On Semi-weakly n-Hyponormal Weighted Shifts

Semi-weak n-hyponormality is defined and studied using the notion of positive determinant partition. Several examples related to semi-weakly n-hyponormal weighted shifts are discussed. In particular, it is proved that there exists a semi-weakly three-hyponormal weighted shift W (alpha) with alpha (0) = alpha (1) < alpha (2) which is not two-hyponormal, which illustrates the gaps between various weak subnormalities.

5a-Butyl-1,3,8,10-tetra-chloro-7,13-bis-(4-nitro-benzo-yl)-5a,6a,12a,12b-tetra-hydro-7H,13H-thieno[2,3-b:4,5-b']bis-(1,4-benzoxazine)

The title compound, C(34)H(24)Cl(4)N(4)O(8)S, is a linear penta-cyclic system formed of two substituted benzoxazinyl groups fused to 2-n-butyl-tetra-hydro-thio-phene. The oxazine ring, which is fused to the n-butyl-substituted side of the thio-phene ring, is in a boat conformation. The other fused oxazine ring and the tetra-hydro-thiene ring are each in an envelope conformation. The bridgehead C atom alpha to both the S and N atoms forms the flap of each envelope. This results in a twist of the penta-cyclic system such that the dihedral angle between the terminal dichloro-benzene rings is 82.92 (8)°. In the crystal, inversion-related mol-ecules form a weakly hydrogen-bonded dimer, with two C-H⋯O inter-actions between an H atom on the oxazine ring and an amide O atom. Additionally, C-H⋯O inter-actions occur between an H atom on a screw-related nitro-benzene ring and an O atom on the nitro-benzene ring of one mol-ecule. One of the Cl atoms and the butyl group are disordered over two sets of sites with occupancy ratios of 0.94 (2):0.06 (2) and 0.624 (4):0.376 (4), respectively.

Pervasive Algebras and Maximal Subalgebras.

A uniform algebra A on its Shilov boundary X is maximal if A is not C(X) and no uniform algebra is strictly contained between A and C(X) . It is essentially pervasive if A is dense in C(F) whenever F is a proper closed subset of the essential set of A. If A is maximal, then it is essentially pervasive and proper. We explore the gap between these two concepts. We show: (1) If A is pervasive and proper, and has a nonconstant unimodular element, then A contains an infinite descending chain of pervasive subalgebras on X . (2) It is possible to find a compact Hausdorff space X such that there is an isomorphic copy of the lattice of all subsets of N in the family of pervasive subalgebras of C(X). (3) In the other direction, if A is strongly logmodular, proper and pervasive, then it is maximal. (4) This fails if the word “strongly” is removed. We discuss examples involving Dirichlet algebras, A(U) algebras, Douglas algebras, and subalgebras of H∞(D), and develop new results that relate pervasiveness, maximality, and relative maximality to support sets of representing measures.

Zeros of Weakly Holomorphic Modular Forms of Levels 2 and 3

Let M-k(#)(N) be the space of weakly holomorphic modular forms for Gamma(0)(N) that are holomorphic at all cusps except possibly at infinity. We study a canonical basis for M-k(#)(2) and M-k(#)(3) and prove that almost all modular forms in this basis have the property that the majority of their zeros in a fundamental domain lie on a lower boundary arc of the fundamental domain.