For an nonnegative matrix , an isomorphism is obtained between the lattice of initial subsets (of ) for and the lattice of -invariant faces of the nonnegative orthant . Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If leaves invariant a polyhedral cone , then for each distinguished eigenvalue of for , there is a chain of distinct -invariant join-irreducible faces of , each containing in its relative interior a generalized eigenvector of corresponding to (referred to as semi-distinguished -invariant faces associated with ), where is the maximal order of distinguished generalized eigenvectors of corresponding to , but there is no such chain with more than members. We introduce the important new concepts of semi-distinguished -invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.
For modern processing of ceramics at the nanoscale, the influence of interparticle interactions in the suspended state becomes increasingly important. The Hamaker 2 program has been developed for the rapid prediction of these interactions, allowing us to gain important understanding of the often delicate balance of forces in ceramic powder suspensions. This article discusses the theoretical foundation of the implemented models and shows the benefit of this predictive approach applied to mullite production by colloidal methods. 相似文献
The stability of the shapes of crystal growth face and dissolution face in a two-dimensional mathematical model of crystal growth from solution under microgravity is studied.It is proved that the stable shapes of crystal growth face and dissolution face do exist,which are suitably shaped curves with their upper parts inclined backward properly.The stable shapes of crystal growth faces and dissolution faces are calculated for various values of parameters,Ra,Pr and Sc.It is shown that the stronger the convection relative to the diffusion in solution is,the more backward the upper parts of the stable crystal growth face and dissolution face are inclined.The orientation and the shape of dissolution face hardly affect the stable shape of crystal growth face and vice versa. 相似文献