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 .

     

Discrete spectra of -algebras and orthogonally closed submodules in Hilbert -modules

Let and be -algebras and let be an --imprimitivity bimodule. Then it is shown that if the spectrum of (resp. of ) is discrete, then every closed --submodule of is orthogonally closed in , and conversely that if (resp. ) is a -space and if every closed --submodule of is orthogonally closed in , then (resp. ) is discrete.

     

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).

     

Blocks with -power character degrees

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

     

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.

     

Symplectic tori in homotopy 's

This short note presents a simple construction of nonisotopic symplectic tori representing the same primitive homology class in the symplectic -manifold , obtained by knot surgery on the rational elliptic surface with the left-handed trefoil knot . has the simplest homotopy type among simply-connected symplectic -manifolds known to exhibit such a property.

     

-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.

     

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.

     

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 .

     

Strong local homogeneity and coset spaces

We prove that for every homogeneous and strongly locally homogeneous separable metrizable space there is a metrizable compactification of such that, among other things, for all there is a homeomorphism such that . This implies that is a coset space of some separable metrizable topological group .

     

-algebras whose torsion part is not cyclic

We consider algebras over a Dedekind domain with the property and generalize Schultz' structure theory of the case to Dedekind domains. We construct examples of mixed -algebras, which are non-split extensions of the submodule of elements infinitely divisible by the relevant prime ideals. This is also new in the case .

     

Definite regular quadratic forms over

Let be a power of an odd prime, and be the ring of polynomials over a finite field of elements. A quadratic form over is said to be regular if globally represents all polynomials that are represented by the genus of . In this paper, we study definite regular quadratic forms over . It is shown that for a fixed , there are only finitely many equivalence classes of regular definite primitive quadratic forms over , regardless of the number of variables. Characterizations of those which are universal are also given.

     

Real

The famous problem involves applying two maps: and to positive integers. If is even, one applies , if it is odd, one applies . The conjecture states that each trajectory of the system arrives to the periodic orbit . In this paper, instead of choosing each time which map to apply, we allow ourselves more freedom and apply both and independently of . That is, we consider the action of the free semigroup with generators and on the space of positive real numbers. We prove that this action is minimal (each trajectory is dense) and that the periodic points are dense. Moreover, we give a full characterization of the group of transformations of the real line generated by and .

     

Almost everywhere convergence of series in

We answer positively a question of J. Rosenblatt (1988), proving the existence of a sequence with , such that for every dynamical system and , converges almost everywhere. A similar result is obtained in the real variable context.

     

Borel classes of sets of extreme and exposed points in

It is known that the sets of extreme and exposed points of a convex Borel subset of are Borel. We show that for there exist convex subsets of such that the sets of their extreme and exposed points coincide and are of arbitrarily high Borel class. On the other hand, we show that the sets of extreme and of exposed points of a convex set of additive Borel class are of ambiguous Borel class . For proving the latter-mentioned results we show that the union of the open and the union of the closed segments of are of the additive Borel class if is a convex set of additive Borel class .

     

The classical monotone convergence theorem of Beppo Levi fails in noncommutative -spaces

Let be a complex Hilbert space and let be a von Neumann algebra over equipped with a faithful, normal state . Then is a prehilbert space with respect to the inner product , whose completion is given by the Gelfand-Naimark-Segal representation theorem, according to which there exist a one-to-one -homomorphism of into the algebra of all bounded linear operators acting on and a cyclic, separating vector such that for all . Given any separable Hilbert space , we construct a faithful, normal state on and an increasing sequence of positive operators acting on such that is bounded, but fails to converge both bundlewise and in -norm. We also present an example of an increasing sequence of positive operators which has a subsequence converging both bundlewise and in -norm, but the whole sequence fails to converge in either sense. Finally, we observe that our results are linked to a previous one by R. V. Kadison.

     

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.

     

Filling analytic sets by the derivatives of -smooth bumps

If is an infinite-dimensional Banach space, with separable dual, and is an analytic set such that any point can be reached from  by a continuous path contained (except for the point ) in the interior of , then is the range of the derivative of a -smooth function on  with bounded nonempty support.

     

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.