-bounding and -induction

Working in the base theory of , we show that for all , the bounding principle for -formulas ( ) is equivalent to the induction principle for -formulas ( ). This partially answers a question of J. Paris.

     

There is no separable universal -factor

Gromov constructed uncountably many pairwise nonisomorphic discrete groups with Kazhdan's property . We will show that no separable -factor can contain all these groups in its unitary group. In particular, no separable -factor can contain all separable -factors in it. We also show that the full group -algebras of some of these groups fail the lifting property.

     

Minimal degrees which are but not

We give a short proof of the existence of minimal Turing degrees which are but not .

     

Isometric copies of and in Orlicz spaces equipped with the Orlicz norm

Criteria in order that an Orlicz space equipped with the Orlicz norm contains a linearly isometric copy (or an order linearly isometric copy) of (or ) are given.

     

-adic -basis and regular local ring

We introduce the concept of -adic -basis as an extension of the concept of -basis. Let be a regular local ring of prime characteristic and a ring such that . Then we prove that is a regular local ring if and only if there exists an -adic -basis of and is Noetherian.

     

Morava -theory of extraspecial -groups

We compute the Morava -theory of some extraspecial 2-groups and associated compact groups.

     

An ultrafilter with property

Let be a -supercompact cardinal. We show that carries a normal ultrafilter with a property introduced by Menas. With it we give a transparent proof of Kamo's theorem that carries a normal ultrafilter with the partition property.

     

-additive families and the invariance of Borel classes

We prove that any -additive family of sets in an absolutely Souslin metric space has a -discrete refinement provided every partial selector set for is -discrete. As a corollary we obtain that every mapping of a metric space onto an absolutely Souslin metric space, which maps -sets to -sets and has complete fibers, admits a section of the first class. The invariance of Borel and Souslin sets under mappings with complete fibers, which preserves -sets, is shown as an application of the previous result.

     

-actions with

Suppose that is a smooth -action on a closed smooth -dimensional manifold such that all Stiefel-Whitney classes of the tangent bundle to each connected component of the fixed point set vanish in positive dimension. This paper shows that if 2^k\dim F$"> and each -dimensional part possesses the linear independence property, then bounds equivariantly, and in particular, is the best possible upper bound of if is nonbounding.

     

Banach spaces embedding isometrically into when

For we give examples of Banach spaces isometrically embedding into but not into any with

     

A separable non-remainder of

We prove that there is a compact separable continuum that (consistently) is not a remainder of the real line.

     

Robust transitivity and topological mixing for -flows

We prove that nontrivial homoclinic classes of -generic flows are topologically mixing. This implies that given , a nontrivial -robustly transitive set of a vector field , there is a -perturbation of such that the continuation of is a topologically mixing set for . In particular, robustly transitive flows become topologically mixing after -perturbations. These results generalize a theorem by Bowen on the basic sets of generic Axiom A flows. We also show that the set of flows whose nontrivial homoclinic classes are topologically mixing is not open and dense, in general.

     

Pseudocompact spaces and -spaces

Let be a completely regular Hausdorff space, and let be the space of continuous real-valued functions on endowed with the compact-open topology. We find various equivalent conditions for to be a -space, resolving an old question of Jarchow and consolidating work by Jarchow, Mazon, McCoy and Todd. Included are analytic characterizations of pseudocompactness and an example that shows that, for , Grothendieck's -spaces do not coincide with Jarchow's -spaces. Any such example necessarily answers a thirty-year-old question on weak barrelledness properties for , our original motivation.

     

Pseudofree -actions on surfaces

In this paper, we give a weak classification of locally linear pseudofree actions of the cyclic group of order on a surface, and prove the existence of such an action which cannot be realized as a smooth action on the standard smooth surface.

     

Biliaison classes of curves in

We characterize the curves in that are minimal in their biliaison class. Such curves are exactly the curves that do not admit an elementary descending biliaison. As a consequence we have that every curve in can be obtained from a minimal one by means of a finite sequence of ascending elementary biliaisons.

     

A hereditarily subspace of without the Schur property

Let p_1 > p_2 > \cdots > 1$">. We construct an easily determined -symmetric basic sequence in , which spans a hereditarily subspace without the Schur property. An immediate consequence is the existence of hereditarily subspaces of without the Schur property.

     

Sharply -transitive groups with point stabilizer of exponent or

Using the fact that all groups of exponent are nilpotent, we show that every sharply -transitive permutation group whose point stabilizer has exponent or is finite.

     

An optimal Poincaré inequality in for convex domains

For convex domains with diameter we prove

for any with zero mean value on . We also show that the constant in this inequality is optimal.

     

The projective -character bounds the order of a -base

All spaces below are Tychonov. We define the projective - character of a space as the supremum of the values where ranges over all (Tychonov) continuous images of . Our main result says that every space has a -base whose order is ; that is, every point in is contained in at most -many members of the -base. Since for compact , this is a significant generalization of a celebrated result of Shapirovskii.

     

interval maps not Borel conjugate to any map

We show that there exist interval maps that are not Borel conjugate to any map. These examples can be chosen to be topologically mixing and , for any finite, arbitrarily large .