Mathematical methods in small-angle scattering data analysis

Applications of modern mathematical methods to the problems of small-angle scattering data treatment and interpretation are considered. Special possibilities in data treatment, namely simultaneous treatment of data sets obtained with different experimental conditions and joint data processing for the anomalous-dispersion experiments, are presented. The methods are further developments of the indirect method based on the regularization technique [Svergun, Semenyuk & Feigin (1988). Acta Cryst. A44, 244-250]. Recent improvements in the shape-determination technique based on the multipole expansion theory [Stuhrmann (1970). Z. Phys. Chem. (Frankfurt am Main), 72, 177-184, 185-198] are described. A new method is proposed for the determination of positioning and mutual orientation of subunits in complex particles using the spherical harmonics technique.