Hypersurfaces in a sphere with constant mean curvature

Let be a closed hypersurface of constant mean curvature immersed in the unit sphere . Denote by the square of the length of its second fundamental form. If , is a small hypersphere in . We also characterize all with .

     

On invariants dual to the Bass numbers

Let be a commutative Noetherian ring, and let be an -module. In earlier papers by Bass (1963) and Roberts (1980) the Bass numbers were defined for all primes and all integers by use of the minimal injective resolution of . It is well known that . On the other hand, if is finitely generated, the Betti numbers are defined by the minimal free resolution of over the local ring . In an earlier paper of the second author (1995), using the flat covers of modules, the invariants were defined by the minimal flat resolution of over Gorenstein rings. The invariants were shown to be somehow dual to the Bass numbers. In this paper, we use homologies to compute these invariants and show that

for any cotorsion module . Comparing this with the computation of the Bass numbers, we see that is replaced by and the localization is replaced by (which was called the colocalization of at the prime ideal by Melkersson and Schenzel).

     

On the von Neumann-Jordan constant for Banach spaces

Let be the von Neumann-Jordan constant for a Banach space . It is known that for any Banach space ; and is a Hilbert space if and only if . We show that: (i) If is uniformly convex, is less than two; and conversely the condition implies that admits an equivalent uniformly convex norm. Hence, denoting by the infimum of all von Neumann-Jordan constants for equivalent norms of , is super-reflexive if and only if . (ii) If , (the same value as that of -space), is of Rademacher type and cotype for any with , where ; the converse holds if is a Banach lattice and is finitely representable in or .

     

Rudin's orthogonality problem and the Nevanlinna counting function

Let be a holomorphic function taking the open unit disk into itself. We show that the set of nonnegative powers of is orthogonal in if and only if the Nevanlinna counting function of , , is essentially radial. As a corollary, we obtain that the orthogonality of for a univalent implies for some constant . We also show that if is orthogonal, then the closure of must be a disk.

     

Commensurators of parabolic subgroups of Coxeter groups

Let be a Coxeter system, and let be a subset of . The subgroup of generated by is denoted by and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of in is the subgroup of in such that has finite index in both and . The subgroup can be decomposed in the form where is finite and all the irreducible components of are infinite. Let be the set of in such that for all . We prove that the commensurator of is . In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and is its own commensurator if and only if .

     

Volumes of hyperbolic Haken manifolds, II

We show that if is a closed hyperbolic 3-manifold and if has a non-abelian free quotient, then the volume of is greater than . If, in addition, contains no genus- surface groups, then the volume of is greater than . Using these results we show that if there are infinitely many primitive homology classes in which are not represented by fibroids, then the volume of is greater than .

     

On functions arising as potentials on spaces of homogeneous type

On a space of homogeneous type we consider functions in , , which are potentials of order of functions. We show that these functions belong to the class of smooth functions of Calderón-Scott. This result has applications to tangential convergence.

     

Sharp estimates for the Bochner-Riesz operator of negative order in

The Bochner-Riesz operator on of order is defined by

where denotes the Fourier transform and if , and if . We determine all pairs such that on of negative order is bounded from to . To be more precise, we prove that for the estimate holds if and only if , where

We also obtain some weak-type results for .

     

Semi-free actions of zero-dimensional compact groups on Menger compacta

Let be the -dimensional universal Menger compactum, a -set in and a metrizable zero-dimensional compact group with the unit. It is proved that there exists a semi-free -action on such that is the fixed point set of every . As a corollary, it follows that each compactum with can be embedded in as the fixed point set of some semi-free -action on .

     

A note on GK dimension of skew polynomial extensions

Let be a finitely generated commutative domain over an algebraically closed field , an algebra endomorphism of , and a -derivation of . Then if and only if is locally algebraic in the sense that every finite dimensional subspace of is contained in a finite dimensional -stable subspace.

Similarly, if is a finitely generated field over , a -endomorphism of , and a -derivation of , then if and only if is an automorphism of finite order.