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Data-based detection and quantification of causation in complex, nonlinear dynamical systems is of paramount importance to science, engineering, and beyond. Inspired by the widely used methodology in recent years, the cross-map-based techniques, we develop a general framework to advance towards a comprehensive understanding of dynamical causal mechanisms, which is consistent with the natural interpretation of causality. In particular, instead of measuring the smoothness of the cross-map as conventionally implemented, we define causation through measuring the scaling law for the continuity of the investigated dynamical system directly. The uncovered scaling law enables accurate, reliable, and efficient detection of causation and assessment of its strength in general complex dynamical systems, outperforming those existing representative methods. The continuity scaling-based framework is rigorously established and demonstrated using datasets from model complex systems and the real world.

We extend two results about the ordinary continued fraction expansion to best simultaneous Diophantine approximations of vectors or matrices. The first is Levy-Khintchin Theorem about the almost sure growth rate of the denominators of the convergents. The second is a Theorem of Bosma, Hendrik and Wiedijk about the almost sure limit distribution of the sequence of products q_n d(q_n θ,Z) where the qn's are the denominators of the convergents associated with the real number θ by the ordinary continued fraction algorithm. Beside these two main results, we show that when d≥2, for almost all vectors θ∈R^d, lim inf_{n→∞} q_{n+k} d(q_n θ,Z^d)=0 for all positive integers k, where (q_n)_{n∈N} is the sequence of best approximation denominators of θ.

We study the effective fronts of first order front propagations in two dimensions (n=2) in the periodic setting. Using PDE-based approaches, we show that for every α∈(0,1), the class of centrally symmetric polygons with rational vertices and nonempty interior is admissible as effective fronts for given front speeds in C^{1,α}(T^2,(0,∞)). This result can also be formulated in the language of stable norms corresponding to periodic metrics in T^2. Similar results were known long time ago when n≥3 for front speeds in C^∞(T^n,(0,∞)). Due to topological restrictions, the two dimensional case is much more subtle. In fact, the effective front is C^1, which cannot be a polygon, for given C^{1,1}(T^2,(0,∞)) front speeds. Our regularity requirements on front speeds are hence optimal. To the best of our knowledge, this is the first time that polygonal effective fronts have been constructed in two dimensions.

We study the periodic homogenization of first order front propagations. Based on PDE methods, we provide a simple proof that, for n ≥ 3, the class of centrally symmetric polytopes with rational coordinates and nonempty interior is admissible as effective fronts, which was also established by I. Babenko and F. Balacheff [Sur la forme de la boule unit\'e de la norme stable unidimensionnelle, Manuscripta Math. 119 (2006) 347--358] and M. Jotz [Hedlund metrics and the stable norm, Diff. Geometry Appl. 27 (2009) 543--550] in the form of stable norms as an extension of G. A. Hedlund's classical result [Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. Math. 33 (1932) 719--739]. Besides, we obtain the optimal convergence rate of the homogenization problem for this class.