On a certain family of generalized Laguerre polynomials

Following the work of Schur and Coleman, we prove the generalized Laguerre polynomial is irreducible over the rationals for all n?1 and has Galois group A_n if n+1 is an odd square, and S_n otherwise. We also show that for certain negative integer values of α and certain congruence classes of n modulo 8, the splitting field of L_n^(α)(x) can be embedded in a double cover.     

Circumscribed Polygons of Small Area

Given any plane strictly convex region K and any positive integer n≥3, there exists an inscribed 2n-gon Q _2n and a circumscribed n-gon P _n such that

Central configurations consist of two layer twisted regular polygons

In this paper, we study the existence of a special twisted regular polygonal central configuration in R₃.     

Rotational polygon overlap minimization and compaction

An effective and fast algorithm is given for rotational overlap minimization: given an overlapping layout of polygons P₁, P₂, P₃, …, P_k in a container polygon Q, translate and rotate the polygons to diminish their overlap to a local minimum. A (local) overlap minimum has the property that any perturbation of the polygons increases the overlap. Overlap minimization is modified to create a practical algorithm for compaction: starting with a non-overlapping layout in a rectangular container, plan a non-overlapping motion that diminishes the length or area of the container to a local minimum. Experiments show that both overlap minimization and compaction work well in practice and are likely to be useful in industrial applications.     

Stability estimates for h-p spectral element methods for general elliptic problems on curvilinear domains

In this paper we show that the h-p spectral element method developed in [3,8,9] applies to elliptic problems in curvilinear polygons with mixed Neumann and Dirichlet boundary conditions provided that the Babuska-Brezzi inf-sup conditions are satisfied. We establish basic stability estimates for a non-conforming h-p spectral element method which allows for simultaneous mesh refinement and variable polynomial degree. The spectral element functions are non-conforming if the boundary conditions are Dirichlet. For problems with mixed boundary conditions they are continuous only at the vertices of the elements. We obtain a stability estimate when the spectral element functions vanish at the vertices of the elements, which is needed for parallelizing the numerical scheme. Finally, we indicate how the mesh refinement strategy and choice of polynomial degree depends on the regularity of the coefficients of the differential operator, smoothness of the sides of the polygon and the regularity of the data to obtain the maximum accuracy achievable. An erratum to this article is available at .     

理想材料零件实体模型自动添加支撑技术研究

在一些RP工艺中,给各种形状的零件添加支撑对于完成零件的制造和保证零件的精度非常重要,但目前尚无很好的自动添加支撑算法能够应用于一般的情况．为此提出两种自动添加支撑算法：曲线追踪法和标志位追踪法．前者主要是在切片前基于追踪得到的采样点利用蒙面法重构支撑面片;而后者则是在切片后通过多边形的布尔运算逐层计算各切层端面的实际轮廓和综合轮廓．这两种算法形成了一种新的自动添加支撑方案．实例证明了这种方案的有效性和便利性．