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https://www.selleckchem.com/products/ly-411575.html Nanoscale pattern formation on the surface of a solid that is bombarded with a broad ion beam is studied for angles of ion incidence, θ, just above the threshold angle for ripple formation, θ_c. We carry out a systematic expansion in powers of the small parameter ε≡(θ-θ_c)^1/2 and retain all terms up to a given order in ε. In the case of two diametrically opposed, obliquely incident beams, the equation of motion close to threshold and at sufficiently long times is rigorously shown to be a particular version of the anisotropic Kuramoto-Sivashinsky equation. We also determine the long-time, near-threshold scaling behavior of the rippled surface's wavelength, amplitude, and transverse correlation length for this case. When the surface is bombarded with a single obliquely incident beam, linear dispersion plays a crucial role close to threshold and dramatically alters the behavior highly ordered ripples can emerge at sufficiently long times and solitons can propagate over the solid surface. A generalized crater function formalism that rests on a firm mathematical footing is developed and is used in our derivations of the equations of motion for the single and dual beam cases.A bredge (bridge-edge) in a network is an edge whose deletion would split the network component on which it resides into two separate components. Bredges are vulnerable links that play an important role in network collapse processes, which may result from node or link failures, attacks, or epidemics. Therefore, the abundance and properties of bredges affect the resilience of the network to these collapse scenarios. We present analytical results for the statistical properties of bredges in configuration model networks. Using a generating function approach based on the cavity method, we calculate the probability P[over ̂](e∈B) that a random edge e in a configuration model network with degree distribution P(k) is a bredge (B). We also calculate the joi

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