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This paper is concerned with the metric properties of the generalized continued fractions (GCFϵ) with the parameter function ϵ(kn), where kn is the nth partial quotient of the GCFϵ expansion. When -1 < ϵ(kn) ≤ 1, Zhong [Metrical properties for a class of continued fractions with increasing digits, J. Number Theory 128 (2008) 1506-1515] obtained the following metrical properties: which are entirely unrelated to the choice of ϵ(kn) ∈ (-1, 1]. Here we deal with the case of ϵ(k) = c(k + 1) with constant c ∈ (0, ∞). It is proved that: which change with the real c ∈ (0, ∞). Note that as c → 0, it indicates that when c → 0, the GCFϵ has the same metrical property as the case of -1 < ϵ(kn) ≤ 1. [ABSTRACT FROM AUTHOR]

Numerical simulations of quasi-static indentation and low velocity impact of low density polymethacrylimide (PMI) Rohacell 51 WF foam using indenters with different nose shapes (conical, truncated-conical, hemi-spherical and flat) were carried out using the finite element code LS-DYNA. A 2D axisymmetric model was generated. A strain-rate dependent material model and r-adaptive remeshing were used for low velocity impact simulations. Numerical predictions matched the available experimental data very well. Moreover, the predicted resistance force closely matched the empirical results. The results demonstrated the ability of the model to reproduce the deformation mechanisms of the penetration process of Rohacell 51 WF foam. [ABSTRACT FROM AUTHOR]

It is well known that every x ∈ (0, 1] can be expanded into an infinite Lüroth series with the form of where dn(x) ≥ 2 and is called the nth digits of x for each n ≥ 1. In [ Representations of Real Numbers by Infinite Series, Lecture Notes in Mathematics, Vol. 502 (Springer, New York, 1976)], Galambos showed that for Lebesgue almost all x ∈ (0, 1], , where Ln(x) = {d1(x), ..., dn(x)} denotes the largest digit among the first n ones of x. In this paper, we consider the Hausdorff dimension of the set for any α ≥ 0. [ABSTRACT FROM AUTHOR]

For x ∈ I, let [A1(x), A2(x), ...] be the continued fraction expansions over the field of Laurent series, write Ln(x) ≔ { A1(x), A2(x), ..., An(x)}, which is called the largest degree of partial quotients. In this paper, we give an iterated logarithm type theorem for Ln(x), and by which, we get that for P-almost all x ∈ I, . Also the Hausdorff dimensions of the related exceptional sets are determined. [ABSTRACT FROM AUTHOR]