In this paper, we show that as τ → √−1∞, any zero of the Lam´e function converges to either ∞ or a finite point p satisfying Rep = 1 2 and e2πip being an algebraic number. Our proof is based on studying a special family of simply-periodic KdV potentials with period 1, i.e. algebro-geometric simply-periodic solutions of the KdV hierarchy. We show that except the pole 0, all other poles of such KdV potentials locate on the line Rez = 1 2. We also compute explicitly the eigenvalue setofthecorresponding L2[0,1] eigenvalueproblemforsuchKdVpotentials, thus extends Takemura’s works [26, 27]. Our main idea is to apply the classification result for simply-periodic KdV potentials by Gesztesy, Unterkofler and Weikard [11] and the Darboux transformation.

We study the two main types of trajectories of the ABC flow in the near-integrable regime: spiral orbits and edge orbits. The former are helical orbits which are perturbations of similar orbits that exist in the integrable regime, while the latter exist only in the nonintegrable regime. We prove existence of ballistic (i.e., linearly growing) spiral orbits by using the contraction mapping principle in the Hamiltonian formulation, and we also find and analyze ballistic edge orbits. We discuss the relationship of existence of these orbits with questions concerning front propagation in the presence of flows, in particular the question of linear (i.e., maximal possible) front speed enhancement rate for ABC flows.

We show that there exist $(d-1)$ -Ahlfors regular compact sets $E \subset\mathbb{R}^{d}$ , $d\geq2$ such that for any $t< d-1$ , we have $$\sup_{T} \frac{\mathcal{H}^{d-1}(E\cap T)}{w(T)^{t}}< \infty $$ where the supremum is over all tubes $T$ with width $w(T) >0$ . This settles a question of T. Orponen. The sets we construct are random Cantor sets, and the method combines geometric and probabilistic estimates on the intersections of these random Cantor sets with affine subspaces.

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 → +∞.

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