A sequential property of and a covering property of Hurewicz

has the monotonic sequence selection property if there is for each , and for every sequence where for each is a sequence converging pointwise monotonically to , a sequence such that for each is a term of , and converges pointwise to . We prove a theorem which implies for metric spaces that has the monotonic sequence selection property if, and only if, has a covering property of Hurewicz.

     

Every nonreflexive subspace of fails the fixed point property

The main result of this paper is that every nonreflexive subspace of fails the fixed point property for closed, bounded, convex subsets of and nonexpansive (or contractive) mappings on . Combined with a theorem of Maurey we get that for subspaces of , is reflexive if and only if has the fixed point property. For general Banach spaces the question as to whether reflexivity implies the fixed point property and the converse question are both still open.

     

-bundles over aspherical surfaces and 4-dimensional geometries

Melvin has shown that closed 4-manifolds that arise as -bundles over closed, connected aspherical surfaces are classified up to diffeomorphism by the Stiefel-Whitney classes of the associated bundles. We show that each such 4-manifold admits one of the geometries or [depending on whether or ]. Conversely a geometric closed, connected 4-manifold of type or is the total space of an -bundle over a closed, connected aspherical surface precisely when its fundamental group is torsion free. Furthermore the total spaces of -bundles over closed, connected aspherical surfaces are all geometric. Conversely a geometric closed, connected 4-manifold is the total space of an -bundle if and only if where is torsion free.

     

All maps of type are boundary maps

Let be a continuous map of an interval into itself having periodic points of period for all and no other periods. It is shown that every neighborhood of contains a map such that the set of periods of the periodic points of is finite. This answers a question posed by L. S. Block and W. A. Coppel.

     

An extension of the Rabinowitz bifurcation theorem to Lipschitz potential operators in Hilbert spaces

The main result of the paper is an extension of the bifurcation theorem of Rabinowitz to equations with continuous jointly in and of class . We also prove a bifurcation theorem for critical points of the function which is just continuous and changes at an isolated minimum (in ) to isolated maximum when passes, say, zero. The proofs of the theorems, as well as the the theorems themselves, are new, in certain important aspects, even when applied to smooth functions.

     

Some Schrödinger operators with dense point spectrum

Given any sequence of positive energies and any monotone function on with , , we can find a potential on such that are eigenvalues of and .

     

A dimension result arising from the -spectrum of a measure

We give a rigorous proof of the following heuristic result: Let be a Borel probability measure and let be the -spectrum of . If is differentiable at , then the Hausdorff dimension and the entropy dimension of equal . Our result improves significantly some recent results of a similar nature; it is also of particular interest for computing the Hausdorff and entropy dimensions of the class of self-similar measures defined by maps which do not satisfy the open set condition.

     

A Cantor-Lebesgue theorem with variable ``coefficients'

If is a lacunary sequence of integers, and if for each , and are trigonometric polynomials of degree then must tend to zero for almost every whenever does. We conjecture that a similar result ought to hold even when the sequence has much slower growth. However, there is a sequence of integers and trigonometric polynomials such that tends to zero everywhere, even though the degree of does not exceed for each . The sequence of trigonometric polynomials tends to zero for almost every , although explicit formulas are developed to show that the sequence of corresponding conjugate functions does not. Among trigonometric polynomials of degree with largest Fourier coefficient equal to , the smallest one ``at' is while the smallest one ``near' is unknown.

     

Densities with the mean value property for harmonic functions in a Lipschitz domain

Let be a bounded domain in , , and let . We consider positive functions on such that for all bounded harmonic functions on . We determine Lipschitz domains having such with .

     

Perturbations of the Haar wavelet

Let be given. For any we construct a function having the following properties: (a) has support in . (b) . (c) If denotes the Haar function and , then . (d) generates an affine Riesz basis whose frame bounds (which are given explicitly) converge to as .

     

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 .

     

On invertibility in non-selfadjoint operator algebras

Let be a complete commutative subspace lattice on a Hilbert space. When is purely atomic, we give a necessary and sufficient condition for for every in , where and denote the spectrum of in and respectively. In addition, we discuss the properties of the spectra and the invertibility conditions for operators in .

     

Commutative group algebras of -summable abelian groups

In this note we study the commutative modular and semisimple group rings of -summable abelian -groups, which group class was introduced by R. Linton and Ch. Megibben. It is proved that is -summable if and only if is -summable, provided is an abelian group and is a commutative ring with 1 of prime characteristic , having a trivial nilradical. If is a -summable -group and the group algebras and over a field of characteristic are -isomorphic, then is a -summable -group, too. In particular provided is totally projective of a countable length.

Moreover, when is a first kind field with respect to and is -torsion, is -summable if and only if is a direct sum of cyclic groups.

     

Deformations of dihedral representations

G. Burde proved (1990) that the representation space of two-bridge knot groups is one-dimensional. The same holds for all torus knot groups. The aim of this note is to prove the following:
Given a knot we denote by its twofold branched covering space. Assume that there is a prime number such that . Then there exist representations of the knot group onto the binary dihedral group and these representations are smooth points on a one-dimensional curve of representations into .

     

Covering by complements of subspaces, II

Let be an -dimensional vector space over an algebraically closed field . Define to be the least positive integer for which there exists a family of -dimensional subspaces of such that every -dimensional subspace of has at least one complement among the 's. Using algebraic geometry we prove that .

     

Complemented isometric copies of in dual Banach spaces

Let be a real or complex Banach space and . Then contains a -complemented, isometric copy of if and only if contains a -complemented, isometric copy of if and only if contains a subspace -asymptotic to .

     

No submaximal topology on a countable set is -complementary

Two -topologies and given on the same set , are called transversal if their union generates the discrete topology on . The topologies and are -complementary if they are transversal and their intersection is the cofinite topology on . We establish that for any connected Tychonoff topology there exists a connected Tychonoff transversal one. Another result is that no -complementary topology exists for the maximal topology constructed by van Douwen on the rational numbers. This gives a negative answer to Problem 162 from Open Problems in Topology (1990).

     

A new subcontinuum of

We present a method for describing all indecomposable subcontinua of . This method enables us to construct in a new subcontinuum of .

We also show that the nontrivial layers of standard subcontinua can be described by our method. This allows us to construct a layer with a proper dense -subset and bring the number of (known) nonhomeomorphic subcontinua of to 14.

     

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 .

     

Factorisation in nest algebras

We give a necessary and sufficient condition on an operator for the existence of an operator in the nest algebra of a continuous nest satisfying (resp. . We also characterise the operators in which have the following property: For every continuous nest there exists an operator in satisfying (resp. .