Zero divisors and

Let be a discrete group, the group ring of over and the Lebesgue space of with respect to Haar measure. It is known that if is torsion free elementary amenable, and , then . We will give a sufficient condition for this to be true when , and in the case we will give sufficient conditions for this to be false when .

     

A note on Greenberg's conjecture and the abc conjecture

For any totally real number field and any prime number , Greenberg's conjecture for asserts that the Iwasawa invariants and are both zero. For a fixed real abelian field , we prove that the conjecture is ``affirmative' for infinitely many (which split in if we assume the abc conjecture for .

     

Improving the metric in an open manifold with nonnegative curvature

The soul theorem states that any open Riemannian manifold with nonnegative sectional curvature contains a totally geodesic compact submanifold such that is diffeomorphic to the normal bundle of . In this paper we show how to modify into a new metric so that:

1. has nonnegative sectional curvature and soul .
2. The normal exponential map of is a diffeomorphism.
3. splits as a product outside of a compact set.

As a corollary we obtain that any such is diffeomorphic to the interior of a convex set in a compact manifold with nonnegative sectional curvature.

     

The primitive -Frobenius groups

Let be a fixed prime. A finite primitive permutation group with every two-point stabilizer being a -group is called a primitive -Frobenius group. Using our earlier results on -intersection subgroups, we give a complete classification of the primitive -Frobenius groups.

     

Automorphic-differential identities and actions of pointed coalgebras on rings

In this paper, we prove the following two results which generalize the theorem concerning automorphic-differential endomorphisms asserted by J. Bergen. Let be a ring, its left Martindale quotient ring and a right ideal of having no nonzero left annihilator. (1) Let be a pointed coalgebra which measures such that the group-like elements of act as automorphisms of . If is prime and for , then . Furthermore, if the action of extends to and if such that , then . (2) Let be an endomorphism of given as a sum of composition maps of left multiplications, right multiplications, automorphisms and skew-derivations. If is semiprime and , then .

     

A counterexample to a question of R. Haydon, E. Odell and H. Rosenthal

We give an example of a compact metric space , an open dense subset of , and a sequence in which is pointwise convergent to a non-continuous function on , such that for every there exists with for all , yet is equivalent to the unit vector basis of the James quasi-reflexive space of order 1. Thus does not embed isomorphically in the closed linear span of . This answers in the negative a question asked by H. Haydon, E. Odell and H. Rosenthal.

     

-convexity of Orlicz-Bochner spaces

A characterization of -convexity of arbitrary Banach space is given. Moreover, it is proved that the Orlicz-Bochner function space is P-convex if and only if both spaces and are -convex. In particular, the Lebesgue-Bochner space with is -convex iff is -convex.

     

A weak-type inequality of subharmonic functions

We prove the weak-type inequality , , between a non-negative subharmonic function and an -valued smooth function , defined on an open set containing the closure of a bounded domain in a Euclidean space , satisfying , and , where is a constant. Here is the harmonic measure on with respect to 0. This inequality extends Burkholder's inequality in which and , a Euclidean space.

     

On the Deleted Product Criterion For Embeddability in

For a space let . Let act on and on by exchanging factors and antipodes respectively. We present a new short proof of the following theorem by Weber: For an -polyhedron and , if there exists an equivariant map , then is embeddable in . We also prove this theorem for a peanian continuum and . We prove that the theorem is not true for the 3-adic solenoid and .

     

On the suspension order of

It is shown that the suspension order of the -fold cartesian product of real projective -space is less than or equal to the suspension order of the -fold symmetric product of and greater than or equal to , where and satisfy and . In particular has suspension order , and for fixed the suspension orders of the spaces are unbounded while their stable suspension orders are constant and equal to .

     

A version of Zabrodsky's lemma

Zabrodsky's Lemma says: Suppose given a fibrant space and a homotopy fiber sequence with connected. If the map which is induced by is a weak equivalence, then is a weak equivalence. This has been generalized by Bousfield. We improve on Bousfield's generalization and give some applications.

     

groups and quasi-equivalence

A torsion-free abelian group is if every map from a pure subgroup of into lifts to an endomorphism of The class of groups has been extensively studied, resulting in a number of nice characterizations. We obtain some characterizations for the class of homogeneous groups, those homogeneous groups such that, for pure in every has a lifting to a quasi-endomorphism of An irreducible group is if and only if every pure subgroup of each of its strongly indecomposable quasi-summands is strongly indecomposable. A group is if and only if every endomorphism of is an integral multiple of an automorphism. A group has minimal test for quasi-equivalence ( if whenever and are quasi-isomorphic pure subgroups of then and are equivalent via a quasi-automorphism of For homogeneous groups, we show that in almost all cases the and properties coincide.

     

-integral bases of a cubic field

A -integral basis of a cubic field is determined for each rational prime , and then an integral basis of and its discriminant are obtained from its -integral bases.

     

An estimate on the distortion of the logarithmic capacity

Let be a Fuchsian group. We show that the existence of a set on with no -equivalent points and positive logarithmic capacity does not imply that is of convergence type.

     

Spectral averaging and the Krein spectral shift

We provide a new proof of a theorem of Birman and Solomyak that if with trace class and , then , where is the Krein spectral shift from to . Our main point is that this is a simple consequence of the formula .

     

A characterization for spaces of sections

The space of smooth sections of a bundle over a compact smooth manifold can be equipped with a manifold structure, called an -manifold, where represents the Fréchet algebra of real valued smooth functions on . We prove that the -manifold structure characterizes the spaces of sections of bundles over and its open subspaces. We also describe the -maps between -manifolds.

     

Extendibility of homogeneous polynomials on Banach spaces

We study the -homogeneous polynomials on a Banach space that can be extended to any space containing . We show that there is an upper bound on the norm of the extension. We construct a predual for the space of all extendible -homogeneous polynomials on and we characterize the extendible 2-homogeneous polynomials on when is a Hilbert space, an -space or an -space.

     

Ditkin's condition for certain Beurling algebras

Let be a locally compact abelian group. A function is said to be a weight if it is locally bounded, Borel measurable and submultiplicative. We call a weight on semi-bounded if there exist a constant and a subsemigroup with such that

for all Using functional analytic methods, we show that all Beurling algebras whose defining weight is semi-bounded satisfy Ditkin's condition.

     

A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs

For any undirected graph , let be the graph parameter introduced by Colin de Verdière. In this paper we show that if and only if is linklessly embeddable (in ). This forms a spectral characterization of linklessly embeddable graphs, and was conjectured by Robertson, Seymour, and Thomas. A key ingredient is a Borsuk-type theorem on the existence of a pair of antipodal linked -spheres in certain mappings . This result might be of interest in its own right. We also derive that for each linklessly embeddable graph , where is the graph parameter introduced by van der Holst, Laurent, and Schrijver. (It is the largest dimension of any subspace of such that for each nonzero , the positive support of induces a nonempty connected subgraph of .)