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We demonstrate a one-dimensional magnetic system can exhibit a Cantor-type spectrum using an example of a chain graph with coupling at the vertices exposed to a magnetic field perpendicular to the graph plane and varying along the chain. If the field grows linearly with an irrational slope, measured in terms of the flux through the loops of the chain, we demonstrate the character of the spectrum relating it to the almost Mathieu operator.

We study an implicit predictor-corrector iteration process for finitely many asymptotically quasi-nonexpansive self-mappings on a nonempty closed convex subset of a Banach space. We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. In the case is a uniformly convex Banach space and the mappings are asymptotically nonexpansive, we verify the weak (resp., strong) convergence of this iteration process to a common fixed point of these mappings if Opial's condition is satisfied (resp., one of these mappings is semicompact). Our results improve and extend earlier and recent ones in the literature.

As a continuation of our series works on the Boltzmann equation without angular cutoff assumption, in this part, the global existence of solution to the Cauchy problem in the whole space is proved in some suitable weighted Sobolev spaces for hard potential when the solution is a small perturbation of a global equilibrium.

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