Preprint No. A-00-13

Ehrherd Behrends, Fritz Mädler

Transformations into optimal parallelism in euclidean spaces (or: how to explain the shape of the electron-density distribution inside a crystal)

Abstract: Let ${\bf x}_{1},\dots,{\bf x}_{n}$ and ${\bf y}_{1},\dots,{\bf y}_{n}$ be vectors in the finite dimensional euclidean space ${\Bbb R}^d$. We investigate the problem of how one can find a $U$ such that $\sum_{i}\dotsangle U{\bf x}_{i},{\bf y}_{i}\rangle$ is maximal when $U$ runs through the orthogonal group or the special orthogonal group, i.e., we are looking for a $U$ such that the $U{\bf x}_{1},\dots,U{\bf x}_{n}$ and the ${\bf y}_{1},\dots,{\bf y}_{n}$ are ``as parallel as possible''.

For $d=3$ this problem arises, for instance, from the data analysis of crystallographic diffraction experiments on orientationally disordered systems: the ${\bf x}$'s stand for the atom positions of fragments $M_{1}$ of the crystal structure, the ${\bf y}$'s are taken from the set $M_{2}$ of maxima of the electron-density distribution reconstructed from diffraction data, and one must know the above transformations $U$ in order to determine the $M_{1}$-$M_{2}$-configurations of minimum distance since they are responsible for the shape of the density distribution.

It turns out that one can associate a $d\times d$-matrix $G$ with this problem in such a way that the relevant $U$ are precisely those for which the trace of $GU$ is maximal. Using this transformation we are able to provide an explicit solution by means of the singular value decomposition of $G$.

Several further topics in connection with this problem are also discussed. In particular we investigate in which cases the optimal position is unique, and we study generalizations to nondiscrete situations and to the infinite dimensional setting.

Keywords: Euclidean space, (special) orthogonal group, singular value decomposition, nuclear operator

Mathematics Subject Classification (MSC91): 15A18, 46C05, 47B10, 82D25

Language: ENG

Available: Pr-A-00-13.ps, Pr-A-00-13.ps.gz

Contact: Ehrhard Behrends, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (behrends@math.fu-berlin.de)

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