Lightface -indescribable cardinals

-absoluteness for forcing means that for any forcing , . `` inaccessible to reals' means that for any real , . To measure the exact consistency strength of `` -absoluteness for forcing and is inaccessible to reals', we introduce a weak version of a weakly compact cardinal, namely, a (lightface) -indescribable cardinal; has this property exactly if it is inaccessible and .

     

Cofinality changes required for a large set of unapproachable ordinals below

In , assume that is a strong limit cardinal and . Let be the set of approachable ordinals less than . An open question of M. Foreman is whether can be non-stationary in some and preserving extension of . It is shown here that if is such an outer model, then is infinite, for each positive integer .

     

Tensor products of -weakly closed nest algebra submodules

In this paper we prove that for any unital -weakly closed algebra which is -weakly generated by finite-rank operators in , every -weakly closed -submodule has . In the case of nest algebras, if are nests, we obtain the following -fold tensor product formula:

where each is the -weakly closed Alg -submodule determined by an order homomorphism from into itself.

     

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.

     

(APD)-property of -algebras by extensions of -algebras with (APD)

A unital -algebra is said to have the (APD)-property if every nonzero element in has the approximate polar decomposition. Let be a closed ideal of . Suppose that and have (APD). In this paper, we give a necessary and sufficient condition that makes have (APD). Furthermore, we show that if and or is a simple purely infinite -algebra, then has (APD).

     

planes in

We establish a homeomorphism between the moduli space of ordered -tuples of 2-dimensional linear subspaces (mod ) and the quotient by simultaneous conjugation of a certain open subset . For , this leads to an explicit computation of the moduli space of central 2-arrangements in mod and its subspace of those classes that contain a complex hyperplane arrangement.

     

-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.

     

-bounded groups and other topological groups with strong combinatorial properties

We construct several topological groups with very strong combinatorial properties. In particular, we give simple examples of subgroups of (thus strictly -bounded) which have the Menger and Hurewicz properties but are not -compact, and show that the product of two -bounded subgroups of may fail to be -bounded, even when they satisfy the stronger property . This solves a problem of Tkacenko and Hernandez, and extends independent solutions of Krawczyk and Michalewski and of Banakh, Nickolas, and Sanchis. We also construct separable metrizable groups of size continuum such that every countable Borel -cover of contains a -cover of .

     

Exact multiplicity result for the perturbed scalar curvature problem in

Let denote the closure of in the norm Let and define the constants and Let We consider the following problem for

We show an exact multiplicity result for for all small .

     

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.

     

Serre duality for non-commutative -bundles

Let be a smooth scheme of finite type over a field , let be a locally free -bimodule of rank , and let be the non-commutative symmetric algebra generated by . We construct an internal functor, , on the category of graded right -modules. When has rank 2, we prove that is Gorenstein by computing the right derived functors of . When is a smooth projective variety, we prove a version of Serre Duality for using the right derived functors of .

     

A note on -bases of a regular affine domain extension

Let be a tower of commutative rings where is a regular affine domain over an algebraically closed field of prime characteristic and is a regular domain. Suppose has a -basis over and . For a subset of whose elements satisfy a certain condition on linear independence, let be a set of maximal ideals of such that is a -basis of over . We shall characterize this set in a geometrical aspect.

     

Real rank and squaring mappings for unital -algebras

It is proved that if is a compact Hausdorff space of Lebesgue dimension , then the squaring mapping , defined by , is open if and only if . Hence the Lebesgue dimension of can be detected from openness of the squaring maps . In the case it is proved that the map , from the selfadjoint elements of a unital -algebra into its positive elements, is open if and only if is isomorphic to for some compact Hausdorff space with .

     

``Lebesgue measure' on , II

Let be the set of real numbers, and define . We construct a complete measure space where the -algebra contains the Borel subsets of , and is a translation-invariant measure such that for any measurable rectangle , if , then , where is Lebesgue measure on . The measure is not -finite. We prove three Fubini theorems, namely, the Fubini theorem, the mean Fubini-Jensen theorem, and the pointwise Fubini-Jensen theorem. Finally, as an application of the measure , we construct, via selfadjoint operators on , a ``Schrödinger model' of the canonical commutation relations: , , .

     

Hochschild cohomology of Frobenius algebras

Let be a field, a finite-dimensional Frobenius -algebra and , the Nakayama automorphism of with respect to a Frobenius homomorphism . Assume that has finite order and that has a primitive -th root of unity . Consider the decomposition of , obtained by defining , and the decomposition of the Hochschild cohomology of , obtained from the decomposition of . In this paper we prove that and that if the decomposition of is strongly -graded, then acts on 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.

     

Representation of measures with polynomial denseness in , , and its application to determinate moment problems

It has been proved that algebraic polynomials are dense in the space , , iff the measure is representable as with a finite non-negative Borel measure and an upper semi-continuous function such that is a dense subset of the space    as equipped with the seminorm . The similar representation ( ) with the same and ( , and is also a dense

subset of ) corresponds to all those measures (supported by ) that are uniquely determined by their moments on ( ). The proof is based on de Branges' theorem (1959) on weighted polynomial approximation. A more general question on polynomial denseness in a separable Fréchet space in the sense of Banach has also been examined.

     

The Laplacian subalgebra of is a strongly singular masa

In this paper, we present a new class of strongly singular maximal abelian subalgebras living inside the -folded tensor product of the free group factor with itself (). The notions of strongly singular masas in type factors and that of asymptotic homomorphism were introduced by A. Sinclair and R. Smith. One of their first examples was the Laplacian subalgebra of the free group factor, generated by the sum of words of length 1 in . This subalgebra was known to be a singular masa. Using the results of A. Sinclair and R. Smith, we show that the unique trace-preserving conditional expectation onto the Laplacian subalgebra of is an asymptotic homomorphism, and hence the Laplacian subalgebra is a strongly singular masa for every .

     

On the normal bundle of submanifolds of

Let be a submanifold of dimension of the complex projective space . We prove results of the following type.i) If is irregular and , then the normal bundle is indecomposable. ii) If is irregular, and , then is not the direct sum of two vector bundles of rank . iii) If , and is decomposable, then the natural restriction map is an isomorphism (and, in particular, if is embedded Segre in , then is indecomposable). iv) Let and , and assume that is a direct sum of line bundles; if assume furthermore that is simply connected and is not divisible in . Then is a complete intersection. These results follow from Theorem 2.1 below together with Le Potier's vanishing theorem. The last statement also uses a criterion of Faltings for complete intersection. In the case when this fact was proved by M. Schneider in 1990 in a completely different way.

     

Multiplicative bijections of

Let be a compact Hausdorff space which satisfies the first axiom of countability, let and let , be the set of all continuous functions from to If , ,is a bijective multiplicative map, then there exist a homeomorphism and a continuous map such that for all and for all