We prove an analogue of Lang's conjecture on divisible groups for polynomial dynamical systems over number fields. In our setting, the role group is taken by small orbit a point $\alpha$ where $f$ given \begin{align*} \mathcal{S}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) n}(\alpha) \text{ some } n \mathbb{Z}_{\geq 0}\}. \end{align*} Our main theorem classification algebraic relations ...

Many popular curve estimators based on smoothing have diicul-ties caused by boundary eeects. These eeects are visually disturbing in practice and can play a dominant role in theoretical analysis. Local polynomial regression smoothers are known to correct boundary eeects automatically. Some analogs are implemented for density estimation and the resulting estimators also achieve automatic boundar...

This paper presents a general framework for the construction of piecewise-polynomial local interpolants with given smoothness and approximation order, defined on non-uniform knot partitions. We design such splines through a suitable combination of polynomial interpolants with either polynomial or rational, compactly supported blending functions. In particular, when the blending functions are ra...

The essentially non-oscillatory (ENO) method is an efficient high order numerical method for solving hyperbolic conservation laws designed to reduce the Gibbs oscillations, if existent, by adaptively choosing the local stencil for the interpolation. The original ENO method is constructed based on the polynomial interpolation and the overall rate of convergence provided by the method is uniquely...

Multivariate local polynomial fitting is applied to the multivariate linear heteroscedastic regression model. Firstly, the local polynomial fitting is applied to estimate heteroscedastic function, then the coefficients of regression model are obtained by using generalized least squares method. One noteworthy feature of our approach is that we avoid the testing for heteroscedasticity by improvin...