Unitary interpolants and factorization indices of matrix functions

For an $n\times n$ bounded matrix function $\Phi$ we study unitary interpolants $U$, i.e., unitary-valued functions $U$ such that $\hat U(j)=\hat\Phi(j)$, $j<0$. We are looking for unitary interpolants $U$ for which the Toeplitz operator $T_U$ is Fredholm. We give a new approach based on superoptimal singular values and thematic factorizations. We describe Wiener--Hopf factorization indices of $U$ in terms of superoptimal singular values of $\Phi$ and thematic indices of $\Phi-F$, where $F$ is a superoptimal approximation of $\Phi$ by bounded analytic matrix functions. The approach essentially relies on the notion of a monotone thematic factorization introduced in [AP]. In the last section we discuss hereditary properties of unitary interpolants. In particular, for matrix functions $\Phi$ of class H^\be+C we study unitary interpolants $U$ of class $QC$.Comment: 20 page

Cosmogenic Mn-53 in meteorites

The distributions of the Mn-53 contents in 106 nonantarctic and 112 antarctic chondrites were analyzed. A correlation analysis of these distributions showed that the correlation coefficient is a maximum r sub max = 0.75 plus or minus 0.03) when the histogram for antarctic chondrites is displaced by 40 plus or minus 8 dpm kg-1Fe towards lower Mn-53 contents. The distribution of Mn-53 saturated contents in nonantarctic ordinary chondrites was investigated as a function of their radiation age (T). It is found that the Mn-53 average content is higher by (21 plus or minus 9)% in H-chondrites with radiation age T approximately 12 Myr than in thse with T 12 Myr. This effect can be attributed to the fact that a considerable proportion of H-chondrites with T 12 Myr originates from a comet or from the objects of Chiron like, with the orbits more inclined to the ecliptic plane and/or more extended, which caused their irradiation by cosmic rays of higher intensity