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We prove that the solutions to the discrete nonlinear Schrödinger equation with non-local algebraically decaying coupling converge strongly in$$L^2({\mathbb {R}}^2)$$$L^2\left(R^2\right)$to those of the continuum fractional nonlinear Schrödinger equation, as the discretization parameter tends to zero. The proof relies on sharp dispersive estimates that yield the Strichartz estimates that are uniform in the discretization parameter. An explicit computation of the leading term of the oscillatory integral asymptotics is used to show that the best constants of a family of dispersive estimates blow up as the non-locality parameter$$\alpha \in (1,2)$$$\alpha \in (1,2)$approaches the boundaries.

 

The advent of Persistent Memory (PM) devices enables systems to actively persist information at low costs, including program state traditionally in volatile memory. However, this trend poses a reliability challenge in which multiple classes of soft faults that go away after restart in traditional systems turn into hard (recurring) faults in PM systems. In this paper, we first characterize this rising problem with an empirical study of 28 real-world bugs. We analyze how they cause hard faults in PM systems. We then propose Arthas, a tool to effectively recover PM systems from hard faults. Arthas checkpoints PM states via fine-grained versioning and uses program slicing of fault instructions to revert problematic PM states to good versions. We evaluate Arthas on 12 real-world hard faults from five large PM systems. Arthas successfully recovers the systems for all cases while discarding 10× less data on average compared to state-of-the-art checkpoint-rollback solutions.