基张量和张量基的概念及其分析

通常张量有整体表示式和分量（指标）表示式，但这二者尚不是全面而完整的。因此，须将矢量中的基矢量和矢量基的概念推广到基张量和张量基。为此，需以基矢量及张量外积为基础，导出基张量和张量基，任一张量可表达为诸基张量的线性和。据此，可以阐明并矢正是二矢量的外积，并建立了其与张量的对应关系，这样的认识对于物理定律的张量方程及其应用有着重要的意义。     

Estimating Particle Shape and Orientation Using Volume Tensors

We develop statistical procedures for estimating shape and orientation of arbitrary three‐dimensional particles. We focus on the case where particles cannot be observed directly, but only via sections. Volume tensors are used for describing particle shape and orientation, and we derive stereological estimators of the tensors. These estimators are combined to provide consistent estimators of the moments of the so‐called particle cover density. The covariance structure associated with the particle cover density depends on the orientation and shape of the particles. For instance, if the distribution of the typical particle is invariant under rotations, then the covariance matrix is proportional to the identity matrix. We develop a non‐parametric test for such isotropy. A flexible Lévy‐based particle model is proposed, which may be analysed using a generalized method of moments in which the volume tensors enter. The developed methods are used to study the cell organization in the human brain cortex.     

Stereological inference on mean particle shape from vertical sections

It was a major breakthrough when design-based stereological methods for vertical sections were developed by Adrian Baddeley and coworkers in the 1980s. Most importantly, it was shown how to estimate in a design-based fashion surface area from observations in random vertical sections with uniform position and uniform rotation around the vertical axis. The great practical importance of these developments is due to the fact that some biostructures can only be recognised on vertical sections. Later, local design-based estimation of mean particle volume from vertical sections was developed. In the present paper, we review these important advances in stereology. Quite recently, vertical sections have gained renewed interest, since it has been shown that mean particle shape can be estimated from such sections. These new developments are also reviewed in the present paper.