Extension of covariant derivative(Ⅰ): From component form to objective form

This paper extends the covariant derivative un der curved coordinate systems in 3D Euclid space. Based on the axiom of the covariant form invariability, the classical covariant derivative that can only act on components is ex tended to the generalized covariant derivative that can act on any geometric quantity including base vectors, vectors and tensors. Under the axiom, the algebra structure of the gen eralized covariant derivative is proved to be covariant dif ferential ring. Based on the powerful operation capabilities and simple analytical properties of the generalized covariant derivative, the tensor analysis in curved coordinate systems is simplified to a large extent.

Extension of covariant derivative(Ⅱ): From flat space to curved space

This paper extends the classical covariant deriva tive to the generalized covariant derivative on curved sur faces. The basement for the extension is similar to the pre vious paper, i.e., the axiom of the covariant form invariabil ity. Based on the generalized covariant derivative, a covari ant differential transformation group with orthogonal duality is set up. Through such orthogonal duality, tensor analy sis on curved surfaces is simplified intensively. Under the covariant differential transformation group, the differential invariabilities and integral invariabilities are constructed on curved surfaces.

Generalized covariant differentiation and axiom-based tensor analysis

This paper reports the new progresses in the axiomatization of tensor anal- ysis, including the thought of axiomatization, the concept of generalized components, the axiom of covariant form invariability, the axiomatized definition, the algebraic structure, the transformation group, and the simple calculation of generalized covariant differentia- tions. These progresses strengthen the tendency of the axiomatization of tensor analysis.

Generalized Covariant Derivative with Respect to Time in Flat Space(Ⅰ):Eulerian Description

This paper reports a new derivative in the Eulerian description in flat space-the generalized covariant derivative with respect to time. The following contents are included:(a) the restricted covariant derivative with respect to time for Eulerian component is defined;(b) the postulate of the covariant form invariability in time field is set up;(c) the generalized covariant derivative with respect to time for generalized Eulerian component is defined;(d) the algebraic structure of the generalized covariant derivative with respect to time is made clear;(e) the covariant differential transformation group in time filed is derived. These progresses reveal the covariant form invariability of Eulerian space and time.

Generalized Covariant Derivative with Respect to Time in Flat Space(Ⅱ):Lagrangian Description

The previous paper reported a new derivative in the Eulerian description in flat space—the generalized covariant derivative of generalized Eulerian component with respect to time. This paper extends the thought from the Eulerian description to the Lagrangian description:on the basis of the postulate of covariant form invariability in time field, we define a new derivative in the Lagrangian description in flat space—the generalized covariant derivative of generalized Lagrangian component with respect to time. Besides, the covariant differential transformation group is set up. The covariant form invariability of Lagrangian space-time is ascertained.

稳健最优不变二次无偏估计

导出了稳健最优不变二次无偏估计、稳健最小范数二次无偏估计、稳健Ｈｅｌｍｅｒｔ估计，并说明了最优不变二次无偏估计、最小范数二次无偏估计以及Ｈｅｌｍｅｒｔ估计等均是稳健最优不变二次无偏估计的特例。