Given a bivariate polynomial $f(x,y)$, let $\phi(y)$ be a power series root of $f(x,y)=0$ with respect to $x$, i.e., $\phi(y)$ is a function of $y$ such that $f(\phi(y),y)=0$. If $\phi(y)$ is analytic at $y=0$, then we have its power series expansion \begin{equation} \phi(y)=\alpha_{0}+\alpha_{1}y+\alpha_{2}y^{2}+\cdots+\alpha_{r}y^{r}+\cdots. \end{equation} Let $\phi^{(k)}(y)$ denote $\phi(y)$ truncated at $y^{k}$, i.e., \begin{equation} \phi^{(k)}(y)=\alpha_{0}+\alpha_{1}y+\alpha_{2}y^{2}+\cdots+\alpha_{k}y^{k}. \end{equation} Then, it is well known that, given initial value $\phi^{(0)}(y)=\alpha_{0}\in\mathbf{C}$, the symbolic Newton's method with the formula \begin{equation} \phi^{(2^{m}-1)}(y)\gets\phi^{(2^{m-1}-1)}(y) -\frac{f(\phi^{(2^{m-1}-1)}(y),y)}{\frac{\partial f}{\partial x}(\phi^{(2^{m-1}-1)}(y),y)} \quad (\mod y^{2^{m}}) \end{equation} computes $\phi^{(2^{m}-1)}(y)$ ($1\le m$) in (2) with quadratic convergence (the roots are computed in the order $\phi^{(0)}(y) \to \phi^{(2^{1}-1)}(y) \to \phi^{(2^{2}-1)}(y) \to \cdots \to \phi^{(2^{m}-1)}(y)$). References [1] and [3] indicate that the symbolic Newton's method can be generalized so that its convergence degree is an arbitrary integer $p$ where its roots are computed in the order $\phi^{(0)}(y) \to \phi^{(p-1)}(y) \to \phi^{(p^{2}-1)}(y) \to \cdots \to \phi^{(p^{m}-1)}(y)$. Although the high degree convergent formula in [1] and [3] requires fewer iterations than the symbolic Newton's method, it may not be efficient as expected, since one iteration of the formula requires more computations than one in the symbolic Newton's method.

In this paper, we combine the polynomial evaluation method in [9] with the formula of arbitrary degree convergence and propose an algorithm that computes the above power series root $\phi^{(k)}(y)$. We analyze the complexity of the algorithm and give the number of multiplications/divisions required to compute a power series root in an explicit form. It is shown that when the degree of polynomial $f(x,y)$ is high, high degree convergent formula is advantageous over the symbolic Newton's method.

In this paper, we investigate a mode selection problem for the Turing patterns generated from small random initial disturbances in one-dimensional reaction-diffusion systems on a sufficiently large domain. For this problem, it is widely accepted that the maximizer of the dispersion relation give rise to the wavenumber to be selected. Even in a small neighborhood of the bifurcation point, our numerical experiments show that this is not always true.

We consider modulated Poisson--Voronoi tessellations, intended as models for telecommunication networks on a nationwide scale. By introducing an algorithm for the simulation of the typical cell of the latter tessellation, we lay the mathematical foundation for such a global analysis. A modulated Poisson--Voronoi tessellation has an intensity which is spatially variable and, hence, is able to provide a broad spectrum of model scenarios. Nevertheless, the considered tessellation model is stationary and we consider the case where the modulation is generated by a Boolean germ-grain model with circular grains. These circular grains may either have a deterministic or random but bounded radius. Furthermore, based on the introduced simulation algorithm for the typical cell and on Neveu's exchange formula for Palm probability measures, we show how to estimate the mean distance from a randomly chosen location to its nearest Voronoi cell nucleus. The latter distance is interpreted as an important basic cost characteristic in telecommunication networks, especially for the computation of more sophisticated functionals later on. Said location is chosen at random among the points of another modulated Poisson process where the modulation is generated by the same Boolean model as for the nuclei. The case of a completely random placement for the considered location is thereby included as a special case. The estimation of the cost functional is performed in a way such that a simulation of the location placement is not necessary. Test methods for the correctness of the algorithm based on tests for random software are briefly discussed. Numerical examples are provided for characteristics of the typical cell as well as for the cost functional. We conclude with some remarks about extensions and modifications of the model regarded in this paper, like modulated Poisson--Delaunay tessellations.

In order to study the buckled states of an elastic ring under uniform pressure, Tadjbakhsh and Odeh [14] introduced an energy functional which is a linear combination of the total squared curvature (elastic energy) and the area enclosed by the ring. We prove that the minimizer of the functional is not a disk when the pressure is large, and its curvature can be expressed by Jacobian elliptic $\cn({}\cdot{})$ function. Moreover, the uniqueness of the minimizer is proven for certain range of the pressure.