### $\mathbf{T}\mathbf{o}\mathbf{p}$ and its adjoint relatives

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Introduction: This article will present just one example of a general construction known as the Bernstein-Gelfand-Gelfand (BGG) resolution. It was the motivating example from two lectures on the BGG resolution given at the 19th Czech Winter School on Geometry and Physics held in Srní in January 1999. This article may be seen as a technical example to go with a more elementary introduction which will appear elsewhere [M. Eastwood, Notices Am. Math. Soc. 46, No. 11, 1368-1376 (1999)]. In fact, there...

Summary: The Ado theorem is a fundamental fact, which has a reputation of being a `strange theorem'. We give its natural proof.

Song, Yin and Zhang (Int. J. Numer. Anal. Model. 4: 127-140, 2007) discovered a remarkable property of oscillatory finite element solutions of one-dimensional convection-diffusion problems that leads to a novel numerical method for the solution of such problems. In the present paper this property is described using several figures, then a simple proof of the phenomenon is given which is much more intuitive than the technical analysis of Song et al.

We present a method that in certain sense stores the inverse of the stiffness matrix in $O(NlogN)$ memory places, where $N$ is the number of degrees of freedom and hence the matrix size. The setup of this storage format requires $O\left({N}^{3/2}\right)$ arithmetic operations. However, once the setup is done, the multiplication of the inverse matrix and a vector can be performed with $O(NlogN)$ operations. This approach applies to the first order finite element discretization of linear elliptic and parabolic problems in triangular domains,...