Normalizers of the congruence subgroups of the Hecke group II

Let . Let be an ideal of and let be the maximal ideal of such that . Then . In particular, if is square free, then is self-normalized in .

     

Collapsibility of and some related CW complexes

Let denote the order complex of the partition lattice. The natural -action on the set induces an -action on . We show that the regular CW complex is collapsible. Even more, we show that is collapsible, where is a suitable type selection of the partition lattice. This allows us to generalize and reprove in a conceptual way several previous results regarding the multiplicity of the trivial character in the -representation on .

     

Statistical extensions of some classical Tauberian theorems

Hardy's well-known Tauberian theorem for Cesàro means says that if the sequence satisfies and , then . In this paper it is shown that the hypothesis can be replaced by the weaker assumption of the statistical limit: st-lim , i.e., for every , . Similarly, the ``one-sided' Tauberian theorem of Landau and Schmidt's Tauberian theorem for the Abel method are extended by replacing and with st-lim and st-lim , respectively. The Hardy-Littlewood Tauberian theorem for Borel summability is also extended by replacing , where is a continuous parameter, with , and further replacing it by -st-lim , where is the Borel matrix method.

     

On Burgess's theorem and related problems

Let be a graph. We determine all graphs which are -like. We also prove that if are graphs, then in order that each -like continuum be -indecomposable for some it is necessary and sufficient that if is a graph, then is not -like for some integer with . This generalizes a well known theorem of Burgess.

     

Weyl's theorem holds for algebraically hyponormal operators

In this note it is shown that if is an ``algebraically hyponormal" operator, i.e., is hyponormal for some nonconstant complex polynomial , then for every , Weyl's theorem holds for , where denotes the set of analytic functions on an open neighborhood of .

     

Injective resolutions of Noetherian rings and cogenerators

We give new construction of injective resolutions of complexes and bimodules. Applying this construction to an injective resolution of a Noetherian ring, we construct a -embedding cogenerator for the category of modules of projective dimension . Moreover, for a Noetherian projective -algebra , we show that satisfies the Auslander condition if and only if the flat dimension of every -module is equal to or larger than the one of the injective hull .

     

Connected sums of manifolds which induce approximate fibrations

Codimension-2 fibrators are -manifolds which automatically induce approximate fibration, in the following sense: given any proper mapping from an -manifold onto a -manifold such that each point-preimage is a copy of the codimension-2 fibrator, is necessarily an approximate fibration. In this paper, we give some answers to the following question: given an -manifold which is a nontrivial connected sum, when is a codimension-2 fibrator?

     

Maximal estimates for the -dimensional Walsh-Fourier series

The -dimensional dyadic martingale Hardy spaces are introduced and it is proved that the maximal operator of the means of a Walsh-Fourier series is bounded from to and is of weak type , provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain that the means of a function converge a.e. to the function in question. Moreover, we prove that the means are uniformly bounded on whenever . Thus, in case , the means converge to in norm. The same results are proved for the conjugate means, too.

     

A characterization of total reflection orders

Let be a Coxeter system with set of reflections . It is known that if is a total reflection order for , then, for each , and its complement are stable under conjugation by . Moreover the upper and lower -conjugates of are still total reflection orders. For any total order on , say that is stable if is stable under conjugation by for each . We prove that if and all orders obtained from by successive lower or upper -conjugations are stable, then is a total reflection order.