A perturbation bound for the generalized polar decomposition

LetA be anm×n complex matrix. A decompositionA=QH is termed ageneralized polar decomposition ofA ifQ is anm×n subunitary matrix (sometimes also called a partial isometry) andH a positive semidefinite Hermitian matrix. It was proved that a nonzero matrixA ^m×n has a unique generalized polar decompositionA=QH with the property (Q ^H )=(H), whereQ ^H denotes the conjugate transpose ofQ and (H) the column space ofH. The main result of this note is a perturbation bound forQ whenA is perturbed.     

Control of mesoporous structure of aerogels derived from cresol-formaldehyde

Polycondensation of a cresol mixture (C(m)) with formaldehyde (F) in basic aqueous solutions leads to formation of highly cross-linked C(m)F aquagels that can be supercritically dried with carbon dioxide to form organic C(m)F aerogels. Aerogels synthesized with different catalyst contents and reactant concentrations are characterized by low-temperature nitrogen adsorption. The present experimental results suggest that the C(m)F aerogels are typical mesoporous materials and have almost no micropores in bulk. The microstructure of the organic C(m)F aerogels can be controlled and tailored effectively by varying synthesis conditions during the initial sol-gel process. C(m)F organic aerogels with specific surface area as high as 627 m(2)/g and corresponding pore volume 2.06 ml/g have been obtained with a dominant pore size of 30 nm. C(m)F organic aerogels with peaky pore size distributions concentrated at 11 nm have also been prepared.