Hochschild cohomology of qq-Schur algebras

We compute the Hochschild cohomology of any block of $q$-Schur algebras. We focus the even part of this Hochschild cohomology ring. To compute the Hochschild cohomology of $q$-Schur algebras, we prove the following two results: first, we construct two graded algebra surjections between the Hochschild cohomologies of quasi-hereditary algebras because all $q$-Schur algebras over a field are quasi-hereditary. Second, we give the graded algebra isomorphism of Hochschild cohomologies by using a certain derived equivalence.Comment: 18 page

Application of Hoffman modulation contrast microscopy coupled with three-wavelength two-beam interferometry to the in situ direct observation of the growth process of a crystal in microgravity

Direct visualization of three dimensional transfer process of both heat and mass around a growing crystal and mono-molecular growth layers on the surface is possible in situ by means of high resolution Hoffman modulation contrast microscopy coupled with three wavelength two beam Mach-Zehnder interferometry. This in situ observation is very suitable for the verification of the growth mechanism of a crystal in a solution or a melt in microgravity