The aim of this paper is to extend the Morse theory of $(\mathrm{\Lambda }M,E)$ with low differentiability and degenerate critical points, where $\mathrm{\Lambda }M$ is the space of $H^1$-closed curves on an $n$-dimensional compact manifold $M$ endowed with a Finsler metric $F:TM\to R$ and $F:\mathrm{\Lambda }M\to R$ is the associated energy integral, or simply the energy.

Dimensions of spaces of multivariate splines remain unknown in general. A computational method to obtain explicit formulas for the dimension of spline spaces on simplicial partitions is described. The method is based on Hilbert series and Hilbert polynomials. It is applied to conjecture the dimension formulas for splines on the Alfeld split of a simplex and on several other partitions.

In this paper, we state the notion of morphisms in the category of abelian crossed modules and prove that this category is equivalent to the category of strict Picard categories and regular symmetric monoidal functors. The theory of obstructions for symmetric monoidal functors and symmetric cohomology groups are applied to show a treatment of the group extension problem of the type of an abelian crossed module.