Hankel determinants, Pad\'e approximations, and irrationality exponents

The irrationality exponent of an irrational number $\xi$, which measures the approximation rate of $\xi$ by rationals, is in general extremely difficult to compute explicitly, unless we know the continued fraction expansion of $\xi$. Results obtained so far are rather fragmentary, and often treated case by case. In this work, we shall unify all the known results on the subject by showing that the irrationality exponents of large classes of automatic numbers and Mahler numbers (which are transcendental) are exactly equal to $2$. Our classes contain the Thue--Morse--Mahler numbers, the sum of the reciprocals of the Fermat numbers, the regular paperfolding numbers, which have been previously considered respectively by Bugeaud, Coons, and Guo, Wu and Wen, but also new classes such as the Stern numbers and so on. Among other ingredients, our proofs use results on Hankel determinants obtained recently by Han.Comment: International Mathematics Research Notices 201

Coniferous Canopy BRF Simulation Based on 3-D Realistic Scene

It is difficulties for the computer simulation method to study radiation regime at large-scale. Simplified coniferous model was investigate d in the present study. It makes the computer simulation methods such as L-systems and radiosity-graphics combined method (RGM) more powerf ul in remote sensing of heterogeneous coniferous forests over a large -scale region. L-systems is applied to render 3-D coniferous forest scenarios: and RGM model was used to calculate BRF (bidirectional refle ctance factor) in visible and near-infrared regions. Results in this study show that in most cases both agreed well. Meanwhiie at a tree and forest level. the results are also good