks for the manoeuvre of Blue and Red forces, and the pattern of engagement between the two. The model therefore integrates fires between dispersed elements, as well as manoeuvre through an internal-to-each-side diffusive interaction. We explain the model with several simple examples, including cases where conservation laws hold. We then apply an optimisation approach where, for a fixed-in-structure adversary, we optimise the internal manoeuvre and external engagement structures where the trade-off between maximising damage on the adversary and minimising own-losses can be examined. In the space of combat outcomes this leads to a sequence of transitions from defeat to stalemate and then to victory for the force with optimised networks. Depending on the trade-off between destruction and self-preservation, the optimised networks develop a number of structures including the appearance of so-called sacrificial nodes, that may be interpreted as feints, manoeuvre hubs, and suppressive fires. We discuss these in light of Manoeuvre Warfare theory.
We provide a graph theoretic construction enabling tree-based hierarchies to be modified into structures with enhanced connectivity and synchronisability. Specifically, the construct transforms trees into members of a family of graphs known as expanders, which we call 'expander-augmented-hierarchies' or trees. We show that this produces graphs with significantly enhanced synchronisation properties in the context of the Kuramoto model of phase oscillators coupled on networks. When considered as organisational structures these networks enjoy both the managability of simple hierarchies with near regular degree distribution, and low critical coupling by the addition of relatively few extra edges. For the expander augmented tree, we examine the synchronisation properties, computed through the time-averaged Kuramoto order parameter over an ensemble of natural frequencies. We compare this with a range of other networks including hierarchies augmented by random matching of the leaf nodes. For these we compute the ratio Q of smallest to largest Laplacian eigenvalues, the smallness of which has been argued to be an indicator of good synchronisability. While not the best of these in Q, the expander-augmentedhierarchy exhibits synchronisability barely distinguishable from others with lower Q within the variance over an ensemble of natural frequencies. However, the expander augmented tree has the advantage that its properties are automatically designed for as opposed to the outcome of a random search for low Q-value graphs that in itself scales very poorly.
We consider the Cauchy problem for the Burgers hierarchy with general time dependent coefficients. The closed form for the Green's function of the corresponding linear equation of arbitrary order N is shown to be a sum of generalised hypergeometric functions. For suitably damped initial conditions we plot the time dependence of the Cauchy problem over a range of N values. For N = 1, we introduce a spatial forcing term. Using connections between the associated second order linear Schrödinger and Fokker-Planck equations, we give closed form expressions for the corresponding Green's functions of the sinked Bessel process with constant drift. We then apply the Green's function to give time dependent profiles for the corresponding forced Burgers Cauchy problem.
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