Symplectic hypersurfaces in

We establish the uniqueness of the symplectic -manifolds which admit low degree symplectic embeddings into . We also discuss the uniqueness of the fundamental group of the complement of such embeddings into arbitrary symplectic -manifolds.

     

Lifts of - and -morphisms to -morphisms

Let be the Hochschild complex of cochains on and let be the space of multivector fields on . In this paper we prove that given any -structure (i.e. Gerstenhaber algebra up to homotopy structure) on , and any -morphism (i.e. morphism of a commutative, associative algebra up to homotopy) between and , there exists a -morphism between and that restricts to . We also show that any -morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a -morphism, using Tamarkin's method for any -structure on . We also show that any two of such -morphisms are homotopic.

     

Smooth actions of

Given a smooth action of on a -dimensional differentiable manifold , for each we associate with ``almost all" oriented orbits of dimension an element of .

     

Hyperelliptic curves over of every -rank without extra automorphisms

We prove that for any pair of integers such that or 0$">, there exists a (hyper)elliptic curve over of genus and -rank whose automorphism group consists of only identity and the (hyper)elliptic involution. As an application, we prove the existence of principally polarized abelian varieties over of dimension and -rank such that .

     

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.

     

Spectral pictures of and

The spectral pictures of products and of Banach space operators are compared; in particular when one of them is `of index zero'.

     

Gromov hyperbolicity of the and metrics

In this note it is shown that the metric is always Gromov hyperbolic, but that the metric is Gromov hyperbolic if and only if has exactly one boundary point. As a corollary we get a new proof for the fact that the quasihyperbolic metric is Gromov hyperbolic in uniform domains.

     

of Hamiltonian manifolds

Let be a connected, compact symplectic manifold equipped with a Hamiltonian action. We prove that, as fundamental groups of topological spaces, , where is the symplectic quotient at any value in the image of the moment map .

     

Remarks on product

Well known results related to the compactness of Hankel operators of one complex variable are extended to little Hankel operators of two complex variables. Critical to these considerations is the result of Ferguson and Lacey (2002) characterizing the boundedness of the little Hankel operators in terms of the product BMO of S.-Y. Chang and R. Fefferman (1985), (1980).

     

-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.

     

Automatic continuity of -derivations on -algebras

Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.

     

A minimal pair of -degrees

We construct a minimal pair of -degrees. We do this by showing the existence of an unbounded nondecreasing function which forces -triviality in the sense that is -trivial if and only if for all , .

     

Cube-approximating bounded wavelet sets in

We prove that for any real expansive matrix , there exists a bounded -dilation wavelet set in the frequency domain (the inverse Fourier transform of whose characteristic function is a band-limited single wavelet in the time domain ). Moreover these wavelet sets can approximate a cube in arbitrarily. This result improves Dai, Larson and Speegle's result about the existence of (basically unbounded) wavelet sets for real expansive matrices.

     

On quasi-complete intersections of codimension

In this paper, we prove that if , , is a locally complete intersection of pure codimension and defined scheme-theoretically by three hypersurfaces of degrees , then for using liaison theory and the Arapura vanishing theorem for singular varieties. As a corollary, a smooth threefold is projectively normal if is defined by three quintic hypersurfaces.

     

The hypersurface over a field of arbitrary characteristic

We develop techniques for computing the AK invariant of domains with arbitrary characteristic. As an example, we show that for any field the ring is not isomorphic to a polynomial ring over .

     

Fixed point theorems in ordered -spaces

The purpose of this paper is to present some fixed point results in ordered L-spaces. Our results generalize and extend a recent result of Ran and Reurings (2004). Some applications to matrix equations are also considered.

     

Lehmer -tuples

Generalizing a classical problem of Lehmer, in this paper we provide an asymptotic result for the number of Lehmer -tuples.

     

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

     

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.

     

On maximal operators on -spheres in

A. Magyar's result on -bounds for a family of operators on -spheres () in is improved to match the corresponding theorem for -spheres.