Row-complete latin squares of every composite order exist

A construction for a row-complete latin square of order n, where n is any odd composite number other than 9, is given in this article. Since row-complete latin squares of order 9 and of even order have previously been constructed, this proves that row-complete latin squares of every composite order exist. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:63–77, 1998     

Ueber die Bestimmung des Cadmiums

Ohne Zusammenfassung     

Multidentate aminophenol ligands prepared with Mannich condensations

Mannich condensations were used to prepare a series of multidentate aminophenol compounds from ethylenediamine, various amounts of paraformaldehyde and 2,4-disubstituted phenols. One of these diaminotetraphenol compounds was reacted with Zn(OAc)₂·2H₂O and the resulting solid state structure of the pentacoordinated Zn(II) compound was determined by single crystal X-ray diffraction analysis.     

Perturbation theory and backward error forAX−XB=C

Because of the special structure of the equationsAX–XB=C the usual relation for linear equations backward error = relative residual does not hold, and application of the standard perturbation result forAx=b yields a perturbation bound involving sep (A, B)^–1 that is not always attainable. An expression is derived for the backward error of an approximate solutionY; it shows that the backward error can exceed the relative residual by an arbitrary factor. A sharp perturbation bound is derived and it is shown that the condition number it defines can be arbitrarily smaller than the sep(A, B)^–1-based quantity that is usually used to measure sensitivity. For practical error estimation using the residual of a computed solution an LAPACK-style bound is shown to be efficiently computable and potentially much smaller than a sep-based bound. A Fortran 77 code has been written that solves the Sylvester equation and computes this bound, making use of LAPACK routines.Nuffield Science Research Fellow. This work was carried out while the author was a visitor at the Institute for Mathematics and its Applications, University of Minnesota.     

Factorizing complex symmetric matrices with positive definite real and imaginary parts

Complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for LU factorization. This result adds to the classes of matrix for which it is known to be safe not to pivot in LU factorization. Block factorization with the pivoting strategy of Bunch and Kaufman is also considered, and it is shown that for such matrices only pivots are used and the same growth factor bound of 2 holds, but that interchanges that destroy band structure may be made. The latter results hold whether the pivoting strategy uses the usual absolute value or the modification employed in LINPACK and LAPACK.