New approaches to structure analysis

There are relationships among the phases and magnitudes of the structure factors that have not been extensively studied regarding their potential for enhancing procedures for structure determination. These relationships arise from a special way of writing the determinantal inequalities that form the necessary and sufficient conditions for a Fourier series to be non-negative. This particular form also lends itself readily to the development of probability measures by the use of the central limit theorem. Higher-order determinants are of interest. The relationships among the phases and magnitudes of the structure factors are algebraic relationships and the focus is on those which retain their reliability, even though the magnitudes of the structure factors contain experimental errors. The future utility of the algebraic relationships depends upon the development of suitable algorithms for solving them to obtain values for the unknown phases. One approach concerns a method for extending the range of least-squares calculations by modifying the defining equations without changing the global minima and by further altering the nature of the minimization function from time to time during the course of the least-squares calculation, while still preserving the global minima. The objective is to smooth the minimization function and alter the remaining false minima from time to time so that the minimization function is not trapped in a false minimum. Some calculations have been made that indicate the nature of the algebraic relationships among the phases and magnitudes and how the results of the calculations are benefited by having large values for the structure-factor magnitudes in the determinants.(ABSTRACT TRUNCATED AT 250 WORDS)