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For m ∈ N and β ∈ (1 , m + 1 ] , we consider the set U m , β consisting of unique β -expansions in { 0 , 1 , ⋯ , m } N ∖ { 0 ∞ , m ∞ }. Let k ∈ { 1 , ⋯ , m } with k > k ‾ where k ‾ : = m − k. We determine the bifurcation value of β 's, below which in any w ∈ U m , β the digit frequencies of k and k ‾ exist and are equal, and above which there are many w ∈ U m , β , consisting of a set of positive dimension, such that the digit frequencies of k and k ‾ in w do not exist. We also determine the bifurcation value of β 's, below which in any w ∈ U m , β the upper and lower frequencies of k are respectively equal to the upper and lower frequencies of k ‾ , and above which there exists c > 0 , such that for any r ∈ (− c , c) , there are many w ∈ U m , β , consisting of a set of positive dimension, such that the difference of the digit frequencies of k and k ‾ in w is exactly equal to r. For a video summary of this paper, please visit https://youtu.be/IJvlWTt5DIQ. [ABSTRACT FROM AUTHOR]

We introduce generalized Thue-Morse numbers of the form π β (θ) : = ∑ n = 1 ∞ θ n β n where β ∈ (1 , m + 1 ] with m ∈ N and θ = (θ n) n ≥ 1 ∈ { 0 , 1 , ⋯ , m } N is a generalized Thue-Morse sequence previously studied by many authors in different terms. This is a natural generalization of the classical Thue-Morse number ∑ n = 1 ∞ t n 2 n where (t n) n ≥ 0 is the well-known Thue-Morse sequence 01101001⋯. We study when θ would be the unique, greedy, lazy, quasi-greedy and quasi-lazy β -expansions of π β (θ) , and generalize a result given by Kong and Li in 2015. In particular we deduce that the shifted Thue-Morse sequence (t n) n ≥ 1 is the unique β -expansion of ∑ n = 1 ∞ t n β n if and only if it is the greedy expansion, if and only if it is the lazy expansion, if and only if it is the quasi-greedy expansion, if and only if it is the quasi-lazy expansion, and if and only if β is no less than the Komornik-Loreti constant. [ABSTRACT FROM AUTHOR]

Graphical abstract Abstract Hypothesis Hydrophobic spheres may exhibit three impact modes after impacting the liquid surface. For the impact of common non-spherical particles in practical processes, particle shape affects the flow of surrounding fluid and forces acting on them, thus the impact behaviors of non-spherical particles may differ from those of spherical particles and remain to be revealed. Simulation The impact of hydrophobic micron ellipsoidal particles is numerically studied by solving the coupled equations of particle motion and surrounding fluid flow combining VOF method and dynamic meshing technique. The motion characteristics of the ellipsoids and fluids, and the main forces acting on the ellipsoids during impact are analyzed. Findings The increase in the axis ratio (AR) of ellipsoid decreases the surface tension and fluid force, resulting in a smaller total force acting on the ellipsoid. Correspondingly, the ellipsoid’s impact behavior changes. An impact-mode phase diagram is presented for the studied ellipsoids. Three impact modes, namely, submergence, rebound, and oscillation, exist when AR > 0.8, while only submergence and oscillation exist when AR ≤ 0.8. The critical velocities decrease with the increase in AR , which is well illustrated by the analysis of energy conversion under critical conditions. [ABSTRACT FROM AUTHOR]

Although the static flotation of elongated cylinders is well understood, their flotation behavior under dynamic conditions after contacting a liquid surface, such as in situations experienced by water-walking insects, still need to be revealed. The motion of elongated horizontal cylinders after zero-velocity contact with a liquid surface is considered, and the motion equation of the cylinder is established and solved to obtain the dynamic flotation conditions. The limiting density ratios for the dynamic flotation of elongated cylinders with different Bond numbers and contact angles are determined. The results show that although increasing the hydrophobicity promotes floatability of the elongated cylinders, there is little effect for contact angles >120°. Based on the energy balance, an asymptotic formula for the limiting density ratio is proposed, which agrees well with the theoretical results. Unlike small spheres, all elongated cylinders with contact angles >0° are observed to exhibit floatability after contacting a liquid surface. In addition, we found that elongated hydrophilic cylinders exhibit better floatability compared to small hydrophilic spheres with the same contact angle and Bond number, which assists small insects float and survive in various water environments. [ABSTRACT FROM AUTHOR]