We prove the existence of a resonance-free region in scattering by a strictly convex obstacle $\mathcal{O}$ with the Robin boundary condition $\partial_{\nu}u+\gamma u|_{\partial\mathcal{O}}=0$ . More precisely, we show that the scattering resonances lie below a cubic curve ℑ$ζ$=−$S$|$ζ$|^{1/3}+$C$. The constant$S$is the same as in the case of the Neumann boundary condition$γ$=0. This generalizes earlier results on cubic pole-free regions obtained for the Dirichlet boundary condition.

In this paper, we will derive some twist criteria for the periodic solution of a periodic scalar Newtonian equation using the third order approximation. As an application to the forced pendulum x ̈ + ω^2 sin x = p(t), we will find an explicit bound P (ω) for the L1 norm, ∥p∥1, of the periodicforcingp(t)usingthefrequencyωasaparametersuchthattheleastamplitudeperiodic solution of the forced pendulum is of twist type when ∥p∥1 < P (ω). The bound P (ω) has the order of O(ω1/2) when ω is bounded away from resonance of orders ≤ 4 and ω → +∞.

We derive the alternating arm exponents of the critical Ising model. We obtain six different patterns of alternating boundary arm exponents which correspond to the boundary conditions (⊖⊕), (⊖ free) and (free free), and the alternating interior arm exponents.

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