Examples of asymptotic Banach spaces

Two examples of asymptotic Banach spaces are given. The first, , has an unconditional basis and is arbitrarily distortable. The second, , does not contain any unconditional basic sequence. Both are spaces of the type of Tsirelson's.

     

The operator on Banach space

The operator on , where is an interval contained in the real line, is considered in many places. In this paper, we attempt to reconsider it in the subspace of containing all even functions, and show that it generates a strongly continuous semigroup. It is interesting that our main conditions seem contradictory to previous ones. It is due to the symmetry of the functions and the different domain of the operator than usual.

     

Bifurcation problems for the

In this paper we consider the bifurcation problem

in with . We show that a continuum of positive solutions bifurcates out from the principal eigenvalue of the problem

Here both functions and may change sign.

     

Polynomial continuity on

A mapping between Banach spaces is said to be polynomially continuous if its restriction to any bounded set is uniformly continuous for the weak polynomial topology. A Banach space has property (RP) if given two bounded sequences , we have that for every polynomial on whenever for every polynomial on ; i.e., the restriction of every polynomial on to each bounded set is uniformly sequentially continuous for the weak polynomial topology. We show that property (RP) does not imply that every scalar valued polynomial on must be polynomially continuous.

     

The

In this paper we analyze the localization of , the fiber of the double suspension map , with respect to . If four cells at the bottom of , the th extended power spectrum of the Moore spectrum, are collapsed to a point, then one obtains a spectrum . Let be the James-Hopf map followed by the collapse map. Then we show that the secondary suspension map has a lifting to the fiber of and this lifting is shown to be a -periodic equivalence, hence an -equivalence.

     

and interpolation between Banach lattices

We study the possibility of obtaining the -norm by an interpolation method starting from a couple of Banach lattice norms. We describe all couples of Banach lattice norms in such that the -norm is a strict interpolation norm between them. Further we consider the possibility of obtaining the -norm by any method which guarantees interpolation of not only linear operators (= bilinear forms on but also of all polylinear forms. Here we show that either one of the initial norms has to be proportional to the -norm, or both have to be weighted -norms.

     

Completely continuous multilinear operators on spaces

Given a -linear operator from a product of spaces into a Banach space , our main result proves the equivalence between being completely continuous, having an -valued separately continuous extension to the product of the biduals and having a regular associated polymeasure. It is well known that, in the linear case, these are also equivalent to being weakly compact, and that, for , being weakly compact implies the conditions above but the converse fails.

     

Nonnormal spaces with countable extent

Examples of spaces are constructed for which is not normal but all closed discrete subsets are countable. A monolithic example is constructed in ZFC and a separable first countable example is constructed using .

     

Asymptotic behavior of -semigroups in Banach spaces

We present optimal estimates for the asymptotic behavior of
strongly continuous semigroups in terms of growth abscissas of the resolvent function of the generator . In particular we give Ljapunov's classical stability condition a definite form for (infinite dimensional) abstract Cauchy problems: The abscissa of boundedness of equals the growth bound of the classical solutions of .

     

The irrationality of

We shall show that the numbers and
are linearly independent over for any natural number . The key is to construct explicit Padé-type approximations using Legendre-type polynomials.

     

The Diophantine equation

An effective method is derived for solving the equation of the title in positive integers and for given completely, and is carried out for all . If is of the form , then there is the solution , ; in the above range, except for with solution , , there are no other solutions.

     

Bohr almost periodic maps into spaces

Let be a locally finite simplicial complex of finite topological dimension. Assume further that is a space where is a group whose only abelian subgroups are infinite cyclic. We prove that a Bohr almost periodic map of the real line into is uniformly homotopic to a periodic map. As a consequence we show that a Bohr almost periodic geodesic on a compact Riemannian manifold of everywhere negative curvature is necessarily periodic.

     

Intersection theory on and elliptic Gromov-Witten invariants

We find a new relation among codimension algebraic cycles in the moduli space , and use this to calculate the elliptic Gromov-Witten invariants of projective spaces and .

     

-convexity of Orlicz-Bochner spaces

A characterization of -convexity of arbitrary Banach space is given. Moreover, it is proved that the Orlicz-Bochner function space is P-convex if and only if both spaces and are -convex. In particular, the Lebesgue-Bochner space with is -convex iff is -convex.

     

Calibrated thin

Let be a compact metric space, and let be a calibrated thin -ideal. Then is . This solves an open problem, which was posed by Kechris, Louveau and Woodin. Using our result we obtain a new proof of Kaufman's theorem concerning -sets and -sets.

     

Gaps in

For a partial order , let denote the statement that for every -increasing -sequence there is a -decreasing -sequence on top of such that is an -gap in . The main result of this paper is that . It is also shown, as a corollary, that but .

     

without sharps

We show that the supremum of the lengths of prewellorderings of the reals can be , with inaccessible to reals, assuming only the consistency of an inaccessible.