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.

     

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 .

     

Relative boundedness and relative compactness for linear operators in Banach spaces

If and are linear operators acting between Banach spaces, we show that compactness of relative to does not in general imply that has -bound zero. We do, however, give conditions under which the above implication is valid.

     

Finite generation properties for fuchsian group von Neumann algebras tensor

We prove that the algebra , a free group with finitely many generators, contains a subnormal operator such that the linear span of the set is weakly dense in . This is the analogue for the factor , finite, of a well known fact about the unilateral shift on a Hilbert space : the linear span of all the monomials is weakly dense in .

We also show that for a suitable space of square summable analytic functions, if is the projection from the Hilbert space of all square summable functions onto and is the unbounded operator of multiplication by on , then the (unbounded) operator is nonzero (with nonzero domain).

     

Annihilating a subspace of with the sign of a continuous function

Let be a -finite, nonatomic, Baire measure space. Let be a finite dimensional subspace of . There is a bounded, continuous function, , defined on , such that

(1) for all , and (2) almost everywhere.

     

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?

     

On a theorem of Barbara Schmid

Let be a finite group and a complex representation. Barbara Schmid has shown that the algebra of invariant polynomial functions on the vector space is generated by homogeneous polynomials of degree at most , where is the largest degree of a generator in a minimal generating set for , and is the complex regular representation of . In this note we give a new proof of this result, and at the same time extend it to fields whose characteristic is larger than , the order of the group .

     

On Euler products associated with noncuspidal metaplectic forms

In this paper, we obtain an Euler product with functional equation associated to a noncuspidal metaplectic form on the double cover of . Zagier's idea of Rankin-Selberg method is used to define the convolution of and the -function.

     

Extensions of holomorphic maps through hypersurfaces and relations to the Hartogs extensions in infinite dimension

A generalization of Kwack's theorem to the infinite dimensional case is obtained. We consider a holomorphic map from into , where is a hypersurface in a complex Banach manifold and is a hyperbolic Banach space. Under various assumptions on , and we show that can be extended to a holomorphic map from into . Moreover, it is proved that an increasing union of pseudoconvex domains containing no complex lines has the Hartogs extension property.

     

A criterion for splitting -algebras in tensor products

The goal of the paper is to prove the following theorem: if , are unital -algebras, simple and nuclear, then any -subalgebra of the -tensor product of and , which contains the tensor product of with the scalar multiples of the unit of , splits in the -tensor product of with some -subalgebra of .

     

Open subgroups of

Let be a locally compact group and let denote the -algebra generated by left translation operators on . Let and be the spaces of almost periodic and weakly almost periodic functionals on the Fourier algebra , respectively. It is shown that if contains an open abelian subgroup, then (1) if and only if is norm dense in ; (2) is a -algebra if is norm dense in , where denotes the set of elements in with compact support. In particular, for any amenable locally compact group which contains an open abelian subgroup, has the dual Bohr approximation property and is a -algebra.

     

Splitting sets in integral domains

Let be an integral domain. A saturated multiplicatively closed subset of is a splitting set if each nonzero may be written as where and for all . We show that if is a splitting set in , then is a splitting set in , a multiplicatively closed subset of , and that is a splitting set in is an lcm splitting set of , i.e., is a splitting set of with the further property that is principal for all and . Several new characterizations and applications of splitting sets are given.

     

On Tate-Shafarevich groups of abelian varieties

Let be a finite Galois extension of number fields with Galois group , let be an abelian variety defined over , and let and denote, respectively, the Tate-Shafarevich groups of over and of over . Assuming that these groups are finite, we derive, under certain restrictions on and , a formula for the order of the subgroup of of -invariant elements. As a corollary, we obtain a simple formula relating the orders of , and when is a quadratic extension and is the twist of by the non-trivial character of .

     

A note on -bases of rings

Let be a tower of rings of characteristic . Suppose that is a finitely presented -module. We give necessary and sufficient conditions for the existence of -bases of over . Next, let be a polynomial ring where is a perfect field of characteristic , and let be a regular noetherian subring of containing such that . Suppose that is a free -module. Then, applying the above result to a tower of rings, we shall show that a polynomial of minimal degree in is a -basis of over .

     

Twin solutions to singular boundary value problems

In this paper we establish the existence of two nonnegative solutions to singular and singular focal boundary value problems. Our nonlinearity may be singular at , and/or .

     

A bound on the reduction number of a primary ideal

Let be a local ring of positive dimension and let be an -primary ideal. We denote the reduction number of by , which is the smallest integer such that for some reduction of In this paper we give an upper bound on in terms of numerical invariants which are related with the Hilbert coefficients of when is Cohen-Macaulay. If , it is known that where denotes the multiplicity of If in Corollary 1.5 we prove where is the first Hilbert coefficient of From this bound several results follow. Theorem 1.3 gives an upper bound on in a more general setting.

     

Curvature restrictions on convex, timelike surfaces in Minkowski 3-space

Suppose that and are Minkowski Gauss curvature and Minkowski mean curvature respectively on a timelike surface that is immersed in Minkowski 3-space . Suppose also that and that is complete as a surface in the underlying Euclidean 3-space . It is shown that neither nor can be bounded away from zero on such a surface .

     

Convergence of paths for pseudo-contractive mappings in Banach spaces

Let be a real Banach space, let be a closed convex subset of , and let , from into , be a pseudo-contractive mapping (i.e. for all and 1)$">. Suppose the space has a uniformly Gâteaux differentiable norm, such that every closed bounded convex subset of enjoys the Fixed Point Property for nonexpansive self-mappings. Then the path , , defined by the equation is continuous and strongly converges to a fixed point of as , provided that satisfies the weakly inward condition.

     

The spectral properties of certain linear operators and their extensions

Let be a Hilbert space with inner-product , and let be a bounded positive operator on which determines an inner-product, . Denote by the completion of with respect to the norm . In this paper, operators having certain relationships with are studied. In particular, if where , then has an extension , and and have essentially the same spectral and Fredholm properties.