Characterizations of isometric isomorphisms between uniform algebras via nonlinear range-preserving properties

Let be a surjective mapping from a uniform algebra on a compact Hausdorff space onto a uniform algebra on a compact Hausdorff space . Suppose that holds for every . Then we have that is an almost isometric isomorphism, which is a generalization of results of Molnár (2002) and Rao and Roy (2005).

     

When does the subadditivity theorem for multiplier ideals hold?

Demailly, Ein and Lazarsfeld proved the subadditivity theorem for multiplier ideals on nonsingular varieties, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals. We prove that, in the two-dimensional case, the subadditivity theorem holds on log terminal singularities. However, in the higher dimensional case, we have several counterexamples. We consider the subadditivity theorem for monomial ideals on toric rings and construct a counterexample on a three-dimensional toric ring.

     

“Molecular Rope Curtain” Type of Liquid Crystals Based on a Sliding Graft Copolymer Having Mobile PEG Side Chains

A sliding graft copolymer having a polyrotaxane main chain and side chains of polyethylene glycol with a molecular weight of 2000 (PEG2000) was found to be a novel “molecular rope curtain” type of liquid crystalline material, which displayed a smectic A mesophase between 51 °C and 250 °C without any rigid mesogenic moieties.     

Extraction of alkali metal picrates by crown ethers having uncharged coordinating flexible side-chains

Crown ethers were synthesized which have a flexible, neutral side-chain capable of providing the crown ether-bound metal with an additional binding site from the axial direction. Liquid—liquid extraction of alkali metal picrates with these crown ethers was studied; the extraction is compared with that of the normal crown ethers as well as those crown ethers with a proton-dissociable side-chain. The neutral side-chain may assist or suppress the ease of metal extraction depending on the site of the ligating atom on the chain. An anionic ligating side-chain interacts favorably with the lighter metals, resulting in a net decrease in the metal extraction selectivity.