Real -flats tangent to quadrics in

Let and denote the dimension and the degree of the Grassmannian , respectively. For each there are (a priori complex) -planes in tangent to general quadratic hypersurfaces in . We show that this class of enumerative problems is fully real, i.e., for there exists a configuration of real quadrics in (affine) real space so that all the mutually tangent -flats are real.

     

-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.

     

Properly -realizable groups

A finitely presented group is said to be properly -realizable if there exists a compact -polyhedron with and whose universal cover has the proper homotopy type of a (p.l.) -manifold with boundary. In this paper we show that, after taking wedge with a -sphere, this property does not depend on the choice of the compact -polyhedron with . We also show that (i) all -ended and -ended groups are properly -realizable, and (ii) the class of properly -realizable groups is closed under amalgamated free products (HNN-extensions) over a finite cyclic group (as a step towards proving that -ended groups are properly -realizable, assuming -ended groups are).

     

Irreducibility of the -classes on smooth rational surfaces

We give a characterization for a -divisor on a smooth rational surface to be irreducible under the assumption that an anticanonical divisor of is nef. Here is nef means for every effective divisor on , and a -divisor is a divisor such that the two numerical conditions hold.

As an application we give explicit examples of blowing up the projective plane at nine points infinitely near such that the obtained surface has an infinite number of -curves. A -curve is a smooth rational curve of self-intersection .

     

A new representation of the Dedekind completion of -spaces

A new representation of the Dedekind completion of is given. We present a necessary and sufficient condition on a compact Hausdorff space for which the Dedekind completion of is , the space of real valued bounded functions on some set .

     

Semi-continuity of metric projections in -direct sums

Let be a proximinal subspace of finite codimension of . We show that is proximinal in and the metric projection from onto is Hausdorff metric continuous. In particular, this implies that the metric projection from onto is both lower Hausdorff semi-continuous and upper Hausdorff semi-continuous.

     

Measures of concordance determined by -invariant measures on

A measure, , on is said to be -invariant if its value for any Borel set is invariant with respect to the symmetries of the unit square. A function, , generated in a certain way by a measure, , on is shown to be a measure of concordance if and only if the generating measure is positive, regular, -invariant, and satisfies certain inequalities. The construction examined here includes Blomqvist's beta as a special case.

     

Embedding as Lipschitz functions

Let be a compact Hausdorff space and let be a lower semicontinuous metric on it. We prove that is fragmented by if, and only if, contains no copy of made up of Lipschitz functions with respect to . As applications we obtain a characterization of Asplund Banach spaces and Radon-Nikodým compacta.

     

Blocks with -power character degrees

Let be a -block of a finite group . If is a -power for all , then is nilpotent.

     

Hopf algebras of dimension

Let be a finite-dimensional Hopf algebra over an algebraically closed field of characteristic 0. If is not semisimple and for some odd integer , then or is not unimodular. Using this result, we prove that if for some odd prime , then is semisimple. This completes the classification of Hopf algebras of dimension .

     

More on convexity numbers of closed sets in

The convexity number of a set is the least size of a family of convex sets with . is countably convex if its convexity number is countable. Otherwise is uncountably convex.

Uncountably convex closed sets in have been studied recently by Geschke, Kubis, Kojman and Schipperus. Their line of research is continued in the present article. We show that for all , it is consistent that there is an uncountably convex closed set whose convexity number is strictly smaller than all convexity numbers of uncountably convex subsets of .

Moreover, we construct a closed set whose convexity number is and that has no uncountable -clique for any 1$">. Here is a -clique if the convex hull of no -element subset of is included in . Our example shows that the main result of the above-named authors, a closed set either has a perfect -clique or the convexity number of is in some forcing extension of the universe, cannot be extended to higher dimensions.

     

Local automorphisms and derivations on certain -algebras

It is shown that continuous -local derivations on -algebras are derivations and surjective -local *-automorphisms on prime -algebras or on -algebras such that the identity element is properly infinite are *-automorphisms.

     

Line arrangements in

If is a hyperbolic manifold and is a simple closed geodesic, then lifts to a collection of lines in acted upon by . In this paper we show that such a collection of lines cannot contain a particular type of subset (called a bad triple) unless has orientation-reversing elements. This fact allows us to extend certain lower bounds on hyperbolic volume to the non-orientable case.

     

The homological determinant of quantum groups of type

Let be a Hecke symmetry depending algebraically on a parameter . We show that the homology of the Koszul complex associated with is one-dimensional when is not a root of unity. A generator of this homology group then induces the homological determinant of the quantum group associated with .

     

Local isometries of

In this paper we study the structure of local isometries on . We show that when is first countable and is uniformly convex and the group of isometries of is algebraically reflexive, the range of a local isometry contains all compact operators. When is also uniformly smooth and the group of isometries of is algebraically reflexive, we show that a local isometry whose adjoint preserves extreme points is a -module map.

     

Spectrally bounded -derivations on Banach algebras

Applying the density theorem on algebras with -derivations, we show that if a -derivation of a unital Banach algebra is spectrally bounded, then . Also, if and only if , where denotes the spectral radius of .

     

Tietze extension theorem for Hilbert -modules

We prove the following generalization of the noncommutative Tietze extension theorem: if is a countably generated Hilbert -module over a -unital -algebra, then the canonical extension of a surjective morphism of Hilbert -modules to extended (multiplier) modules, , is also surjective.

     

Examples of non-formal closed -connected manifolds of dimensions

We construct closed -connected manifolds of dimensions that possess non-trivial rational Massey triple products. We also construct examples of manifolds such that all the cup-products of elements of vanish, while the group is generated by Massey products: such examples are useful for the theory of systols.

     

-invariant subspaces of

In this note, we give a new proof of the characterization of the -invariant subspaces of for in using ideas from approximation theory.     

Blocks of central -group extensions

Let and be finite groups that have a common central -subgroup for a prime number , and let and respectively be -blocks of and induced by -blocks and respectively of and , both of which have the same defect group. We prove that if and are Morita equivalent via a certain special -bimodule, then such a Morita equivalence lifts to a Morita equivalence between and .